(original) (raw)

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b(A)h(set)h(is)e Fw(K)7 b FA(-trivial)36 b(if)h(its)505 865 y(initial)30 b(segmen)m(t)j(pre\014x-free)e(Kolmogoro)m(v)i(complexit)m(y)e(is)g (the)h(same)h(as)f(that)g(of)505 973 y(the)d(sequence)f(of)h(all)e (ones)h(\(up)f(to)i(an)f(additiv)m(e)g(constan)m(t\).)i(Noncomputable)e Fw(K)7 b FA(-)505 1081 y(trivial)33 b(sets)j(exist)e(but)g(are)h(all)f (\001)1716 1048 y Fy(0)1716 1105 y(2)1755 1081 y FA(.)h(The)g(c.e.)h (sets)f(among)g(them)g(are)g(solutions)505 1189 y(to)46 b(P)m(ost's)h(Problem.)d(W)-8 b(e)46 b(explore)f(Nies')g(results)f(on)h (the)g(T)-8 b(uring)44 b(degrees)h(of)505 1297 y(suc)m(h)34 b(sets,)h(whic)m(h)e(form)g(a)i(natural)e(\006)1871 1264 y Fy(0)1871 1321 y(3)1944 1297 y FA(ideal)g(in)g(the)h(c.e.)i(T)-8 b(uring)32 b(degrees,)j(and)505 1405 y(the)41 b(connections)f(b)s(et)m (w)m(een)g Fw(K)7 b FA(-trivialit)m(y)-8 b(,)39 b(lo)m(wness)h (notions,)f(and)h(other)g(forms)505 1513 y(of)e(computational)f(w)m (eakness)h(related)g(to)g(randomness.)f(W)-8 b(e)38 b(then)g(discuss)d (the)505 1621 y(complexit)m(y)27 b(of)h(c.e.)g(sets)g(and)e(a)i (theorem)f(of)h(Kummer)e(whic)m(h)g(c)m(haracterizes)i(c.e.)505 1729 y(degrees)j(con)m(taining)f(complex)g(c.e.)i(sets.)588 1837 y(After)d(this)e(w)m(e)h(turn)f(to)i(other)g(notions)e(of)h (randomness)f(suc)m(h)h(as)h(Sc)m(hnorr)e(ran-)505 1944 y(domness)35 b(and)f(computable)g(randomness,)g(where)g(the)h(notion)g (of)g(what)f(consti-)505 2052 y(tutes)28 b(a)g(random)f(set)h(actually) f(c)m(hanges.)h(Again,)g(there)g(ha)m(v)m(e)g(b)s(een)f(man)m(y)g (recen)m(t)505 2160 y(results)21 b(in)g(this)g(area,)i(suc)m(h)e(as)i (a)f(mac)m(hine)g(c)m(haracterization)h(of)f(Sc)m(hnorr)f(random-)505 2268 y(ness,)i(and)g(b)s(eautiful)d(lo)m(wness)j(c)m(haracterizations)h (related)f(to)g(the)h(h)m(yp)s(erimm)m(une-)505 2376 y(free)38 b(degrees.)h(W)-8 b(e)38 b(also)g(discuss)e(lo)m(wness)h (notions)g(for)g(v)-5 b(arious)37 b(\015a)m(v)m(ors)h(of)g(ran-)505 2484 y(domness.)588 2592 y(Next)f(w)m(e)f(lo)s(ok)f(at)h(arithmetical)e (v)m(ersions)h(of)h(randomness)e(\()p Fw(n)p FA(-randomness\).)505 2700 y(W)-8 b(e)44 b(include)d(a)i(bit)f(of)h(bac)m(kground)g(material) f(here,)h(as)g(it)f(seems)h(not)h(widely)505 2808 y(kno)m(wn,)32 b(and)g(the)h(lo)m(v)m(ely)f(recen)m(t)h(result)f(that)h(a)f(set)h(is)e (2-random)i(\(a)g(relativized)505 2916 y(pre\014x-free)23 b(complexit)m(y)f(notion\))h(iff)f(its)g(initial)e(segmen)m(t)k (complexit)m(y)f(is)f(maximal)505 3024 y(in\014nitely)36 b(often)j(when)f(measured)g(b)m(y)h(plain)e(Kolmogoro)m(v)j(complexit)m (y)-8 b(.)39 b(Man)m(y)505 3132 y(\\t)m(ypical")d(random)f(phenomena)f (only)h(really)f(o)s(ccur)h(for)g(2-)h(or)g(ev)m(en)g(3-random)505 3240 y(sets.)30 b(F)-8 b(or)29 b(instance,)g(if)f(w)m(e)h(concen)m (trate)i(on)d(the)h(left-c.e.)h(reals,)f(then)f(w)m(e)h(get)h(the)505 3348 y(impression)g(that)j(1-random)f(sets)g(lo)s(ok)g(lik)m(e)f(\012)h (and)g(are)g(computationally)f(ric)m(h.)505 3456 y(Ho)m(w)m(ev)m(er,)38 b(2-random)e(sets)g(are)g(computationally)e(v)m(ery)i(w)m(eak,)g(and)f (lo)s(ok)g(m)m(uc)m(h)505 3563 y(more)29 b(lik)m(e)e Fs(low)39 b FA(sets.)29 b(The)f(next)h(section)f(includes)e(sev)m(eral) i(results)f(of)i(Miller)d(and)505 3671 y(Y)-8 b(u,)40 b(whic)m(h)f(explore)g(another)h(measure)f(of)h(relativ)m(e)f (randomness)g(and)g(further)505 3779 y(expand)27 b(on)h(this)e(story)-8 b(,)28 b(sho)m(wing)f(for)g(instance)h(that)g(there)f(is)g(a)h (relationship)d(b)s(e-)505 3887 y(t)m(w)m(een)i(lev)m(els)e(of)h (randomness)e(and)h(initial)e(segmen)m(t)k(complexit)m(y)-8 b(.)26 b(W)-8 b(e)27 b(follo)m(w)e(this)505 3995 y(with)32 b(a)i(lo)s(ok)f(at)h(relativizing)e(randomness)g(b)m(y)h(considering)e (\012)i(as)h(an)f(op)s(erator,)505 4103 y(whic)m(h)f(is)g(complicated)g (b)m(y)h(the)g(fact)h(that)f(it)f(is)g(a)h(c.e.)h(op)s(erator)f(but)f (is)g(not)h(c.e.)505 4211 y(in)c Fs(and)34 b(ab)-5 b(ove)p FA(.)588 4319 y(In)28 b(the)g(last)f(section)h(w)m(e)h(tak)m(e)g(a)g (brief)d(lo)s(ok)h(at)i(e\013ectiv)m(e)h(Hausdor\013)d(dimension)505 4427 y(and)e(related)g(notions)f(of)h(partial)f(randomness.)g(This)f (is)h(a)h(h)m(uge)h(area)f(of)h(researc)m(h,)505 4535 y(particularly)32 b(in)h(computer)i(science,)g(and)e(w)m(e)i(will)d (only)h(men)m(tion)h(a)h(few)f(recen)m(t)505 4643 y(results.)588 4751 y(The)j(topic)g(of)g(this)f(pap)s(er)f(is)h(a)i(mix)e(of)h (computabilit)m(y)e(theory)-8 b(,)38 b(algorithmic)505 4859 y(information)27 b(theory)-8 b(,)29 b(and)f(measure)g(theory)-8 b(.)29 b(Our)f(computabilit)m(y-theoretic)f(no-)505 4967 y(tation)e(generally)e(follo)m(ws)g(Odifreddi)d([108)r(,)k(109)q(])g (and)f(Soare)h([122)r(].)g(W)-8 b(e)25 b(deal)f(with)505 5074 y(sev)m(eral)35 b(degree)g(structures,)f(but)f(when)g(w)m(e)i(men) m(tion)f(degrees)h(without)e(further)p eop %%Page: 4 4 4 3 bop 505 363 a FD(4)319 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(sp)s(eci\014cation,)40 b(w)m(e)h(mean)f(T)-8 b(uring)39 b(degrees.)i(W)-8 b(e)42 b(will)37 b(denote)k(the)g Fw(e)p FA(-th)g(partial)505 649 y(computable)g(function)f(b)m(y)h(\010) 1576 663 y Fx(e)1613 649 y FA(,)g(the)h Fw(e)p FA(-th)f(partial)f (computable)h(function)f(with)505 757 y(oracle)33 b Fw(X)39 b FA(b)m(y)31 b(\010)1077 724 y Fx(X)1077 780 y(e)1144 757 y FA(,)h(and)f(the)h Fw(e)p FA(-th)g(computably)f(en)m(umerable)f (set)j(b)m(y)e Fw(W)3066 771 y Fx(e)3103 757 y FA(.)h(When)505 865 y(w)m(e)40 b(write)e(log)16 b Fw(n)p FA(,)39 b(w)m(e)g(mean)g(the)g (base)g(2)g(logarithm)f(of)h Fw(n)p FA(,)g(rounded)e(up)h(to)h(the)505 973 y(nearest)f(in)m(teger.)h(W)-8 b(e)38 b(will)d(use)i Fw(C)44 b FA(to)38 b(denote)g(plain)e(Kolmogoro)m(v)i(complexit)m(y)-8 b(,)505 1081 y(and)30 b Fw(K)37 b FA(to)31 b(denote)g(pre\014x-free)f (Kolmogoro)m(v)h(complexit)m(y)-8 b(.)588 1189 y(This)29 b(pap)s(er)g(is)h(not)h(a)f(general)h(in)m(tro)s(duction)d(to)j(recen)m (t)h(w)m(ork)f(on)f(algorithmic)505 1297 y(randomness,)k(but)f(rather)h (an)h(attempt)g(to)g(giv)m(e)g(the)f(reader)h(insigh)m(t)d(in)m(to)j (what)505 1405 y(w)m(e)29 b(feel)f(are)g(some)h(of)f(the)g(high)f(p)s (oin)m(ts)g(in)f(the)j(program)e(to)i(understand)d(relativ)m(e)505 1513 y(randomness,)42 b(the)h(Kolmogoro)m(v)g(complexit)m(y)g(of)f (sets,)i(and)e(the)g(relationships)505 1621 y(of)d(these)h(topics)e(to) i(classical)e(computabilit)m(y)f(theory)-8 b(.)40 b(This)d(is)h(a)h (fast-gro)m(wing)505 1729 y(area)28 b(of)f(researc)m(h,)g(and)g(w)m(e)g (ha)m(v)m(e)h(necessarily)d(omitted)i(man)m(y)g(imp)s(ortan)m(t)f (results)505 1837 y(and)35 b(ev)m(en)h(en)m(tire)f(fruitful)d(lines)i (of)h(in)m(v)m(estigation.)h(There)f(is)f(a)h(w)m(ealth)h(of)f(op)s(en) 505 1944 y(questions)h(in)g(this)g(area;)i(w)m(e)g(men)m(tion)e(a)i (few)e(b)s(elo)m(w,)h(but)f(refer)h(the)g(reader)g(to)505 2052 y(Miller)29 b(and)h(Nies)g([94)q(])g(for)g(a)h(more)g (comprehensiv)m(e)f(list.)588 2160 y(Although)20 b(w)m(e)h(do)f(not)h 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y(b)s(o)s(ok)29 b(b)m(y)g(Nies)f([105)r(])h(con)m(tains)g(details) f(on)h(some)g(of)g(the)g(results)f(here,)h(in)f(partic-)505 3132 y(ular)c(where)h(the)g(application)e(of)j(randomness)e(notions)g (in)g(computabilit)m(y)f(theory)505 3240 y(is)37 b(concerned)h(\(for)h (instance)e(lo)m(wness)h(prop)s(erties)e(and)h(priorit)m(y-free)g (solutions)505 3348 y(to)29 b(P)m(ost's)g(problem\).)d(It)i(also)g(con) m(tains)g(a)g(c)m(hapter)g(on)g(formalizing)e(the)i(in)m(tuitiv)m(e)505 3456 y(notion)i(of)h(randomness)e(via)h(e\013ectiv)m(e)i(descriptiv)m (e)d(set)i(theory)-8 b(,)32 b(studied)c(in)h([49)q(].)588 3685 y Fu(x)p Ft(2.)53 b(Sets,)35 b(measure,)f(and)h(martingales.)588 3810 y(2.1.)53 b(Sets)41 b(and)g(measure.)k FA(The)35 b(Can)m(tor)i(space)f(of)g(all)e(in\014nite)g(binary)g(se-)505 3918 y(quences)f(is)e(denoted)h(b)m(y)h(2)1456 3885 y Fx(!)1506 3918 y FA(.)g(This)d(space)j(is)e(endo)m(w)m(ed)i(with)e(the) h(tree)h(top)s(ology)-8 b(,)505 4026 y(whic)m(h)30 b(has)g(as)g(basic)g (clop)s(en)f(sets)1454 4205 y([)p Fw(\033)s FA(])d(:=)f Fr(f)p Fw(X)33 b Fr(2)25 b FA(2)1990 4168 y Fx(!)2066 4205 y FA(:)h Fw(\033)i Fr(\036)d Fw(X)7 b Fr(g)p Fw(;)505 4384 y FA(where)37 b Fw(\033)i Fr(2)d FA(2)1008 4352 y Fx()e Fw(d)p FA(\()p Fw(\033)s FA(0\))d(+)e Fw(d)p FA(\()p Fw(\033)s FA(1\))p Fw(:)505 3282 y FA(A)33 b(\(sup)s(er\)martingale)f Fw(d)h Fs(suc)-5 b(c)g(e)g(e)g(ds)36 b(on)k FA(a)34 b(set)f Fw(A)g FA(if)f(lim)15 b(sup)2608 3304 y Fx(n)p Fq(!1)2811 3282 y Fw(d)p FA(\()p Fw(A)2992 3270 y Fz(\026)3059 3282 y Fw(n)p FA(\))29 b(=)g Fr(1)p FA(.)505 3390 y(W)-8 b(e)40 b(sa)m(y)f(that)g Fw(d)f FA(succeeds)h(on,)f(or)h Fs(c)-5 b(overs)7 b FA(,)39 b(a)g(class)f Fr(A)g(\022)g FA(2)2660 3357 y Fx(!)2749 3390 y FA(if)g Fw(d)g FA(succeeds)h(on)505 3498 y(ev)m(ery)33 b Fw(A)27 b Fr(2)g(A)p FA(.)32 b(The)f Fs(suc)-5 b(c)g(ess)34 b(set)41 b Fw(S)5 b FA([)p Fw(d)p FA(])32 b(of)g Fw(d)g FA(is)f(the)g(class)h(of)g(all)e(sets)i(on)g(whic)m(h)e Fw(d)505 3606 y FA(succeeds.)k(The)f(reader)g(should)e(think)g(of)i(a)h (martingale)e(as)i(a)f(b)s(etting)f(strategy)-8 b(.)505 3714 y(The)30 b(function)e Fw(d)i FA(assigns)e(a)i(p)s(ortion)f(of)g (our)g(capital)h(to)g(b)s(e)f(b)s(et)g(on)h(the)g(string)e Fw(\033)s FA(.)505 3822 y(The)37 b(success)f(set)i(of)e Fw(d)h FA(is)f(th)m(us)g(the)h(collection)f(of)h(sets)g(on)f(whic)m(h)g (this)f(b)s(etting)505 3929 y(strategy)d(allo)m(ws)e(us)g(to)h (increase)f(our)g(capital)g(arbitrarily)d(m)m(uc)m(h.)588 4037 y(The)39 b(follo)m(wing)f(classical)g(result)g(sho)m(ws)h(ho)m(w)h (the)f(concept)h(of)f(a)h(martingale)505 4145 y(relates)31 b(to)g(measurabilit)m(y)-8 b(.)588 4310 y FB(Theorem)34 b FA(2.1)h(\(Ville)29 b([135)r(]\))p FB(.)46 b Fs(F)-7 b(or)36 b(any)g(class)g Fr(A)30 b(\022)g FA(2)2548 4277 y Fx(!)2634 4310 y Fs(the)36 b(fol)5 b(lowing)36 b(state-)505 4418 y(ments)e(ar)-5 b(e)33 b(e)-5 b(quivalent:)563 4545 y FA(\(i\))42 b Fr(A)32 b Fs(has)h(L)-5 b(eb)g(esgue)33 b(me)-5 b(asur)g(e)34 b(zer)-5 b(o,)538 4653 y FA(\(ii\))41 b Fs(Ther)-5 b(e)33 b(exists)g(a)g(martingale)h(that)g(suc)-5 b(c)g(e)g(e)g(ds)34 b(on)f Fr(A)p Fs(.)588 4817 y FA(Martingales)26 b(will)e(pro)m(v)m(e)j(imp)s(ortan)m(t)e(when)g(w)m(e)i(lo)s(ok)e(at)i (re\014nemen)m(ts)f(of)g(classi-)505 4925 y(cal)d(Martin-L\177)-45 b(of)23 b(randomness.)f(Martingales)h(are)g(the)g(k)m(ey)h(to)f(lo)s (oking)f(at)i(measure)505 5033 y(and)29 b(Hausdor\013)g(dimension)e(in) i(small)f(classes)h(suc)m(h)g(as)h(p)s(olynomial)d(time,)j(a)g(fact)505 5141 y(\014rst)g(realized)g(b)m(y)g(Lutz)g([81)q(].)h(\(See)g(also)g (Lutz)f([84)q(].\))p eop %%Page: 6 6 6 5 bop 505 363 a FD(6)319 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y Fu(x)p Ft(3.)53 b(Three)e(approac)m(hes)i(to)e(randomness.)45 b FA(Historically)2860 508 y Fy(1)2898 541 y FA(,)g(there)g(ha)m(v)m(e) 505 649 y(b)s(een)22 b(three)g(main)f(approac)m(hes)i(to)g(the)f (de\014nition)e(of)i(an)h(algorithmically)c(random)505 757 y(sequence.)31 b(They)f(are)h(via)f(what)g(w)m(e)h(call)563 902 y(\(i\))42 b(the)30 b(measure-theoretic)i(paradigm,)538 1010 y(\(ii\))41 b(the)30 b(unpredictabilit)m(y)d(paradigm,)j(and)513 1118 y(\(iii\))40 b(the)30 b(incompressibilit)m(y)c(paradigm.)588 1263 y Ft(3.1.)53 b(The)43 b(measure-theoretic)g(paradigm.)i FA(Among)38 b(the)g(oldest)f(de\014ni-)505 1371 y(tions)31 b(of)g(randomness)e(are)j(those)f(sa)m(ying)g(that)g(a)g(random)g(set)g (should)e(ha)m(v)m(e)j(cer-)505 1479 y(tain)f(sto)s(c)m(hastic)h(prop)s (erties.)e(F)-8 b(or)32 b(instance,)f(a)g(random)g(set)g(should)e(ha)m (v)m(e)k(ab)s(out)505 1587 y(as)c(man)m(y)g(0's)g(as)f(1's.)i(V)-8 b(on)29 b(Mises,)f(in)f(his)g(remark)-5 b(able)28 b(pap)s(er)f([98)q (],)i(de\014ned)e(a)i(no-)505 1695 y(tion)22 b(of)g(randomness)f(based) h(on)g(suc)m(h)g(sto)s(c)m(hastic)h(prop)s(erties,)e(and)h(noted)g (that)h(for)505 1803 y(an)m(y)j Fs(c)-5 b(ountable)34 b FA(collection)25 b(of)h(suc)m(h)f(prop)s(erties)f(a)j(nonempt)m(y)e (notion)g(of)h(random-)505 1910 y(ness)34 b(results.)e(But)i(he)f(did)f (not)i(ha)m(v)m(e)h(a)f(canonical)f(c)m(hoice)h(of)g(suc)m(h)f(a)h (coun)m(table)505 2018 y(collection)39 b(at)h(hand.)e(Later)i(Ch)m(urc) m(h)e(made)h(the)g(connection)g(with)f(the)h(theory)505 2126 y(of)c(computabilit)m(y)e(b)m(y)i(suggesting)f(that)h(one)g (should)e(tak)m(e)j(all)e Fs(c)-5 b(omputable)43 b FA(sto-)505 2234 y(c)m(hastic)d(prop)s(erties.)d(Martin-L\177)-45 b(of)38 b(then)g(noted)h(that)g(these)g(are)g(a)g(sp)s(ecial)e(kind)505 2342 y(of)32 b(measure)f(zero)h(subsets)e(of)h(2)1622 2309 y Fx(!)1673 2342 y FA(,)g(and)g(that)h(a)f(more)g(general)g(and)g (smo)s(oth)g(de\014-)505 2450 y(nition)g(could)g(b)s(e)g(obtained)g(b)m (y)h(considering)e(all)h(e\013ectiv)m(ely)h(measure)g(zero)h(sets.)505 2558 y(W)-8 b(e)36 b(discuss)d(this)h(approac)m(h)h(of)g(Martin-L\177) -45 b(of)34 b(b)s(elo)m(w.)h(F)-8 b(or)35 b(more)g(discussion)d(and)505 2666 y(references)45 b(on)f(the)g(original)e(sto)s(c)m(hasticit)m(y)j (approac)m(h)g(see)f(Am)m(b)s(os-Spies)f(and)505 2774 y(Ku)m(\024)-43 b(cera)32 b([1)q(].)588 2882 y(The)21 b(measure-theoretic)h(paradigm)e(is)g(that)h(the)g(random)f(sets)i (should)d(b)s(e)h(those)505 2990 y(with)34 b(no)h Fs(e\013e)-5 b(ctively)36 b(r)-5 b(ar)g(e)44 b FA(prop)s(erties.)34 b(If)g(a)h(prop)s(ert)m(y)g(constitutes)g(an)g(e\013ectiv)m(e)505 3098 y(n)m(ull)29 b(set,)i(then)f(a)h(random)e(set)i(should)e(not)h(ha) m(v)m(e)i(this)d(prop)s(ert)m(y)-8 b(.)588 3206 y(A)29 b(collection)f(of)h(sets)g(that)g(is)f(e\013ectiv)m(ely)h(en)m (umerated)g(is)e(a)i(\006)2792 3173 y Fy(0)2792 3230 y(1)2831 3206 y Fs(-class)p FA(.)g(W)-8 b(e)30 b(can)505 3314 y(represen)m(t)i(a)f(\006)1041 3281 y Fy(0)1041 3338 y(1)1080 3314 y FA(-class)g Fw(U)41 b FA(as)1540 3246 y Fp(S)1615 3341 y Fx(\033)r Fq(2)p Fx(W)1786 3314 y FA([)p Fw(\033)s FA(])31 b(for)g(some)h(pre\014x-free)e(c.e.)j(set)e (of)h(strings)505 3429 y Fw(W)13 b FA(.)34 b(W)-8 b(e)36 b(sa)m(y)e(that)h Fw(W)47 b FA(is)33 b(a)h Fs(pr)-5 b(esentation)44 b FA(of)34 b Fw(U)10 b FA(.)34 b(Whenev)m(er)h(w)m(e)g(men)m(tion)f(a)g (\006)3325 3396 y Fy(0)3325 3454 y(1)3364 3429 y FA(-)505 3537 y(class)d Fw(U)10 b FA(,)31 b(w)m(e)g(assume)f(w)m(e)h(ha)m(v)m(e) h(a)f(\014xed)f(presen)m(tation)h Fw(W)43 b FA(of)31 b Fw(U)10 b FA(,)30 b(and)h(iden)m(tify)e Fw(U)505 3645 y FA(with)k Fw(W)13 b FA(.)33 b(So,)g(for)h(instance,)f(for)g Fw(\033)h Fr(2)c FA(2)1916 3612 y Fx()e Fw(d)890 616 y Fq(0)913 649 y FA(\()p Fw(\033)s FA(\))33 b(for)e(all)f Fw(\033)s FA(.)i(The)f(existence)h(of) f(an)h(optimal)e(c.e.)j(sup)s(ermartingale)505 757 y(is)24 b(also)g(implicit)d(in)i(Levin's)h(construction)g(of)g(a)h(univ)m (ersal)d(c.e.)k(semimeasure)e(\(see)505 865 y(Zv)m(onkin)32 b(and)f(Levin)h([143)r(]\).)h(The)f(fact)h(that)g(there)g(is)f(no)g (optimal)g(c.e.)h(martin-)505 973 y(gale)40 b(is)f(one)h(of)g(the)f (reasons)h(to)g(consider)f(sup)s(ermartingales.)f(\(This)g(fact)i(w)m (as)505 1081 y(implicit)28 b(in)h([143)q(];)i(see)g([33)q(])g(for)f(a)h (pro)s(of.\))588 1206 y Ft(3.3.)53 b(The)38 b(incompressibilit)m(y)g (paradigm.)45 b FA(A)33 b(third)e(approac)m(h)j(to)f(de\014n-)505 1314 y(ing)f(the)h(notion)f(of)h(a)g(random)f(set)i(is)d(the)i(one)g (essen)m(tially)f(due)g(to)i(Kolmogoro)m(v)505 1421 y([64)r(].)j(Here)h (w)m(e)g(regard)f(a)h Fs(string)46 b FA(as)37 b(random)g(iff)f(it)h (has)g(no)g(short)g(description,)505 1529 y(that)h(is,)f(there)g(is)g (no)g(short)g(program)g(to)h(generate)g(the)g(string,)e(meaning)h(that) 505 1637 y(the)30 b(only)e(w)m(a)m(y)i(to)g(generate)g(it)f(is)f(essen) m(tially)g(to)i(hardwire)d(it)h(in)m(to)h(the)h(mac)m(hine.)505 1745 y(\(As)d(opp)s(osed)e(to,)h(e.g.,)i(101010)g(rep)s(eated)e(1000)i (times,)e(whic)m(h)e(can)i(b)s(e)f(generated)505 1853 y(b)m(y)32 b(a)g(short)g(program.\))g(W)-8 b(e)33 b(then)e(use)h(this)f (idea)g(to)h(de\014ne)g(randomness)e(of)i(sets)505 1961 y(b)m(y)e(considering)e(the)h(lengths)g(of)h(shortest)g(descriptions)e (of)h(its)g(initial)e(segmen)m(ts.)505 2069 y(W)-8 b(e)38 b(men)m(tion)e(only)f(a)i(few)f(basic)f(results)g(ab)s(out)h(Kolmogoro) m(v)h(complexit)m(y;)g(for)505 2177 y(more)31 b(on)f(the)h(sub)5 b(ject,)30 b(see)h(Li)f(and)f(Vit\023)-45 b(an)m(yi)30 b([79)q(])h(or)f(Calude)f([14)q(].)588 2293 y Ft(3.3.1.)48 b Fs(Plain)39 b(Kolmo)-5 b(gor)g(ov)43 b(c)-5 b(omplexity.)47 b FA(Fix)37 b(a)h(univ)m(ersal)e(T)-8 b(uring)36 b(mac)m(hine)505 2401 y Fw(U)10 b FA(.)33 b(Giv)m(en)g(a)g(string)f Fw(\033)h Fr(2)c FA(2)1460 2368 y Fx()f Fr(j)p Fw(\033)s Fr(j)33 b(\000)f Fw(k)s FA(.)49 b(\(This)e(de\014ni-)505 3402 y(tion)37 b(will)e(only)h(b)s(e)h(used)f(in)g(this)g(section,)h(and)g (should)e(not)j(b)s(e)e(confused)h(with)505 3509 y(the)43 b(notion)f(of)g Fw(n)p FA(-randomness)f(w)m(e)i(will)c(in)m(tro)s(duce) i(in)g(Section)h(12.\))i(An)e(easy)505 3617 y(coun)m(ting)f(argumen)m (t)g(sho)m(ws)g(that)g(random)f(strings)g(exist:)h(F)-8 b(or)42 b(eac)m(h)g Fw(n)p FA(,)e(there)505 3731 y(are)667 3663 y Fp(P)763 3690 y Fx(n)p Fq(\000)p Fx(k)r Fq(\000)p Fy(1)763 3758 y Fx(i)p Fy(=0)1009 3731 y FA(2)1054 3698 y Fq(\000)p Fx(i)1180 3731 y FA(=)i(2)1338 3698 y Fx(n)p Fq(\000)p Fx(k)1505 3731 y Fr(\000)27 b FA(1)41 b(programs)f(of)h (length)f Fw(<)i(n)26 b Fr(\000)h Fw(k)s FA(,)41 b(so)f(there)h(are)505 3846 y(at)k(least)f(2)901 3813 y Fx(n)977 3846 y Fr(\000)29 b FA(2)1122 3813 y Fx(n)p Fq(\000)p Fx(k)1292 3846 y FA(+)g(1)44 b(man)m(y)g Fw(k)s FA(-random)f(strings)g(of)h(length)f Fw(n)p FA(.)g(F)-8 b(or)45 b(ev)m(ery)505 3954 y Fw(k)s FA(,)g(the)g(set)g(of)f Fw(k)s FA(-random)g(strings)f(is)h(an)g Fs(immune)52 b FA(\005)2489 3921 y Fy(0)2489 3978 y(1)2572 3954 y FA(set,)46 b(i.e.,)e(it)g(do)s(es)g(not)505 4062 y(con)m(tain)37 b(an)m(y)f(in\014nite)d(c.e.)k(subsets.)f(As)f(a)i (function,)d Fw(C)42 b FA(is)35 b(not)h(computable.)g(If)505 4169 y Fw(m)p FA(\()p Fw(x)p FA(\))26 b(=)f(min)o Fr(f)p Fw(C)7 b FA(\()p Fw(y)s FA(\))25 b(:)h Fw(y)i Fz(>)d Fw(x)p Fr(g)p FA(,)30 b(then)f Fw(m)g FA(is)f(un)m(b)s(ounded)f(\(b)s (ecause)j(w)m(e)g(ev)m(en)m(tually)505 4277 y(run)23 b(out)h(of)h(short)e(programs\),)i(but)e(gro)m(ws)i(slo)m(w)m(er)f (than)g(an)m(y)g(partial)f(computable)505 4385 y(function.)588 4493 y(W)-8 b(e)33 b(w)m(ould)e(lik)m(e)g(to)i(extend)e(the)h (de\014nition)e(of)i(randomness)e(for)i(\014nite)f(strings)505 4601 y(to)i(a)g(de\014nition)d(for)i(in\014nite)e(strings.)h(Naiv)m (ely)-8 b(,)33 b(w)m(e)g(could)e(de\014ne)g(a)i(set)g Fw(A)f FA(to)h(b)s(e)505 4709 y(random)40 b(iff)f(there)i(is)f(a)g Fw(k)k FA(suc)m(h)c(that)h(ev)m(ery)g Fw(\033)46 b Fr(\036)c Fw(A)e FA(is)g Fw(k)s FA(-random.)g(Ho)m(w)m(ev)m(er,)505 4817 y(Martin-L\177)-45 b(of)26 b(sho)m(w)m(ed)g(that)h(suc)m(h)e(sets) i(do)e(not)h(exist!)g(This)f(can)h(b)s(e)f(seen)h(using)e(the)505 4925 y(follo)m(wing)i(argumen)m(t)h(of)g(Katse\013)h([59)q(].)f(Let)h Fw(\033)2102 4939 y Fy(0)2141 4925 y Fw(;)15 b(\033)2233 4939 y Fy(1)2273 4925 y Fw(;)g(:)g(:)g(:)43 b FA(b)s(e)26 b(an)h(e\013ectiv)m(e)i(listing)24 b(of)505 5033 y(all)i(strings,)f (with)g Fr(j)p Fw(\033)1223 5047 y Fx(n)1270 5033 y Fr(j)h FA(=)f(log)16 b Fw(n)p FA(.)26 b(If)g Fw(A)1835 5021 y Fz(\026)1899 5033 y Fw(m)f FA(=)g Fw(\033)2152 5047 y Fx(n)2198 5033 y FA(,)i(then)f(from)g(the)g Fs(length)34 b FA(of)26 b Fw(A)3277 5021 y Fz(\026)3340 5033 y Fw(n)505 5141 y FA(w)m(e)37 b(can)g(reco)m(v)m(er)i Fw(A)1239 5129 y Fz(\026)1313 5141 y Fw(m)p FA(.)d(Th)m(us,)g(to)h(generate)h Fw(A)2307 5129 y Fz(\026)2381 5141 y Fw(n)p FA(,)e(w)m(e)h(need)f(only) g(generate)p eop %%Page: 9 9 9 8 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)929 b FD(9)505 541 y FA(the)33 b(string)e Fw(\034)42 b FA(suc)m(h)31 b(that)i Fw(A)1507 529 y Fz(\026)1573 541 y Fw(n)28 b FA(=)f Fw(\033)1806 555 y Fx(n)1853 541 y Fw(\034)42 b FA(and)32 b(compute)g Fw(n)f FA(from)h Fr(j)p Fw(\034)10 b Fr(j)29 b FA(=)e Fw(n)21 b Fr(\000)g FA(log)c Fw(n)p FA(,)505 649 y(whic)m(h)30 b(giv)m(es)g(us)g Fw(\033)1157 663 y Fx(n)1204 649 y FA(.)g(This)f(sho)m(ws)h(that)h(for)f(ev)m(ery)h Fw(A)p FA(,)1251 795 y Fr(9)1302 758 y Fq(1)1377 795 y Fw(n)15 b FA([)p Fw(C)7 b FA(\()p Fw(A)1672 783 y Fz(\026)1735 795 y Fw(n)p FA(\))25 b Fz(6)g Fw(n)20 b Fr(\000)g FA(log)c Fw(n)k FA(+)g Fw(O)s FA(\(1\)])p Fw(:)588 941 y FA(The)32 b(basic)f(in)m(tuition)e(for)j(what)f(go)s(es)i(wrong)e(in)f(trying)h (to)i(use)e(plain)f(Kolmo-)505 1049 y(goro)m(v)39 b(complexit)m(y)e(to) h(de\014ne)e(randomness)g(for)h(sets)g(is)g(that)g(the)h(Kolmogoro)m(v) 505 1157 y(complexit)m(y)30 b(of)h Fw(\034)40 b FA(should)28 b(b)s(e)h(the)h(length)g(of)g(the)g(shortest)h(string)e Fw(\033)k FA(suc)m(h)d(that)h Fw(\034)505 1265 y FA(can)h(b)s(e)e (obtained)g(from)g(the)h Fs(bits)39 b FA(of)31 b Fw(\033)s FA(.)g(The)f Fs(length)39 b FA(of)31 b Fw(\033)j FA(seems)d(to)g(giv)m (e)h(an)e(ad-)505 1373 y(ditional)c(log)17 b Fw(n)27 b FA(man)m(y)h(bits)f(of)h(information.)e(This)g(idea)i(is)f (explicitly)e(used)i(ab)s(o)m(v)m(e)505 1481 y(to)33 b(demonstrate)f(that,)h(using)e(plain)e(Kolmogoro)m(v)k(complexit)m(y) -8 b(,)32 b(w)m(e)h(will)c(alw)m(a)m(ys)505 1589 y(get)i(complexit)m(y) e(oscillations)e(in)h(the)i(initial)c(segmen)m(ts)k(of)g(a)f(set.)h (Levin)f([76)q(,)g(77)q(],)505 1697 y(G\023)-45 b(acs)41 b([45)r(],)f(and)f(Chaitin)f([20)r(,)i(22)q(])g(in)m(tro)s(duced)e (metho)s(ds)i(to)g(get)i(around)d(this)505 1805 y(problem,)29 b(as)i(w)m(e)g(no)m(w)f(discuss.)588 1912 y(It)24 b(should)d(b)s(e)i (noted)g(that)h(there)f Fs(is)31 b FA(no)m(w)23 b(a)h(plain)d (complexit)m(y)i(c)m(haracterization)505 2020 y(of)31 b(1-randomness,)f(giv)m(en)h(recen)m(tly)f(b)m(y)g(Miller)f(and)h(Y)-8 b(u)30 b([95)q(].)588 2182 y FB(Theorem)k FA(3.5)h(\(Miller)29 b(and)h(Y)-8 b(u)30 b([95)q(]\))p FB(.)47 b Fs(A)35 b(set)h Fw(A)g Fs(is)g FA(1)p Fs(-r)-5 b(andom)39 b(iff)c(for)i(every)505 2296 y(c)-5 b(omputable)35 b(function)d Fw(g)37 b Fs(such)32 b(that)1804 2228 y Fp(P)1900 2323 y Fx(n)1962 2296 y FA(2)2007 2263 y Fq(\000)p Fx(g)r Fy(\()p Fx(n)p Fy(\))2226 2296 y Fw(<)25 b Fr(1)p Fs(,)1382 2444 y Fw(C)7 b FA(\()p Fw(A)1582 2432 y Fz(\026)1645 2444 y Fw(n)p FA(\))25 b Fz(>)g Fw(n)20 b Fr(\000)g Fw(g)s FA(\()p Fw(n)p FA(\))h Fr(\000)f Fw(O)s FA(\(1\))p Fw(:)588 2606 y Ft(3.3.2.)48 b Fs(Pr)-5 b(e\014x-fr)g(e)g(e)33 b(Kolmo)-5 b(gor)g(ov)36 b(c)-5 b(omplexity.)47 b FA(Call)29 b(a)i(T)-8 b(uring)28 b(mac)m(hine)j Fw(M)40 b FA(a)505 2714 y Fs(pr)-5 b(e\014x-fr)g(e)g(e) 36 b FA(\(or)29 b Fs(self-delimiting)8 b FA(\))28 b Fs(machine)36 b FA(if)27 b(dom)o(\()p Fw(M)10 b FA(\))26 b Fr(\022)f FA(2)2651 2681 y Fx()f Fr(j)p Fw(\033)s Fr(j)c FA(+)f(log)d Fr(j)p Fw(\033)s Fr(j)k FA(+)e Fw(c)p FA(.)588 1323 y(The)30 b(follo)m(wing)f(is)h(a)g (fundamen)m(tal)g(result)f(ab)s(out)h Fw(K)7 b FA(.)588 1484 y FB(Theorem)34 b FA(3.6)h(\(Coun)m(ting)30 b(Theorem,)g(Chaitin)f ([20)q(]\))p FB(.)579 1610 y FA(\(i\))41 b(max)900 1536 y Fp(\010)953 1610 y Fw(K)7 b FA(\()p Fw(\033)s FA(\))26 b(:)g Fr(j)p Fw(\033)s Fr(j)g FA(=)f Fw(n)1521 1536 y Fp(\011)1599 1610 y FA(=)g Fw(n)19 b FA(+)h Fw(K)7 b FA(\()p Fw(n)p FA(\))20 b Fr(\006)g Fw(O)s FA(\(1\))p Fs(.)553 1733 y FA(\(ii\))716 1656 y Fp(\014)716 1710 y(\014)746 1733 y Fr(f)p Fw(\033)45 b FA(:)d Fr(j)p Fw(\033)s Fr(j)g FA(=)f Fw(n)g Fr(^)h Fw(K)7 b FA(\()p Fw(\033)s FA(\))42 b Fz(6)f Fw(n)26 b FA(+)g Fw(K)7 b FA(\()p Fw(n)p FA(\))27 b Fr(\000)f Fw(r)s Fr(g)2376 1656 y Fp(\014)2376 1710 y(\014)2448 1733 y Fz(6)41 b FA(2)2605 1700 y Fx(n)p Fq(\000)p Fx(r)r Fy(+)p Fx(O)r Fy(\(1\))2941 1733 y Fs(,)h(wher)-5 b(e)42 b(the)716 1841 y(c)-5 b(onstant)35 b Fw(O)s FA(\(1\))e Fs(do)-5 b(es)34 b(not)f(dep)-5 b(end)34 b(on)f Fw(n)f Fs(and)i Fw(r)s Fs(.)588 2002 y FA(W)-8 b(e)25 b(are)e(no)m(w)g(in)f(a) i(p)s(osition)d(to)j(de\014ne)e(randomness)g(for)h(sets)g(in)f(terms)h (of)g(initial)505 2110 y(segmen)m(t)32 b(complexit)m(y)-8 b(.)588 2272 y FB(Definition)35 b FA(3.7)h(\(Levin)44 b([76)q(],)56 b(Chaitin)44 b([20)q(]\))p FB(.)i FA(A)d(set)g Fw(A)g FA(is)f Fs(L)-5 b(evin-Chaitin)505 2380 y(r)g(andom)47 b FA(\(or)37 b(Kolmogoro)m(v-Levin-Chaitin)e(random\))i(if)f(there)h (is)f(a)h(constan)m(t)h Fw(c)505 2487 y FA(suc)m(h)30 b(that)h Fw(K)7 b FA(\()p Fw(A)1120 2475 y Fz(\026)1183 2487 y Fw(n)p FA(\))25 b Fz(>)g Fw(n)20 b Fr(\000)g Fw(c)31 b FA(for)f(ev)m(ery)h Fw(n)p FA(.)588 2649 y(Again)g(w)m(e)f(arriv)m(e) h(at)g(the)f(same)h(concept)g(of)g(randomness)e(as)i(ab)s(o)m(v)m(e.) 588 2810 y FB(Theorem)j FA(3.8)h(\(Sc)m(hnorr,)30 b(see)h(Chaitin)e ([20)q(]\))p FB(.)46 b Fs(A)32 b(set)h Fw(A)26 b Fr(2)g FA(2)2774 2777 y Fx(!)2857 2810 y Fs(is)33 b(Martin-L\177)-46 b(of)505 2918 y(r)-5 b(andom)35 b(iff)e(it)f(is)h(L)-5 b(evin-Chaitin)33 b(r)-5 b(andom.)588 3079 y FA(The)46 b(pro)s(of)f(of)h(Theorem)f(3.8)i(giv)m(en)f(b)s(elo)m(w)f(uses)h(the)g (e\013ectiv)m(e)h(v)m(ersion)e(of)505 3187 y(Kraft's)e(Inequalit)m(y)f ([65)q(],)h(whic)m(h)f(is)g(a)h(fundamen)m(tal)e(to)s(ol)i(in)f(this)f (area.)j(This)505 3295 y(v)m(ersion)32 b(is)g(usually)e(kno)m(wn)i(as)h (the)g(Kraft-Chaitin)e(Theorem,)h(as)h(it)f(app)s(ears)g(in)505 3403 y(Chaitin)c([20)q(],)j(but)e(according)h(to)h(G\023)-45 b(acs,)31 b(it)e(app)s(eared)g(earlier)g(in)g(Levin's)g(disser-)505 3511 y(tation)36 b([75)q(].)f(W)-8 b(e)36 b(retain)f(the)g(terminology) f(\\Kraft-Chaitin")g(for)h(this)f(theorem)505 3619 y(and)29 b(certain)f(asso)s(ciated)i(concepts)f(de\014ned)f(b)s(elo)m(w)g(for)h (the)g(sak)m(e)h(of)f(terminolog-)505 3727 y(ical)i(consistency)h(with) e(man)m(y)h(of)h(the)g(pap)s(ers)e(w)m(e)i(discuss.)e(If)h Fw(U)42 b FA(is)30 b(a)i(pre\014x-free)505 3834 y(mac)m(hine)h(then)f (the)h(op)s(en)f(set)h(presen)m(ted)f(b)m(y)h(the)g(domain)e(of)i Fw(U)42 b FA(is)32 b(measurable.)505 3942 y(The)27 b(Kraft-Chaitin)e (Theorem)i(is)f(a)h(kind)e(of)j(con)m(v)m(erse)g(to)g(this)e(fact,)i (and)e(implies)505 4050 y(that)38 b(eac)m(h)f(left-c.e.)h(real)e(is)g (the)g(measure)h(of)f(the)h(domain)f(of)g(some)h(pre\014x-free)505 4158 y(mac)m(hine.)588 4319 y FB(Theorem)d FA(3.9)h(\(Levin)30 b([75)q(],)h(Chaitin)d([20)q(]\))p FB(.)46 b Fs(L)-5 b(et)24 b Fr(h)p Fw(d)2475 4333 y Fx(i)2504 4319 y Fw(;)15 b(\034)2584 4333 y Fx(i)2612 4319 y Fr(i)2647 4333 y Fx(i)p Fq(2)p Fx(!)2792 4319 y Fs(b)-5 b(e)24 b(a)f(c)-5 b(omputable)505 4427 y(se)g(quenc)g(e)28 b(of)f(p)-5 b(airs)29 b(\(which)f(we)g(c)-5 b(al)5 b(l)37 b FA(requests)p Fs(\),)27 b(with)i Fw(d)2432 4441 y Fx(i)2485 4427 y Fr(2)c Fo(N)40 b Fs(and)28 b Fw(\034)2875 4441 y Fx(i)2929 4427 y Fr(2)c FA(2)3059 4394 y Fx()24 b FA(1.)k(In)f(this)f(case)i Fw(x)1759 1889 y Fx(n)p Fy(+1)1923 1922 y FA(is)f(the)g(same)h(as)f Fw(x)2550 1889 y Fx(n)2624 1922 y FA(except)i(for)e(p)s(ositions)505 2051 y Fw(j;)15 b(:)g(:)g(:)33 b(;)15 b(d)807 2065 y Fx(n)p Fy(+1)945 2051 y FA(,)36 b(where)g(w)m(e)h(ha)m(v)m(e)g Fw(x)1682 2013 y Fx(n)p Fy(+1)1682 2078 y Fx(j)1854 2051 y FA(=)e(0)i(and)e Fw(x)2276 2018 y Fx(n)p Fy(+1)2276 2073 y Fx(m)2448 2051 y FA(=)g(1)i(for)f Fw(j)k(<)35 b(m)g Fz(6)g Fw(d)3232 2065 y Fx(n)p Fy(+1)3369 2051 y FA(.)505 2178 y(Let)41 b Fw(\033)730 2192 y Fx(n)p Fy(+1)909 2178 y FA(=)g Fw(\026)1076 2145 y Fx(n)1076 2204 y(j)1123 2178 y FA(0)1168 2145 y Fx(d)1204 2154 y Fl(n)p Fn(+1)1324 2145 y Fq(\000)p Fx(j)1415 2178 y FA(.)g(F)-8 b(or)41 b Fw(m)g(<)g(j)46 b FA(or)40 b Fw(m)i(>)f(d)2372 2192 y Fx(n)p Fy(+1)2509 2178 y FA(,)g(let)f Fw(\026)2771 2145 y Fx(n)p Fy(+1)2771 2201 y Fx(m)2949 2178 y FA(=)h Fw(\026)3116 2145 y Fx(n)3116 2201 y(m)3183 2178 y FA(,)f(and)505 2302 y(for)45 b Fw(j)54 b(<)48 b(m)h Fz(6)f Fw(d)1164 2316 y Fx(n)p Fy(+1)1302 2302 y FA(,)c(let)h Fw(\026)1572 2269 y Fx(n)p Fy(+1)1572 2324 y Fx(m)1757 2302 y FA(=)k Fw(\026)1932 2269 y Fx(n)1932 2327 y(j)1979 2302 y FA(0)2024 2269 y Fx(m)p Fq(\000)p Fx(j)t Fq(\000)p Fy(1)2268 2302 y FA(1.)c(Then)f Fw(S)2691 2316 y Fx(n)p Fy(+1)2876 2302 y FA(=)49 b Fr(f)p Fw(\033)3093 2316 y Fx(i)3170 2302 y FA(:)g Fw(i)g Fz(6)505 2424 y Fw(n)21 b FA(+)g(1)p Fr(g)i([)d(f)p Fw(\026)967 2391 y Fx(n)p Fy(+1)967 2447 y Fx(m)1133 2424 y FA(:)28 b Fw(x)1238 2391 y Fx(n)p Fy(+1)1238 2447 y Fx(m)1403 2424 y FA(=)g(1)p Fr(g)33 b FA(is)e(the)h(same)h(as)f Fw(S)2275 2438 y Fx(n)2354 2424 y FA(except)h(that)f Fw(\026)2894 2391 y Fx(n)2894 2450 y(j)2973 2424 y FA(is)f(replaced)505 2534 y(b)m(y)d(a)g(pairwise)d (incomparable)h(set)i(of)g(sup)s(erstrings)d(of)i Fw(\026)2513 2501 y Fx(n)2513 2560 y(j)2560 2534 y FA(.)h(This)e(clearly)g(ensures) 505 2644 y(that)31 b Fw(S)758 2658 y Fx(n)p Fy(+1)926 2644 y FA(is)e(pre\014x-free.)588 2752 y(This)20 b(completes)j(the)f (de\014nition)d(of)j(the)g Fw(\033)2037 2766 y Fx(i)2065 2752 y FA(.)g(Eac)m(h)h(\014nite)e(subset)g(of)h Fr(f)p Fw(\033)3016 2766 y Fy(0)3056 2752 y Fw(;)15 b(\033)3148 2766 y Fy(1)3188 2752 y Fw(;)g(:)g(:)g(:)h Fr(g)505 2860 y FA(is)35 b(con)m(tained)h(in)f(some)h Fw(S)1420 2874 y Fx(n)1467 2860 y FA(,)g(and)f(is)g(hence)h(pre\014x-free.)f(Th)m(us)g (the)h(whole)f(set)h(is)505 2968 y(pre\014x-free.)31 b(Since)f(the)h Fw(\033)1408 2982 y Fx(i)1467 2968 y FA(are)h(c)m(hosen)f(e\013ectiv)m(ely)-8 b(,)33 b(w)m(e)e(can)g (de\014ne)g(a)g(pre\014x-free)505 3076 y(mac)m(hine)f Fw(M)41 b FA(b)m(y)30 b(letting)g Fw(M)10 b FA(\()p Fw(\033)1589 3090 y Fx(i)1618 3076 y FA(\))25 b(=)g Fw(\034)1814 3090 y Fx(i)1872 3076 y FA(for)31 b(eac)m(h)g Fw(i)p FA(.)1067 b Fr(a)588 3213 y FA(W)-8 b(e)38 b(call)e(an)h(e\013ectiv)m(ely)h(en)m (umerated)f(set)g(of)g(requests)g Fr(h)p Fw(d)2686 3227 y Fx(i)2715 3213 y Fw(;)15 b(\034)2795 3227 y Fx(i)2823 3213 y Fr(i)2858 3227 y Fx(i)p Fq(2)p Fx(!)3016 3213 y FA(suc)m(h)37 b(that)505 3255 y Fp(P)601 3350 y Fx(i)645 3323 y FA(2)690 3290 y Fq(\000)p Fx(d)781 3300 y Fl(i)837 3323 y Fz(6)25 b FA(1)31 b(a)f Fs(Kr)-5 b(aft-Chaitin)34 b(set)p FA(.)c(The)g Fs(weight)39 b FA(of)30 b(this)f(set)i(is)2803 3255 y Fp(P)2899 3350 y Fx(i)2942 3323 y FA(2)2987 3290 y Fq(\000)p Fx(d)3078 3300 y Fl(i)3109 3323 y FA(.)g(As)f(an)505 3431 y(illustration)j(of)i(the)g(use)g(of)g(the)g(Kraft-Chaitin)e (Theorem,)i(w)m(e)h(giv)m(e)g(a)f(pro)s(of)f(of)505 3539 y(Sc)m(hnorr's)c(Theorem)g(3.8.)588 3676 y FB(Pr)n(oof)k(of)g(Theorem)f (3.8.)42 b FA(\(Only)28 b(if)7 b(\))28 b(Let)h Fw(U)38 b FA(b)s(e)28 b(the)h(univ)m(ersal)e(pre\014x-free)505 3784 y(mac)m(hine)d(relativ)m(e)h(to)g(whic)m(h)e Fw(K)31 b FA(is)23 b(de\014ned.)h(Let)h Fw(R)2284 3799 y Fx(k)2352 3784 y FA(=)2447 3715 y Fp(S)2523 3784 y Fr(f)p FA([)p Fw(\033)s FA(])i(:)e Fw(K)7 b FA(\()p Fw(\033)s FA(\))26 b Fz(6)f Fr(j)p Fw(\033)s Fr(j)8 b(\000)g Fw(k)s Fr(g)p FA(.)505 3891 y(Notice)36 b(that)g Fr(f)p Fw(R)1111 3906 y Fx(k)1154 3891 y Fr(g)1199 3906 y Fx(k)r Fq(2)p Fx(!)1370 3891 y FA(is)e(a)h(uniformly)d(c.e.)k(sequence)f(of)g(\006)2686 3858 y Fy(0)2686 3916 y(1)2725 3891 y FA(-classes.)g(W)-8 b(e)36 b(no)m(w)505 3999 y(sho)m(w)31 b(that)g(it)f(is)f(a)i (Martin-L\177)-45 b(of)30 b(test.)588 4107 y(Let)37 b Fw(P)815 4122 y Fx(k)894 4107 y FA(b)s(e)f(the)g(set)g(of)h Fw(\033)i FA(suc)m(h)d(that)g Fw(K)7 b FA(\()p Fw(\033)s FA(\))36 b Fz(6)e Fr(j)p Fw(\033)s Fr(j)25 b(\000)f Fw(k)39 b FA(but)c Fw(K)7 b FA(\()p Fw(\034)j FA(\))36 b Fw(>)e Fr(j)p Fw(\034)10 b Fr(j)25 b(\000)e Fw(k)505 4215 y FA(for)k(all)f Fw(\034)36 b Fr(\036)24 b Fw(\033)s FA(.)k(Then)e Fw(P)1335 4230 y Fx(k)1405 4215 y FA(is)g(pre\014x-free)h(and)f Fw(R)2162 4230 y Fx(k)2230 4215 y FA(=)2326 4147 y Fp(S)2402 4242 y Fx(\033)r Fq(2)p Fx(P)2536 4254 y Fl(k)2579 4215 y FA([)p Fw(\033)s FA(].)i(F)-8 b(urthermore,)27 b(for)505 4330 y(eac)m(h)32 b Fw(\033)c Fr(2)d Fw(P)934 4345 y Fx(k)977 4330 y FA(,)30 b(there)h(is)e(a)i(string)e Fw(\033)1745 4297 y Fq(\003)1815 4330 y FA(suc)m(h)g(that)i Fw(U)10 b FA(\()p Fw(\033)2378 4297 y Fq(\003)2418 4330 y FA(\))26 b(=)f Fw(\033)33 b FA(and)d Fr(j)p Fw(\033)2917 4297 y Fq(\003)2957 4330 y Fr(j)25 b Fz(6)g Fr(j)p Fw(\033)s Fr(j)c(\000)e Fw(k)s FA(.)505 4444 y(Since)30 b Fw(U)40 b FA(is)29 b(pre\014x-free,)1392 4376 y Fp(P)1488 4471 y Fx(\033)r Fq(2)p Fx(P)1622 4483 y Fl(k)1679 4444 y FA(2)1724 4411 y Fq(\000j)p Fx(\033)1841 4388 y Fj(\003)1878 4411 y Fq(j)1927 4444 y Fz(6)2023 4376 y Fp(P)2119 4471 y Fx(U)7 b Fy(\()p Fx(\034)h Fy(\))-12 b Fq(#)2310 4444 y FA(2)2355 4411 y Fq(\000j)p Fx(\034)8 b Fq(j)2518 4444 y Fz(6)25 b FA(1.)31 b(So)754 4654 y Fw(\026)p FA(\()p Fw(R)913 4669 y Fx(k)956 4654 y FA(\))26 b(=)1133 4568 y Fp(X)1113 4765 y Fx(\033)r Fq(2)p Fx(P)1247 4777 y Fl(k)1301 4654 y FA(2)1346 4617 y Fq(\000j)p Fx(\033)r Fq(j)1512 4654 y Fz(6)1629 4568 y Fp(X)1608 4765 y Fx(\033)r Fq(2)p Fx(P)1742 4777 y Fl(k)1796 4654 y FA(2)1841 4617 y Fq(\000)p Fy(\()p Fq(j)p Fx(\033)1985 4593 y Fj(\003)2022 4617 y Fq(j)p Fy(+)p Fx(k)r Fy(\))2192 4654 y FA(=)f(2)2333 4617 y Fq(\000)p Fx(k)2467 4568 y Fp(X)2446 4765 y Fx(\033)r Fq(2)p Fx(P)2580 4777 y Fl(k)2634 4654 y FA(2)2679 4617 y Fq(\000j)p Fx(\033)2796 4593 y Fj(\003)2832 4617 y Fq(j)2881 4654 y Fz(6)g FA(2)3022 4617 y Fq(\000)p Fx(k)3120 4654 y Fw(:)588 4925 y FA(Th)m(us)38 b Fr(f)p Fw(R)941 4940 y Fx(k)985 4925 y Fr(g)1030 4940 y Fx(k)r Fq(2)p Fx(!)1205 4925 y FA(is)g(a)i(Martin-L\177)-45 b(of)39 b(test.)h(No)m(w)g(if)e Fw(A)h FA(is)f(Martin-L\177)-45 b(of)39 b(random,)505 5033 y(then)28 b Fw(A)35 b(=)-55 b Fr(2)889 4965 y Fp(T)965 5060 y Fx(k)1023 5033 y Fw(R)1092 5048 y Fx(k)1134 5033 y FA(,)28 b(so)g(there)g(is)e(a)i Fw(k)j FA(suc)m(h)c(that)i Fw(K)7 b FA(\()p Fw(A)2374 5021 y Fz(\026)2437 5033 y Fw(n)p FA(\))25 b Fw(>)g(n)15 b Fr(\000)g Fw(k)30 b FA(for)d(all)g Fw(n)p FA(,)g(and)505 5141 y(th)m(us)j Fw(A)h FA(is)e(Levin-Chaitin)f(random.)p eop %%Page: 12 12 12 11 bop 505 363 a FD(12)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FA(\(If)7 b(\))31 b(Let)f Fr(f)p Fw(U)1026 556 y Fx(k)1070 541 y Fr(g)1115 556 y Fx(k)r Fq(2)p Fx(!)1281 541 y FA(b)s(e)f(the)i(univ)m(ersal)d(Martin-L\177)-45 b(of)30 b(test.)h(As)f(discussed)e(ab)s(o)m(v)m(e,)505 649 y(w)m(e)j(iden)m(tify)e(eac)m(h)j Fw(U)1237 664 y Fx(k)1310 649 y FA(with)d(a)h(particular)f(\014xed)h(presen)m(tation)g (of)h Fw(U)2916 664 y Fx(k)2959 649 y FA(.)f(Let)1226 798 y Fw(L)25 b FA(=)g Fr(fhj)p Fw(\033)s Fr(j)d(\000)e Fw(k)s(;)15 b(\033)s Fr(i)26 b FA(:)g Fr(9)p Fw(k)i Fz(>)d FA(1)15 b([)p Fw(\033)29 b Fr(2)c Fw(U)2500 813 y Fy(2)p Fx(k)2578 798 y FA(])p Fr(g)p Fw(:)505 946 y FA(Then)30 b Fw(L)g FA(is)f(a)i(Kraft-Chaitin)e(set,)i(since)f(it)f(is)h(clearly)f (c.e.)j(and)747 1022 y Fp(X)749 1219 y Fx(k)r Fk(>)p Fy(1)931 1022 y Fp(X)894 1218 y Fx(\033)r Fq(2)p Fx(U)1031 1230 y Fn(2)p Fl(k)1115 1108 y FA(2)1160 1070 y Fq(\000j)p Fx(\033)r Fq(j)p Fy(+)p Fx(k)1420 1108 y FA(=)1516 1022 y Fp(X)1517 1219 y Fx(k)r Fk(>)p Fy(1)1662 1108 y FA(2)1707 1070 y Fx(k)1751 1108 y Fw(\026)p FA(\()p Fw(U)1903 1123 y Fy(2)p Fx(k)1981 1108 y FA(\))25 b Fz(6)2137 1022 y Fp(X)2139 1219 y Fx(k)r Fk(>)p Fy(1)2284 1108 y FA(2)2329 1070 y Fx(k)2372 1108 y FA(2)2417 1070 y Fq(\000)p Fy(2)p Fx(k)2575 1108 y FA(=)2671 1022 y Fp(X)2673 1219 y Fx(k)r Fk(>)p Fy(1)2818 1108 y FA(2)2863 1070 y Fq(\000)p Fx(k)2986 1108 y FA(=)g(1)p Fw(:)505 1356 y FA(So)h(b)m(y)f(the)g(Kraft-Chaitin)f (Theorem)h(there)g(is)f(a)i Fw(c)f FA(suc)m(h)g(that)h Fw(K)7 b FA(\()p Fw(\033)s FA(\))26 b Fz(6)f Fr(j)p Fw(\033)s Fr(j)10 b(\000)g Fw(k)j FA(+)d Fw(c)505 1464 y FA(for)26 b(all)f Fw(k)k Fz(>)c FA(1)h(and)f Fw(\033)k Fr(2)c Fw(U)1406 1479 y Fy(2)p Fx(k)1484 1464 y FA(.)h(No)m(w)h(if)e Fw(A)h FA(is)f(Levin-Chaitin)e(random)i(then)h(there)g(is)505 1572 y(a)j Fw(k)j FA(suc)m(h)c(that)h Fw(K)7 b FA(\()p Fw(A)1268 1560 y Fz(\026)1331 1572 y Fw(n)p FA(\))25 b Fz(>)g Fw(n)16 b Fr(\000)g Fw(k)j FA(+)d Fw(c)29 b FA(for)f(all)f Fw(n)p FA(,)h(whic)m(h)f(implies)f(that)j Fw(A)36 b(=)-56 b Fr(2)25 b Fw(U)3291 1587 y Fy(2)p Fx(k)3369 1572 y FA(,)505 1680 y(and)30 b(hence)h(that)g Fw(A)f FA(is)g(Martin-L\177)-45 b(of)30 b(random.)1222 b Fr(a)588 1843 y FB(Remark.)45 b FA(W)-8 b(e)44 b(will)c(often)j(use)g(the)g (Kraft-Chaitin)e(Theorem)i(in)f(conjunc-)505 1951 y(tion)33 b(with)f(Kleene's)h(Recursion)e(Theorem)i(\(see)h([122)r(,)f(Theorem)g (I)s(I.3.1]\).)h(If)f(w)m(e)505 2059 y(uniformly)27 b(en)m(umerate)j (Kraft-Chaitin)e(sets)h Fw(K)2186 2073 y Fx(n)2263 2059 y FA(for)g Fw(n)c Fr(2)g Fo(N)42 b FA(and)29 b(use)g(the)g(Kraft-)505 2167 y(Chaitin)h(theorem)j(to)g(obtain)f(corresp)s(onding)e (pre\014x-free)h(mac)m(hines)h Fw(M)3088 2181 y Fx(n)3135 2167 y FA(,)h(then,)505 2275 y(since)h(the)h(pro)s(of)e(of)i(the)f (Kraft-Chaitin)f(Theorem)h(is)f(e\013ectiv)m(e,)j(the)f(Recursion)505 2383 y(Theorem)40 b(implies)c(that)k(there)g(is)e(an)h Fw(n)g FA(suc)m(h)g(that)h Fw(M)2501 2397 y Fx(n)2588 2383 y FA(has)f(co)s(ding)f(constan)m(t)505 2491 y Fw(n)p FA(.)g(Th)m(us,)f(when)g(w)m(e)h(build)d(a)j(pre\014x-free)f(mac)m (hine)g Fw(M)48 b FA(via)37 b(the)h(Kraft-Chaitin)505 2599 y(Theorem,)27 b(w)m(e)h(can)f(assume)g(w)m(e)h(kno)m(w)f(the)g(co) s(ding)f(constan)m(t)j Fw(c)e FA(of)g Fw(M)37 b FA(in)26 b(adv)-5 b(ance)505 2707 y(and)27 b(use)g(it)f(in)g(en)m(umerating)h (our)g(requests)g(\(as)g(long)g(as)g(the)h(en)m(umeration)e(yields)505 2815 y(a)31 b(Kraft-Chaitin)e(set)i(for)f(an)m(y)g(v)-5 b(alue)30 b(of)h Fw(c)p FA(\).)588 2978 y(Recen)m(tly)-8 b(,)35 b(Miller)d(and)h(Y)-8 b(u)34 b([95)q(])g(ha)m(v)m(e)h(pro)m(v)m (ed)f(signi\014can)m(t)f(generalizations)g(of)505 3086 y(Sc)m(hnorr's)d(Theorem)h(3.8,)h(whic)m(h)e(can)h(b)s(e)f(in)m (terpreted)g(as)i(sa)m(ying)e(that)i Fs(not)i(only)505 3194 y(is)e(a)g(set)g FA(1)p Fs(-r)-5 b(andom)34 b(iff)d(its)h(initial) g(se)-5 b(gment)33 b(c)-5 b(omplexity)33 b(is)f(always)h(ab)-5 b(ove)33 b Fw(n)p Fs(,)e(but)505 3302 y(the)h(initial)f(se)-5 b(gment)31 b(c)-5 b(omplexity)33 b(of)e FA(1)p Fs(-r)-5 b(andom)33 b(sets)e(is)g(\\wel)5 b(l)31 b(ab)-5 b(ove")31 b Fw(n)f Fs(most)i(of)505 3410 y(the)h(time)p FA(.)588 3573 y FB(Theorem)h FA(3.10)i(\(Miller)29 b(and)g(Y)-8 b(u)31 b([95)q(]\))p FB(.)46 b Fs(L)-5 b(et)33 b Fw(A)g Fs(b)-5 b(e)32 b FA(1)p Fs(-r)-5 b(andom.)563 3714 y FA(\(i\))701 3645 y Fp(P)797 3740 y Fx(n)859 3714 y FA(2)904 3681 y Fx(n)p Fq(\000)p Fx(K)5 b Fy(\()p Fx(A)1147 3669 y Fk(\026)1176 3681 y Fx(n)p Fy(\))1275 3714 y Fw(<)25 b Fr(1)p Fs(.)538 3832 y FA(\(ii\))41 b Fs(F)-7 b(or)33 b(any)h(function)e Fw(f)42 b Fs(such)33 b(that)1882 3763 y Fp(P)1978 3858 y Fx(n)2040 3832 y FA(2)2085 3799 y Fq(\000)p Fx(f)7 b Fy(\()p Fx(n)p Fy(\))2309 3832 y FA(=)25 b Fr(1)p Fs(,)1399 3982 y Fr(9)1450 3944 y Fq(1)1524 3982 y Fw(n)15 b FA([)p Fw(K)7 b FA(\()p Fw(A)1831 3970 y Fz(\026)1894 3982 y Fw(n)p FA(\))26 b Fw(>)f(n)19 b FA(+)h Fw(f)10 b FA(\()p Fw(n)p FA(\)])p Fw(:)588 4145 y FB(Pr)n(oof.)41 b FA(P)m(art)36 b(\(ii\))e(follo)m(ws)g(easily)g (from)g(part)h(\(i\).)g(W)-8 b(e)36 b(giv)m(e)f(a)h(pro)s(of)e(of)h (part)505 4253 y(\(i\))c(due)e(to)j(Nies.)e(Let)1240 4420 y Fw(d)p FA(\()p Fw(\033)s FA(\))d(=)1537 4334 y Fp(X)1535 4529 y Fx(\034)8 b Fk(4)p Fx(\033)1686 4420 y FA(2)1731 4383 y Fq(j)p Fx(\034)g Fq(j\000)p Fx(K)d Fy(\()p Fx(\034)j Fy(\))2047 4420 y FA(+)2141 4334 y Fp(X)2138 4525 y Fx(\033)r Fq(\036)p Fx(\034)2290 4420 y FA(2)2335 4383 y Fq(j)p Fx(\033)r Fq(j\000)p Fx(K)d Fy(\()p Fx(\034)j Fy(\))2635 4420 y Fw(:)505 4688 y FA(It)34 b(is)e(easy)i(to)f(c)m(hec)m(k)i(that)f Fw(d)f FA(is)f(a)i(c.e.)g (martingale,)f(and)g(that)2744 4620 y Fp(P)2840 4715 y Fx(n)2903 4688 y FA(2)2948 4655 y Fx(n)p Fq(\000)p Fx(K)5 b Fy(\()p Fx(A)3191 4643 y Fk(\026)3220 4655 y Fx(n)p Fy(\))3324 4688 y Fz(6)505 4798 y FA(lim)15 b(sup)784 4820 y Fx(n)846 4798 y Fw(d)p FA(\()p Fw(A)1022 4786 y Fz(\026)1085 4798 y Fw(n)p FA(\).)31 b(But)f(this)f(limsup)f(is)h (\014nite,)h(since)g Fw(A)g FA(is)f(1-random.)292 b Fr(a)588 4925 y FA(P)m(art)32 b(\(i\))f(of)g(Theorem)g(3.10)i(is)d(kno)m(wn)g (as)h(the)h(Ample)e(Excess)h(Theorem,)g(and)505 5033 y(impro)m(v)m(es)f(a)f(result)g(of)g(Chaitin)e([22)r(],)j(who)e(sho)m (w)m(ed)i(that)g(if)e Fw(A)i FA(is)e(1-random)i(then)505 5141 y(lim)632 5155 y Fx(n)694 5141 y Fw(K)7 b FA(\()p Fw(A)906 5129 y Fz(\026)969 5141 y Fw(n)p FA(\))q Fr(\000)q Fw(n)24 b FA(=)h Fr(1)p FA(.)c(P)m(art)g(\(ii\))f(had)g(b)s(een)g(pro)m (v)m(ed)h(for)f Fs(c)-5 b(omputable)29 b FA(functions)p eop %%Page: 13 13 13 12 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(13)505 541 y Fw(f)40 b FA(b)m(y)29 b(Solo)m(v)-5 b(a)m(y)31 b([126)r(].)f(It)g(is)f(easy)h(to)h(c)m(hec)m(k)h(that)e(if) f Fw(A)h FA(is)f(not)h(1-random,)g(then)g(\(i\))505 649 y(and)d(\(ii\))g(fail,)f(so)i(these)f(conditions)f(are)i(actually)f Fs(e)-5 b(quivalent)36 b FA(to)28 b(1-randomness.)588 834 y(In)f(this)e(section)i(w)m(e)g(ha)m(v)m(e)h(seen)f(that)h(for)e (certain)h(natural)f(notions)g(of)h(random-)505 942 y(ness,)g(v)-5 b(arious)26 b(approac)m(hes)i(to)g(the)f(de\014nition)e(of)i (randomness)f(lead)g(to)i(the)f(same)505 1050 y(class,)36 b(the)g(1-random)g(sets.)g(Later)h(w)m(e)f(will)d(lo)s(ok)j(at)g (certain)g(criticisms)e(of)i(this)505 1158 y(notion)e(and)g(v)-5 b(ariations)34 b(generated)h(b)m(y)g(suc)m(h)f(criticisms.)e(W)-8 b(e)36 b(no)m(w)e(turn)g(to)h(our)505 1266 y(\014rst)30 b(approac)m(h)h(to)g(calibrating)e(randomness.)588 1473 y Fu(x)p Ft(4.)53 b(Solo)m(v)-6 b(a)m(y)47 b(reducibilit)m(y)-9 b(,)46 b(and)g(a)f(c)m(haracterization)h(of)g(1-random)505 1581 y(left-c.e.)31 b(reals.)46 b FA(Notice)28 b(that)f(a)h (consequence)f(of)h(the)f(Kraft-Chaitin)e(Theorem)505 1689 y(is)i(that)h Fs(a)j(r)-5 b(e)g(al)31 b(is)f(left-c.e.)f(iff)h(it) g(is)g(the)h(me)-5 b(asur)g(e)31 b(of)f(the)h(domain)g(of)g(a)f(pr)-5 b(e\014x-fr)g(e)g(e)505 1797 y(machine)p FA(.)27 b(Th)m(us,)e(in)f (this)h(setting,)g(left-c.e.)i(reals)e(o)s(ccup)m(y)h(the)g(same)g (place)f(as)h(c.e.)505 1905 y(sets)31 b(do)f(in)g(classical)f (computabilit)m(y)g(theory)-8 b(.)588 2013 y(W)g(e)28 b(ha)m(v)m(e)f(not)f(y)m(et)h(seen)f(an)g(example)g(of)g(a)g(1-random)g (set.)h(Since)e(the)h(univ)m(ersal)505 2121 y(Martin-L\177)-45 b(of)39 b(test)g(de\014nes)e(an)i(e\013ectiv)m(ely)g(n)m(ull)d(set,)k (the)e(collection)g(of)h(Martin-)505 2229 y(L\177)-45 b(of)29 b(random)f(reals)g(has)h(measure)f(1.)i(One)e(migh)m(t)g(w)m (ell)g(ask)h(to)h(what)e(exten)m(t)i(they)505 2337 y(resem)m(ble)45 b(one)g(another,)g(and)f(ho)m(w)g(suc)m(h)h(resem)m(blance)f(migh)m(t)h (b)s(e)f(measured.)505 2445 y(Since)33 b(for)h(eac)m(h)h Fw(k)i FA(the)d(set)g Fw(P)1544 2460 y Fx(k)1618 2445 y FA(=)c Fr(f)p Fw(A)i FA(:)f Fr(8)p Fw(n)15 b FA([)p Fw(K)7 b FA(\()p Fw(A)2284 2433 y Fz(\026)2353 2445 y Fw(n)p FA(\))30 b Fz(>)h Fw(n)22 b Fr(\000)g Fw(k)s FA(])p Fr(g)35 b FA(is)e(a)h(\005)3142 2412 y Fy(0)3142 2469 y(1)3181 2445 y FA(-class)505 2553 y(con)m(taining)43 b(only)g(1-random)g(sets,)i(there)e(are)h(1-random)f(sets)h(of)g(lo)m (w)f(T)-8 b(uring)505 2661 y(degree)46 b(\(b)m(y)g(the)f(Lo)m(w)h (Basis)f(Theorem)g([57)q(]\),)h(and)f(1-random)g(left-c.e.)i(reals)505 2769 y(\(since)41 b(the)h(leftmost)f(path)g(of)g(a)g(\005)1795 2736 y Fy(0)1795 2793 y(1)1835 2769 y FA(-class)g(is)f(a)i(left-c.e.)g (real\).)g(On)e(the)h(other)505 2877 y(hand,)30 b(w)m(e)h(ha)m(v)m(e)g (the)g(follo)m(wing)e(result.)588 3039 y FB(Theorem)34 b FA(4.1)h(\(Ku)m(\024)-43 b(cera)32 b([66)q(]\))p FB(.)46 b Fs(If)31 b(a)g FA(1)p Fs(-r)-5 b(andom)33 b(set)d Fw(A)h Fs(has)h(c.e.)d(de)-5 b(gr)g(e)g(e,)32 b(then)505 3147 y Fw(A)26 b Fr(\021)670 3161 y Fi(T)750 3147 y Fr(;)795 3114 y Fq(0)819 3147 y Fs(.)588 3309 y FA(The)k(most)h(famous)f (explicitly)e(de\014ned)h(1-random)i(set)f(is)g(Chaitin's)e(\012)i([20) q(]:)1362 3474 y(\012)25 b(:=)1596 3387 y Fp(X)1574 3589 y Fx(U)7 b Fy(\()p Fx(\033)r Fy(\))-12 b Fq(#)1765 3474 y FA(2)1810 3436 y Fq(\000j)p Fx(\033)r Fq(j)1977 3474 y FA(=)25 b Fw(\026)p FA(\(dom\()p Fw(U)10 b FA(\)\))p Fw(;)505 3724 y FA(where)30 b Fw(U)41 b FA(is)29 b(a)i(univ)m(ersal)e (pre\014x-free)g(mac)m(hine.)i(This)d(is)i(the)g Fs(halting)k(pr)-5 b(ob)g(ability)505 3832 y FA(of)31 b Fw(U)10 b FA(.)30 b(Notice)i(that)f(\012)f(is)f(a)i(left-c.e.)h(real.)588 3940 y(Here)f(is)e(a)i(short)f(pro)s(of)f(that)i(\012)f(is)f(1-random.) h(It)g(follo)m(ws)g(from)f(Theorem)h(4.1,)505 4048 y(and)41 b(is)g(not)h(hard)f(to)h(c)m(hec)m(k)h(directly)-8 b(,)41 b(that)i(\012)g Fr(\021)2320 4062 y Fy(T)2419 4048 y Fr(;)2464 4015 y Fq(0)2488 4048 y FA(,)f(so)g(in)e(particular)g(\012)h (is)505 4156 y(not)36 b(rational.)f(Th)m(us)f(for)i(eac)m(h)g Fw(n)f FA(there)h(is)e(an)i Fw(s)f FA(with)f(\012)2542 4170 y Fx(s)2612 4144 y Fz(\026)2683 4156 y Fw(n)g FA(=)f(\012)2975 4144 y Fz(\026)3047 4156 y Fw(n)p FA(,)i(where)505 4265 y(\012)571 4279 y Fx(s)633 4265 y FA(:=)754 4196 y Fp(P)850 4291 y Fx(U)898 4299 y Fl(s)931 4291 y Fy(\()p Fx(\033)r Fy(\))-12 b Fq(#)1072 4265 y FA(2)1117 4232 y Fq(\000j)p Fx(\033)r Fq(j)1258 4265 y FA(.)29 b(W)-8 b(e)30 b(build)c(a)j (pre\014x-free)g(mac)m(hine)f Fw(M)10 b FA(.)29 b(By)h(the)f(Recursion) 505 4379 y(Theorem,)23 b(w)m(e)g(can)g(assume)f(w)m(e)h(kno)m(w)g(its)f (co)s(ding)g(constan)m(t)i Fw(c)e FA(in)g Fw(U)10 b FA(.)23 b(Whenev)m(er)g(at)505 4487 y(a)30 b(stage)g Fw(s)e FA(w)m(e)i(ha)m(v)m (e)g Fw(U)1285 4501 y Fx(s)1322 4487 y FA(\()p Fw(\034)10 b FA(\))26 b(=)f(\012)1630 4501 y Fx(s)1691 4475 y Fz(\026)1755 4487 y Fw(n)j FA(for)h(some)g Fw(\034)39 b FA(suc)m(h)28 b(that)i Fr(j)p Fw(\034)10 b Fr(j)26 b Fw(<)f(n)16 b Fr(\000)h Fw(c)29 b FA(\(whic)m(h)505 4595 y(means)f(that)g Fw(K)1052 4609 y Fx(U)1112 4595 y FA(\(\012)1213 4609 y Fx(s)1275 4583 y Fz(\026)1338 4595 y Fw(n)p FA(\))d Fw(<)g(n)15 b Fr(\000)g Fw(c)p FA(\),)28 b(w)m(e)g(c)m(ho)s(ose)h(a)f (string)e Fw(\026)i FA(not)g(in)e(the)i(range)g(of)505 4703 y Fw(U)567 4717 y Fx(s)639 4703 y FA(and)34 b(declare)g Fw(M)10 b FA(\()p Fw(\034)g FA(\))34 b(=)e Fw(\026)p FA(.)i(Since)g Fw(M)44 b FA(is)34 b(co)s(ded)g(in)g Fw(U)44 b FA(with)34 b(co)s(ding)f(constan)m(t)505 4811 y Fw(c)p FA(,)h(there)f(m)m(ust)g(b)s(e)f(a)h Fw(\027)39 b FA(suc)m(h)33 b(that)g Fr(j)p Fw(\027)6 b Fr(j)30 b Fz(6)f Fr(j)p Fw(\034)10 b Fr(j)22 b FA(+)g Fw(c)30 b(<)f(n)k FA(and)f Fw(U)10 b FA(\()p Fw(\027)c FA(\))30 b(=)f Fw(M)10 b FA(\()p Fw(\034)g FA(\))30 b(=)g Fw(\026)p FA(.)505 4925 y(Since)f Fw(\026)35 b(=)-55 b Fr(2)25 b FA(rng)q(\()p Fw(U)1138 4939 y Fx(s)1175 4925 y FA(\),)31 b(it)e(follo)m(ws)g(that)h Fw(\027)41 b(=)-55 b Fr(2)25 b FA(dom\()p Fw(U)2281 4939 y Fx(s)2318 4925 y FA(\),)31 b(so)f(\012)19 b Fr(\000)f FA(\012)2760 4939 y Fx(s)2822 4925 y Fz(>)25 b FA(2)2963 4892 y Fq(\000j)p Fx(\027)t Fq(j)3126 4925 y Fw(>)g FA(2)3267 4892 y Fq(\000)p Fx(n)3369 4925 y FA(,)505 5033 y(and)j(hence)f(\012) 1020 5021 y Fz(\026)1083 5033 y Fw(n)e Fr(6)p FA(=)g(\012)1325 5047 y Fx(s)1387 5021 y Fz(\026)1450 5033 y Fw(n)p FA(.)j(This)e(pro)s (cedure)g(ensures)h(that)h(if)f Fr(j)p Fw(\034)10 b Fr(j)26 b Fw(<)f(n)15 b Fr(\000)g Fw(c)27 b FA(then)505 5141 y Fw(U)10 b FA(\()p Fw(\034)g FA(\))26 b Fr(6)p FA(=)f(\012)910 5129 y Fz(\026)973 5141 y Fw(n)p FA(,)31 b(whence)f Fw(K)1479 5155 y Fx(U)1538 5141 y FA(\(\012)1664 5129 y Fz(\026)1727 5141 y Fw(n)p FA(\))25 b Fz(>)g Fw(n)20 b Fr(\000)g Fw(c)31 b FA(for)f(all)f Fw(n)p FA(.)p eop %%Page: 14 14 14 13 bop 505 363 a FD(14)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FA(Of)27 b(course,)h(the)g(de\014nition)d(of)j(\012)f(dep)s(ends) e(on)j(the)g(c)m(hoice)g(of)g(univ)m(ersal)d(pre\014x-)505 649 y(free)45 b(mac)m(hine,)f(so)g(w)m(e)g(should)e(really)h(sa)m(y)i (that)g(\012)e(is)g Fs(a)52 b FA(halting)42 b(probabilit)m(y)-8 b(,)505 757 y(rather)21 b(than)f Fs(the)28 b FA(halting)19 b(probabilit)m(y)-8 b(,)19 b(as)i(it)f(is)g(commonly)g(referred)g(to.)h (Ho)m(w)m(ev)m(er,)505 865 y(the)40 b(analog)f(of)g(\012)g(in)e (classical)i(computabilit)m(y)e(is)h(the)h(halting)f(problem)f Fr(;)3235 832 y Fq(0)3299 865 y FA(:=)505 973 y Fr(f)p Fw(i)42 b FA(:)e(\010)754 987 y Fx(i)782 973 y FA(\()p Fw(i)p FA(\))26 b Fr(#g)p FA(,)41 b(and)e(w)m(e)g(usually)f(talk)h(ab)s (out)g Fs(the)47 b FA(halting)38 b(problem,)g(although)505 1081 y(that)i(situation)e(is)h(analogous,)g(in)f(that)i(the)g (de\014nition)d(of)i Fr(;)2710 1048 y Fq(0)2773 1081 y FA(dep)s(ends)e(on)i(the)505 1189 y(c)m(hoice)24 b(of)e(en)m (umeration)g(of)h(the)f(partial)f(computable)h(functions.)f(What)j (allo)m(ws)d(us)505 1297 y(to)32 b(disregard)d(this)g(en)m (umeration-dep)s(endence)h(is)g(Myhill's)e(Theorem)i(\(see)i([122)q(,) 505 1405 y(Theorem)e(I)s(I.4.6]\),)h(whic)m(h)d(sa)m(ys)i(that)g(all)f (halting)f(problems)g(are)i(essen)m(tially)e(the)505 1513 y(same,)j(since)f(they)h(are)f(all)g(equiv)-5 b(alen)m(t)30 b(mo)s(dulo)e(a)j(v)m(ery)g(strong)f(reducibilit)m(y)-8 b(.)588 1621 y(Solo)m(v)j(a)m(y)31 b([126)r(])e(recognized)h(this)f (situation,)g(and)g(sough)m(t)h(to)g(in)m(tro)s(duce)f(appro-)505 1729 y(priate)21 b(reducibilities)16 b(to)21 b(establish)f(an)g(analog) h(to)h(Myhill's)c(Theorem.)j(As)f(w)m(e)h(no)m(w)505 1837 y(discuss,)26 b(Solo)m(v)-5 b(a)m(y's)28 b(program)e(has)h(b)s (een)f(recen)m(tly)h(realized)f(b)m(y)h(the)g(join)m(t)f(w)m(ork)h(of) 505 1944 y(sev)m(eral)35 b(authors.)g(Our)e(starting)i(p)s(oin)m(t)e (is)h(the)h(notion)f(of)h(Solo)m(v)-5 b(a)m(y)35 b(reducibilit)m(y)-8 b(,)505 2052 y(or)31 b(domination,)e(in)m(tro)s(duced)g(b)m(y)h(Solo)m (v)-5 b(a)m(y)31 b(in)e(his)g(man)m(uscript)g([126)r(].)588 2194 y FB(Definition)35 b FA(4.2)h(\(Solo)m(v)-5 b(a)m(y)31 b([126)r(]\))p FB(.)46 b FA(W)-8 b(e)44 b(sa)m(y)f(that)g(a)g(real)f Fw(\013)h FA(is)f Fs(Solovay)k(r)-5 b(e-)505 2302 y(ducible)43 b FA(to)38 b Fw(\014)j FA(\(or)c(that)g Fw(\014)42 b Fs(dominates)k Fw(\013)p FA(\),)37 b(and)f(write)g Fw(\013)g Fz(6)2667 2316 y Fy(S)2745 2302 y Fw(\014)5 b FA(,)37 b(if)f(there)g(are)h(a)505 2410 y(constan)m(t)30 b Fw(c)e FA(and)g(a)g(partial)f(computable)g(function)g Fw(f)37 b FA(so)29 b(that)f(for)g(all)f Fw(q)h Fr(2)d Fo(Q)43 b FA(with)505 2518 y Fw(q)29 b(<)24 b(\014)5 b FA(,)1547 2626 y Fw(\013)21 b Fr(\000)f Fw(f)10 b FA(\()p Fw(q)s FA(\))25 b Fw(<)g(c)p FA(\()p Fw(\014)h Fr(\000)20 b Fw(q)s FA(\))p Fw(:)588 2768 y FA(One)34 b(w)m(a)m(y)g(to)h(lo)s(ok)e (at)i(this)d(de\014nition)g(is)h(that)h(a)g(sequence)g(of)g(rationals)f (con-)505 2876 y(v)m(erging)39 b(to)h Fw(\014)j FA(can)c(b)s(e)f(used)g (to)i(generate)g(one)f(con)m(v)m(erging)g(to)h Fw(\013)f FA(at)g(the)g(same)505 2984 y(rate)34 b(or)e(faster.)i(Indeed,)e(if)f (w)m(e)i(ha)m(v)m(e)h(increasing)d(computable)i(sequences)f(of)h(ra-) 505 3092 y(tionals)39 b Fr(f)p Fw(r)894 3106 y Fx(n)941 3092 y Fr(g)986 3106 y Fx(n)p Fq(2)p Fx(!)1166 3092 y FA(and)g Fr(f)p Fw(q)1438 3106 y Fx(n)1485 3092 y Fr(g)1530 3106 y Fx(n)p Fq(2)p Fx(!)1710 3092 y FA(con)m(v)m(erging)h(to)g Fw(\013)g FA(and)f Fw(\014)5 b FA(,)39 b(resp)s(ectiv)m(ely)-8 b(,)40 b(then)505 3200 y Fw(f)10 b FA(\()p Fw(q)636 3214 y Fx(n)683 3200 y FA(\))28 b Fr(#)41 b FA(for)g(all)e Fw(n)p FA(,)i(and)f(for)h(eac)m(h)h Fw(n)e FA(w)m(e)i(can)f(e\013ectiv) m(ely)h(\014nd)d(a)i Fw(k)j FA(suc)m(h)d(that)505 3308 y Fw(f)10 b FA(\()p Fw(q)636 3322 y Fx(n)683 3308 y FA(\))27 b Fw(<)f(r)883 3323 y Fx(k)952 3308 y Fw(<)h(\013)p FA(.)32 b(This)d(observ)-5 b(ation)31 b(yields)f(the)h(follo)m(wing)f(c)m (haracterization)i(of)505 3416 y(Solo)m(v)-5 b(a)m(y)32 b(reducibilit)m(y)-8 b(.)588 3558 y FB(Lemma)34 b FA(4.3)i(\(Calude,)29 b(Coles,)h(Hertling,)g(and)g(Khoussaino)m(v)54 b([16)r(]\))p FB(.)46 b Fs(F)-7 b(or)24 b(left-)505 3666 y(c.e.)32 b(r)-5 b(e)g(als)34 b Fw(\013)e Fs(and)h Fw(\014)5 b Fs(,)33 b(let)f Fr(f)p Fw(r)1483 3680 y Fx(n)1530 3666 y Fr(g)1575 3680 y Fx(n)p Fq(2)p Fx(!)1748 3666 y Fs(and)h Fr(f)p Fw(q)2010 3680 y Fx(n)2057 3666 y Fr(g)2102 3680 y Fx(n)p Fq(2)p Fx(!)2275 3666 y Fs(b)-5 b(e)32 b(incr)-5 b(e)g(asing)33 b(c)-5 b(omputable)34 b(se-)505 3774 y(quenc)-5 b(es)32 b(of)h(r)-5 b(ationals)35 b(c)-5 b(onver)g(ging)33 b(to)g Fw(\013)g Fs(and)g Fw(\014)5 b Fs(,)33 b(r)-5 b(esp)g(e)g(ctively.)34 b(Then)e Fw(\013)26 b Fz(6)3153 3788 y Fi(S)3222 3774 y Fw(\014)38 b Fs(iff)505 3882 y(ther)-5 b(e)39 b(exist)e(a)h(total)h(c)-5 b(omputable)39 b(function)e Fw(g)k Fs(and)d(a)g(c)-5 b(onstant)39 b Fw(c)f Fs(such)f(that)i(for)505 3990 y(al)5 b(l)34 b Fw(n)p Fs(,)1520 4098 y Fw(\013)21 b Fr(\000)f Fw(r)1731 4116 y Fx(g)r Fy(\()p Fx(n)p Fy(\))1894 4098 y Fw(<)25 b(c)p FA(\()p Fw(\014)h Fr(\000)20 b Fw(q)2273 4112 y Fx(n)2319 4098 y FA(\))p Fw(:)588 4243 y FA(Solo)m(v)-5 b(a)m(y)45 b([126)r(])f(observ)m(ed)f(that)i(this)d(\\analytic")j(v)m (ersion)e(of)h(m-reducibilit)m(y)505 4351 y(could)36 b(b)s(e)g(used)g(to)h(extend)f(man)m(y)h(results)e(ab)s(out)h(\012)g (to)i(a)f(class)f(of)g(left-c.e.)i(re-)505 4459 y(als)30 b(with)f(a)h(mac)m(hine-indep)s(enden)m(t)e(de\014nition,)g(namely)h (the)i(\012)p Fs(-like)p FA(,)e(or)h Fs(Solovay)505 4567 y(c)-5 b(omplete)p FA(,)31 b(left-c.e.)f(reals,)e(whic)m(h)f(are)i (those)g(left-c.e.)h(reals)e Fw(\013)h FA(suc)m(h)f(that)h Fw(\014)i Fz(6)3268 4581 y Fy(S)3336 4567 y Fw(\013)505 4675 y FA(for)j(all)f(left-c.e.)j(reals)d Fw(\014)5 b FA(.)35 b(\(It)f(is)g(not)g(hard)f(to)i(sho)m(w)f(that)h(\012)e(is)g (\012-lik)m(e.\))h(In)g(par-)505 4783 y(ticular,)29 b(if)f(a)i (left-c.e.)g(real)f(is)g(\012-lik)m(e,)g(then)g(it)g(is)f(1-random.)i (This)d(result)h(follo)m(ws)505 4891 y(from)i(the)h(follo)m(wing)e (prop)s(ert)m(y)g(of)i(Solo)m(v)-5 b(a)m(y)31 b(reducibilit)m(y)-8 b(.)588 5033 y FB(Lemma)34 b FA(4.4)i(\(Solo)m(v)-5 b(a)m(y)31 b([126)r(]\))p FB(.)46 b Fs(If)39 b Fw(\013)f Fz(6)2007 5047 y Fi(S)2089 5033 y Fw(\014)45 b Fs(then)40 b Fw(K)7 b FA(\()p Fw(\013)2609 5021 y Fz(\026)2684 5033 y Fw(n)p FA(\))38 b Fz(6)f Fw(K)7 b FA(\()p Fw(\014)3133 5021 y Fz(\026)3209 5033 y Fw(n)p FA(\))25 b(+)505 5141 y Fw(O)s FA(\(1\))p Fs(.)p eop %%Page: 15 15 15 14 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(15)588 541 y FB(Pr)n(oof)34 b(Sketch.)40 b FA(The)k(pro)s(of)g (relies)g(on)h(the)g(follo)m(wing)e(fact)j(observ)m(ed)f(b)m(y)505 649 y(Solo)m(v)-5 b(a)m(y:)37 b Fs(F)-7 b(or)38 b(e)-5 b(ach)39 b Fw(d)e Fs(ther)-5 b(e)38 b(is)f(a)h Fw(k)j Fs(such)c(that,)h(for)g(al)5 b(l)38 b Fw(n)33 b Fz(>)h FA(1)j Fs(and)i(al)5 b(l)37 b Fw(\033)n(;)15 b(\034)48 b Fs(of)505 757 y(length)36 b Fw(n)f Fs(with)h Fr(j)p FA(0)p Fw(:\033)26 b Fr(\000)c FA(0)p Fw(:\034)10 b Fr(j)31 b Fw(<)f(d)p FA(2)1700 724 y Fq(\000)p Fx(n)1802 757 y Fs(,)35 b(we)h(have)f Fr(j)p Fw(K)7 b FA(\()p Fw(\034)j FA(\))23 b Fr(\000)f Fw(K)7 b FA(\()p Fw(\033)s FA(\))p Fr(j)31 b Fz(6)e Fw(k)s Fs(.)k FA(Solo)m(v)-5 b(a)m(y's)505 865 y(observ)g(ation)39 b(\(whic)m(h)f(is)g(also)h(true)f(for)h Fw(C)7 b FA(\))39 b(follo)m(ws)f(easily)g(from)g(the)h(fact)h(that)505 973 y(there)31 b(are)g(only)e Fw(O)s FA(\()p Fw(d)p FA(\))i(man)m(y)g (suc)m(h)f Fw(\034)40 b FA(for)30 b(a)h(\014xed)f Fw(\033)s FA(.)588 1081 y(No)m(w)h(let)f Fw(f)39 b FA(and)30 b Fw(c)g FA(b)s(e)f(as)i(in)d(De\014nition)h(4.2.)i(Let)g Fw(\014)2404 1095 y Fx(n)2476 1081 y FA(=)25 b(0)p Fw(:)p FA(\()p Fw(\014)2760 1069 y Fz(\026)2823 1081 y Fw(n)p FA(\).)30 b(Since)f Fw(\014)3256 1095 y Fx(n)3333 1081 y FA(is)505 1195 y(rational)k(and)f Fw(\014)27 b Fr(\000)22 b Fw(\014)1248 1209 y Fx(n)1325 1195 y Fw(<)29 b FA(2)1470 1162 y Fq(\000)p Fy(\()p Fx(n)p Fy(+1\))1717 1195 y FA(,)34 b(w)m(e)f(ha)m(v)m(e)h Fw(\013)23 b Fr(\000)e Fw(f)10 b FA(\()p Fw(\014)2438 1209 y Fx(n)2485 1195 y FA(\))30 b Fw(<)g(c)p FA(2)2735 1162 y Fq(\000)p Fy(\()p Fx(n)p Fy(+1\))2982 1195 y FA(.)j(Th)m(us,)g(b)m(y)505 1303 y(the)k(observ)-5 b(ation,)36 b Fw(K)7 b FA(\()p Fw(\013)1394 1291 y Fz(\026)1467 1303 y Fw(n)p FA(\))35 b Fz(6)f Fw(K)7 b FA(\()p Fw(f)j FA(\()p Fw(\014)1957 1317 y Fx(n)2004 1303 y FA(\)\))25 b(+)f Fw(O)s FA(\(1\))37 b(.)f(Since)f Fw(f)46 b FA(is)35 b(computable,)505 1411 y(this)30 b(implies)d(that)k Fw(K)7 b FA(\()p Fw(\013)1391 1399 y Fz(\026)1454 1411 y Fw(n)p FA(\))25 b Fz(6)g Fw(K)7 b FA(\()p Fw(\014)1866 1399 y Fz(\026)1929 1411 y Fw(n)p FA(\))20 b(+)g Fw(O)s FA(\(1\).)997 b Fr(a)588 1551 y FA(Using)29 b(the)h(Kraft-Chaitin)d (Theorem,)j(Calude,)e(Hertling,)h(Khoussaino)m(v,)f(and)505 1659 y(W)-8 b(ang)34 b([17)q(])e(pro)m(v)m(ed)h(that)g(if)e Fw(\013)i FA(is)e(\012-lik)m(e,)h(then)f Fw(\013)i FA(is)e(the)i (halting)e(probabilit)m(y)e(of)505 1767 y(a)h(univ)m(ersal)e (pre\014x-free)g(mac)m(hine.)i(W)-8 b(e)30 b(giv)m(e)g(a)g(short)f(pro) s(of)f(based)h(on)g(a)h(c)m(harac-)505 1875 y(terization)h(of)f(Solo)m (v)-5 b(a)m(y)32 b(reducibilit)m(y)27 b(b)m(y)j(Do)m(wney)-8 b(,)32 b(Hirsc)m(hfeldt,)d(and)h(Nies)g([37)q(].)588 2055 y FB(Theorem)k FA(4.5)h(\(Calude,)30 b(Hertling,)g(Khoussaino)m (v,)f(and)h(W)-8 b(ang)31 b([17)r(]\))p FB(.)46 b Fs(L)-5 b(et)37 b Fw(\013)505 2163 y Fs(b)-5 b(e)33 b(a)f(left-c.e.)g(r)-5 b(e)g(al)34 b(such)e(that)i FA(\012)25 b Fz(6)1750 2177 y Fi(S)1819 2163 y Fw(\013)p Fs(.)33 b(Then)f Fw(\013)h Fs(is)g(a)f(halting)i(pr)-5 b(ob)g(ability.)34 b(That)505 2278 y(is,)f(ther)-5 b(e)33 b(is)g(a)g(universal)g(pr)-5 b(e\014x-fr)g(e)g(e)34 b(machine)2220 2255 y Fp(b)2206 2278 y Fw(U)43 b Fs(such)33 b(that)g Fw(\026)p FA(\(dom\()3013 2255 y Fp(b)3000 2278 y Fw(U)11 b FA(\)\))26 b(=)f Fw(\013)p Fs(.)588 2458 y FB(Pr)n(oof.)41 b FA(Let)26 b Fw(U)35 b FA(b)s(e)24 b(a)i(univ)m(ersal)d(pre\014x-free)h(mac)m(hine)h(with)f (\012)g(=)h Fw(\026)p FA(\(dom\()p Fw(U)10 b FA(\)\).)505 2566 y(In)45 b([37)q(],)h(it)f(is)f(sho)m(wn)h(that)h(\012)j Fz(6)1747 2580 y Fy(S)1841 2566 y Fw(\013)c FA(implies)e(that)j(there)f (are)h(sequences)f(of)505 2674 y(rationals)30 b(0)c(=)f(\012)1113 2688 y Fy(0)1178 2674 y Fw(<)g FA(\012)1340 2688 y Fy(1)1405 2674 y Fw(<)g Fr(\001)15 b(\001)g(\001)47 b FA(and)30 b(0)c(=)f Fw(\013)2055 2688 y Fy(0)2121 2674 y Fw(<)g(\013)2275 2688 y Fy(1)2340 2674 y Fw(<)g Fr(\001)15 b(\001)g(\001)47 b FA(con)m(v)m(erging)32 b(to)f(\012)f(and)505 2782 y Fw(\013)p FA(,)39 b(resp)s(ectiv)m(ely)-8 b(,)37 b(and)g(a)h(constan)m (t)h Fw(c)f FA(suc)m(h)f(that)i(\012)2348 2796 y Fx(s)p Fy(+1)2499 2782 y Fr(\000)25 b FA(\012)2661 2796 y Fx(s)2735 2782 y Fw(<)37 b FA(2)2888 2749 y Fx(c)2923 2782 y FA(\()p Fw(\013)3016 2796 y Fx(s)p Fy(+1)3169 2782 y Fr(\000)24 b Fw(\013)3322 2796 y Fx(s)3359 2782 y FA(\))505 2890 y(for)30 b(all)f Fw(s)p FA(.)h(Assume)f(w)m(e)h(ha)m(v)m(e)i(c)m(hosen) e Fw(c)g FA(large)g(enough)g(so)g(that)h Fw(\013)20 b FA(+)f(2)2962 2857 y Fq(\000)p Fx(c)3077 2890 y Fw(<)25 b FA(1)30 b(and)505 2998 y(2)550 2965 y Fq(\000)p Fx(c)666 2998 y Fw(<)25 b(\013)p FA(.)588 3105 y(No)m(w)43 b Fw(\014)48 b FA(=)c Fw(\013)28 b FA(+)f(2)1250 3073 y Fq(\000)p Fx(c)1340 3105 y FA(\(1)i Fr(\000)e FA(\012\))41 b(is)g(a)h(left-c.e.)g (real,)g(so)f(b)m(y)h(the)f(Kraft-Chaitin)505 3213 y(Theorem)f(there)h (is)e(a)h(pre\014x-free)g(mac)m(hine)g Fw(M)50 b FA(suc)m(h)40 b(that)g Fw(\026)p FA(\(dom\()p Fw(M)10 b FA(\)\))43 b(=)e Fw(\014)5 b FA(,)505 3321 y(and)30 b(w)m(e)h(can)g(assume)f(that) h(there)g(is)e(a)i(string)f Fw(\032)g FA(suc)m(h)g(that)h Fr(j)p Fw(\032)p Fr(j)26 b FA(=)f Fw(c)31 b FA(and)f Fw(M)10 b FA(\()p Fw(\032)p FA(\))g Fr(#)p FA(.)505 3441 y(De\014ne)41 b(a)g(pre\014x-free)g(mac)m(hine)1703 3418 y Fp(b)1690 3441 y Fw(U)51 b FA(b)m(y)40 b(letting)2251 3418 y Fp(b)2238 3441 y Fw(U)10 b FA(\()p Fw(\033)s FA(\))43 b(=)f Fw(M)10 b FA(\()p Fw(\033)s FA(\))42 b(if)e Fw(\033)45 b Fr(6)p Fz(<)d Fw(\032)f FA(and)519 3538 y Fp(b)505 3561 y Fw(U)10 b FA(\()p Fw(\032\034)g FA(\))31 b(=)e Fw(U)10 b FA(\()p Fw(\034)g FA(\).)34 b(Then)1379 3538 y Fp(b)1366 3561 y Fw(U)43 b FA(is)32 b(univ)m(ersal,)g(since)g(it)h (co)s(des)g Fw(U)10 b FA(,)33 b(and)f Fw(\026)p FA(\(dom\()3164 3538 y Fp(b)3151 3561 y Fw(U)10 b FA(\)\))31 b(=)505 3669 y Fw(\026)p FA(\(dom\()p Fw(M)10 b FA(\)\))22 b Fr(\000)e FA(2)1128 3636 y Fq(\000)p Fx(c)1218 3669 y FA(\(1)h Fr(\000)f FA(\012\))25 b(=)g Fw(\013)p FA(.)1624 b Fr(a)588 3809 y FA(Ku)m(\024)-43 b(cera)32 b(and)e(Slaman)f (\014nished)f(the)i(story)h(b)m(y)f(pro)m(ving)g(the)g(follo)m(wing.) 588 3989 y FB(Theorem)k FA(4.6)h(\(Ku)m(\024)-43 b(cera)32 b(and)e(Slaman)f([68)q(]\))p FB(.)46 b Fs(Every)24 b FA(1)p Fs(-r)-5 b(andom)27 b(left-c.e.)c(r)-5 b(e)g(al)505 4097 y(is)33 b FA(\012)p Fs(-like.)588 4277 y FB(Pr)n(oof)h(Sketch.)40 b FA(Supp)s(ose)26 b(that)j Fw(\013)f FA(is)f(a)h(1-random)g(left-c.e.) h(real)f(and)f Fw(\014)33 b FA(is)27 b(a)505 4385 y(left-c.e.)33 b(real.)f(W)-8 b(e)33 b(need)e(to)i(sho)m(w)e(that)i Fw(\014)f Fz(6)2094 4399 y Fy(S)2165 4385 y Fw(\013)p FA(.)g(W)-8 b(e)33 b(en)m(umerate)g(a)f(Martin-L\177)-45 b(of)505 4493 y(test)29 b Fr(f)p Fw(R)794 4507 y Fx(n)842 4493 y Fr(g)887 4507 y Fx(n)p Fq(2)p Fx(!)1056 4493 y FA(in)e(stages.)i(Let)g Fr(f)p Fw(\013)1715 4507 y Fx(s)1752 4493 y Fr(g)1797 4507 y Fx(s)p Fq(2)p Fx(!)1956 4493 y FA(and)f Fr(f)p Fw(\014)2227 4507 y Fx(s)2264 4493 y Fr(g)2309 4507 y Fx(s)p Fq(2)p Fx(!)2468 4493 y FA(b)s(e)f (increasing)g(sequences)505 4601 y(of)32 b(rationals)e(con)m(v)m (erging)i(to)g Fw(\013)f FA(and)g Fw(\014)5 b FA(,)32 b(resp)s(ectiv)m(ely)-8 b(.)31 b(A)m(t)h(stage)g Fw(s)p FA(,)f(if)f Fw(\013)3050 4615 y Fx(s)3114 4601 y Fr(2)c Fw(R)3270 4615 y Fx(n;s)3369 4601 y FA(,)505 4709 y(do)31 b(nothing,)f(and)g(otherwise)g(put)g(\()p Fw(\013)1833 4723 y Fx(s)1870 4709 y Fw(;)15 b(\013)1968 4723 y Fx(s)2026 4709 y FA(+)20 b(2)2162 4676 y Fq(\000)p Fx(n)2265 4709 y FA(\()p Fw(\014)2351 4723 y Fx(s)p Fy(+1)2499 4709 y Fr(\000)g Fw(\014)2641 4723 y Fx(t)2666 4731 y Fl(s)2704 4709 y FA(\)\))31 b(in)m(to)g Fw(R)3059 4723 y Fx(n)3106 4709 y FA(,)g(where)505 4817 y Fw(t)538 4831 y Fx(s)611 4817 y FA(is)36 b(the)g(last)h(stage)g(at)h(whic)m(h)d(something)h(w)m (as)g(put)g(in)m(to)g Fw(R)2725 4831 y Fx(n)2772 4817 y FA(.)h(No)m(w)g Fw(\026)p FA(\()p Fw(R)3206 4831 y Fx(n)3253 4817 y FA(\))f Fw(<)505 4925 y FA(2)550 4892 y Fq(\000)p Fx(n)653 4925 y Fw(\014)g(<)31 b FA(2)887 4892 y Fq(\000)p Fx(n)990 4925 y FA(,)j(and)f(th)m(us)h Fr(f)p Fw(R)1547 4939 y Fx(n)1594 4925 y Fr(g)1639 4939 y Fx(n)p Fq(2)p Fx(!)1814 4925 y FA(is)f(a)i(Martin-L\177)-45 b(of)34 b(test.)h(As)f Fw(\013)g FA(is)f(1-random,)505 5033 y(there)d(is)f(an)h Fw(n)f FA(suc)m(h)g(that)i Fw(\013)36 b(=)-56 b Fr(2)25 b Fw(R)1677 5047 y Fx(n)1724 5033 y FA(,)30 b(whic)m(h)f(implies)e(that)j Fw(\014)g Fz(6)2695 5047 y Fy(S)2764 5033 y Fw(\013)g FA(with)e(constan)m(t)505 5141 y(2)550 5108 y Fx(n)598 5141 y FA(.)2716 b Fr(a)p eop %%Page: 16 16 16 15 bop 505 363 a FD(16)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FA(Theorem)27 b(4.6)i(giv)m(es)e(great)i(insigh)m(t)c(in)m(to)i (the)h(structure)f(of)g(the)g(1-random)g(left-)505 649 y(c.e.)40 b(reals.)d(All)g(that)i(is)e(needed)g(for)h Fw(\013)g FA(to)h(b)s(e)e(1-random)h(is)f(that)i Fw(K)7 b FA(\()p Fw(\013)3120 637 y Fz(\026)3196 649 y Fw(n)p FA(\))38 b Fz(>)505 757 y Fw(n)15 b Fr(\000)g Fw(O)s FA(\(1\).)29 b(But)f(of)g(course)g Fw(K)7 b FA(\()p Fw(\013)1660 745 y Fz(\026)1723 757 y Fw(n)p FA(\))28 b(can)g(b)s(e)f(near)h Fw(n)15 b FA(+)g Fw(K)7 b FA(\()p Fw(n)p FA(\).)27 b(In)g(fact,)i(w)m (e)g(kno)m(w)505 865 y(that)35 b(the)f(complexit)m(y)f(of)h(the)g (initial)d(segmen)m(ts)k(of)f(\012)f(m)m(ust)h(oscillate)f(near)h(this) 505 973 y(b)s(ound,)c(and,)g(indeed,)g(b)m(y)h(the)g(w)m(ork)g(of)g (Solo)m(v)-5 b(a)m(y)32 b(men)m(tioned)e(after)i(the)f(pro)s(of)f(of) 505 1081 y(Theorem)39 b(3.10,)h(all)d(1-random)i(sets)f(exhibit)f(suc)m (h)h(oscillations.)f(The)h(Ku)m(\024)-43 b(cera-)505 1189 y(Slaman)36 b(Theorem)g(sa)m(ys)h(that)g(all)e(1-random)i (left-c.e.)h(reals)d(exhibit)g(the)i Fs(same)505 1297 y FA(pattern)45 b(of)g(complexit)m(y)g(oscillation,)e(and)h(that)i(in)d (a)j(strong)e(sense,)h(there)g(is)505 1405 y(essen)m(tially)30 b(only)f(one)i(1-random)f(left-c.e.)i(real.)588 1513 y(Results)j(suc)m(h)g(as)h(the)f(ab)s(o)m(v)m(e)i(motiv)-5 b(ate)36 b(us)f(to)h(understand)e(the)h(structure)g(of)505 1621 y(left-c.e.)42 b(reals)e(under)f Fz(6)1398 1635 y Fy(S)1441 1621 y FA(.)i(Naturally)-8 b(,)40 b(this)f(reducibilit)m(y) e(giv)m(es)k(rise)e(to)i(equiv-)505 1729 y(alence)f(classes,)g(called)f Fs(Solovay)j(de)-5 b(gr)g(e)g(es)p FA(.)41 b(W)-8 b(e)41 b(denote)f(the)g(Solo)m(v)-5 b(a)m(y)40 b(degree)g(of)505 1837 y Fw(\013)g FA(b)m(y)g(deg)877 1858 y Fy(S)920 1837 y FA(\()p Fw(\013)p FA(\).)h(It)f(w)m(as)g(observ)m(ed)f(b)m(y)h(Solo)m (v)-5 b(a)m(y)40 b(and)f(others,)h(suc)m(h)g(as)g(Calude,)505 1944 y(Hertling,)c(Khoussaino)m(v,)f(and)g(W)-8 b(ang)37 b([17)r(],)f(that)h(the)f(Solo)m(v)-5 b(a)m(y)37 b(degrees)f(of)g (left-)505 2052 y(c.e.)g(reals)f(form)f(an)h(upp)s(er)e(semilattice,)h (with)g(the)h(join)f(op)s(eration)g(induced)f(b)m(y)505 2160 y(addition)39 b(\(or)h(equiv)-5 b(alen)m(tly)d(,)40 b(m)m(ultiplication\);)e(that)j(is,)e(deg)2688 2182 y Fy(S)2731 2160 y FA(\()p Fw(\013)p FA(\))28 b Fr(_)e FA(deg)3112 2182 y Fy(S)3155 2160 y FA(\()p Fw(\014)5 b FA(\))43 b(=)505 2268 y(deg)643 2290 y Fy(S)686 2268 y FA(\()p Fw(\013)28 b FA(+)e Fw(\014)5 b FA(\))43 b(=)e(deg)1288 2290 y Fy(S)1331 2268 y FA(\()p Fw(\013)28 b Fr(\001)f Fw(\014)5 b FA(\).)41 b(Do)m(wney)-8 b(,)41 b(Hirsc)m(hfeldt,)e(and)h (Nies)g([37)q(])g(sho)m(w)m(ed)505 2376 y(that)k(this)e(upp)s(er)f (semilattice)i(is)f(distributiv)m(e,)f(and)h(established)f(some)j(of)f (its)505 2484 y(structural)30 b(prop)s(erties.)588 2630 y FB(Theorem)k FA(4.7)h(\(Do)m(wney)-8 b(,)33 b(Hirsc)m(hfeldt,)c(and)g (Nies)h([37)r(]\))p FB(.)563 2756 y FA(\(i\))42 b Fs(The)26 b(Solovay)i(de)-5 b(gr)g(e)g(es)27 b(of)f(left-c.e.)f(r)-5 b(e)g(als)28 b(ar)-5 b(e)27 b(dense.)f(That)h(is,)f(if)g Fw(\013)f(<)3121 2770 y Fi(S)3190 2756 y Fw(\014)32 b Fs(ar)-5 b(e)701 2863 y(left-c.e.)31 b(r)-5 b(e)g(als,)34 b(then)f(ther)-5 b(e)34 b(is)f(a)g(left-c.e.)e(r)-5 b(e)g(al)34 b Fw(\015)k Fs(such)33 b(that)g Fw(\013)26 b(<)3004 2877 y Fi(S)3073 2863 y Fw(\015)31 b(<)3222 2877 y Fi(S)3291 2863 y Fw(\014)5 b Fs(.)538 2971 y FA(\(ii\))41 b Fs(If)j Fw(\014)51 b(<)981 2985 y Fi(S)1071 2971 y Fw(\013)c(<)1247 2985 y Fi(S)1338 2971 y FA(\012)c Fs(ar)-5 b(e)45 b(left-c.e.)f(r)-5 b(e)g(als,)45 b(then)g(ther)-5 b(e)45 b(exist)g(left-c.e.)e(r)-5 b(e)g(als)701 3079 y Fw(\015)748 3093 y Fy(1)802 3079 y Fr(j)827 3093 y Fi(S)887 3079 y Fw(\015)934 3093 y Fy(2)1010 3079 y Fs(such)37 b(that)h Fw(\014)g(<)1571 3093 y Fi(S)1647 3079 y Fw(\015)1694 3093 y Fy(1)1734 3079 y Fw(;)15 b(\015)1821 3093 y Fy(2)1897 3079 y Fs(and)38 b Fw(\013)33 b FA(=)g Fw(\015)2320 3093 y Fy(1)2382 3079 y FA(+)23 b Fw(\015)2523 3093 y Fy(2)2563 3079 y Fs(.)36 b(In)h(other)h(wor)-5 b(ds,)38 b(ev-)701 3187 y(ery)30 b(inc)-5 b(omplete)32 b(Solovay)g(de)-5 b(gr)g(e)g(e)31 b(of)g(left-c.e.)e(r)-5 b(e)g(als)32 b(splits)f(over)g(e)-5 b(ach)31 b(lesser)701 3295 y(de)-5 b(gr)g(e)g(e.)513 3403 y FA(\(iii\))40 b Fs(If)30 b Fw(\013)g Fs(and)i Fw(\014)j Fs(ar)-5 b(e)31 b(left-c.e.)e(r)-5 b(e)g(als)32 b(such)f(that)g FA(\012)25 b(=)g Fw(\013)15 b FA(+)g Fw(\014)36 b Fs(then)30 b(either)h Fw(\013)26 b Fr(\021)3260 3417 y Fi(S)3329 3403 y FA(\012)701 3511 y Fs(or)37 b Fw(\014)i Fr(\021)983 3525 y Fi(S)1060 3511 y FA(\012)p Fs(.)d(In)i(other)g(wor)-5 b(ds,)39 b(the)e(c)-5 b(omplete)39 b(Solovay)f(de)-5 b(gr)g(e)g(e)38 b(of)g(left-c.e.)701 3619 y(r)-5 b(e)g(als)34 b(do)-5 b(es)34 b(not)f(split)g(in)g(the)g (Solovay)h(de)-5 b(gr)g(e)g(es)34 b(of)f(left-c.e.)e(r)-5 b(e)g(als.)3009 3586 y Fy(4)588 3765 y FA(Items)32 b(\(ii\))f(and)f (\(iii\))g(ab)s(o)m(v)m(e)j(demonstrate)f(that)g(1-random)f(left-c.e.)i (reals)e(are)505 3873 y(qualitativ)m(ely)h(di\013eren)m(t)g(from)g(all) g(other)h(left-c.e.)h(reals)e(in)g(the)g(sense)h(that)g(they)505 3981 y(cannot)k(b)s(e)e(split)f(in)m(to)i(t)m(w)m(o)h(left-c.e.)g (reals)e(of)h(lesser)f(Solo)m(v)-5 b(a)m(y)37 b(degree.)g(It)e(is)g (im-)505 4089 y(p)s(ortan)m(t)30 b(to)h(realize)e(that)i(this)d(is)h (only)g(true)h(of)g(left-c.e.)h(reals.)e(T)-8 b(o)31 b(see)f(this,)f(note)505 4197 y(that)h(if)e(\012)d(=)g Fw(:a)1043 4211 y Fy(0)1083 4197 y Fw(a)1131 4211 y Fy(1)1186 4197 y Fw(:)15 b(:)g(:)45 b FA(and)28 b(w)m(e)i(let)f Fw(\013)d FA(=)f Fw(:a)2027 4211 y Fy(0)2067 4197 y FA(0)p Fw(a)2160 4211 y Fy(2)2200 4197 y FA(0)p Fw(a)2293 4211 y Fy(4)2332 4197 y FA(0)15 b Fw(:)g(:)g(:)46 b FA(and)29 b Fw(\014)h FA(=)25 b Fw(:)p FA(0)p Fw(a)3014 4211 y Fy(1)3055 4197 y FA(0)p Fw(a)3148 4211 y Fy(3)3188 4197 y FA(0)15 b Fw(:)g(:)g(:)h FA(,)505 4305 y(then)30 b(clearly)e(neither) h Fw(\013)h FA(nor)f Fw(\014)34 b FA(can)c(b)s(e)f(1-random,)h(y)m(et)h Fw(\013)19 b FA(+)f Fw(\014)30 b FA(=)25 b(\012.)k(But)h Fw(\013)g FA(and)505 4413 y Fw(\014)43 b FA(are)38 b(not)f(left-c.e.)i (The)e(fact)h(that)g(addition)e(induces)g(the)h(join)g(op)s(eration)g (on)505 4521 y(left-c.e.)c(reals)e(leads)g(to)i(another)e(c)m (haracterization)i(of)f(Solo)m(v)-5 b(a)m(y)32 b(reducibilit)m(y)d(on) 505 4628 y(left-c.e.)j(reals.)588 4774 y FB(Theorem)i FA(4.8)h(\(Do)m(wney)-8 b(,)15 b(Hirsc)m(hfeldt,)g(and)31 b(Nies)f([37)q(]\))p FB(.)46 b Fs(L)-5 b(et)24 b Fw(\013)g Fs(and)g Fw(\014)29 b Fs(b)-5 b(e)23 b(left-)505 4882 y(c.e.)35 b(r)-5 b(e)g(als.)38 b(Then)e Fw(\013)31 b Fz(6)1322 4896 y Fi(S)1396 4882 y Fw(\014)41 b Fs(iff)35 b(ther)-5 b(e)37 b(exist)f(a)g(c)-5 b(onstant)37 b Fw(c)f Fs(and)h(a)f(left-c.e.)e(r)-5 b(e)g(al)37 b Fw(\015)p 505 4958 499 4 v 588 5018 a Fn(4)623 5050 y Fv(According)22 b(to)g(Ku)n(\024)-36 b(cera)22 b(\(p)r(ersonal)h(comm)n(unication\),)e (item)g(\(iii\))i(in)f(Theorem)f(4.7)i(had)f(b)r(een)505 5141 y(pro)n(v)n(ed)j(earlier)i(b)n(y)e(Dem)n(uth.)p eop %%Page: 17 17 17 16 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(17)505 541 y Fs(such)33 b(that)1718 691 y Fw(c\014)e FA(=)25 b Fw(\013)c FA(+)f Fw(\015)5 b(:)588 876 y FA(Before)30 b(w)m(e)f(lea)m(v)m(e)h(the)e(Solo)m(v)-5 b(a)m(y)30 b(degrees)f(of)f(left-c.e.)i(reals,)e(w)m(e)h(note)g(that)g(their)505 984 y(structure)h(is)g(quite)g(complicated.)588 1170 y FB(Theorem)k FA(4.9)h(\(Do)m(wney)-8 b(,)15 b(Hirsc)m(hfeldt,)g(and) 31 b(LaF)-8 b(orte)32 b([35)q(]\))p FB(.)47 b Fs(The)35 b(\014rst-or)-5 b(der)505 1277 y(the)g(ory)35 b(of)e(the)g(Solovay)h (de)-5 b(gr)g(e)g(es)34 b(of)f(left-c.e.)e(r)-5 b(e)g(als)34 b(is)f(unde)-5 b(cidable.)505 1463 y FA(The)39 b(pro)s(of)f(of)h (Theorem)g(4.9)h(uses)f(Nies')f(metho)s(d)h(of)g(in)m(terpreting)f (e\013ectiv)m(ely)505 1571 y(dense)26 b(Bo)s(olean)h(algebras)f(\(see)h ([101)r(]\),)g(together)h(with)d(a)i(tec)m(hnical)f(construction)505 1679 y(of)31 b(a)g(certain)f(class)g(of)h(\(strongly\))f(c.e.)i(reals.) 588 1787 y(Calude)39 b(and)h(Nies)g([18)q(])h(pro)m(v)m(ed)g(that)g (the)f(1-random)g(left-c.e.)i(reals)e(are)h(all)505 1895 y(wtt-complete.)f(This)d(also)h(follo)m(ws)f(from)h(the)h(result)e(in)g (Do)m(wney)-8 b(,)40 b(Hirsc)m(hfeldt,)505 2003 y(and)33 b(LaF)-8 b(orte)36 b([34)q(])d(that)h(if)f Fw(\014)39 b FA(is)32 b(a)i(left-c.e.)h(real)e(and)g Fw(\013)h FA(is)e(a)i (strongly)f(c.e.)i(real,)505 2111 y(then)e Fw(\013)e Fz(6)875 2125 y Fy(S)948 2111 y Fw(\014)38 b FA(implies)30 b Fw(\013)h Fz(6)1508 2125 y Fy(wtt)1647 2111 y Fw(\014)39 b FA(\(and)32 b(ev)m(en)i Fw(\013)d Fz(6)2321 2125 y Fy(sw)2433 2111 y Fw(\014)5 b FA(,)34 b(whic)m(h)e(will)e(b)s(e)j (de\014ned)505 2218 y(in)i(Section)i(5.1\).)h(If)e(w)m(e)g(com)m(bine)h (this)e(result)g(with)g(the)i(follo)m(wing)e(theorem)h(of)505 2326 y(Dem)m(uth,)d(and)d(the)i(fact)h(that)f(if)e(a)i(1-random)g(set)g (has)f(c.e.)i(T)-8 b(uring)30 b(degree)i(then)505 2434 y(it)i(is)f(T)-8 b(uring)32 b(complete,)j(w)m(e)f(see)h(that)f(while)e (a)i(1-random)g(left-c.e.)h(real)f(is)f(wtt-)505 2542 y(complete,)38 b(it)e(is)g(tt-incomparable)g(with)f(all)h (noncomputable,)f(incomplete)h(c.e.)505 2650 y(sets.)31 b(In)f(particular,)f(no)h(1-random)h(left-c.e.)g(real)f(can)h(b)s(e)f (tt-complete.)588 2836 y FB(Theorem)k FA(4.10)i(\(Dem)m(uth\))p FB(.)46 b Fs(If)37 b Fw(A)f Fs(is)g FA(1)p Fs(-r)-5 b(andom)39 b(and)e Fw(B)g Fz(6)2767 2850 y Fi(tt)2850 2836 y Fw(A)f Fs(is)g(nonc)-5 b(om-)505 2944 y(putable,)34 b(then)f(ther)-5 b(e)33 b(is)g(a)g FA(1)p Fs(-r)-5 b(andom)35 b(set)e Fw(C)e Fr(\021)2168 2958 y Fi(T)2248 2944 y Fw(B)5 b Fs(.)505 3129 y FA(Pro)s(ofs)22 b(of)h(Dem)m(uth's)g(Theorem)f(can)h(b) s(e)f(found)f(in)g(Kautz)i([60)q(,)f(Theorem)g(IV.3.16])505 3237 y(and)30 b(Do)m(wney)h(and)f(Hirsc)m(hfeldt)f([33)q(].)588 3472 y Fu(x)p Ft(5.)53 b(Other)47 b(reducibilities)i(that)e(calibrate)h (randomness.)e FA(In)41 b([34)q(],)h(a)505 3580 y(transitiv)m(e)j (preordering)e Fz(6)i FA(on)g(sets)h(w)m(as)f(said)g(to)h(b)s(e)e(a)i Fs(me)-5 b(asur)g(e)47 b(of)g(r)-5 b(elative)505 3688 y(r)g(andomness)41 b FA(if)29 b(it)h(satis\014es)g(the)h Fs(Solovay)j(pr)-5 b(op)g(erty)8 b FA(:)1051 3875 y(If)30 b Fw(A)25 b Fz(6)g Fw(B)5 b FA(,)30 b(then)g Fw(K)7 b FA(\()p Fw(A)1880 3863 y Fz(\026)1943 3875 y Fw(n)p FA(\))25 b Fz(6)g Fw(K)7 b FA(\()p Fw(B)2372 3863 y Fz(\026)2435 3875 y Fw(n)p FA(\))20 b(+)g Fw(O)s FA(\(1\))p Fw(:)505 4062 y FA(This)39 b(view)g(of)h(what)g(constitutes)h(a)f(measure)g(of)g (relativ)m(e)h(randomness)e(is)g(to)s(o)505 4169 y(restrictiv)m(e,)30 b(as)f(it)g(is)f(tailored)h(to)h(reducibilities)25 b(motiv)-5 b(ated)29 b(b)m(y)g(the)h(incompress-)505 4277 y(ibilit)m(y)19 b(approac)m(h)j(to)h(randomness,)e(so)h(w)m(e)g(will)d(instead)i(use)g (the)h(term)g Fw(K)7 b Fs(-me)-5 b(asur)g(e)505 4385 y(of)27 b(r)-5 b(elative)27 b(r)-5 b(andomness)p FA(.)26 b(\(In)d(Section)g(5.3)h(w)m(e)g(will)d(see)j(a)f(reducibilit)m(y)d (motiv)-5 b(ated)505 4493 y(b)m(y)31 b(the)f(unpredictabilit)m(y)d (approac)m(h)k(to)g(randomness.\))588 4601 y(Solo)m(v)-5 b(a)m(y)41 b(reducibilit)m(y)c(is)i(a)h Fw(K)7 b FA(-measure)40 b(of)g(relativ)m(e)g(randomness,)f(but)g(it)g(is)505 4709 y(certainly)34 b(not)g(the)h(only)e(one.)i(Moreo)m(v)m(er,)i(it)d (b)s(eha)m(v)m(es)h(reasonably)f(only)f(on)h(the)505 4817 y(left-c.e.)e(reals;)d(it)h(is)e(quite)i(easy)g(to)h(construct)f (a)g(real)g Fw(\013)g FA(and)f(a)h Fs(c)-5 b(omputable)39 b FA(real)505 4925 y Fw(\014)h FA(with)33 b Fw(\014)k Fo(\012)966 4939 y Fy(S)1041 4925 y Fw(\013)p FA(.)e(Ev)m(en)g(on)f (the)g(left-c.e.)i(reals,)e(S-reducibilit)m(y)d(is)i(to)s(o)i(\014ne)f (and)505 5033 y(uniform)i(\(as)h(w)m(e)h(will)d(see\),)j(and)f(badly)f (fails)g(to)i(capture)f(relativ)m(e)h(complexit)m(y)505 5141 y(exactly)-8 b(.)p eop %%Page: 18 18 18 17 bop 505 363 a FD(18)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y Ft(5.1.)53 b Fz(6)874 555 y Fh(sw)1010 541 y Ft(and)40 b Fz(6)1288 555 y Fh(rK)1389 541 y Ft(.)46 b FA(In)34 b([34)q(],)i(Do)m(wney)-8 b(,)37 b(Hirsc)m(hfeldt,)d(and)h(LaF)-8 b(orte)37 b(in)m(tro-)505 649 y(duced)42 b(another)g Fw(K)7 b FA(-measure)42 b(of)h(relativ)m(e)f(randomness)f(called)h Fs(sw-r)-5 b(e)g(ducibility)505 757 y FA(\(strong)31 b(w)m(eak)h(truth)d(table)h(reducibilit)m(y\).)588 921 y FB(Definition)35 b FA(5.1)p FB(.)47 b FA(W)-8 b(e)31 b(sa)m(y)f(that)g Fw(A)g FA(is)f Fs(sw-r)-5 b(e)g(ducible)37 b FA(to)31 b Fw(B)5 b FA(,)29 b(and)h(write)e Fw(A)e Fz(6)3312 935 y Fy(sw)505 1028 y Fw(B)5 b FA(,)36 b(if)e(there)i(is)f (a)h(functional)e(\000)h(suc)m(h)g(that)i(\000)2138 996 y Fx(B)2232 1028 y FA(=)d Fw(A)h FA(and)g(the)h(use)f(of)h(\000)3112 996 y Fx(B)3173 1028 y FA(\()p Fw(n)p FA(\))f(is)505 1136 y(b)s(ounded)29 b(b)m(y)h Fw(n)20 b FA(+)g Fw(O)s FA(\(1\).)588 1300 y(Again)k(it)f(is)f(not)i(di\016cult)d(to)j(pro)m(v) m(e)g(that)g Fz(6)2099 1314 y Fy(sw)2205 1300 y FA(is)f(a)g Fw(K)7 b FA(-measure)24 b(of)f(relativ)m(e)h(ran-)505 1408 y(domness.)31 b(This)e(reducibilit)m(y)e(is)j(quite)g(close)h(to)g (one)g(considered)f(b)m(y)h(Csima)e([26)q(])505 1516 y(and)k(Soare)g([123)q(])g(in)f(connection)h(with)e(w)m(ork)i(of)g (Nabuto)m(vsky)g(and)g(W)-8 b(ein)m(b)s(erger)505 1623 y([100)r(])26 b(in)e(di\013eren)m(tial)g(geometry)-8 b(,)27 b(the)f(di\013erence)e(b)s(eing)g(that)i(the)g(use)f(is)g(b)s (ounded)505 1731 y(b)m(y)g Fw(n)f FA(in)f(their)g(case.)j(Lewis)d(and)h (Barmpalias)g([78)q(])g(ha)m(v)m(e)i(recen)m(tly)f(giv)m(en)f(an)h(in)m (ter-)505 1839 y(esting)j(c)m(haracterization)h(of)f(sw-reducibilit)m (y)c(in)i(terms)i(of)f(Lipsc)m(hitz)g(con)m(tin)m(uit)m(y)-8 b(.)588 1947 y(Do)m(wney)g(,)30 b(Hirsc)m(hfeldt,)c(and)i(LaF)-8 b(orte)29 b([34)q(])f(sho)m(w)m(ed)g(that)h Fz(6)2695 1961 y Fy(sw)2805 1947 y FA(agrees)g(with)e Fz(6)3352 1961 y Fy(S)505 2055 y FA(on)37 b(the)h Fs(str)-5 b(ongly)47 b FA(c.e.)38 b(reals,)f(but)g(the)g(t)m(w)m(o)i(notions)d(are)i (incomparable)d(on)i(the)505 2163 y(left-c.e.)g(reals,)e(in)g(the)g (sense)h(that)g(there)g(exist)f(left-c.e.)i(reals)e Fw(\013)p FA(,)h Fw(\014)5 b FA(,)36 b Fw(\015)5 b FA(,)36 b Fw(\016)j FA(with)505 2271 y Fw(\013)g Fz(6)673 2285 y Fy(S)754 2271 y Fw(\014)k FA(but)38 b Fw(\013)g Fo(\012)1190 2285 y Fy(sw)1311 2271 y Fw(\014)5 b FA(,)38 b(and)g Fw(\015)43 b Fz(6)1776 2285 y Fy(sw)1897 2271 y Fw(\016)f FA(but)37 b Fw(\015)43 b Fo(\012)2314 2285 y Fy(S)2395 2271 y Fw(\016)s FA(.)c(F)-8 b(urthermore,)39 b(if)e Fw(\013)h FA(is)f(a)505 2379 y(noncomputable)30 b(left-c.e.)i(real,)f(then)f(there)h(is)f(a)h (noncomputable)f(strongly)g(c.e.)505 2487 y(real)41 b Fw(\014)47 b Fz(6)862 2501 y Fy(sw)987 2487 y Fw(\013)p FA(,)41 b(but)f(this)f(is)h Fs(not)49 b FA(true)41 b(in)e(general)h (for)h Fz(6)2616 2501 y Fy(S)2659 2487 y FA(,)f(as)h(sho)m(wn)f(b)m(y)g (the)505 2595 y(follo)m(wing)29 b(theorem,)i(whic)m(h)e(is)h(pro)m(v)m (ed)h(b)m(y)f(a)h(gap/co-gap)h(argumen)m(t.)588 2758 y FB(Theorem)i FA(5.2)h(\(Do)m(wney)-8 b(,)33 b(Hirsc)m(hfeldt,)c(and)g (LaF)-8 b(orte)33 b([34)q(]\))p FB(.)46 b Fs(Ther)-5 b(e)32 b(exists)g(a)505 2866 y(nonc)-5 b(omputable)40 b(left-c.e.)c(r)-5 b(e)g(al)39 b Fw(\013)e Fs(such)h(that)g(al)5 b(l)38 b(str)-5 b(ongly)39 b(c.e.)d(r)-5 b(e)g(als)39 b(S-b)-5 b(elow)38 b Fw(\013)505 2974 y Fs(ar)-5 b(e)34 b(c)-5 b(omputable.)588 3137 y FA(By)38 b(and)e(large,)h(ho)m(w)m(ev)m (er,)i(sw-reducibilit)m(y)33 b(is)j(v)m(ery)h(badly)f(b)s(eha)m(v)m (ed,)h(as)g(wit-)505 3245 y(nessed)30 b(b)m(y)h(the)f(next)h(t)m(w)m(o) g(theorems.)588 3408 y FB(Theorem)j FA(5.3)h(\(Do)m(wney)-8 b(,)15 b(Hirsc)m(hfeldt,)g(and)31 b(LaF)-8 b(orte)32 b([34)q(]\))p FB(.)47 b Fs(The)26 b(sw-de)-5 b(gr)g(e)g(es)505 3516 y(of)33 b(left-c.e.)f(r)-5 b(e)g(als)34 b(do)g(not)f(form)g(an)h (upp)-5 b(er)33 b(semilattic)-5 b(e.)588 3680 y FB(Theorem)34 b FA(5.4)h(\(Y)-8 b(u)31 b(and)f(Ding)g([139)q(]\))p FB(.)47 b Fs(Ther)-5 b(e)46 b(is)g(no)g(sw-c)-5 b(omplete)48 b(left-c.e.)505 3788 y(r)-5 b(e)g(al.)34 b(Thus)e(the)g(analo)-5 b(g)34 b(of)e(the)g(Ku)n(\024)-44 b(cer)-5 b(a-Slaman)34 b(The)-5 b(or)g(em)34 b(4.6)e(c)-5 b(annot)33 b(hold)h(for)505 3896 y(sw-r)-5 b(e)g(ducibility.)588 4059 y FA(Theorem)41 b(5.4)g(sa)m(ys)g(that)g(while)d(the)j(initial)c(segmen)m(t)42 b(complexit)m(y)e(of)h(all)e(1-)505 4167 y(random)30 b(left-c.e.)h(reals)e(is)g(the)h(same,)h(there)f(is)f(no)h(natural)f (uniform)f(w)m(a)m(y)j(to)f(get)505 4275 y(the)h Fs(bits)38 b FA(of)30 b(one)h(v)m(ersion)f(of)g(\012)g(from)g(those)h(of)g (another.)588 4383 y(On)k(the)g(other)h(hand,)f(there)g(is)g(something) f(w)m(e)i(can)g(sa)m(y)g(ab)s(out)f(sw-hardness)505 4491 y(with)29 b(resp)s(ect)i(to)g(c.e.)h Fs(sets)p FA(.)588 4654 y FB(Theorem)i FA(5.5)h(\(Do)m(wney)d(and)e(Hirsc)m(hfeldt)15 b([33]\))p FB(.)46 b Fs(L)-5 b(et)23 b Fw(\013)h Fs(b)-5 b(e)23 b(a)h FA(1)p Fs(-r)-5 b(andom)26 b(left-)505 4762 y(c.e.)43 b(r)-5 b(e)g(al.)44 b(Then)f Fw(B)49 b Fz(6)1335 4776 y Fi(sw)1459 4762 y Fw(\013)43 b Fs(for)h(any)f(c.e.)f(set)h Fw(B)5 b Fs(.)42 b(Thus,)i(not)f(only)h(is)f FA(\012)f Fs(wtt-)505 4870 y(c)-5 b(omplete,)34 b(but)f(it)f(is)h(sw-har)-5 b(d)35 b(for)e(c.e.)f(sets.)588 5033 y FB(Pr)n(oof.)41 b FA(Since)22 b Fw(\013)i FA(is)e(1-random,)h(there)h(is)e(a)h Fw(d)g FA(suc)m(h)g(that)h Fw(K)7 b FA(\()p Fw(\013)2805 5021 y Fz(\026)2868 5033 y Fw(n)p FA(\))25 b Fw(>)g(n)6 b Fr(\000)g Fw(d)22 b FA(for)505 5141 y(all)h Fw(n)p FA(.)h(W)-8 b(e)24 b(build)d(a)j(pre\014x-free)f(mac)m(hine)h Fw(M)34 b FA(using)22 b(the)i(Kraft-Chaitin)e(Theorem.)p eop %%Page: 19 19 19 18 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(19)505 541 y FA(By)30 b(the)g(Recursion)e(Theorem,)i(w)m(e)g(can)f (assume)h(w)m(e)g(kno)m(w)f(the)h(co)s(ding)e(constan)m(t)505 649 y Fw(c)g FA(of)f Fw(M)37 b FA(in)25 b(the)j(univ)m(ersal)d (pre\014x-free)h(mac)m(hine)h Fw(U)10 b FA(.)27 b(Whenev)m(er)h(w)m(e)f (see)h Fw(n)c(>)h(c)13 b FA(+)g Fw(d)505 757 y FA(en)m(ter)22 b Fw(B)k FA(at)c(stage)g Fw(s)p FA(,)f(w)m(e)h(en)m(umerate)g(a)g (request)f Fr(h)p Fw(n)r Fr(\000)r Fw(c)r Fr(\000)r Fw(d;)15 b(\013)2591 771 y Fx(s)2652 745 y Fz(\026)2716 757 y Fw(n)p Fr(i)p FA(.)21 b(The)g(total)h(cost)505 867 y(of)29 b(these)g(requests)f(is)g(b)s(ounded)e(b)m(y)1772 799 y Fp(P)1868 894 y Fx(n>c)p Fy(+)p Fx(d)2106 867 y FA(2)2151 834 y Fx(n)p Fq(\000)p Fx(c)p Fq(\000)p Fx(d)2401 867 y FA(=)f(1,)k(so)f(the)h(h)m(yp)s(otheses)f(of)505 975 y(the)e(Kraft-Chaitin)d(Theorem)i(are)g(satis\014ed.)f(Th)m(us)g(for)h (all)f Fw(n)h(>)g(c)10 b FA(+)g Fw(d)p FA(,)24 b(if)g Fw(n)h FA(en)m(ters)505 1083 y Fw(B)h FA(at)21 b(stage)i Fw(s)p FA(,)d(then)h Fw(K)7 b FA(\()p Fw(\013)1388 1097 y Fx(s)1450 1071 y Fz(\026)1513 1083 y Fw(n)p FA(\))25 b Fz(6)g Fw(K)1801 1097 y Fx(M)1881 1083 y FA(\()p Fw(\013)1974 1097 y Fx(s)2036 1071 y Fz(\026)2099 1083 y Fw(n)p FA(\))q(+)q Fw(c)h Fz(6)f Fw(n)q Fr(\000)q Fw(c)q Fr(\000)q Fw(d)q FA(+)q Fw(c)h FA(=)f Fw(n)q Fr(\000)q Fw(d)p FA(,)c(whic)m(h)505 1191 y(implies)g(that)j Fw(\013)1081 1179 y Fz(\026)1144 1191 y Fw(n)h Fr(6)p FA(=)g Fw(\013)1378 1205 y Fx(s)1440 1179 y Fz(\026)1503 1191 y Fw(n)p FA(.)f(So)f(to)i(compute)f Fw(B)5 b FA(\()p Fw(n)p FA(\))23 b(for)g Fw(n)i(>)g(c)7 b FA(+)g Fw(d)p FA(,)24 b(it)f(is)g(enough)505 1299 y(to)35 b(run)d(the)i(appro)m(ximation)e(of)i Fw(\013)g FA(un)m(til)e(a)i (stage)h Fw(s)f FA(suc)m(h)f(that)h Fw(\013)2865 1287 y Fz(\026)2934 1299 y Fw(n)c FA(=)h Fw(\013)3179 1313 y Fx(s)3246 1287 y Fz(\026)3315 1299 y Fw(n)p FA(,)505 1407 y(and)c(then)g Fw(n)e Fr(2)g Fw(B)32 b FA(iff)26 b Fw(n)f Fr(2)g Fw(B)1493 1421 y Fx(s)1529 1407 y FA(.)j(Since)f(the)g (use)g(of)h(this)e(computation)i(is)e Fw(n)p FA(,)h(it)g(is)g(an)505 1515 y(sw-reduction.)2302 b Fr(a)588 1640 y FA(The)34 b(ab)s(o)m(v)m(e)h(pro)s(of)e(can)h(easily)f(b)s(e)g(mo)s(di\014ed)f (to)j(sho)m(w)e(that)i(if)e Fw(\013)h FA(is)f(a)h(left-c.e.)505 1748 y(real)28 b(suc)m(h)f(that)h Fr(8)p Fw(n)15 b FA([)p Fw(K)7 b FA(\()p Fw(\013)1424 1736 y Fz(\026)1487 1748 y Fw(n)p FA(\))25 b Fw(>)g("n)p FA(])j(for)f(some)h Fw(")e(>)f FA(0,)j(then)f Fw(\013)h FA(is)e(wtt-complete.)588 1856 y(There)k(ha)m(v)m(e)g(b)s(een)f(sev)m(eral)h(recen)m(t)h(results)d(on) i(sw-reducibilit)m(y)-8 b(.)26 b(F)-8 b(or)31 b(instance,)505 1964 y(Barmpalias)c([5])h(sho)m(w)m(ed)f(that)h(there)g(are)g(no)f (sw-maximal)f(c.e.)i(sets,)g(and)f(Barm-)505 2072 y(palias)43 b(and)h(Lewis)f([6)q(])h(sho)m(w)m(ed)g(that)h(there)f(are)h(left-c.e.) g(reals)f(that)h(are)f(not)505 2180 y(sw-b)s(elo)m(w)27 b(an)m(y)h(1-random)g(left-c.e.)h(real)e(\(cf.)i(the)f(commen)m(t)h (follo)m(wing)d(Theorem)505 2288 y(12.1\).)41 b(It)e(is)e(not)i(kno)m (wn)f(whether)g(there)g(is)g(a)h(maximal)e(sw-degree,)i(although)505 2396 y(Lewis)34 b(and)f(Barmpalias)h([78)q(])g(sho)m(w)m(ed)h(that)g (no)f(1-random)g(set)h(can)g(ha)m(v)m(e)g(max-)505 2504 y(imal)d(sw-degree.)i(On)f(the)g(other)h(hand,)e(they)i(also)f(sho)m(w) m(ed)h(that)g(there)f(are)h(sw-)505 2612 y(degrees)k(deg)968 2634 y Fy(sw)1051 2612 y FA(\()p Fw(A)p FA(\))g(that)f(are)g Fs(quasi-maximal)p FA(,)i(in)c(the)i(sense)g(that)g(if)f Fw(A)g Fz(6)3203 2626 y Fy(sw)3321 2612 y Fw(B)505 2720 y FA(then)h Fw(B)j Fr(\021)899 2734 y Fy(T)990 2720 y Fw(A)p FA(.)d(Indeed,)f(they)g(sho)m(w)m(ed)h(that)h(ev)m(ery)f (1-random)g(set)g(has)f(quasi-)505 2828 y(maximal)30 b(sw-degree.)588 2936 y(Both)36 b(S-reducibilit)m(y)31 b(and)j(sw-reducibilit)m(y)d(are)k(uniform)e(in)g(a)i(w)m(a)m(y)h(that) f(rel-)505 3043 y(ativ)m(e)f(initial-segmen)m(t)e(complexit)m(y)g(is)g (not.)i(Motiv)-5 b(ated)34 b(b)m(y)e(this)g(idea,)h(Do)m(wney)-8 b(,)505 3151 y(Hirsc)m(hfeldt,)30 b(and)f(LaF)-8 b(orte)32 b([34)r(])e(in)m(tro)s(duced)f(the)i(follo)m(wing)d(notion.)588 3305 y FB(Definition)35 b FA(5.6)p FB(.)47 b FA(Let)39 b Fw(A)h FA(and)e Fw(B)44 b FA(b)s(e)38 b(sets.)i(W)-8 b(e)40 b(sa)m(y)g(that)g Fw(B)j FA(is)c Fs(r)-5 b(elative)41 b(K-)505 3413 y(r)-5 b(e)g(ducible)38 b FA(\(rK-reducible\))28 b(to)i Fw(A)p FA(,)g(and)g(write)f Fw(B)g Fz(6)2292 3427 y Fy(rK)2404 3413 y Fw(A)p FA(,)h(if)f(there)h(exist)f(a)i(partial)505 3521 y(computable)26 b(binary)e(function)g Fw(f)35 b FA(and)25 b(a)i(constan)m(t)g Fw(k)i FA(suc)m(h)c(that)h(for)g(eac)m(h) h Fw(n)e FA(there)505 3629 y(is)30 b(a)h Fw(j)f Fz(6)25 b Fw(k)34 b FA(for)c(whic)m(h)f Fw(f)10 b FA(\()p Fw(A)1499 3617 y Fz(\026)1562 3629 y Fw(n;)15 b(j)5 b FA(\))10 b Fr(#)p FA(=)27 b Fw(B)1986 3617 y Fz(\026)2049 3629 y Fw(n)p FA(.)588 3783 y FB(Theorem)34 b FA(5.7)h(\(Do)m(wney)-8 b(,)33 b(Hirsc)m(hfeldt,)c(and)g(LaF)-8 b(orte)33 b([34)q(]\))p FB(.)563 3908 y FA(\(i\))42 b Fz(6)772 3922 y Fi(rK)893 3908 y Fs(is)32 b(a)h Fw(K)7 b Fs(-me)-5 b(asur)g(e)34 b(of)e(r)-5 b(elative)34 b(r)-5 b(andomness.)538 4016 y FA(\(ii\))41 b Fs(If)32 b Fw(A)26 b Fz(6)961 4030 y Fi(sw)1066 4016 y Fw(B)5 b Fs(,)32 b(then)h Fw(A)26 b Fz(6)1567 4030 y Fi(rK)1681 4016 y Fw(B)5 b Fs(.)538 4124 y FA(\(ii\))41 b Fs(If)32 b Fw(\013)h Fs(and)h Fw(\014)j Fs(ar)-5 b(e)34 b(left-c.e.)e(r)-5 b(e)g(als)34 b(and)f Fw(\013)26 b Fz(6)2176 4138 y Fi(S)2245 4124 y Fw(\014)5 b Fs(,)33 b(then)g Fw(\013)26 b Fz(6)2719 4138 y Fi(rK)2833 4124 y Fw(\014)5 b Fs(.)513 4232 y FA(\(iii\))40 b Fs(A)32 b(left-c.e.)f(r)-5 b(e)g(al)34 b Fw(\013)f Fs(is)g(rK-c)-5 b(omplete)34 b(iff)e(it)h(is)f FA(1)p Fs(-r)-5 b(andom.)515 4340 y FA(\(iv\))42 b Fs(If)32 b Fw(A)26 b Fz(6)961 4354 y Fi(rK)1074 4340 y Fw(B)37 b Fs(then)d Fw(A)25 b Fz(6)1547 4354 y Fi(T)1627 4340 y Fw(B)5 b Fs(.)588 4493 y FA(The)23 b(most)h(in)m(teresting)e(c)m(haracterization)i(of)g(rK-reducibilit)m (y)19 b(\(and)k(the)g(reason)505 4601 y(for)29 b(its)g(name\))h(is)e (giv)m(en)h(b)m(y)g(the)h(follo)m(wing)d(result,)i(whic)m(h)f(sho)m(ws) h(that)g(there)h(is)e(a)505 4709 y(v)m(ery)e(natural)f(sense)g(in)f (whic)m(h)h(rK-reducibilit)m(y)c(is)k(an)g(exact)i(measure)f(of)f (relativ)m(e)505 4817 y(randomness.)30 b(The)g(pre\014x-free)g (complexit)m(y)h Fw(K)7 b FA(\()p Fw(\034)36 b Fr(j)26 b Fw(\033)s FA(\))31 b(of)g Fw(\034)40 b Fs(r)-5 b(elative)34 b(to)j Fw(\033)d FA(is)29 b(the)505 4925 y(length)h(of)h(the)g (shortest)g(string)e Fw(\026)h FA(suc)m(h)g(that)i Fw(U)2207 4892 y Fx(\033)2253 4925 y FA(\()p Fw(\026)p FA(\))26 b(=)g Fw(\034)10 b FA(,)30 b(where)g Fw(U)41 b FA(is)29 b(a)i(pre\014x-)505 5033 y(free)j(mac)m(hine)g(that)g(is)f(univ)m (ersal)f(with)g(resp)s(ect)i(to)h(an)m(y)f(oracle.)g(W)-8 b(e)35 b(can)f(de\014ne)505 5141 y Fw(C)7 b FA(\()p Fw(\034)36 b Fr(j)25 b Fw(\033)s FA(\))31 b(analogously)-8 b(.)p eop %%Page: 20 20 20 19 bop 505 363 a FD(20)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FB(Theorem)34 b FA(5.8)h(\(Do)m(wney)-8 b(,)33 b(Hirsc)m(hfeldt,)c(and)g(LaF)-8 b(orte)33 b([34)q(]\))p FB(.)46 b Fs(L)-5 b(et)48 b Fw(A)h Fs(and)g Fw(B)505 649 y Fs(b)-5 b(e)40 b(sets.)f(Then)h Fw(A)d Fz(6)1257 663 y Fi(rK)1383 649 y Fw(B)43 b Fs(iff)c Fw(K)7 b FA(\()p Fw(A)1843 637 y Fz(\026)1918 649 y Fw(n)37 b Fr(j)g Fw(B)2183 637 y Fz(\026)2258 649 y Fw(n)p FA(\))g Fz(6)g Fw(O)s FA(\(1\))j Fs(\(or,)g(e)-5 b(quivalently,)505 757 y Fw(C)7 b FA(\()p Fw(A)706 745 y Fz(\026)769 757 y Fw(n)25 b Fr(j)g Fw(B)998 745 y Fz(\026)1061 757 y Fw(n)p FA(\))g Fz(6)g Fw(O)s FA(\(1\))p Fs(\).)588 924 y FA(The)g(rK-degrees)h(ha)m(v) m(e)h(man)m(y)e(of)h(the)g(same)f(nice)g(structural)g(prop)s(erties)f (as)h(the)505 1032 y(S-degrees.)588 1200 y FB(Theorem)34 b FA(5.9)h(\(Do)m(wney)-8 b(,)33 b(Hirsc)m(hfeldt,)c(and)g(LaF)-8 b(orte)33 b([34)q(]\))p FB(.)563 1329 y FA(\(i\))42 b Fs(The)32 b(rK-de)-5 b(gr)g(e)g(es)33 b(of)f(left-c.e.)f(r)-5 b(e)g(als)33 b(form)g(an)g(upp)-5 b(er)33 b(semilattic)-5 b(e)33 b(with)f(le)-5 b(ast)701 1437 y(de)g(gr)g(e)g(e)33 b(that)h(of)f(the)g(c)-5 b(omputable)34 b(sets)f(and)h(highest)f(de)-5 b(gr)g(e)g(e)34 b(that)g(of)f FA(\012)p Fs(.)538 1545 y FA(\(ii\))41 b Fs(F)-7 b(or)28 b(left-c.e.)d(r)-5 b(e)g(als)29 b Fw(\013)e Fs(and)h Fw(\014)5 b Fs(,)27 b(we)f(have)i FA(deg)2227 1567 y Fi(rK)2316 1545 y FA(\()p Fw(\013)p FA(\))8 b Fr(_)g FA(deg)2659 1567 y Fi(rK)2748 1545 y FA(\()p Fw(\014)d FA(\))26 b(=)f(deg)3134 1567 y Fi(rK)3222 1545 y FA(\()p Fw(\013)8 b FA(+)701 1653 y Fw(\014)d FA(\))p Fs(.)513 1761 y FA(\(iii\))40 b Fs(The)33 b(rK-de)-5 b(gr)g(e)g(es)33 b(of)g(left-c.e.)f(r)-5 b(e)g(als)34 b(ar)-5 b(e)33 b(dense.)515 1869 y FA(\(iv\))42 b Fs(F)-7 b(or)42 b(any)g(rK-de)-5 b(gr)g(e)g(es)42 b Ft(a)f Fw(<)g Ft(b)g Fw(<)g FA(deg)2072 1891 y Fi(rK)2161 1869 y FA(\(\012\))g Fs(of)h(left-c.e.)e(r)-5 b(e)g(als,)43 b(ther)-5 b(e)43 b(ar)-5 b(e)701 1977 y(rK-de)g(gr)g(e)g(es)35 b Ft(c)1197 1991 y Fh(0)1277 1977 y Fs(and)g Ft(c)1501 1991 y Fh(1)1581 1977 y Fs(of)g(left-c.e.)e(r)-5 b(e)g(als)37 b(such)d(that)i Ft(a)28 b Fw(<)h Ft(c)2853 1991 y Fh(0)2898 1977 y Fw(;)15 b Ft(c)2984 1991 y Fh(1)3059 1977 y Fw(<)28 b Ft(b)35 b Fs(and)701 2085 y Ft(c)747 2099 y Fh(0)812 2085 y Fr(_)20 b Ft(c)939 2099 y Fh(1)1010 2085 y FA(=)k Ft(b)p Fs(.)505 2252 y FA(Of)30 b(course,)g(Theorem)g(4.7)h(\(iii\))e(implies)e(that)k (deg)2286 2274 y Fy(rK)2373 2252 y FA(\(\012\))f(cannot)h(b)s(e)e (split)g(in)f(the)505 2360 y(rK-degrees)k(of)f(left-c.e.)h(reals.)f(W) -8 b(e)32 b(do)f(not)g(kno)m(w)g(whether)g(the)g(rK-degrees)g(are)505 2468 y(distributiv)m(e.)37 b(The)h(theories)g(of)h(the)g(sw-)f(and)g (rK-degrees)h(ha)m(v)m(e)h(not)e(y)m(et)i(b)s(een)505 2576 y(pro)m(v)m(ed)31 b(to)g(b)s(e)f(undecidable,)e(though)i(this)g(m) m(ust)g(surely)f(b)s(e)g(the)i(case.)588 2700 y Ft(5.2.)53 b(The)24 b(basic)g(measures)f Fz(6)1767 2714 y Fx(K)1858 2700 y Ft(and)h Fz(6)2120 2714 y Fx(C)2179 2700 y Ft(.)45 b FA(Of)20 b(course,)h(w)m(e)g(are)g(particularly)505 2808 y(in)m(terested)38 b(in)e(the)i(measure)f(of)h(relativ)m(e)f (complexit)m(y)h Fs(de\014ne)-5 b(d)48 b FA(b)m(y)37 b(the)h(Solo)m(v)-5 b(a)m(y)505 2916 y(prop)s(ert)m(y:)30 b(W)-8 b(e)32 b(sa)m(y)f(that)g Fw(A)f FA(is)g Fw(K)7 b Fs(-r)-5 b(e)g(ducible)37 b FA(to)31 b Fw(B)5 b FA(,)30 b(and)g(write)f Fw(A)d Fz(6)2913 2930 y Fx(K)3006 2916 y Fw(B)5 b FA(,)30 b(if)1359 3071 y Fw(K)7 b FA(\()p Fw(A)1571 3059 y Fz(\026)1635 3071 y Fw(n)p FA(\))25 b Fz(6)g Fw(K)7 b FA(\()p Fw(B)2064 3059 y Fz(\026)2127 3071 y Fw(n)p FA(\))20 b(+)g Fw(O)s FA(\(1\))p Fw(:)505 3226 y FA(The)31 b(preordering)e Fz(6)1250 3240 y Fx(C)1340 3226 y FA(is)h(de\014ned)f(analogously)-8 b(.)32 b(Note)g(that)f Fz(6)2745 3240 y Fx(K)2844 3226 y FA(is)f(not)h(really)f(a)505 3334 y Fs(r)-5 b(e)g(ducibility)8 b FA(,)35 b(but)d(simply)e(a)j (transitiv)m(e)g(preordering)e(measuring)h(relativ)m(e)h(com-)505 3442 y(plexit)m(y)-8 b(.)31 b(This)d(is)i(b)s(est)g(seen)g(b)m(y)g(the) h(follo)m(wing)e(result.)588 3609 y FB(Theorem)34 b FA(5.10)i(\(Y)-8 b(u,)31 b(Ding,)f(and)g(Do)m(wney)h([141)r(]\))p FB(.)46 b Fr(jf)p Fw(A)26 b FA(:)f Fw(A)h Fz(6)2876 3623 y Fx(K)2969 3609 y FA(\012)p Fr(gj)g FA(=)f(2)3272 3576 y Fq(@)3315 3585 y Fn(0)3354 3609 y Fw(:)588 3776 y FA(In)k(Theorem)h(5.10,)h(w)m (e)f(can)g(replace)f(\012)g(b)m(y)h(an)m(y)g(1-random)f(set.)i(In)d (Section)i(13)505 3884 y(w)m(e)d(will)d(see)j(that)f(Jo)s(e)h(Miller)d (has)i(pro)m(v)m(ed)g(that)h(the)g Fw(K)7 b FA(-degree)27 b(of)f(an)m(y)h(1-random)505 3992 y(set)e(is)f(coun)m(table.)h(\(This)e (is)h(consisten)m(t,)h(since)f(there)g(is)g(no)g(natural)g(join)f(op)s (erator)505 4100 y(on)30 b(the)h Fw(K)7 b FA(-degrees,)31 b(or)f(indeed)f(an)m(y)h(join)f(op)s(erator)i(at)g(all;)e(see)i (Corollary)d(13.4.\))588 4208 y(In)34 b(Section)g(6)g(w)m(e)h(will)c (see)k(that)f Fz(6)1842 4222 y Fx(K)1944 4208 y FA(do)s(es)g(not)g (imply)e Fz(6)2648 4222 y Fy(T)2703 4208 y FA(,)i(ev)m(en)h(on)f(the)g (c.e.)505 4316 y(sets.)d(In)m(terestingly)-8 b(,)31 b Fz(6)1326 4330 y Fx(C)1415 4316 y FA(do)s(es)f(imply)e Fz(6)1946 4330 y Fy(T)2031 4316 y FA(on)i(the)h(left-c.e.)h(reals.)588 4483 y FB(Theorem)i FA(5.11)i(\(Stephan)15 b(\(p)s(ersonal)29 b(comm)m(unication\);)15 b(see)31 b([33)q(]\))p FB(.)46 b Fs(If)27 b Fw(\013)g Fs(and)505 4591 y Fw(\014)38 b Fs(ar)-5 b(e)34 b(left-c.e.)d(r)-5 b(e)g(als)34 b(such)f(that)h Fw(\013)26 b Fz(6)1834 4605 y Fx(C)1918 4591 y Fw(\014)5 b Fs(,)33 b(then)g Fw(\013)25 b Fz(6)2391 4605 y Fi(T)2472 4591 y Fw(\014)5 b Fs(.)588 4758 y FA(Theorem)23 b(5.11)g(generalizes)g (an)f(old)f(result)h(of)g(Chaitin)e([20)r(])i(\(whic)m(h)g(generalizes) 505 4866 y(an)31 b(ev)m(en)g(older)e(result)h(of)g(Lo)m(v)m(eland)h ([80)q(]\).)588 5033 y FB(Theorem)j FA(5.12)i(\(Chaitin)28 b([20)r(]\))p FB(.)46 b Fs(Supp)-5 b(ose)34 b(that)f(either)g Fw(C)7 b FA(\()p Fw(A)2833 5021 y Fz(\026)2896 5033 y Fw(n)p FA(\))26 b Fz(6)e Fw(C)7 b FA(\()p Fw(n)p FA(\))20 b(+)505 5141 y Fw(d)44 b Fs(for)f(an)g(in\014nite)g(c)-5 b(omputable)45 b(set)e(of)g Fw(n)f Fs(or)i Fw(C)7 b FA(\()p Fw(A)2408 5129 y Fz(\026)2490 5141 y Fw(n)p FA(\))44 b Fz(6)f FA(log)17 b Fw(n)27 b FA(+)g Fw(d)44 b Fs(for)f(an)p eop %%Page: 21 21 21 20 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(21)505 541 y Fs(in\014nite)39 b(c)-5 b(omputable)40 b(set)f(of)g Fw(n)p Fs(.)f(Then)h Fw(A)f Fs(is)h(c)-5 b(omputable.)40 b(F)-7 b(urthermor)i(e,)41 b(for)e(a)505 649 y(given)32 b(c)-5 b(onstant)35 b Fw(d)p Fs(,)e(ther)-5 b(e)33 b(ar)-5 b(e)34 b(only)f Fw(O)s FA(\(2)1943 616 y Fx(d)1984 649 y FA(\))g Fs(many)h(such)e Fw(A)p Fs(.)588 824 y FA(Before)25 b(w)m(e)f(turn)e(to)i(the)g(v)m(ery)f(in)m (teresting)g(relationship)e(of)i Fz(6)2731 838 y Fx(K)2823 824 y FA(to)h Fz(6)2998 838 y Fy(T)3053 824 y FA(,)f(w)m(e)h(lo)s(ok) 505 932 y(at)36 b(the)e(structure)g(of)h(the)g(left-c.e.)h(reals)e (under)f Fz(6)2319 946 y Fx(K)2387 932 y FA(.)h(There)g(are)h(ob)m (vious)f(simi-)505 1040 y(larities)28 b(b)s(et)m(w)m(een)h(Theorems)g (4.7)h(and)f(5.9.)h(Do)m(wney)g(and)e(Hirsc)m(hfeldt)g([33)q(])h(ha)m (v)m(e)505 1148 y(pro)m(v)m(ed)i(the)g(follo)m(wing)e(generalization,)h (whic)m(h)f(applies)f(to)k Fz(6)2694 1162 y Fx(K)2792 1148 y FA(in)d(particular.)588 1322 y FB(Theorem)34 b FA(5.13)i(\(Do)m(wney)31 b(and)f(Hirsc)m(hfeldt)f([33)q(]\))p FB(.)46 b Fs(L)-5 b(et)28 b Fz(6)2724 1336 y Fx(Q)2812 1322 y Fs(b)-5 b(e)27 b(any)i Fw(K)7 b Fs(-me)-5 b(a-)505 1433 y(sur)g(e)27 b(of)g(r)-5 b(elative)28 b(r)-5 b(andomness)29 b(with)f(a)f FA(\006)1931 1400 y Fy(0)1931 1457 y(3)1997 1433 y Fs(de\014nition)g(on)g(the)g(left-c.e.)f(r)-5 b(e)g(als,)28 b(such)505 1541 y(that)33 b(the)f(le)-5 b(ast)33 b Fw(Q)p Fs(-de)-5 b(gr)g(e)g(e)32 b(of)g(left-c.e.)e(r)-5 b(e)g(als)34 b(c)-5 b(ontains)33 b(the)f(c)-5 b(omputable)33 b(r)-5 b(e)g(als,)33 b(the)505 1649 y(top)39 b Fw(Q)p Fs(-de)-5 b(gr)g(e)g(e)39 b(of)f(left-c.e.)f(r)-5 b(e)g(als)40 b(is)e(that)h(of)f FA(\012)p Fs(,)g(and)h FA(+)f Fs(induc)-5 b(es)38 b(a)g(join)h(on)f(the)505 1757 y Fw(Q)p Fs(-de)-5 b(gr)g(e)g(es)34 b(of)f(left-c.e.)e(r)-5 b(e)g(als.)34 b(Then)f(the)g(fol)5 b(lowing)34 b(hold.)563 1892 y FA(\(i\))42 b Fs(The)33 b Fw(Q)p Fs(-de)-5 b(gr)g(e)g(es)33 b(of)g(left-c.e.)e(r)-5 b(e)g(als)34 b(ar)-5 b(e)34 b(dense.)538 2000 y FA(\(ii\))41 b Fs(F)-7 b(or)37 b(any)g Fw(Q)p Fs(-de)-5 b(gr)g(e)g(es)37 b Ft(a)32 b Fw(<)g Ft(b)h Fw(<)e FA(deg)1985 2022 y Fx(Q)2045 2000 y FA(\(\012\))36 b Fs(of)h(left-c.e.)f(r)-5 b(e)g(als,)38 b(ther)-5 b(e)37 b(ar)-5 b(e)37 b Fw(Q)p Fs(-)701 2108 y(de)-5 b(gr)g(e)g(es)44 b Ft(c)1065 2122 y Fh(0)1152 2108 y Fs(and)g Ft(c)1385 2122 y Fh(1)1473 2108 y Fs(of)f(left-c.e.)f (r)-5 b(e)g(als)44 b(such)f(that)h Ft(a)f Fw(<)g Ft(c)2815 2122 y Fh(0)2860 2108 y Fw(;)15 b Ft(c)2946 2122 y Fh(1)3036 2108 y Fw(<)43 b Ft(b)g Fs(and)701 2216 y Ft(c)747 2230 y Fh(0)812 2216 y Fr(_)20 b Ft(c)939 2230 y Fh(1)1010 2216 y FA(=)k Ft(b)p Fs(.)588 2390 y FA(A)m(t)41 b(a)f(talk)g(b)m(y)g (the)g(\014rst)f(author)g(in)g(Heidelb)s(erg,)f(in)h(Ma)m(y)i(2003,)g (Alexander)505 2498 y(Shen)27 b(p)s(oin)m(ted)g(out)h(that)h(a)f (natural)f(measure)g(of)i(relativ)m(e)f(randomness)e(could)h(b)s(e)505 2606 y(obtained)32 b(b)m(y)g(replacing)f(the)h(constan)m(t)h(in)e(the)h (de\014nition)e(of)i Fw(K)7 b FA(-reducibilit)m(y)29 b(b)m(y)505 2714 y Fw(O)s FA(\(log)17 b Fw(n)p FA(\).)25 b(That)f(is,)g(he)g(suggested)h(considering)e(the)h(ordering)g (de\014ned)f(b)m(y)h(letting)505 2822 y Fw(A)i Fz(6)f Fw(B)31 b FA(if)26 b Fw(K)7 b FA(\()p Fw(A)1087 2810 y Fz(\026)1151 2822 y Fw(n)p FA(\))25 b Fz(6)g Fw(K)7 b FA(\()p Fw(B)1580 2810 y Fz(\026)1643 2822 y Fw(n)p FA(\))13 b(+)g Fw(O)s FA(\(log)k Fw(n)p FA(\).)27 b(The)f(reason)h(for) g(this)e(suggestion)505 2930 y(is)k(that)h(v)-5 b(arious)29 b(approac)m(hes)h(to)h(de\014ning)d(relativ)m(e)h(randomness)g(are)h (equiv)-5 b(alen)m(t)505 3038 y(up)29 b(to)i(a)f(log)g(factor,)i(and)d (hence)h(this)f(de\014nition)e(w)m(ould)i(b)s(e)g(indep)s(enden)m(t)f (of)i(the)505 3146 y(c)m(hoice)24 b(of)f(approac)m(h.)h(W)-8 b(e)24 b(will)c(not)j(discuss)e(this)h(line)g(of)h(researc)m(h)g(here,) g(but)g(p)s(oin)m(t)505 3254 y(to)i(a)f(pap)s(er)f(b)m(y)h(Cherno)m(v,) g(Muc)m(hnik,)f(Romashc)m(henk)m(o,)i(Shen,)e(and)g(V)-8 b(ereshc)m(hagin)505 3362 y([24)r(].)30 b(Notice)i(that)f(this)e (ordering)g(is)g(still)g(\006)2056 3329 y Fy(0)2056 3386 y(3)2095 3362 y FA(.)588 3470 y(The)d(measures)g Fz(6)1228 3484 y Fx(C)1313 3470 y FA(and)f Fz(6)1556 3484 y Fx(K)1650 3470 y FA(seemed)i(at)g(\014rst)e(di\016cult)f(to)j(deal)e(with)g (directly)-8 b(,)505 3578 y(and)37 b(ev)m(en)h(no)m(w)g(there)f(is)f(m) m(uc)m(h)i(ab)s(out)f(them)g(that)h(is)e(not)i(kno)m(wn.)f(In)g(view)f (of)505 3686 y(Theorem)27 b(5.10,)i(it)e(w)m(as)h(not)f(ev)m(en)h (clear)g(whether)e(there)i(are)f(uncoun)m(tably)f(man)m(y)505 3793 y Fw(K)7 b FA(-degrees.)35 b(This)e(question)g(w)m(as)h(recen)m (tly)h(solv)m(ed)f(b)m(y)f(sho)m(wing)h(that)g(while)e(the)505 3901 y(cardinalit)m(y)f(of)i(the)g(collection)f(of)h(sets)f Fw(K)7 b FA(-b)s(elo)m(w)32 b(a)h(giv)m(en)g(set)g(can)f(b)s(e)g (large,)h(its)505 4009 y(measure)e(is)e(alw)m(a)m(ys)i(small.)588 4184 y FB(Theorem)j FA(5.14)i(\(Y)-8 b(u,)31 b(Ding,)f(and)g(Do)m(wney) h([141)r(]\))p FB(.)46 b Fs(F)-7 b(or)27 b(any)f(set)h Fw(B)5 b Fs(,)25 b(we)h(have)505 4292 y Fw(\026)p FA(\()p Fr(f)p Fw(A)33 b FA(:)g Fw(A)f Fz(6)970 4306 y Fx(K)1071 4292 y Fw(B)5 b Fr(g)p FA(\))32 b(=)g(0)p Fs(.)37 b(Henc)-5 b(e)36 b(ther)-5 b(e)37 b(ar)-5 b(e)37 b(unc)-5 b(ountably)38 b(many)f Fw(K)7 b Fs(-de)-5 b(gr)g(e)g(es)37 b(of)505 4400 y FA(1)p Fs(-r)-5 b(andom)35 b(sets.)588 4574 y FA(Using)24 b(Theorem)g(5.14,)i(Y)-8 b(u)24 b(and)g(Ding)g([140)q(])h (established)d(the)j(follo)m(wing)d(result.)588 4751 y FB(Theorem)34 b FA(5.15)i(\(Y)-8 b(u)31 b(and)e(Ding)h([140)r(]\))p FB(.)46 b Fs(Ther)-5 b(e)28 b(ar)-5 b(e)28 b FA(2)2549 4718 y Fq(@)2592 4727 y Fn(0)2658 4751 y Fs(many)h Fw(K)7 b Fs(-de)-5 b(gr)g(e)g(es)27 b(of)505 4859 y FA(1)p Fs(-r)-5 b(andom)35 b(sets.)588 5033 y FA(It)30 b(w)m(as)f(later)g(noticed)g (that)g(this)f(result)g(follo)m(ws)g(directly)g(from)g(Theorem)h(5.14) 505 5141 y(b)m(y)j(Silv)m(er's)e(Theorem)h([121)r(])h(that)g(an)m(y)g (coanalytic)g(equiv)-5 b(alence)31 b(relation)f(on)i(2)3344 5108 y Fx(!)p eop %%Page: 22 22 22 21 bop 505 363 a FD(22)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(with)30 b(uncoun)m(tably)g(man)m(y)g(equiv)-5 b(alence)31 b(classes)f(has)h(con)m(tin)m(uum)f(man)m(y)h(equiv)-5 b(a-)505 649 y(lence)24 b(classes.)h(Of)f(course,)g(it)g(also)g(follo)m (ws)f(from)h(Miller's)e(result)h(\(Theorem)i(13.9\))505 757 y(that)31 b(the)g Fw(K)7 b FA(-degree)31 b(of)g(an)m(y)f(1-random)h (set)g(is)e(coun)m(table.)588 865 y(The)c(follo)m(wing)e(are)j (examples)f(of)g(basic)f(questions)g(ab)s(out)h(the)g Fw(K)7 b FA(-degrees)26 b(that)505 973 y(remain)35 b(op)s(en.)g(\(A)h Fs(minimal)i(p)-5 b(air)47 b FA(is)34 b(a)i(pair)e(of)i(degrees)g Ft(a)p Fw(;)15 b Ft(b)35 b FA(suc)m(h)g(that)h(if)f Ft(c)g FA(is)505 1081 y(b)s(elo)m(w)30 b(b)s(oth)g Ft(a)g FA(and)g Ft(b)p FA(,)g(then)g Ft(c)c FA(=)f Ft(0)p FA(.\))588 1266 y FB(Question)33 b FA(5.16)p FB(.)47 b FA(Are)24 b(there)g(minimal)d(pairs)h(of)i Fw(K)7 b FA(-degrees)24 b(of)g(left-c.e.)h(reals?)505 1374 y(Do)32 b(the)e Fw(K)7 b FA(-degrees)31 b(of)g(left-c.e.)g(reals)f(form)g(a)h(lattice?)588 1559 y(F)-8 b(or)24 b(reducibilities)18 b(suc)m(h)k(as)g Fz(6)1643 1573 y Fy(S)1709 1559 y FA(and)g Fz(6)1949 1573 y Fy(rK)2035 1559 y FA(,)g(the)h(existence)g(of)f(minimal)e(pairs) h(fol-)505 1667 y(lo)m(ws)j(from)g(the)h(existence)f(of)h(minimal)c (pairs)i(in)g(the)h(T)-8 b(uring)23 b(degrees.)i(A)f(minimal)505 1775 y(pair)j(of)h Fw(K)7 b FA(-degrees)29 b(\(not)g(con)m(taining)e (left-c.e.)j(reals\))d(w)m(as)i(recen)m(tly)f(constructed)505 1883 y(b)m(y)j(Csima)e(and)h(Mon)m(talb\023)-45 b(an)30 b([27)q(],)h(using)e(Theorem)h(6.4)i(b)s(elo)m(w.)588 1990 y(In)i(Do)m(wney)h(and)f(Hirsc)m(hfeldt)f([33)q(])i(it)f(is)f(sho) m(wn)h(that)h(neither)f(the)g(S-degrees)505 2098 y(nor)j(the)g (rK-degrees)g(of)g(left-c.e.)i(reals)d(form)h(a)g(lattice,)h(b)m(y)e(a) i(straigh)m(tforw)m(ard)505 2206 y(adaptation)30 b(of)f(Jo)s(c)m(kusc)m (h's)g(pro)s(of)f([54)q(])h(of)g(the)g(corresp)s(onding)e(fact)j(for)f (the)g(wtt-)505 2314 y(degrees)i(of)g(left-c.e.)h(reals.)588 2422 y(There)38 b(are)g(a)g(n)m(um)m(b)s(er)e(of)i(exciting)f(recen)m (t)i(results)e(on)g Fz(6)2677 2436 y Fx(K)2783 2422 y FA(and)g Fz(6)3038 2436 y Fx(C)3135 2422 y FA(due)g(to)505 2530 y(Liang)30 b(Y)-8 b(u)31 b(and)e(Jo)s(e)i(Miller.)d(W)-8 b(e)32 b(will)c(discuss)g(some)j(of)g(these)f(in)f(Section)i(13.)588 2655 y Ft(5.3.)53 b(Other)34 b(w)m(a)m(ys)h(to)g(compare)g(randomness.) 45 b FA(It)31 b(is)e(p)s(ossible)f(to)j(de\014ne)505 2763 y(measures)36 b(of)f(relativ)m(e)h(randomness)e(based)i(on)f (other)h(approac)m(hes)g(to)g(random-)505 2871 y(ness.)46 b(In)f(unpublished)40 b(w)m(ork,)46 b(Do)m(wney)-8 b(,)47 b(Gri\016ths,)d(and)h(Hirsc)m(hfeldt)f(studied)505 2978 y Fs(sup)-5 b(ermartingale)39 b(r)-5 b(e)g(ducibility)p FA(,)36 b(where)e Fw(A)f Fz(6)2101 2992 y Fy(su)2204 2978 y Fw(B)39 b FA(if)34 b Fw(d)p FA(\()p Fw(B)2588 2966 y Fz(\026)2658 2978 y Fw(n)p FA(\))e(=)g Fw(O)s FA(\()p Fw(d)p FA(\()p Fw(A)3174 2966 y Fz(\026)3244 2978 y Fw(n)p FA(\)\),)505 3086 y(where)44 b Fw(d)g FA(is)f(an)h (optimal)f(c.e.)i(sup)s(ermartingale,)e(as)h(de\014ned)f(in)f(Section)i (3.2.)505 3194 y(Clearly)-8 b(,)45 b(there)g(is)g(a)h(greatest)h (su-degree,)f(consisting)e(of)h(the)h(1-random)f(sets)505 3302 y(\(whic)m(h)h(implies)f(that)i Fz(6)1427 3316 y Fy(su)1545 3302 y FA(is)f(not)h(a)g Fw(K)7 b FA(-measure)47 b(of)g(relativ)m(e)g(randomness\).)505 3410 y(Do)m(wney)-8 b(,)30 b(Gri\016ths,)d(and)h(Hirsc)m(hfeldt)e(sho)m(w)m(ed)j(that)f (the)h(computable)e(sets)i(form)505 3518 y(the)h(least)g(su-degree,)h (and)e(that)h(addition)e(induces)g(a)i(join)f(on)g(the)h(su-degrees)g (of)505 3626 y(left-c.e.)e(reals.)f(Th)m(us)e(Theorem)h(5.13)i(applies) d(to)i Fz(6)2329 3640 y Fy(su)2400 3626 y FA(.)g(It)g(is)e(not)i(kno)m (wn)f(whether)505 3734 y(there)k(is)f(an)g(exact)i(c)m(haracterization) g(of)f Fz(6)2030 3748 y Fy(su)2130 3734 y FA(in)e(terms)i(of)f(initial) e(segmen)m(t)k(com-)505 3842 y(plexit)m(y)-8 b(.)588 3950 y(It)24 b(w)m(ould)e(b)s(e)g(in)m(teresting)h(to)h(de\014ne)e(a)i (measure)f(of)g(relativ)m(e)h(randomness)e(based)505 4058 y(on)31 b(the)f(measure-theoretic)i(approac)m(h.)588 4292 y Fu(x)p Ft(6.)53 b Fw(K)7 b Ft(-trivialit)m(y)-9 b(,)29 b(P)m(ost's)h(Problem,)f(and)g(generalizing)i(the)d(Ku)m(\024) -49 b(cera-)505 4400 y(Slaman)34 b(Theorem.)588 4524 y(6.1.)53 b Fw(K)7 b Ft(-trivial)27 b(sets.)45 b FA(W)-8 b(e)25 b(return)e(to)h(the)g(fascinating)e(in)m(terrelationship)f(b)s (e-)505 4632 y(t)m(w)m(een)j Fz(6)826 4646 y Fx(K)915 4632 y FA(and)e Fz(6)1155 4646 y Fy(T)1210 4632 y FA(.)g(A)g(natural)f (question)g(is)g(whether)g Fz(6)2504 4646 y Fx(K)2594 4632 y FA(implies)e Fz(6)2965 4646 y Fy(T)3020 4632 y FA(.)j(W)-8 b(e)23 b(ha)m(v)m(e)505 4740 y(seen)35 b(that)h(if)d Fw(A)g Fz(6)1168 4754 y Fx(C)1260 4740 y Fr(;)i FA(then)f Fw(A)h FA(m)m(ust)g(b)s(e)f(computable.)g(Using)g(a)i(relativization) 505 4848 y(of)31 b(the)g(metho)s(d)e(of)i(the)f(pro)s(of)g(of)h(this)e (fact,)i(Chaitin)e(sho)m(w)m(ed)h(the)h(follo)m(wing.)588 5033 y FB(Theorem)j FA(6.1)h(\(Chaitin)29 b([21)q(]\))p FB(.)46 b Fs(If)35 b Fw(K)7 b FA(\()p Fw(A)2112 5021 y Fz(\026)2179 5033 y Fw(x)p FA(\))29 b Fz(6)g Fw(K)7 b FA(\()p Fw(x)p FA(\))23 b(+)e Fw(O)s FA(\(1\))p Fs(,)36 b(then)f Fw(A)30 b Fz(6)3340 5047 y Fi(T)505 5141 y Fr(;)550 5108 y Fq(0)574 5141 y Fs(.)p eop %%Page: 23 23 23 22 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(23)588 541 y FA(Surprisingly)-8 b(,)37 b(w)m(e)42 b(cannot)f(replace)g Fr(;)1942 508 y Fq(0)2007 541 y FA(b)m(y)f Fr(;)i FA(in)d(the)i(ab)s(o)m(v)m(e)h(result.)e(That)h(is,) 505 649 y(although)32 b Fw(A)g FA(ma)m(y)h(lo)s(ok)f(iden)m(tical)f(to) i(the)f(computable)g(sets)g(in)f(terms)h(of)g(initial)505 757 y(segmen)m(t)26 b(pre\014x-free)f(complexit)m(y)-8 b(,)25 b(w)m(e)g(cannot)h(conclude)e(that)i Fw(A)f FA(is)f(computable,) 505 865 y(ev)m(en)31 b(for)g(c.e.)g(sets)g Fw(A)p FA(.)588 973 y(W)-8 b(e)26 b(sa)m(y)f(that)g(a)g(set)f Fw(A)h FA(is)e Fw(K)7 b Fs(-trivial)34 b FA(if)23 b Fw(A)i Fz(6)2091 987 y Fx(K)2185 973 y Fr(;)p FA(.)g(Solo)m(v)-5 b(a)m(y)25 b([126)r(])f(w)m(as)h(the)f(\014rst)g(to)505 1081 y(construct)35 b(a)h(noncomputable)d Fw(K)7 b FA(-trivial)33 b(set;)i(this)f (construction)g(w)m(as)h(adapted)505 1189 y(to)28 b(the)g(case)g(of)f (c.e.)h(sets)g(b)m(y)f(Zam)m(b)s(ella)f([142)q(])h(\(see)i(also)e (Calude)e(and)i(Coles)g([15)q(]\).)505 1297 y(In)k([38)q(],)h(Do)m (wney)-8 b(,)34 b(Hirsc)m(hfeldt,)c(Nies,)h(and)g(Stephan)g(ga)m(v)m(e) j(a)e(new)f(construction)505 1405 y(of)d(a)f(noncomputable)f Fw(K)7 b FA(-trivial)26 b(c.e.)i(set,)g(whic)m(h)e(w)m(e)i(presen)m(t)f (b)s(elo)m(w.)g(\(A)g(similar)505 1513 y(construction)43 b(had)f(b)s(een)f(pro)s(duced)g(indep)s(enden)m(tly)f(b)m(y)i(Kummer)g (in)f(unpub-)505 1621 y(lished)31 b(w)m(ork.\))j(As)f(w)m(e)g(will)d (later)j(see,)h(this)e(construction)h(giv)m(es)g(a)g(priorit)m(y-free,) 505 1729 y(and)d(ev)m(en)h(a)g(requiremen)m(t-free,)f(solution)f(to)i (P)m(ost's)h(Problem.)588 1874 y FB(Theorem)i FA(6.2)h(\(Zam)m(b)s (ella)55 b([142)q(],)31 b(after)g(Solo)m(v)-5 b(a)m(y)57 b([126)q(]\))p FB(.)46 b Fs(Ther)-5 b(e)44 b(is)f(a)g(non-)505 1982 y(c)-5 b(omputable)35 b(c.e.)d(set)g Fw(A)h Fs(such)g(that)h Fw(K)7 b FA(\()p Fw(A)1994 1970 y Fz(\026)2057 1982 y Fw(n)p FA(\))25 b Fz(6)g Fw(K)7 b FA(\()p Fw(n)p FA(\))21 b(+)f Fw(O)s FA(\(1\))p Fs(.)588 2127 y FB(Remark.)45 b FA(While)30 b(the)h(pro)s(of)g(b)s(elo)m(w)g(is)f(easy)-8 b(,)33 b(it)e(is)g(sligh)m(tly)e(hard)i(to)h(see)g(wh)m(y)505 2235 y(it)41 b(w)m(orks.)h(So,)g(b)m(y)f(w)m(a)m(y)i(of)e(motiv)-5 b(ation,)42 b(supp)s(ose)e(that)i(w)m(e)g(w)m(ere)g(to)g(ask)m(ed)g(to) 505 2342 y(\\pro)m(v)m(e")24 b(that)f Fr(;)f FA(has)g(the)g(same)h (initial)c(segmen)m(t)k(complexit)m(y)f(complexit)m(y)g(as)g Fw(!)s FA(.)g(A)505 2450 y(complicated)k(w)m(a)m(y)h(to)f(do)g(this)f (w)m(ould)g(b)s(e)g(to)h(build)d(our)j(o)m(wn)g(pre\014x-free)f(mac)m (hine)505 2558 y Fw(M)48 b FA(whose)38 b(only)f(job)h(is)f(to)i (compute)f(initial)e(segmen)m(ts)j(of)f Fr(;)p FA(.)h(The)e(idea)h(w)m (ould)505 2666 y(b)s(e)29 b(that)i(if)d(the)i(univ)m(ersal)e (pre\014x-free)h(mac)m(hine)h Fw(U)39 b FA(con)m(v)m(erges)32 b(to)e(1)2895 2633 y Fx(n)2972 2666 y FA(on)g(input)d Fw(\033)505 2774 y FA(then)32 b Fw(M)10 b FA(\()p Fw(\033)s FA(\))j Fr(#)p FA(=)29 b(0)1140 2741 y Fx(n)1187 2774 y FA(.)j(Notice)h(that,)g(in)d(fact,)j(using)e(the)h(Kraft-Chaitin)e (Theorem)505 2882 y(it)d(w)m(ould)e(b)s(e)h(enough)g(to)i(build)23 b Fw(M)37 b Fs(implicitly)p FA(,)28 b(en)m(umerating)e(the)g(length)g (request)505 2991 y Fr(hj)p Fw(\033)s Fr(j)p Fw(;)15 b FA(0)730 2958 y Fx(n)779 2991 y Fr(i)p FA(.)23 b(W)-8 b(e)23 b(are)f(guaran)m(teed)h(that)1800 2923 y Fp(P)1896 3018 y Fx(\034)8 b Fq(2)p Fy(dom)o(\()p Fx(M)f Fy(\))2264 2991 y FA(2)2309 2958 y Fq(\000j)p Fx(\034)h Fq(j)2472 2991 y Fz(6)2568 2923 y Fp(P)2664 3018 y Fx(\033)r Fq(2)p Fy(dom\()p Fx(U)f Fy(\))3016 2991 y FA(2)3061 2958 y Fq(\000j)p Fx(\033)r Fq(j)3228 2991 y Fz(6)25 b FA(1,)505 3106 y(and)d(hence)g(the)g(Kraft-Chaitin)e(Theorem)i(applies.)e(Note)k (also)d(that)i(w)m(e)f(could,)g(for)505 3214 y(con)m(v)m(enience)29 b(and)e(as)h(w)m(e)g(do)g(in)e(the)i(main)f(construction,)g(use)h(a)g (string)e(of)i(length)505 3323 y Fr(j)p Fw(\033)s Fr(j)21 b FA(+)f(1,)31 b(in)e(whic)m(h)g(case)j(w)m(e)f(w)m(ould)e(ensure)g (that)2261 3254 y Fp(P)2357 3349 y Fx(\034)8 b Fq(2)p Fy(dom\()p Fx(M)f Fy(\))2726 3323 y FA(2)2771 3290 y Fq(\000j)p Fx(\034)h Fq(j)2934 3323 y Fw(<)25 b FA(1)p Fw(=)p FA(2.)588 3474 y FB(Pr)n(oof)34 b(of)g(Theorem)f(6.2.)42 b FA(W)-8 b(e)26 b(will)c(build)f(a)k(noncomputable)f(c.e.)h(set)g Fw(A)g FA(in)505 3582 y(place)c(of)f Fr(;)h FA(in)e(the)i(remark)f (and,)g(as)h(ab)s(o)m(v)m(e,)h(w)m(e)f(will)d(sla)m(vishly)g(follo)m(w) i(the)g(univ)m(ersal)505 3690 y(pre\014x-free)41 b(mac)m(hine)h Fw(U)51 b FA(on)41 b Fw(n)g FA(in)g(the)g(sense)h(that)g(whenev)m(er)f Fw(U)52 b FA(en)m(umerates,)505 3798 y(at)47 b(stage)f Fw(s)p FA(,)g(a)f(shorter)h Fw(\033)i FA(with)c Fw(U)10 b FA(\()p Fw(\033)s FA(\))52 b(=)e Fw(n)p FA(,)45 b(w)m(e)h(will)c(en)m (umerate)47 b(a)e(request)505 3906 y Fr(hj)p Fw(\033)s Fr(j)23 b FA(+)d(1)p Fw(;)15 b(A)912 3920 y Fx(s)978 3894 y Fz(\026)1043 3906 y Fw(n)p Fr(i)32 b FA(for)f(our)h(mac)m(hine)f Fw(M)10 b FA(.)32 b(T)-8 b(o)32 b(mak)m(e)h Fw(A)f FA(noncomputable,)f (w)m(e)h(will)505 4014 y(also)e(sometimes)h(mak)m(e)g Fw(A)1433 4028 y Fx(s)1495 4002 y Fz(\026)1558 4014 y Fw(n)25 b Fr(6)p FA(=)g Fw(A)1802 4028 y Fx(s)p Fy(+1)1954 4002 y Fz(\026)2017 4014 y Fw(n)p FA(.)30 b(Then)f(for)h(eac)m(h)i Fw(j)j FA(with)29 b Fw(n)c Fz(6)g Fw(j)31 b Fz(6)25 b Fw(s)p FA(,)505 4122 y(for)43 b(the)h(curren)m(tly)e(shortest)h(string) g Fw(\033)1910 4136 y Fx(j)1989 4122 y FA(computing)g Fw(j)5 b FA(,)44 b(w)m(e)f(will)e(also)i(need)g(to)505 4232 y(en)m(umerate)c(a)f(request)g Fr(hj)p Fw(\033)1474 4246 y Fx(j)1512 4232 y Fr(j)p Fw(;)15 b(A)1645 4246 y Fx(s)p Fy(+1)1810 4220 y Fz(\026)1886 4232 y Fw(j)5 b Fr(i)39 b FA(for)e Fw(M)10 b FA(.)39 b(The)e(construction)h(w)m(orks) g(b)m(y)505 4340 y(making)30 b(this)f(extra)i(measure)g(added)e(to)j (the)e(domain)f(of)i Fw(M)40 b FA(small.)588 4448 y(W)-8 b(e)32 b(are)f(ready)f(to)h(de\014ne)f Fw(A)p FA(:)588 4620 y Fw(A)c FA(:=)f Fr(f)p Fw(n)g FA(:)h Fr(9)p Fw(e)15 b Fr(9)p Fw(s)g FA([)p Fw(W)1307 4634 y Fx(e;s)1416 4620 y Fr(\\)k Fw(A)1564 4634 y Fx(s)1627 4620 y FA(=)25 b Fr(;)35 b(^)g Fw(n)25 b(>)g FA(2)p Fw(e)36 b Fr(^)f Fw(n)25 b Fr(2)g Fw(W)2546 4634 y Fx(e;s)2670 4620 y Fr(^)2323 4701 y Fp(X)2280 4897 y Fx(n)p Fk(6)p Fx(j)t Fk(6)p Fx(s)2512 4787 y FA(2)2557 4750 y Fq(\000)p Fx(K)2672 4758 y Fl(s)2706 4750 y Fy(\()p Fx(j)t Fy(\))2822 4787 y Fw(<)g FA(2)2963 4750 y Fq(\000)p Fy(\()p Fx(e)p Fy(+2\))3200 4787 y FA(])p Fr(g)p Fw(;)505 5033 y FA(where)h Fw(W)850 5047 y Fx(e;s)966 5033 y FA(is)f(the)i(stage)g Fw(s)f FA(appro)m(ximation)g(to)h(the)f Fw(e)p FA(-th)h(c.e.)h(set)e Fw(W)2931 5047 y Fx(e)2994 5033 y FA(and)g Fw(K)3244 5047 y Fx(s)3281 5033 y FA(\()p Fw(j)5 b FA(\))505 5141 y(is)30 b(the)g(stage)i Fw(s)e FA(appro)m(ximation)f(to)j(the)e Fw(K)7 b FA(-complexit)m(y)30 b(of)h Fw(j)5 b FA(.)p eop %%Page: 24 24 24 23 bop 505 363 a FD(24)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FA(Clearly)30 b Fw(A)i FA(is)e(c.e.)j(Since)1499 473 y Fp(P)1594 568 y Fx(j)t Fk(>)p Fx(m)1763 541 y FA(2)1808 508 y Fq(\000)p Fx(K)5 b Fy(\()p Fx(j)t Fy(\))2051 541 y FA(go)s(es)32 b(to)g(zero)g(as)g Fw(m)f FA(increases,)g(if)g Fw(W)3358 555 y Fx(e)505 658 y FA(is)k(in\014nite)f(then)i Fw(A)24 b Fr(\\)f Fw(W)1391 672 y Fx(e)1463 658 y Fr(6)p FA(=)34 b Fr(;)p FA(.)i(Since)f Fw(A)h FA(is)f(also)h(coin\014nite,)f (this)g(implies)e(that)505 766 y Fw(A)j FA(is)f(noncomputable.)g (Finally)-8 b(,)35 b(the)h(extra)h(measure)f(put)f(in)m(to)h(the)g (domain)e(of)505 874 y Fw(M)10 b FA(,)30 b(b)s(ey)m(ond)e(one)h(half)e (of)i(that)h(whic)m(h)d(en)m(ters)j(the)f(domain)e(of)i Fw(U)10 b FA(,)29 b(is)f(b)s(ounded)f(b)m(y)505 914 y Fp(P)601 1009 y Fx(e)653 982 y FA(2)698 949 y Fq(\000)p Fy(\()p Fx(e)p Fy(+2\))963 982 y FA(\(corresp)s(onding)f(to)i(at)g (most)f(one)h(initial)c(segmen)m(t)29 b(c)m(hange)f(for)f(eac)m(h)505 1090 y Fw(e)p FA(\),)32 b(whence)836 1206 y Fp(X)725 1407 y Fx(\033)r Fq(2)p Fy(dom\()p Fx(M)7 b Fy(\))1093 1292 y FA(2)1138 1255 y Fq(\000j)p Fx(\033)r Fq(j)1305 1292 y Fz(6)1501 1206 y Fp(X)1401 1407 y Fx(\033)r Fq(2)p Fy(dom\()p Fx(U)g Fy(\))1749 1292 y FA(2)1794 1255 y Fq(\000)p Fy(\()p Fq(j)p Fx(\033)r Fq(j)p Fy(+1\))2100 1292 y FA(+)2191 1206 y Fp(X)2240 1397 y Fx(e)2338 1292 y FA(2)2383 1255 y Fq(\000)p Fy(\()p Fx(e)p Fy(+2\))2645 1292 y Fz(6)2751 1231 y FA(1)p 2751 1272 46 4 v 2751 1355 a(2)2827 1292 y(+)2928 1231 y(1)p 2928 1272 V 2928 1355 a(2)3008 1292 y(=)25 b(1)p Fw(:)505 1556 y FA(So)h(the)f (Kraft-Chaitin)f(Theorem)h(applies,)f(and)h Fw(M)35 b FA(is)25 b(a)h(w)m(ell-de\014ned)d(pre\014x-free)505 1664 y(mac)m(hine.)31 b(Th)m(us)e Fw(K)7 b FA(\()p Fw(A)1330 1652 y Fz(\026)1393 1664 y Fw(n)p FA(\))25 b Fz(6)g Fw(K)7 b FA(\()p Fw(n)p FA(\))20 b(+)g Fw(O)s FA(\(1\).)1203 b Fr(a)588 1795 y FA(The)34 b(ab)s(o)m(v)m(e)h(pro)s(of)e(can)h(b)s(e)g (mo)s(di\014ed)d(to)k(pro)m(v)m(e)g(the)f(result,)f(due)g(to)i(Muc)m (hnik)505 1903 y(\(see)23 b([12)q(]\),)f(that)g(there)g(exists)f(a)h (noncomputable)f(c.e.)i(set)f Fw(A)f FA(that)h(is)f Fs(low)k(for)g Fw(K)7 b FA(,)22 b(in)505 2011 y(the)i(sense)f(that)h Fw(K)7 b FA(-complexit)m(y)24 b(relativized)e(to)i Fw(A)g FA(is)e(the)i(same)f(as)h Fw(K)7 b FA(-complexit)m(y)-8 b(,)505 2119 y(up)25 b(to)i(an)f(additiv)m(e)g(constan)m(t,)h(i.e.,)g Fw(K)7 b FA(\()p Fw(\033)s FA(\))25 b Fz(6)g Fw(K)2175 2086 y Fx(A)2232 2119 y FA(\()p Fw(\033)s FA(\))12 b(+)g Fw(O)s FA(\(1\).)27 b(Suc)m(h)f(an)g Fw(A)g FA(is)f(b)s(oth)505 2227 y Fw(K)7 b FA(-trivial)40 b(and)h(lo)m(w)h(for)g(1-randomness)f (\(whic)m(h)g(will)e(b)s(e)i(formally)f(de\014ned)h(in)505 2335 y(Section)34 b(7,)h(but)f(means)g(that)h(relativizing)d(the)j (de\014nition)d(of)i(1-randomness)g(to)505 2443 y Fw(A)25 b FA(do)s(es)f(not)g(c)m(hange)i(the)e(class)h(of)f(1-random)g(sets\).) i(W)-8 b(e)25 b(will)d(later)i(see)h(that)g(these)505 2551 y(concepts)32 b(are)f(all)f(equiv)-5 b(alen)m(t,)31 b(as)g(sho)m(wn)f(b)m(y)h(the)g(w)m(ork)g(of)g(Nies)g(\(see)h(Sections) e(7)505 2658 y(and)g(8\).)588 2766 y(Clearly)i(the)g(ab)s(o)m(v)m(e)i (pro)s(of)e(also)g(admits)g(man)m(y)g(v)-5 b(ariations.)32 b(F)-8 b(or)34 b(instance,)e(w)m(e)505 2874 y(can)23 b(mak)m(e)h Fw(A)f FA(promptly)e(simple,)g(or)i(b)s(elo)m(w)e(an)m(y)i (nonzero)g(c.e.)h(degree.)g(W)-8 b(e)24 b(cannot)505 2982 y(con)m(trol)29 b(the)g(jump)e(or)h(mak)m(e)i Fw(A)e FA(T)-8 b(uring)27 b(complete,)i(since)f(the)h Fw(K)7 b FA(-trivial)26 b(sets)j(are)505 3090 y(nonhigh,)k(as)i(sho)m(wn)e(b)m (y)i(Do)m(wney)-8 b(,)36 b(Hirsc)m(hfeldt,)d(Nies,)h(and)g(Stephan)f ([38)q(])i(\(and)505 3198 y(in)29 b(fact)j(lo)m(w,)e(as)h(sho)m(wn)f(b) m(y)g(Nies)g([103)q(]\);)i(see)f(Sections)f(6.2,)h(7,)g(and)f(8.)588 3306 y(As)21 b(w)m(e)g(will)c(see)k(in)e(Section)h(6.2,)i(the)f (construction)f(ab)s(o)m(v)m(e)h(automatically)f(yields)505 3414 y(a)37 b(T)-8 b(uring)35 b(incomplete)h(c.e.)h(set.)h(It)e(is)g (th)m(us)g(an)g Fs(injury-fr)-5 b(e)g(e)43 b FA(solution)35 b(to)i(P)m(ost's)505 3522 y(Problem.)21 b(It)g(is)g(not,)h(ho)m(w)m(ev) m(er,)h Fs(priority-fr)-5 b(e)g(e)p FA(,)23 b(in)e(that)h(it)f(dep)s (ends)e(on)i(an)h(ordering)505 3630 y(of)29 b(the)g(simplicit)m(y)d (requiremen)m(ts,)i(with)f(stronger)i(requiremen)m(ts)e(allo)m(w)m(ed)i (to)g(use)505 3738 y(up)34 b(more)i(of)f(the)g(domain)f(of)h(the)g(mac) m(hine)g Fw(M)10 b FA(.)35 b(W)-8 b(e)37 b(can)e(do)g(metho)s (dologically)505 3846 y(b)s(etter)i(b)m(y)f(giving)f(a)i(priorit)m (y-free)e(solution)g(to)i(P)m(ost's)g(Problem,)e(in)g(the)i(sense)505 3954 y(that)32 b(no)e(explicit)f(diagonalization)g(\(suc)m(h)i(as)g (that)g(of)g Fw(W)2521 3968 y Fx(e)2588 3954 y FA(ab)s(o)m(v)m(e\))h(o) s(ccurs)e(in)f(the)505 4062 y(construction)i(of)h(the)g(incomplete)e (c.e.)j(set,)f(and)f(therefore)h(the)f(construction)g(of)505 4169 y(this)25 b(set)i(\(as)f(opp)s(osed)f(to)i(the)f(v)m (eri\014cation)g(that)g(it)f(is)g(noncomputable\))h(do)s(es)f(not)505 4277 y(dep)s(end)c(on)g(an)h(ordering)f(of)h(requiremen)m(ts.)f(W)-8 b(e)23 b(no)m(w)f(sk)m(etc)m(h)h(this)e(metho)s(d,)h(whic)m(h)505 4385 y(is)27 b(due)h(to)g(Do)m(wney)-8 b(,)30 b(Hirsc)m(hfeldt,)c (Nies,)i(and)g(Stephan)f([38)q(],)h(and)g(is)f(rather)g(more)505 4493 y(lik)m(e)33 b(that)h(of)g(Solo)m(v)-5 b(a)m(y's)34 b(original)e(pro)s(of)h(of)g(the)h(existence)g(of)g(a)g(noncomputable) 505 4601 y Fw(K)7 b FA(-trivial)29 b(set.)588 4709 y(Let)k(us)e (reconsider)g(the)h(k)m(ey)h(idea)e(in)g(the)h(pro)s(of)f(of)h(Theorem) g(6.2.)h(A)m(t)g(certain)505 4817 y(stages)38 b(w)m(e)f(wish)e(to)i(c)m (hange)h(an)e(initial)e(segmen)m(t)k(of)e Fw(A)h FA(for)f(the)h(sak)m (e)g(of)g(diago-)505 4925 y(nalization.)32 b(Our)e(metho)s(d)i(is)f(to) i(mak)m(e)g(sure)e(that)i(the)f(total)h(measure)f(added)g(to)505 5033 y(the)k(domain)e(of)h(our)g(mac)m(hine)f Fw(M)46 b FA(\(whic)m(h)34 b(pro)m(v)m(es)i(the)f Fw(K)7 b FA(-trivialit)m(y)33 b(of)j Fw(A)p FA(\))f(due)505 5141 y(to)k(suc)m(h)e(c)m(hanges)h(is)e (b)s(ounded)g(b)m(y)h(1.)h(Supp)s(ose,)e(on)h(the)h(other)f(hand,)g (that)h(w)m(e)p eop %%Page: 25 25 25 24 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(25)505 541 y FA(w)m(ere)39 b(fortunate)g(enough)f(to)h(ha)m(v)m(e) g(the)g(univ)m(ersal)d(mac)m(hine)i(itself)f(\\co)m(v)m(er")k(the)505 649 y(measure)31 b(needed)g(for)f(these)i(c)m(hanges.)g(That)f(is,)f (supp)s(ose)f(w)m(e)i(w)m(ere)h(at)f(a)h(stage)g Fw(s)505 757 y FA(where)g(w)m(e)h(desired)d(to)j(put)e Fw(n)h FA(in)m(to)g Fw(A)1838 771 y Fx(s)p Fy(+1)1987 757 y Fr(\000)21 b Fw(A)2147 771 y Fx(s)2184 757 y FA(,)32 b(and)g(at)g(that)h(v)m(ery)g(stage)g Fw(K)3244 771 y Fx(s)3281 757 y FA(\()p Fw(j)5 b FA(\))505 865 y(c)m(hanged)39 b(for)e(all)f Fw(j)43 b Fr(2)37 b(f)p Fw(n;)15 b(:)g(:)g(:)32 b(;)15 b(s)p Fr(g)p FA(.)38 b(That)f(means)h(that)g Fs(in)h(any)h(c)-5 b(ase)45 b FA(w)m(e)38 b(w)m(ould)505 973 y(need)25 b(to)g(en)m (umerate)h(new)e(requests)g(describing)f Fw(A)2274 987 y Fx(s)p Fy(+1)2426 961 y Fz(\026)2489 973 y Fw(j)30 b FA(for)25 b(all)e Fw(j)31 b Fr(2)25 b(f)p Fw(n;)15 b(:)g(:)g(:)32 b(;)15 b(s)p Fr(g)p FA(,)505 1081 y(whether)27 b(or)g(not)g(these)g(initial)e(segmen)m(ts)j(c)m(hange.)g(Th)m(us)e(at) h(that)h(v)m(ery)f(stage,)i(w)m(e)505 1189 y(could)h(also)g(c)m(hange)i Fw(A)1299 1203 y Fx(s)1361 1177 y Fz(\026)1424 1189 y Fw(j)k FA(for)30 b(all)f Fw(j)i Fr(2)25 b(f)p Fw(n;)15 b(:)g(:)g(:)32 b(;)15 b(s)p Fr(g)31 b FA(at)g(no)f(extra)h(cost.)588 1297 y(Notice)26 b(that)e(w)m(e)h(w)m(ould)e(not)h(need)g(to)h(cop)m(y) g(the)f(univ)m(ersal)e(pre\014x-free)i(mac)m(hine)505 1405 y Fw(U)37 b FA(at)28 b(ev)m(ery)g(stage.)h(W)-8 b(e)28 b(could)f(en)m(umerate)g(a)h(collection)f(of)g(stages)i Fw(t)2908 1419 y Fy(0)2947 1405 y Fw(;)15 b(t)3020 1419 y Fy(1)3059 1405 y Fw(;)g(:)g(:)g(:)44 b FA(and)505 1513 y(only)31 b(up)s(date)f Fw(M)41 b FA(at)32 b(stages)g Fw(t)1555 1527 y Fx(i)1583 1513 y FA(.)f(Th)m(us,)g(for)f(the)i(luc)m (ky)e(situation)g(outlined)f(ab)s(o)m(v)m(e)505 1621 y(to)38 b(o)s(ccur,)f(w)m(e)g(w)m(ould)e(only)h(need)g(the)h(appro)m (ximation)f(to)h Fw(K)7 b FA(\()p Fw(j)e FA(\))38 b(to)f(c)m(hange)h (for)505 1729 y(all)32 b Fw(j)j Fr(2)29 b(f)p Fw(n;)15 b(:)g(:)g(:)32 b(;)15 b(t)1146 1743 y Fx(s)1183 1729 y Fr(g)33 b FA(at)g(some)h(stage)g Fw(u)e FA(with)g Fw(t)2167 1743 y Fx(s)2233 1729 y Fz(6)d Fw(u)g Fz(6)g Fw(t)2547 1743 y Fx(s)p Fy(+1)2674 1729 y FA(.)k(This)e(observ)-5 b(ation)505 1837 y(w)m(ould)26 b(seem)g(to)h(allo)m(w)f(a)h(greater)g (p)s(ossibilit)m(y)c(for)j(this)f(luc)m(ky)h(situation)f(to)i(o)s (ccur,)505 1944 y(since)j(man)m(y)h(stages)g(can)g(o)s(ccur)f(b)s(et)m (w)m(een)h Fw(t)2040 1958 y Fx(s)2107 1944 y FA(and)f Fw(t)2317 1958 y Fx(s)p Fy(+1)2444 1944 y FA(.)588 2052 y(The)42 b(k)m(ey)h(p)s(oin)m(t)e(in)g(this)h(discussion)d(is)j(the)g (follo)m(wing.)f(Let)i Fw(t)2858 2066 y Fy(0)2897 2052 y Fw(;)15 b(t)2970 2066 y Fy(1)3010 2052 y Fw(;)g(:)g(:)g(:)59 b FA(b)s(e)41 b(a)505 2160 y(computable)j(collection)g(of)g(stages.)h Fs(Supp)-5 b(ose)47 b(that)f(we)g(c)-5 b(onstruct)46 b(a)g(set)f Fw(A)j FA(=)505 2200 y Fp(S)581 2295 y Fx(s)633 2268 y Fw(A)701 2282 y Fx(t)726 2290 y Fl(s)799 2268 y Fs(so)36 b(that)h(for)f Fw(n)29 b Fz(6)h Fw(t)1473 2282 y Fx(s)1510 2268 y Fs(,)35 b(if)g Fw(A)1732 2282 y Fx(t)1757 2291 y Fl(s)p Fn(+1)1902 2256 y Fz(\026)1970 2268 y Fw(n)30 b Fr(6)p FA(=)g Fw(A)2224 2282 y Fx(t)2249 2290 y Fl(s)2317 2256 y Fz(\026)2385 2268 y Fw(n)35 b Fs(then)h Fw(K)2757 2282 y Fx(t)2782 2290 y Fl(s)2820 2268 y FA(\()p Fw(j)5 b FA(\))31 b Fw(<)f(K)3141 2282 y Fx(t)3166 2291 y Fl(s)p Fn(+1)3281 2268 y FA(\()p Fw(j)5 b FA(\))505 2376 y Fs(for)29 b(al)5 b(l)29 b Fw(j)k Fs(with)c Fw(n)c Fz(6)g Fw(j)31 b Fz(6)25 b Fw(t)1408 2390 y Fx(s)1445 2376 y Fs(.)j(Then)g Fw(A)g Fs(is)g Fw(K)7 b Fs(-trivial.)26 b FA(W)-8 b(e)26 b(are)g(no)m(w)g(ready)f(to)h(de\014ne)505 2484 y Fw(A)31 b FA(in)e(a)i(priorit)m(y-free)e(w)m(a)m(y)-8 b(.)588 2614 y FB(A)35 b(Priority-Free)f(Solution)e(to)i(Post's)g(Pr)n (oblem.)40 b FA(Let)28 b Fw(t)2961 2628 y Fy(0)3000 2614 y Fw(;)15 b(t)3073 2628 y Fy(1)3113 2614 y Fw(;)g(:)g(:)g(:)43 b FA(b)s(e)505 2722 y(a)25 b(collection)f(of)h(stages)g(suc)m(h)f(that) h Fw(t)1756 2736 y Fx(i)1808 2722 y FA(as)g(a)g(function)e(of)h Fw(i)h FA(dominates)f(all)f(primitiv)m(e)505 2830 y(recursiv)m(e)29 b(functions.)f(\(Actually)-8 b(,)30 b(w)m(e)f(do)g(not)h(need)f Fw(i)c Fr(7!)h Fw(t)2556 2844 y Fx(i)2613 2830 y FA(to)k(b)s(e)e(quite) h(this)f(fast)505 2938 y(gro)m(wing;)36 b(see)g(b)s(elo)m(w)f(for)g (more)g(details.\))g(A)m(t)i(eac)m(h)f(stage)h Fw(u)p FA(,)f(let)f Fr(f)p Fw(a)2948 2952 y Fx(i;u)3071 2938 y FA(:)f Fw(i)g Fr(2)f Fw(!)s Fr(g)505 3053 y FA(list)p 657 2980 69 4 v 29 w Fw(A)726 3067 y Fx(u)771 3053 y FA(.)d(De\014ne)1389 3170 y Fw(A)1457 3184 y Fx(t)1482 3193 y Fl(s)p Fn(+1)1622 3170 y FA(=)25 b Fw(A)1786 3184 y Fx(t)1811 3192 y Fl(s)1869 3170 y Fr([)20 b(f)p Fw(a)2043 3184 y Fx(n;t)2131 3192 y Fl(s)2169 3170 y Fw(;)15 b(:)g(:)g(:)i(;)e(t) 2404 3184 y Fx(s)2441 3170 y Fr(g)p Fw(;)505 3306 y FA(where)42 b Fw(n)f FA(is)g(the)g(least)h(n)m(um)m(b)s(er)f Fz(6)j Fw(t)1861 3320 y Fx(s)1939 3306 y FA(suc)m(h)d(that)i Fw(K)2441 3320 y Fx(t)2466 3329 y Fl(s)p Fn(+1)2581 3306 y FA(\()p Fw(j)5 b FA(\))45 b Fw(<)f(K)2930 3320 y Fx(t)2955 3328 y Fl(s)2993 3306 y FA(\()p Fw(j)5 b FA(\))43 b(for)f(all)505 3414 y Fw(j)50 b Fr(2)43 b(f)p Fw(n;)15 b(:)g(:)g(:)32 b(;)15 b(t)1046 3428 y Fx(s)1083 3414 y Fr(g)p FA(.)42 b(\(Naturally)-8 b(,)41 b(if)f(no)i(suc)m(h)f Fw(n)f FA(exists,)i Fw(A)2567 3428 y Fx(t)2592 3437 y Fl(s)p Fn(+1)2750 3414 y FA(=)i Fw(A)2933 3428 y Fx(t)2958 3436 y Fl(s)2996 3414 y FA(.\))e(Requir-)505 3522 y(ing)c(the)g(complexit)m (y)g(c)m(hange)i(for)e(all)f Fw(j)44 b Fr(2)38 b(f)p Fw(n;)15 b(:)g(:)g(:)31 b(;)15 b(t)2418 3536 y Fx(s)2455 3522 y Fr(g)p FA(,)39 b(rather)f(than)g(just)g Fw(j)44 b Fr(2)505 3630 y(f)p Fw(a)598 3644 y Fx(n;t)686 3652 y Fl(s)724 3630 y Fw(;)15 b(:)g(:)g(:)32 b(;)15 b(t)974 3644 y Fx(s)1011 3630 y Fr(g)p FA(,)27 b(ensures)f(that)h Fw(A)f FA(is)f(coin\014nite,)h(since)f(for)i(eac)m(h)g Fw(n)f FA(there)h(are)f(only)505 3738 y(\014nitely)j(man)m(y)h Fw(s)g FA(suc)m(h)g(that)h Fw(K)1610 3752 y Fx(t)1635 3761 y Fl(s)p Fn(+1)1751 3738 y FA(\()p Fw(n)p FA(\))25 b Fw(<)g(K)2074 3752 y Fx(t)2099 3760 y Fl(s)2137 3738 y FA(\()p Fw(n)p FA(\).)588 3846 y(Note)45 b(that)f(there)g(is)e(no)i (priorit)m(y)d(used)i(in)f(the)i(de\014nition)d(of)i Fw(A)p FA(.)h(It)g(is)e(lik)m(e)505 3954 y(the)e(Dekk)m(er)g (de\014ciency)e(set)i(or)f(the)g(so-called)g(\\dump)e(set")j(\(see)g (Soare)f([122)r(],)505 4062 y(Theorem)31 b(V.2.5\).)588 4169 y(It)h(remains)e(to)i(pro)m(v)m(e)g(that)g Fw(A)g FA(is)e(noncomputable.)h(By)g(the)h(Recursion)e(Theo-)505 4277 y(rem,)j(w)m(e)h(can)f(build)d(a)k(pre\014x-free)e(T)-8 b(uring)32 b(mac)m(hine)h Fw(M)43 b FA(and)32 b(kno)m(w)h(the)g(co)s (ding)505 4385 y(constan)m(t)42 b Fw(c)f FA(of)f Fw(M)50 b FA(in)40 b(the)g(univ)m(ersal)f(pre\014x-free)g(mac)m(hine)h Fw(U)10 b FA(.)41 b(That)f(is,)g(if)f(w)m(e)505 4493 y(declare)44 b Fw(M)10 b FA(\()p Fw(\033)s FA(\))48 b(=)g Fw(j)h FA(then)43 b(w)m(e)h(will)d(ha)m(v)m(e)k Fw(U)10 b FA(\()p Fw(\034)g FA(\))48 b(=)f Fw(j)i FA(for)44 b(some)g Fw(\034)54 b FA(suc)m(h)43 b(that)505 4601 y Fr(j)p Fw(\034)10 b Fr(j)44 b Fz(6)e Fr(j)p Fw(\033)s Fr(j)28 b FA(+)f Fw(c)p FA(.)42 b(Note)g(further)e(that)h(if)f(w)m(e)i(put)e Fw(\033)k FA(in)m(to)d(the)g(domain)f(of)h Fw(M)51 b FA(at)505 4709 y(stage)40 b Fw(t)779 4723 y Fx(s)815 4709 y FA(,)e(then)g Fw(\034)48 b FA(will)35 b(b)s(e)i(in)f(the)i (domain)f(of)h Fw(U)48 b FA(b)m(y)38 b(stage)h Fw(t)2727 4723 y Fx(s)p Fy(+1)2879 4709 y Fr(\000)25 b FA(1.)38 b(\(This)e(is)505 4817 y(wh)m(y)k(w)m(e)h(required)d Fw(i)k Fr(7!)f Fw(t)1455 4831 y Fx(i)1523 4817 y FA(to)g(dominate)f (the)g(primitiv)m(e)e(recursiv)m(e)i(functions.)505 4925 y(In)34 b(fact,)h(w)m(e)f(only)f(need)h(this)f(function)f(to)j (dominate)e(the)i(o)m(v)m(erhead)g(of)f(the)g(Re-)505 5033 y(cursion)d(Theorem;)h(that)g(is,)g(w)m(e)g(only)f(need)h(the)g (prop)s(ert)m(y)f(that)i(if)e Fw(\033)k FA(en)m(ters)d(the)505 5141 y(domain)k(of)h Fw(M)47 b FA(at)38 b(stage)g Fw(t)1470 5155 y Fx(s)1507 5141 y FA(,)f(then)f(there)h(is)f(a)h Fw(\034)47 b FA(suc)m(h)37 b(that)g Fr(j)p Fw(\034)10 b Fr(j)37 b Fz(6)f Fr(j)p Fw(\033)s Fr(j)25 b FA(+)f Fw(c)37 b FA(and)p eop %%Page: 26 26 26 25 bop 505 363 a FD(26)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y Fw(U)567 555 y Fx(t)592 564 y Fl(s)p Fn(+1)703 555 y Fq(\000)p Fy(1)798 541 y FA(\()p Fw(\034)10 b FA(\))g Fr(#)p FA(=)26 b Fw(M)10 b FA(\()p Fw(\033)s FA(\).)27 b(The)f(use)g(of)h(a)g(fast)f(gro)m(wing)h(sequence)f(of)h(stages)h(w)m (as)e(the)505 649 y(k)m(ey)32 b(insigh)m(t)d(in)g(Solo)m(v)-5 b(a)m(y's)31 b(original)e(construction.\))588 757 y(No)m(w)f(the)f(pro) s(of)f(lo)s(oks)g(lik)m(e)h(that)g(of)g(Theorem)g(6.2.)h(W)-8 b(e)28 b(dev)m(ote)g(2)2896 724 y Fq(\000)p Fx(e)3015 757 y FA(of)f(the)g(do-)505 865 y(main)d(of)h(our)f(mac)m(hine)h Fw(M)35 b FA(to)25 b(making)f(sure)g(that)i Fw(A)f FA(satis\014es)f (the)h Fw(e)p FA(-th)g(simplicit)m(y)505 974 y(requiremen)m(t.)i(When)g (w)m(e)g(see)h Fw(a)1617 988 y Fx(n;t)1705 996 y Fl(s)1770 974 y FA(o)s(ccur)e(in)g Fw(W)2200 988 y Fx(e;t)2278 996 y Fl(s)2315 974 y FA(,)i(where)2627 906 y Fp(P)2723 1001 y Fx(n)p Fk(6)p Fx(j)t Fk(6)p Fx(t)2934 1009 y Fl(s)2986 974 y FA(2)3031 941 y Fq(\000)p Fx(K)3146 949 y Fl(t)3170 957 y(s)3207 941 y Fy(\()p Fx(j)t Fy(\))3324 974 y Fw(<)505 1105 y FA(2)550 1072 y Fq(\000)p Fy(\()p Fx(e)p Fy(+)p Fx(c)p Fy(+1\))873 1105 y FA(,)h(w)m(e)h(pro)m(vide)e(shorter)h Fw(M)1778 1119 y Fx(t)1803 1127 y Fl(s)1870 1105 y FA(descriptions)e (of)i(all)f Fw(j)35 b FA(with)28 b Fw(n)d Fz(6)g Fw(j)31 b Fz(6)25 b Fw(t)3248 1119 y Fx(s)3313 1105 y FA(so)505 1213 y(that)31 b Fw(K)779 1227 y Fx(t)804 1236 y Fl(s)p Fn(+1)920 1213 y FA(\()p Fw(j)5 b FA(\))26 b Fw(<)f(K)1231 1227 y Fx(t)1256 1235 y Fl(s)1294 1213 y FA(\()p Fw(j)5 b FA(\))32 b(for)e(all)f(suc)m(h)h Fw(j)5 b FA(.)32 b(The)e(cost)h(of)f (this)g(c)m(hange)h(is)f(b)s(ounded)505 1324 y(b)m(y)h(2)677 1291 y Fq(\000)p Fx(e)769 1324 y FA(,)f(and)g Fw(a)1049 1338 y Fx(n;t)1137 1346 y Fl(s)1205 1324 y FA(will)e(en)m(ter)j Fw(A)1675 1338 y Fx(t)1700 1347 y Fl(s)p Fn(+1)1815 1324 y FA(,)f(as)h(required.)1004 b Fr(a)588 1450 y FA(While)34 b(the)i(ab)s(o)m(v)m(e)g(pro)s(of)e(do)s(es)h(mak)m(e)h(use)f(of)g(an)g (ordering)f(of)h(the)g(simplicit)m(y)505 1558 y(requiremen)m(ts,)d(it)g (do)s(es)f(so)i(only)e(in)g(the)h(v)m(eri\014cation)g(of)h(the)f(fact)h (that)g Fw(A)f FA(is)f(not)505 1666 y(computable,)23 b(and)g(not)g(in)f(the)h(construction)g(of)g Fw(A)p FA(,)h(whic)m(h)d (remains)h(priorit)m(y-free.)588 1774 y(One)37 b(remark)-5 b(able)36 b(fact)i(ab)s(out)f(the)g Fw(K)7 b FA(-trivial)35 b(sets)i(is)f(that)h(there)g(are)h(few)e(of)505 1881 y(them)31 b(for)f(an)m(y)h(giv)m(en)f(witnessing)e(constan)m(t)k(\(cf.) f(Theorem)f(5.12\).)588 2046 y FB(Theorem)k FA(6.3)h(\(Zam)m(b)s(ella) 30 b([142)q(]\))p FB(.)46 b Fs(F)-7 b(or)33 b(e)-5 b(ach)33 b Fw(d)f Fs(ther)-5 b(e)33 b(ar)-5 b(e)33 b Fw(O)s FA(\(2)2892 2013 y Fx(d)2933 2046 y FA(\))f Fs(many)h(sets)505 2154 y Fw(A)g Fs(such)g(that)h Fw(K)7 b FA(\()p Fw(A)1211 2142 y Fz(\026)1274 2154 y Fw(n)p FA(\))26 b Fz(6)f Fw(K)7 b FA(\()p Fw(n)p FA(\))20 b(+)g Fw(d)32 b Fs(for)i(al)5 b(l)33 b Fw(n)p Fs(.)505 2316 y FA(Pro)s(ofs)d(of)h(this)e(theorem)i (can)g(b)s(e)e(found)g(in)g([38)r(])h(and)g([33)q(].)588 2424 y(Recen)m(tly)-8 b(,)42 b(Csima)c(and)h(Mon)m(talb\023)-45 b(an)40 b([27)r(])g(ha)m(v)m(e)h(sho)m(wn)e(that)h(there)g(is)f(a)h (gap)505 2532 y(b)s(et)m(w)m(een)30 b(the)f(initial)e(segmen)m(t)j (complexit)m(y)f(of)g Fw(K)7 b FA(-trivial)27 b(sets)i(and)g(that)h(of) f(non-)505 2640 y Fw(K)7 b FA(-trivial)25 b(sets,)i(mirroring)d(the)i (gap)h(in)e(initial)e(segmen)m(t)28 b(complexit)m(y)e(b)s(et)m(w)m(een) h(1-)505 2748 y(random)22 b(sets)g(and)f(non-1-random)h(sets)h (illustrated)c(b)m(y)j(results)f(suc)m(h)h(as)g(Theorem)505 2856 y(3.10.)588 3018 y FB(Theorem)34 b FA(6.4)h(\(Csima)30 b(and)g(Mon)m(talb\023)-45 b(an)30 b([27)q(]\))p FB(.)47 b Fs(Ther)-5 b(e)41 b(is)f(a)h(nonde)-5 b(cr)g(e)g(asing)505 3126 y(unb)g(ounde)g(d)41 b(function)f Fw(f)48 b Fs(such)40 b(that)g(if)f Fw(A)h Fs(is)f(not)h Fw(K)7 b Fs(-trivial)40 b(then)f Fw(K)7 b FA(\()p Fw(A)3121 3114 y Fz(\026)3196 3126 y Fw(n)p FA(\))38 b Fw(>)505 3234 y(K)7 b FA(\()p Fw(n)p FA(\))21 b(+)f Fw(f)10 b FA(\()p Fw(n)p FA(\))19 b Fr(\000)h Fw(O)s FA(\(1\))p Fs(.)505 3396 y FA(As)31 b(men)m(tioned)f(ab)s(o)m(v)m(e,)h(Csima)f(and)f(Mon)m(talb\023)-45 b(an)31 b([27)q(])g(used)e(this)g(result)h(to)h(sho)m(w)505 3504 y(that)g(there)g(is)e(a)i(minimal)d(pair)h(of)h Fw(K)7 b FA(-degrees.)588 3629 y Ft(6.2.)53 b Fw(K)7 b Ft(-trivial)45 b(sets)g(solv)m(e)h(P)m(ost's)g(Problem.)g FA(While)38 b(w)m(e)i(will)c(see)k(im-)505 3737 y(pro)m(v)m(emen)m(ts) 33 b(on)e(the)h(follo)m(wing)e(results)g(when)h(w)m(e)h(consider)e (Nies')h(w)m(ork)h(in)e(Sec-)505 3845 y(tions)25 b(7)h(and)f(8,)h(w)m (e)g(remark)f(that)h Fw(K)7 b FA(-trivial)23 b(sets)j(are)g (necessarily)e(T)-8 b(uring)24 b(incom-)505 3953 y(plete,)k(and)f (indeed)g(not)h(of)g(high)e(degree,)j(and)e(hence)h(form)f(a)h (somewhat)g(natural)505 4061 y(solution)h(to)i(P)m(ost's)h(Problem.)588 4223 y FB(Theorem)i FA(6.5)h(\(Do)m(wney)-8 b(,)33 b(Hirsc)m(hfeldt,)c (Nies,)h(and)g(Stephan)f([38)q(]\))p FB(.)47 b Fs(If)i(a)g(set)505 4331 y Fw(A)33 b Fs(is)g Fw(K)7 b Fs(-trivial)32 b(then)h Fw(A)g Fs(is)g(T)-7 b(uring)32 b(inc)-5 b(omplete.)34 b(Inde)-5 b(e)g(d,)34 b Fw(A)f Fs(is)f(not)i(even)e(high.)505 4493 y FA(W)-8 b(e)31 b(discuss)c(the)j(pro)s(of)e(of)h(this)f(theorem) i(in)e(Section)h(8.)h(The)e(original)g(pro)s(of)g(can)505 4601 y(b)s(e)i(found)f(in)g([38)q(].)588 4709 y(Do)m(wney)-8 b(,)41 b(Hirsc)m(hfeldt,)e(Nies,)g(and)g(Stephan)f([38)q(])i(pro)m(v)m (ed)g(that)g(the)f(class)g(of)505 4817 y Fw(K)7 b FA(-trivial)37 b(sets)i(is)f(closed)h(under)e(wtt-reduction)i(and)f Fr(\010)p FA(.)g(The)h(former)f(will)e(b)s(e)505 4925 y(impro)m(v)m(ed)i(to)h(T)-8 b(uring)37 b(reduction)g(in)g(Section)h (8.)h(As)g(Nies)f(has)g(remark)m(ed,)h(this)505 5033 y(means)34 b(that)g(the)g Fw(K)7 b FA(-trivial)32 b(c.e.)i(sets)g(are)g (the)g(only)f(kno)m(wn)g(natural)f(non)m(trivial)505 5141 y(\006)571 5108 y Fy(0)571 5165 y(3)641 5141 y FA(ideal)d(in)g (the)i(\(c.e.\))h(T)-8 b(uring)29 b(degrees.)p eop %%Page: 27 27 27 26 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(27)588 541 y FA(Note)33 b(also)e(that,)g(since)f Fz(6)1508 555 y Fx(C)1598 541 y FA(implies)e Fz(6)1978 555 y Fy(T)2064 541 y FA(for)j(c.e.)h(sets,)f(the)g(existence)h(of)f (non-)505 649 y(computable)f Fw(K)7 b FA(-trivial)29 b(c.e.)i(sets)g(means)f(that)h Fz(6)2265 663 y Fx(K)2364 649 y FA(do)s(es)f(not)g(imply)f Fz(6)3057 663 y Fx(C)3115 649 y FA(.)588 803 y FB(Question)k FA(6.6)p FB(.)47 b FA(Do)s(es)29 b Fz(6)1526 817 y Fx(C)1612 803 y FA(imply)d Fz(6)1936 817 y Fx(K)2032 803 y FA(\(for)i(the)g(left-c.e.)h(reals\)?)f (W)-8 b(e)29 b(conjec-)505 911 y(ture)i(that)g(the)f(answ)m(er)g(is)g (no.)588 1064 y Ft(6.3.)53 b(Generalizing)30 b(Ku)m(\024)-49 b(cera-Slaman.)45 b FA(Theorem)24 b(4.6)i(implies)d(that)i(there)505 1172 y(is)g(essen)m(tially)g(only)g(one)h(1-random)g(left-c.e.)h(real,) f(and)f(it)g(is)g(\012.)h(So)g(the)g(strongest)505 1280 y(p)s(ossible)f(extension)h(for)h(the)g(left-c.e.)h(reals)e(of)h(the)g (fact)h(that)f(there)g(are)g(noncom-)505 1388 y(putable)c Fw(K)7 b FA(-trivial)23 b(sets)h(w)m(ould)f(b)s(e)h(that)g(if)f Fw(\013)i FA(is)e(a)i(nonrandom)e(left-c.e.)i(real)f(then)505 1495 y(there)32 b(is)e(a)i(left-c.e.)g(real)f Fw(\014)h Fr(\021)1561 1509 y Fx(K)1655 1495 y Fw(\013)g FA(suc)m(h)f(that)h Fw(\014)f Fo(\012)2302 1509 y Fy(T)2384 1495 y Fw(\013)p FA(.)h(The)e(follo)m(wing)g(theorem)505 1603 y(clari\014es)f(the)i (situation.)588 1757 y FB(Theorem)j FA(6.7)h(\(Do)m(wney)d(and)e(Y)-8 b(ang)31 b([43)q(]\))p FB(.)46 b Fs(Supp)-5 b(ose)34 b(that)g(a)f(left-c.e.)e(r)-5 b(e)g(al)34 b Fw(\013)505 1865 y Fs(has)41 b(the)e(pr)-5 b(op)g(erty)42 b(that)f(for)f(every)f (left-c.e.)f(r)-5 b(e)g(al)40 b Fw(\014)j Fz(6)2464 1879 y Fx(K)2569 1865 y Fw(\013)p Fs(,)d(we)f(have)h Fw(\014)i Fz(6)3216 1879 y Fi(T)3308 1865 y Fw(\013)p Fs(.)505 1972 y(Then)33 b Fw(\013)g Fs(is)g(T)-7 b(uring)33 b(c)-5 b(omplete.)505 2126 y FA(Notice)38 b(that)g(there)f(are)h(suc)m(h)f (left-c.e.)h(reals)f Fw(\013)p FA(.)h(Indeed)e(w)m(e)h(can)h(tak)m(e)g Fw(\013)f FA(=)g Fr(;)3346 2093 y Fq(0)3369 2126 y FA(,)505 2234 y(since)30 b Fw(\014)h Fz(6)881 2248 y Fy(wtt)1016 2234 y Fr(;)1061 2201 y Fq(0)1115 2234 y FA(for)f(ev)m(ery)h(left-c.e.) h(real)e Fw(\014)5 b FA(.)588 2441 y Fu(x)p Ft(7.)53 b(Lo)m(wness)h(for)h FA(1)p Ft(-randomness.)46 b FA(In)g(this)h (section,)g(w)m(e)h(discuss)e(rela-)505 2549 y(tivized)27 b(randomness)f(and)h(lo)m(wness)g(prop)s(erties)f(for)i(the)f(class)h (of)f(1-random)h(sets.)588 2657 y(Relativized)34 b(randomness)f(w)m(as) h(studied)f(b)m(y)h(sev)m(eral)h(authors,)f(including)d(v)-5 b(an)505 2765 y(Lam)m(balgen)33 b([72)q(],)g(Kurtz)e([71)q(],)i(and)f (Kautz)g([60)q(].)h(The)f(de\014nition)e(of)i(1-random-)505 2873 y(ness)27 b(relativ)m(e)h(to)g Fw(A)f FA(is)g(obtained)f(b)m(y)h (substituting)f(\\c.e.")j(b)m(y)e(\\)p Fw(A)p FA(-c.e.")j(in)c (De\014ni-)505 2981 y(tion)21 b(3.1.)i(That)e(is,)f Fw(X)29 b FA(is)20 b(1-random)h(relativ)m(e)h(to)f Fw(A)h FA(if)e(there)h(is)g (no)g(uniformly)d Fw(A)p FA(-c.e.)505 3089 y(sequence)38 b Fr(f)p Fw(U)996 3103 y Fx(i)1025 3089 y Fr(g)1070 3103 y Fx(i)p Fq(2)p Fx(!)1229 3089 y FA(of)f(\006)1405 3056 y Fx(A)1405 3113 y Fy(1)1462 3089 y FA(-classes)g(with)f Fw(\026)p FA(\()p Fw(U)2154 3103 y Fx(i)2183 3089 y FA(\))h Fz(6)f FA(2)2407 3056 y Fq(\000)p Fx(i)2528 3089 y FA(suc)m(h)h(that)h Fw(X)44 b Fr(2)3160 3021 y Fp(T)3236 3116 y Fx(i)3279 3089 y Fw(U)3341 3103 y Fx(i)3369 3089 y FA(.)505 3197 y(W)-8 b(e)29 b(can)g(similarly)24 b(relativize)j(notions)g(suc)m(h)h (as)g(c.e.)h(martingale)e(and)g(pre\014x-free)505 3305 y(Kolmogoro)m(v)32 b(complexit)m(y)-8 b(,)31 b(and)f(obtain)h(the)f (relativized)g(v)m(ersions)g(of)h(Theorems)505 3413 y(3.4)h(and)e(3.8.) 588 3520 y(In)24 b(computabilit)m(y)e(theory)j(a)f(set)h Fw(A)f FA(is)g(called)f Fs(low)35 b FA(if)23 b Fw(A)2493 3487 y Fq(0)2542 3520 y Fz(6)2613 3534 y Fy(T)2693 3520 y Fr(;)2738 3487 y Fq(0)2762 3520 y FA(,)h(where)g Fw(A)3136 3487 y Fq(0)3183 3520 y FA(is)f(the)505 3628 y(halting)29 b(problem)g(relativized)g(to)h Fw(A)p FA(;)h(that)g(is,)e(if)g(the)h (complexit)m(y)g(of)h(the)f(halting)505 3736 y(problem)i(do)s(es)g(not) h(increase)f(when)g(relativized)g(to)h Fw(A)g FA(\(and)f(hence)h(the)g (class)f(of)505 3844 y(\001)581 3811 y Fy(0)581 3869 y(2)652 3844 y FA(sets)h(do)s(es)e(not)h(c)m(hange)h(when)e (relativized)f(to)j Fw(A)p FA(\).)f(In)f(complexit)m(y)h(theory)-8 b(,)33 b(if)505 3952 y(a)h(class)f Fr(C)38 b FA(has)33 b(a)h(de\014nition)d(that)i(relativizes,)g(a)g(set)h Fw(A)g FA(is)e(called)g Fs(low)k(for)44 b Fr(C)38 b FA(\(or)505 4062 y Fr(C)5 b Fs(-low)10 b FA(\))39 b(if)e Fr(C)43 b FA(=)38 b Fr(C)1148 4029 y Fx(A)1205 4062 y FA(.)g(So)g(the)g(lo)m(w) g(sets)g(from)g(computabilit)m(y)e(theory)j(are)f(those)505 4169 y(that)f(are)f(lo)m(w)g(for)g(the)g(class)f(of)h(\001)1745 4136 y Fy(0)1745 4194 y(2)1820 4169 y FA(sets.)h(Similarly)-8 b(,)33 b(a)j(set)h Fw(A)f FA(suc)m(h)f(that)i(ev)m(ery)505 4277 y(1-random)24 b(set)g(is)f(1-random)h(relativ)m(e)g(to)g Fw(A)g FA(is)f(called)g Fs(low)k(for)g(the)g FA(1)p Fs(-r)-5 b(andom)29 b(sets)p FA(,)505 4385 y(or)i Fs(low)i(for)h FA(1)p Fs(-r)-5 b(andomness)p FA(.)588 4493 y(Clearly)d(,)39 b(ev)m(ery)i(computable)e(set)h(is)f(lo)m(w)g(for)g(1-randomness.)h(M.) g(v)-5 b(an)39 b(Lam-)505 4601 y(balgen)g(and)f(D.)i(Zam)m(b)s(ella)d (ask)m(ed)j(whether)e(there)h(exist)g(noncomputable)f(sets)505 4709 y(that)27 b(are)f(lo)m(w)g(for)g(1-randomness.)f(\(The)h(question) f(w)m(as)h(\014rst)g(stated)g(in)f(Zam)m(b)s(ella)505 4817 y([142)r(].\))35 b(This)d(question)h(w)m(as)h(raised)f(in)f(the)i (con)m(text)i(of)e(a)g(comparison)f(b)s(et)m(w)m(een)505 4925 y(randomness)20 b(prop)s(erties)e(in)i(classical)f(dynamic)g (systems)i(\(sp)s(eci\014cally)-8 b(,)19 b(Bernoulli)505 5033 y(sequences\))39 b(and)d(computabilit)m(y-theoretic)h(randomness.) g(A)h(result)e(of)i(Kamae)505 5141 y([58)r(])d(sho)m(w)m(ed)h(that)f (the)h(in\014nite)d(binary)g(sequences)j(that)g(ha)m(v)m(e)g(no)f (information)p eop %%Page: 28 28 28 27 bop 505 363 a FD(28)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(ab)s(out)42 b(Bernoulli)e(sequences)i(\(normal)f(sequences\))i (are)g(precisely)d(those)j(with)505 649 y(zero)38 b(en)m(trop)m(y)-8 b(.)39 b(This)c(fact)j(raised)e(the)i(issue)e(of)h(whether)f(a)i (similar)c(c)m(haracteri-)505 757 y(zation)e(exists)e(for)g(sets)i (that)f(ha)m(v)m(e)h(no)e(information)g(ab)s(out)g(1-random)h(sets,)g (and)505 865 y(motiv)-5 b(ated)37 b(the)g(question)f(of)g(the)h (existence)g(of)g(noncomputable)e(sets)i(that)g(are)505 973 y(lo)m(w)31 b(for)f(1-randomness,)g(whic)m(h)f(w)m(as)i(answ)m (ered)f(b)m(y)g(Ku)m(\024)-43 b(cera)32 b(and)d(T)-8 b(erwijn)29 b([69)q(].)588 1153 y FB(Theorem)34 b FA(7.1)h(\(Ku)m(\024) -43 b(cera)32 b(and)e(T)-8 b(erwijn)28 b([69)r(]\))p FB(.)46 b Fs(Ther)-5 b(e)49 b(exists)g(a)g(nonc)-5 b(omput-)505 1261 y(able)33 b(c.e.)f(set)h(that)h(is)e(low)i(for)f FA(1)p Fs(-r)-5 b(andomness.)588 1441 y FB(Pr)n(oof.)41 b FA(It)36 b(is)e(not)i(di\016cult)d(to)j(build)c(a)k(c.e.)g(op)s (erator)g Fw(I)42 b FA(suc)m(h)35 b(that)h Fw(I)3161 1408 y Fx(A)3253 1441 y FA(is)e(a)505 1549 y(univ)m(ersal)29 b(Solo)m(v)-5 b(a)m(y)32 b(test)f(relativ)m(e)g(to)g Fw(A)g FA(for)f(ev)m(ery)i(oracle)f Fw(A)p FA(.)g(\(Here)g(w)m(e)g (think)e(of)505 1657 y(Solo)m(v)-5 b(a)m(y)37 b(tests)f(as)g (collections)f(of)g(basic)g(clop)s(en)g(sets.\))h(Giv)m(en)g(a)f(set)h Fw(A)p FA(,)g(w)m(e)g(can)505 1766 y(attempt)28 b(to)f(co)m(v)m(er)h Fw(I)1235 1733 y Fx(A)1318 1766 y FA(with)d(an)i(unrelativized)d(Solo)m (v)-5 b(a)m(y)27 b(test)g Fw(J)36 b FA(b)m(y)26 b(adding)f([)p Fw(\033)s FA(])i(to)505 1875 y Fw(J)39 b FA(at)31 b(stage)g Fw(s)e FA(whenev)m(er)h([)p Fw(\033)s FA(])h(is)d(in)h Fw(I)1787 1842 y Fx(A)1840 1850 y Fl(s)1907 1875 y FA(at)h(stage)i Fw(s)p FA(.)d(W)-8 b(e)31 b(then)f(ha)m(v)m(e)h Fw(J)j Fr(\023)25 b Fw(I)3146 1842 y Fx(A)3203 1875 y FA(,)30 b(but)505 1983 y(w)m(e)h(also)f(need)g(to)g(build)d Fw(A)j FA(to)h(ensure)e(that)i Fw(\026)p FA(\()2154 1915 y Fp(S)2245 1983 y Fw(J)9 b FA(\))26 b(=)2461 1915 y Fp(P)2557 2010 y Fy([)p Fx(\033)r Fy(])p Fq(2)p Fx(J)2750 1983 y Fw(\026)p FA(\([)p Fw(\033)s FA(]\))g Fw(<)f Fr(1)p FA(,)30 b(and)505 2098 y(hence)35 b Fw(J)44 b FA(is)33 b(a)i(Solo)m(v)-5 b(a)m(y)35 b(test.)h(That)e(is,)g(w)m(e)h(need)f(to)h(build)d Fw(A)i FA(so)h(that)g(the)g(total)505 2206 y(measure)c(of)f(the)h (\\mistak)m(es")g(w)m(e)g(mak)m(e)h(in)d(appro)m(ximating)g Fw(I)2711 2173 y Fx(A)2798 2206 y FA(is)h(not)g(to)s(o)i(big.)588 2314 y(The)23 b(crucial)f(idea)h(comes)h(from)f([69)q(]:)h(Let)g Fw(M)2142 2328 y Fx(s)2179 2314 y FA(\()p Fw(n)p FA(\))f(b)s(e)g(the)g (collection)g(of)g Fw(\033)k FA(whic)m(h)505 2423 y(are)g(in)e Fw(I)802 2390 y Fx(A)855 2398 y Fl(s)918 2423 y FA(at)i(stage)h Fw(s)e FA(with)f(use)h(greater)h(than)f Fw(n)p FA(.)g(If)g Fw(n)35 b(=)-56 b Fr(2)25 b Fw(A)2617 2437 y Fx(s)2680 2423 y FA(and)h(w)m(e)h(en)m(umerate)505 2531 y Fw(n)36 b FA(in)m(to)g Fw(A)p FA(,)h(then)f(the)g(elemen)m(ts)h(of)f Fw(M)1866 2545 y Fx(s)1903 2531 y FA(\()p Fw(n)p FA(\))g(ma)m(y)h(b)s (e)f(in)f Fw(J)e Fr(\000)24 b Fw(I)2734 2498 y Fx(A)2791 2531 y FA(,)36 b(where)g Fw(J)45 b FA(is)35 b(as)505 2639 y(ab)s(o)m(v)m(e,)d(so)d(w)m(e)i(need)e(to)h(ensure)f(that)i Fw(\026)p FA(\()1923 2571 y Fp(S)2014 2639 y Fw(M)2102 2653 y Fx(s)2139 2639 y FA(\()p Fw(n)p FA(\)\))f(for)f Fw(n)c Fr(2)g Fw(A)2701 2653 y Fx(s)p Fy(+1)2847 2639 y Fr(\000)18 b Fw(A)3004 2653 y Fx(s)3071 2639 y FA(is)29 b(small,)505 2747 y(while)j(still)g(making)g Fw(A)i FA(noncomputable.)f (Th)m(us)g(at)h(eac)m(h)h(stage)g Fw(s)p FA(,)e(for)g(the)h(least)505 2855 y Fw(e)d(<)f(s)j FA(suc)m(h)g(that)h Fw(A)1231 2869 y Fx(s)1290 2855 y Fr(\\)22 b Fw(W)1459 2869 y Fx(e;s)1578 2855 y FA(=)30 b Fr(;)p FA(,)k(if)f(there)g(is)g(an)g Fw(n)d Fr(2)g Fw(W)2591 2869 y Fx(e;s)2713 2855 y FA(suc)m(h)j(that)h Fw(n)c(>)g FA(2)p Fw(e)505 2963 y FA(and)g Fw(\026)p FA(\()772 2895 y Fp(S)863 2963 y Fw(M)951 2977 y Fx(s)988 2963 y FA(\()p Fw(n)p FA(\)\))c Fw(<)f FA(2)1315 2930 y Fq(\000)p Fx(e)1407 2963 y FA(,)31 b(then)f(w)m(e)h(put)e(the)i (least)g(suc)m(h)f Fw(n)g FA(in)m(to)g Fw(A)p FA(.)588 3071 y(It)f(is)e(easy)i(to)g(see)g(that)f(suc)m(h)g(an)g(en)m (umeration)g(can)h(happ)s(en)d(at)j(most)g(once)g(for)505 3179 y(eac)m(h)35 b Fw(e)p FA(,)g(and)e(hence)g(the)h(total)h(measure)e (of)h(our)g(mistak)m(es,)g(namely)f(the)h(sets)g(in)505 3287 y Fw(J)27 b Fr(\000)18 b Fw(I)718 3254 y Fx(A)775 3287 y FA(,)30 b(is)e(b)s(ounded)f(b)m(y)1415 3219 y Fp(P)1511 3314 y Fx(e)1563 3287 y FA(2)1608 3254 y Fq(\000)p Fx(e)1700 3287 y FA(,)j(whic)m(h)e(implies)e(that)k Fw(J)38 b FA(is)28 b(a)i(Solo)m(v)-5 b(a)m(y)30 b(test.)h(So)505 3400 y(if)i Fw(X)41 b FA(is)33 b(not)h(1-random)g(relativ)m(e)g(to)g Fw(A)g FA(then)g Fw(X)k Fr(2)2339 3332 y Fp(S)2430 3400 y Fw(I)2477 3367 y Fx(A)2565 3400 y Fr(\022)2667 3332 y Fp(S)2758 3400 y Fw(J)9 b FA(,)34 b(and)f(hence)h Fw(X)505 3508 y FA(is)f(not)h(1-random.)g(This)e(implies)f(that)k Fw(A)f FA(is)f(lo)m(w)g(for)h(1-randomness.)g(Th)m(us)e(the)505 3616 y(only)38 b(thing)g(left)h(to)g(pro)m(v)m(e)h(is)e(that)h Fw(A)g FA(is)f(noncomputable.)g(Since)g Fw(A)g FA(is)g(clearly)505 3724 y(coin\014nite,)30 b(it)g(is)f(enough)h(to)h(sho)m(w)f(that)h(if)f Fw(W)2123 3738 y Fx(e)2190 3724 y FA(is)f(in\014nite)f(then)i Fw(A)21 b Fr(\\)f Fw(W)3052 3738 y Fx(e)3114 3724 y Fr(6)p FA(=)k Fr(;)p FA(.)588 3832 y(Giv)m(en)30 b Fw(e)g FA(suc)m(h)g(that)g Fw(W)1410 3846 y Fx(e)1477 3832 y FA(is)f(in\014nite,)e(let)j Fw(s)g FA(b)s(e)f(a)h(stage)h(suc)m(h)f(that)g(for)g(all)f Fw(i)c(<)g(e)505 3940 y FA(w)m(e)35 b(ha)m(v)m(e)g Fw(A)23 b Fr(\\)f Fw(W)1116 3954 y Fx(i)1175 3940 y Fr(6)p FA(=)31 b Fr(;)g(\))g Fw(A)1543 3954 y Fx(s)1602 3940 y Fr(\\)23 b Fw(W)1772 3954 y Fx(i;s)1883 3940 y Fr(6)p FA(=)31 b Fr(;)p FA(.)j(Supp)s(ose)e(for)i(a)g(con)m(tradiction)g(that)505 4048 y Fw(A)23 b Fr(\\)f Fw(W)765 4062 y Fx(e)831 4048 y FA(=)30 b Fr(;)p FA(.)k(It)g(is)e(easy)i(to)g(build)c(sequences)k(2)p Fw(e)d(<)f(n)2474 4062 y Fy(0)2543 4048 y Fw(<)g(n)2699 4062 y Fy(1)2768 4048 y Fw(<)g(n)2924 4062 y Fy(2)2993 4048 y Fw(<)g Fr(\001)15 b(\001)g(\001)49 b FA(and)505 4156 y Fw(s)33 b Fz(6)f Fw(s)727 4170 y Fy(0)799 4156 y Fw(<)g(s)945 4170 y Fy(1)1017 4156 y Fw(<)g(s)1163 4170 y Fy(2)1235 4156 y Fw(<)g Fr(\001)15 b(\001)g(\001)51 b FA(suc)m(h)34 b(that)i(the)f Fw(M)2154 4170 y Fx(s)2187 4180 y Fl(i)2217 4156 y FA(\()p Fw(n)2307 4170 y Fx(i)2335 4156 y FA(\))g(are)g(pairwise)e(disjoin)m(t)g(and)505 4264 y Fw(n)560 4278 y Fx(i)629 4264 y Fr(2)41 b Fw(W)817 4278 y Fx(e;s)903 4288 y Fl(i)972 4264 y FA(for)e(eac)m(h)j Fw(i)p FA(.)e(Since)f Fw(n)1733 4278 y Fx(i)1812 4264 y Fw(=)-56 b Fr(2)41 b Fw(A)f FA(for)g(all)e Fw(i)p FA(,)j(it)e(m)m (ust)h(b)s(e)f(the)h(case)h(that)505 4372 y Fw(\026)p FA(\()595 4304 y Fp(S)686 4372 y Fw(M)774 4386 y Fx(s)807 4396 y Fl(i)838 4372 y FA(\()p Fw(n)928 4386 y Fx(i)956 4372 y FA(\)\))30 b Fz(>)g FA(2)1202 4339 y Fq(\000)p Fx(e)1327 4372 y FA(for)j(all)f Fw(i)p FA(.)h(But)h Fw(M)1959 4386 y Fx(s)1992 4396 y Fl(i)2022 4372 y FA(\()p Fw(n)2112 4386 y Fx(i)2140 4372 y FA(\))c Fr(\032)g Fw(J)42 b FA(for)33 b(all)f Fw(i)p FA(,)h(so)h Fw(\026)p FA(\()2963 4304 y Fp(S)3054 4372 y Fw(J)9 b FA(\))30 b(=)f Fr(1)p FA(,)505 4480 y(con)m(tradicting)i(the)f(fact)i(that)f Fw(J)39 b FA(is)29 b(a)i(Solo)m(v)-5 b(a)m(y)31 b(test.)992 b Fr(a)588 4620 y FA(Ku)m(\024)-43 b(cera)34 b(and)d(T)-8 b(erwijn)30 b(left)i(op)s(en)g(questions)f(ab)s(out)h(the)g(p)s (ossible)e(complexit)m(y)505 4728 y(of)k(sets)h(that)f(are)h(lo)m(w)e (for)h(1-randomness.)g(As)g(w)m(e)g(will)d(see,)k(this)e(complexit)m(y) g(is)505 4835 y(restricted)27 b(in)e(v)-5 b(arious)25 b(w)m(a)m(ys.)j(F)-8 b(or)27 b(instance,)g(if)e Fw(A)i FA(is)e(lo)m(w)h(for)h(1-randomness)f(then)505 4943 y(it)f(is)f(GL)806 4957 y Fy(1)845 4943 y FA(,)i(meaning)e(that)i Fw(A)1514 4910 y Fq(0)1563 4943 y Fz(6)1634 4957 y Fy(T)1714 4943 y Fw(A)10 b Fr(\010)g(;)1918 4910 y Fq(0)1966 4943 y FA(\(Ku)m(\024)-43 b(cera)27 b([67)q(];)f(see)g(also)f(Corollary)f (7.8\),)505 5051 y(and)30 b(has)g(certain)h(traceabilit)m(y)f(prop)s (erties.)p eop %%Page: 29 29 29 28 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(29)588 541 y FB(Definition)35 b FA(7.2)h(\(Zam)m(b)s(ella,)29 b(see)i([132)r(]\))p FB(.)46 b FA(A)27 b(set)g Fw(A)h FA(is)e Fs(c.e.-tr)-5 b(ac)g(e)g(able)34 b FA(if)26 b(there)505 649 y(is)31 b(a)i(computable)e(function)g Fw(p)h FA(suc)m(h)f(that,)i (for)f(eac)m(h)h(function)e Fw(f)37 b Fz(6)2891 663 y Fy(T)2974 649 y Fw(A)p FA(,)32 b(there)g(is)505 757 y(a)f(computable)f (function)f Fw(h)i FA(\(called)f(a)h Fs(tr)-5 b(ac)g(e)38 b FA(for)30 b Fw(f)10 b FA(\))30 b(satisfying,)f(for)i(all)e Fw(n)p FA(,)563 884 y(\(i\))42 b Fr(j)p Fw(W)812 902 y Fx(h)p Fy(\()p Fx(n)p Fy(\))954 884 y Fr(j)26 b Fz(6)f Fw(p)p FA(\()p Fw(n)p FA(\))30 b(and)538 996 y(\(ii\))41 b Fw(f)10 b FA(\()p Fw(n)p FA(\))24 b Fr(2)h Fw(W)1077 1014 y Fx(h)p Fy(\()p Fx(n)p Fy(\))1220 996 y FA(.)588 1159 y(It)43 b(is)e(not)h(hard)f(to)h(c)m(hec)m(k)i(that)f(the)f(ab)s (o)m(v)m(e)h(de\014nition)d(do)s(es)h(not)h(c)m(hange)i(if)505 1267 y(w)m(e)33 b(replace)g(\\there)g(is)f(a)h(computable)f(function)f Fw(p)p FA(")i(b)m(y)f(\\for)h(ev)m(ery)g(computable)505 1375 y(un)m(b)s(ounded)38 b(function)g Fw(p)p FA(")i(\(cf.)h(Prop)s (osition)d(11.2\).)k(Since)d(one)h(can)g(uniformly)505 1483 y(en)m(umerate)f(all)e(c.e.)i(traces)g(for)f(a)g(\014xed)f(b)s (ound)f Fw(p)p FA(,)i(there)g(is)f(a)h(univ)m(ersal)e(trace)505 1591 y(with)26 b(b)s(ound)f Fw(p)p FA(,)i(that)g(is,)f(one)h(that)h (traces)g(eac)m(h)g(function)d Fw(f)35 b Fz(6)2711 1605 y Fy(T)2791 1591 y Fw(A)27 b FA(on)g(almost)g(all)505 1699 y(inputs.)588 1862 y FB(Theorem)34 b FA(7.3)h(\(T)-8 b(erwijn)29 b(and)h(Zam)m(b)s(ella)f([134)r(]\))p FB(.)46 b Fs(If)25 b Fw(A)h Fs(is)g(low)g(for)h FA(1)p Fs(-r)-5 b(andom-)505 1970 y(ness)33 b(then)g Fw(A)g Fs(is)g(c.e.-tr)-5 b(ac)g(e)g(able.)505 2133 y FA(Nies)32 b([102)q(])g(has)g(recen)m(tly)g (impro)m(v)m(ed)f(this)f(result)h(b)m(y)h(replacing)e(c.e.-traceabilit) m(y)505 2241 y(with)f(a)i(stronger)g(prop)s(ert)m(y)f(called)f (jump-traceabilit)m(y;)g(see)i([102)r(])f(for)h(details.)588 2349 y(T)-8 b(erwijn)39 b(and)h(Zam)m(b)s(ella)f(ask)m(ed)i(whether)f (all)g(sets)g(that)i(are)e(lo)m(w)h(for)f(1-ran-)505 2457 y(domness)j(are)h(\001)1124 2424 y Fy(0)1124 2482 y(2)1163 2457 y FA(.)g(It)f(w)m(as)h(also)f(noted)h(b)m(y)f(Do)m(wney) -8 b(,)45 b(Hirsc)m(hfeldt,)d(Nies,)h(and)505 2565 y(Stephan)36 b([38)q(])g(that)h(the)f(construction)g(in)f(the)h(pro)s(of)f(of)i (Theorem)f(7.1)h(b)s(ears)e(a)505 2673 y(close)28 b(resem)m(blance)f (to)g(that)h(of)f(a)g Fw(K)7 b FA(-trivial)26 b(set.)h(This)f(is)g(no)g (coincidence,)h(as)g(the)505 2781 y(t)m(w)m(o)32 b(classes)f(are)f(in)f (fact)j(the)e(same!)588 2944 y FB(Theorem)k FA(7.4)h(\(Nies)c([103)q (]\))p FB(.)46 b Fs(A)41 b(set)g Fw(A)g Fs(is)f(low)i(for)g FA(1)p Fs(-r)-5 b(andomness)43 b(iff)e Fw(A)g Fs(is)505 3052 y Fw(K)7 b Fs(-trivial.)588 3216 y FA(Both)36 b(directions)e(of)i (this)e(theorem)h(are)h(rather)f(hard)g(to)h(pro)m(v)m(e,)g (particularly)505 3324 y(the)31 b(\\if)7 b(")30 b(direction,)f(whic)m (h)f(will)g(b)s(e)h(discussed)f(in)h(Section)h(8.)h(The)e(pro)s(of)g (of)i(this)505 3432 y(direction)g(is)f(based)h(on)h(Nies')f(pro)s(of)g (that)h(the)g(class)f(of)h Fw(K)7 b FA(-trivial)29 b(sets)j(is)f (closed)505 3540 y(do)m(wn)m(w)m(ard)i(under)e(T)-8 b(uring)32 b(reducibilit)m(y)-8 b(,)30 b(and)i(is)g(necessarily)g(non)m(uniform,)f (in)g(a)505 3648 y(sense)g(that)g(will)c(b)s(e)j(made)h(precise)e(b)s (elo)m(w.)h(As)h(w)m(e)f(will)e(see)j(in)e(Section)h(8,)h(it)f(can)505 3756 y(b)s(e)j(extended)f(to)i(sho)m(w)f(that)g(ev)m(ery)h Fw(K)7 b FA(-trivial)31 b(set)i(is)f(lo)m(w)h(for)f Fw(K)40 b FA(\(as)33 b(de\014ned)f(in)505 3863 y(Section)f(6.1\).)588 3971 y(The)21 b(pro)s(of)f(of)h(the)g(\\only)f(if)7 b(")21 b(direction)e(of)i(Theorem)g(7.4)h(w)m(en)m(t)f(through)g(v)-5 b(arious)505 4079 y(stages;)24 b(see)e(Nies)g([104)q(])g(for)g (details.)f(One)g(approac)m(h)h(is)f(to)i(con)m(v)m(ert)g(the)f(h)m(yp) s(othesis)505 4187 y(that)29 b Fw(A)f FA(is)f(lo)m(w)h(for)f (1-randomness)h(in)m(to)g(a)g(com)m(binatorial)f(condition,)g(whic)m(h) f(can)505 4295 y(then)41 b(b)s(e)f(used)h(to)g(establish)f(the)h Fw(K)7 b FA(-trivialit)m(y)39 b(of)i Fw(A)p FA(.)g(F)-8 b(or)42 b(instance,)f(an)g(early)505 4403 y(v)m(ersion)36 b(of)g(the)g(pro)s(of)f(used)g(the)h(condition,)e(due)i(to)g(F)-8 b(rank)36 b(Stephan,)f(that)i(for)505 4511 y(some)31 b(\006)799 4478 y Fy(0)799 4535 y(1)869 4511 y FA(op)s(en)e(set)i Fw(R)26 b Fr(\032)f FA(2)1466 4478 y Fx(!)1547 4511 y FA(of)31 b(measure)f(less)g(than)g(1,)1017 4667 y Fr(9)p Fw(b)25 b Fr(2)f Fw(!)18 b Fr(8)p Fw(\033)28 b Fr(2)d FA(2)1554 4630 y Fx(F)-8 b(rom)32 b(Theorem)f(7.4)i(w)m(e)f(can)f(conclude)g(that)h (all)e Fw(K)7 b FA(-trivial)29 b(sets)j(are)g(lo)m(w,)f(as)505 5065 y(w)m(e)g(no)m(w)g(see.)p eop %%Page: 31 31 31 30 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(31)588 541 y FB(Theorem)34 b FA(7.7)h(\(Nies)c(and)e(Stephan)h (\(unpublished\)\))p FB(.)42 b Fs(If)35 b(a)h FA(\001)2865 508 y Fy(0)2865 566 y(2)2940 541 y Fs(set)f Fw(A)h Fs(is)g Fw(B)5 b Fs(-)505 649 y(r)-5 b(andom,)35 b(then)e Fw(B)38 b Fs(is)32 b(GL)1397 663 y Fy(1)1437 649 y Fs(.)588 820 y FB(Pr)n(oof.)41 b FA(Let)24 b Fw(f)10 b FA(\()p Fw(n)p FA(\))22 b(b)s(e)g(the)i(least)f Fw(s)f FA(suc)m(h)h(that)g Fr(8)p Fw(t)i Fz(>)g Fw(s)g FA([)p Fw(A)2590 834 y Fx(t)2645 808 y Fz(\026)2708 820 y Fw(n)g FA(=)g Fw(A)2952 834 y Fx(s)3014 808 y Fz(\026)3077 820 y Fw(n)p FA(].)e(Note)505 940 y(that)28 b Fw(f)35 b Fz(6)850 954 y Fy(T)930 940 y Fr(;)975 907 y Fq(0)999 940 y FA(.)27 b(Let)1232 917 y Fp(b)1211 940 y Fr(R)1288 954 y Fx(e)1353 940 y FA(b)s(e)g(the)g(op)s (en)g(set)h([)p Fw(A)2076 954 y Fx(s)2109 962 y Fl(e)2172 928 y Fz(\026)2235 940 y Fw(e)14 b FA(+)g(1])29 b(where)e Fw(s)2778 954 y Fx(e)2842 940 y FA(is)f(the)i(stage)h(at)505 1060 y(whic)m(h)c(\010)827 1027 y Fx(B)827 1082 y(e)887 1060 y FA(\()p Fw(e)p FA(\))h(con)m(v)m(erges)i(\(or)1591 1037 y Fp(b)1570 1060 y Fr(R)1647 1074 y Fx(e)1709 1060 y FA(=)d Fr(;)h FA(if)e(\010)2020 1027 y Fx(B)2020 1082 y(e)2081 1060 y FA(\()p Fw(e)p FA(\))10 b Fr(")p FA(\).)27 b(Let)f Fr(R)2570 1074 y Fx(i)2624 1060 y FA(=)2720 991 y Fp(S)2796 1086 y Fx(e)p Fk(>)p Fx(i)2947 1037 y Fp(b)2927 1060 y Fr(R)3004 1074 y Fx(e)3041 1060 y FA(.)f(Clearly)-8 b(,)505 1174 y Fr(fR)627 1188 y Fx(i)656 1174 y Fr(g)701 1188 y Fx(i)p Fq(2)p Fx(!)853 1174 y FA(is)29 b(a)h(Martin-L\177)-45 b(of)30 b(test)h(relativ)m(e)f(to)h Fw(B)5 b FA(.)29 b(Since)g Fw(A)36 b(=)-55 b Fr(2)2641 1106 y Fp(T)2717 1201 y Fx(i)2760 1174 y Fr(R)2837 1188 y Fx(i)2865 1174 y FA(,)30 b(only)f(\014nitely)505 1298 y(man)m(y)774 1275 y Fp(b)753 1298 y Fr(R)830 1312 y Fx(e)867 1298 y FA('s)i(con)m(tain)h Fw(A)p FA(.)f(Th)m(us)f Fw(f)10 b FA(\()p Fw(e)p FA(\))26 b Fz(>)g Fw(s)1969 1312 y Fx(e)2037 1298 y FA(for)k(almost)h(all)f Fw(e)h FA(suc)m(h)g(that)g(\010)3139 1265 y Fx(B)3139 1320 y(e)3200 1298 y FA(\()p Fw(e)p FA(\))11 b Fr(#)p FA(.)505 1407 y(So)33 b(using)e Fw(f)42 b FA(and)32 b Fw(B)5 b FA(,)32 b(w)m(e)h(can)g(compute)g(whether)f (\010)2364 1374 y Fx(B)2364 1429 y(e)2425 1407 y FA(\()p Fw(e)p FA(\))14 b Fr(#)p FA(,)34 b(whic)m(h)d(implies)f(that)505 1515 y Fw(B)579 1482 y Fq(0)627 1515 y Fz(6)698 1529 y Fy(T)779 1515 y Fw(B)24 b Fr(\010)c(;)1008 1482 y Fq(0)1032 1515 y FA(.)2282 b Fr(a)588 1648 y FA(Applying)24 b(this)h(result)f (with)h Fw(A)g FA(=)g(\012,)h(w)m(e)h(obtain)e(the)h(follo)m(wing)e (corollary)-8 b(,)26 b(\014rst)505 1755 y(pro)m(v)m(ed)31 b(b)m(y)f(Nies,)h(Stephan,)e(and)h(T)-8 b(erwijn)29 b([107)q(].)588 1926 y FB(Cor)n(ollar)-6 b(y)35 b FA(7.8)g(\(Nies,)c(Stephan,)f(and)g (T)-8 b(erwijn)28 b([107)r(]\))p FB(.)46 b Fs(Every)41 b(set)h(that)h(is)505 2034 y(low)34 b(for)f FA(\012)f Fs(is)h(GL)1136 2048 y Fy(1)1176 2034 y Fs(.)588 2205 y FA(Since)h(all)f Fw(K)7 b FA(-trivial)32 b(sets)j(are)g(\001)1762 2172 y Fy(0)1762 2230 y(2)1801 2205 y FA(,)g(b)m(y)f(Theorem)g(7.4)h(w) m(e)g(ha)m(v)m(e)h(the)e(follo)m(wing)505 2313 y(result.)588 2484 y FB(Cor)n(ollar)-6 b(y)35 b FA(7.9)g(\(Nies)c([103)q(]\))p FB(.)46 b Fs(Every)33 b Fw(K)7 b Fs(-trivial)32 b(set)h(is)g(low.)588 2702 y Fu(x)p Ft(8.)53 b(Characterizing)44 b(the)g Fw(K)7 b Ft(-trivial)43 b(sets.)i FA(In)38 b(this)f(section)i(w)m(e)f(discuss) 505 2810 y(the)j(harder)d(direction)h(of)h(Theorem)g(7.4,)h(along)f (with)f(sev)m(eral)h(related)g(results)505 2918 y(ab)s(out)h(the)g (class)g Fr(K)h FA(of)f Fw(K)7 b FA(-trivial)39 b(sets.)j(These)f (results)e(indicate)h(that)i Fr(K)g FA(is)e(a)505 3026 y(robust)d(class,)g(in)f(the)h(sense)g(that)h(it)e(captures)h(sev)m (eral)h(di\013eren)m(t)e(in)m(tuitiv)m(e)g(no-)505 3134 y(tions)g(of)h(computational)f(w)m(eakness)h(related)g(to)g (randomness,)f(and)g(is)g(of)g(great)505 3242 y(computabilit)m (y-theoretic)30 b(in)m(terest.)588 3350 y(Recall)j(from)g(Section)g (6.1)h(that)g Fw(A)f FA(is)f(lo)m(w)h(for)g Fw(K)40 b FA(if)32 b Fw(K)7 b FA(\()p Fw(\033)s FA(\))30 b Fz(6)g Fw(K)2884 3317 y Fx(A)2941 3350 y FA(\()p Fw(\033)s FA(\))23 b(+)e Fw(O)s FA(\(1\);)505 3457 y(that)39 b(is,)e(using)f Fw(A)i FA(as)g(an)f(oracle)i(do)s(es)e(not)h(help)e(us)h(to)i(reduce)e (the)h(pre\014x-free)505 3565 y(complexit)m(y)e(of)h(an)m(y)f(string)f (\(up)g(to)i(a)g(constan)m(t\).)h(Let)e Fr(M)h FA(denote)f(the)g(class) g(of)505 3673 y(sets)28 b(that)f(are)h(lo)m(w)e(for)h Fw(K)7 b FA(.)27 b(This)e(class)h(w)m(as)i(studied)d(b)m(y)i(Muc)m (hnik;)f(as)h(men)m(tioned)505 3781 y(ab)s(o)m(v)m(e,)32 b(he)e(sho)m(w)m(ed)h(that)g(it)f(con)m(tains)h(a)f(noncomputable)g (set)h(\(see)g([12)q(]\).)588 3891 y(If)39 b Fw(A)g Fz(6)866 3905 y Fy(T)960 3891 y Fw(B)44 b FA(then)38 b Fw(K)1372 3858 y Fx(B)1432 3891 y FA(\()p Fw(\033)s FA(\))j Fz(6)e Fw(K)1792 3858 y Fx(A)1848 3891 y FA(\()p Fw(\033)s FA(\))27 b(+)e Fw(O)s FA(\(1\),)40 b(so)g Fr(M)e FA(is)g(closed)h(do)m(wn)m(w)m (ard)505 3999 y(under)26 b Fz(6)831 4013 y Fy(T)886 3999 y FA(.)h(Since)e(b)s(oth)i(1-randomness)f(and)g Fw(K)7 b FA(-trivialit)m(y)25 b(are)i(de\014ned)f(in)g(terms)505 4107 y(of)44 b(pre\014x-free)e(complexit)m(y)-8 b(,)43 b(if)f(a)i(set)f(is)f(lo)m(w)h(for)g Fw(K)49 b FA(then)43 b(it)g(is)f(b)s(oth)g(lo)m(w)h(for)505 4214 y(1-randomness)30 b(and)g Fw(K)7 b FA(-trivial.)588 4322 y(In)27 b(Nies)g([103)r(],)h (Theorem)f(7.4)h(is)f(pro)m(v)m(ed)g(in)g(t)m(w)m(o)h(separate)h (pieces,)e(b)m(y)g(sho)m(wing)505 4430 y(that)32 b(b)s(oth)e(the)i (class)f(of)g(sets)g(that)h(are)f(lo)m(w)g(for)g(1-randomness)g(and)f Fr(K)j FA(actually)505 4538 y(coincide)41 b(with)e Fr(M)p FA(.)j(W)-8 b(e)42 b(ha)m(v)m(e)g(already)f(discussed)e(the)j(fact)g (that)f(lo)m(wness)g(for)505 4646 y(1-randomness)27 b(implies)e Fw(K)7 b FA(-trivialit)m(y)26 b(\(and)h(will)d(return)j(to)h(it)f(at)h (the)f(end)g(of)h(this)505 4754 y(section,)38 b(when)d(w)m(e)j(discuss) d(bases)h(for)h(1-randomness\).)g(The)f(con)m(v)m(erse)i(follo)m(ws)505 4862 y(immediately)29 b(from)h(the)h(follo)m(wing)d(theorem.)588 5033 y FB(Theorem)34 b FA(8.1)h(\(Nies)c(and)e(Hirsc)m(hfeldt,)55 b(see)31 b(Nies)55 b([103)r(]\))p FB(.)46 b Fs(Every)38 b Fw(K)7 b Fs(-trivial)505 5141 y(set)33 b(is)g(low)g(for)h Fw(K)7 b Fs(.)p eop %%Page: 32 32 32 31 bop 505 363 a FD(32)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FA(The)29 b(pro)s(of)g(of)h(this)e(result)h(is)f(a)i(reasonably)f (straigh)m(tforw)m(ard)g(mo)s(di\014cation)f(of)505 649 y(that)j(of)g(the)f(follo)m(wing)f(theorem.)588 837 y FB(Theorem)34 b FA(8.2)h(\(Nies)c([103)q(]\))p FB(.)46 b Fs(The)26 b(class)g Fr(K)h Fs(is)f(close)-5 b(d)26 b(downwar)-5 b(d)29 b(under)d(T)-7 b(ur-)505 945 y(ing)33 b(r)-5 b(e)g(ducibility.)588 1133 y FA(As)26 b(w)m(e)f(will)e(discuss)g (b)s(elo)m(w,)i(the)g(pro)s(of)g(of)g(Theorem)g(8.2)i(also)e(yields)e (the)j(result)505 1240 y(that)34 b(the)f(construction)f(in)f(the)i(pro) s(of)f(of)h(Theorem)g(6.2)h(is)d(in)h(fact)h(a)h(c)m(haracter-)505 1348 y(ization)j(of)f Fr(K)q FA(,)h(in)f(that)h(it)f(is)f(in)g(essence) j(the)e(only)g(w)m(a)m(y)h(to)h(obtain)e(a)g Fw(K)7 b FA(-trivial)505 1456 y(set!)588 1602 y FB(Pr)n(oof)34 b(Sketch)f(of)h(Theorems)f(8.1)h(and)g(8.2.)42 b FA(W)-8 b(e)26 b(\014rst)d(discuss)g(the)i(eas-)505 1710 y(ier)32 b(pro)s(of)g(that)i(ev)m(ery)f Fw(K)7 b FA(-trivial)31 b(set)j(is)e(T)-8 b(uring)31 b(incomplete,)h(but)g(presen)m(t)h(it)f (as)505 1818 y(a)38 b(\\pro)5 b(jection")38 b(of)f(the)h(pro)s(ofs)e (of)h(Theorems)g(8.1)h(and)f(8.2)h(in)e(Nies)h([103)r(].)g(W)-8 b(e)505 1926 y(then)41 b(brie\015y)e(discuss)g(the)h(mo)s (di\014cations)f(needed)i(to)g(con)m(v)m(ert)h(this)e(argumen)m(t)505 2034 y(in)m(to)31 b(pro)s(ofs)e(of)i(Theorems)f(8.1)h(and)f(8.2.)588 2142 y(Assume)38 b(for)g(a)g(con)m(tradiction)g(that)g Fr(8)p Fw(n)15 b FA([)p Fw(K)7 b FA(\()p Fw(A)2299 2130 y Fz(\026)2375 2142 y Fw(n)p FA(\))37 b Fz(6)h Fw(K)7 b FA(\()p Fw(n)p FA(\))25 b(+)g Fw(b)p FA(])38 b(and)f Fw(A)h FA(is)505 2250 y(T)-8 b(uring)39 b(complete.)i(By)g(Theorem)f (6.1,)i(w)m(e)e(can)h(c)m(ho)s(ose)g(a)g(\001)2739 2217 y Fy(0)2739 2274 y(2)2819 2250 y FA(appro)m(ximation)505 2358 y Fr(f)p Fw(A)618 2372 y Fx(s)656 2358 y Fr(g)701 2372 y Fx(s)p Fq(2)p Fx(!)861 2358 y FA(to)31 b Fw(A)p FA(.)g(W)-8 b(e)31 b(will)c(en)m(umerate)k(requests)f(in)m(to)g(a)h (Kraft-Chaitin)d(set)j Fw(L)e FA(\(in)505 2466 y(fact)k(en)m(umerating) f(at)h(most)f(one)g(request)g Fr(h)p Fw(r)m(;)15 b(n)p Fr(i)33 b FA(for)f(eac)m(h)h Fw(n)28 b Fr(2)f Fw(!)s FA(,)32 b(with)f Fw(r)g Fr(2)c Fw(!)s FA(\).)505 2574 y(The)j Fs(weight)40 b FA(of)30 b Fw(X)j Fr(\022)25 b Fw(!)33 b FA(is)1131 2778 y(wt\()p Fw(X)7 b FA(\))26 b(:=)1531 2692 y Fp(X)1662 2778 y Fr(f)p FA(2)1752 2741 y Fq(\000)p Fx(r)1871 2778 y FA(:)g Fr(h)p Fw(r)m(;)15 b(n)p Fr(i)26 b(2)f Fw(L)46 b Fr(^)f Fw(n)24 b Fr(2)h Fw(X)7 b Fr(g)p Fw(:)505 2982 y FA(This)26 b(w)m(eigh)m(t)j(coincides)d (with)h(the)h(measure)f(of)h(the)g(corresp)s(onding)e(descriptions)505 3090 y(giv)m(en)d(b)m(y)g(the)h(pre\014x-free)e(mac)m(hine)h Fw(M)1863 3105 y Fx(d)1927 3090 y FA(de\014ned)e(b)m(y)i Fw(L)g FA(\(using)f(the)h(Kraft-Chaitin)505 3198 y(Theorem\).)30 b(Roughly)-8 b(,)30 b(w)m(e)g(will)d(en)m(umerate)j(appropriate)f (requests)g Fr(h)p Fw(r)m(;)15 b(n)p Fr(i)31 b FA(in)m(to)f Fw(L)505 3306 y FA(in)g(order)g(to)h(ensure)e(that)j Fw(K)7 b FA(\()p Fw(n)p FA(\))25 b Fz(6)g Fw(r)e FA(+)d Fw(d)p FA(.)31 b(Since)e Fw(A)i FA(is)f Fw(K)7 b FA(-trivial)28 b(via)i Fw(b)p FA(,)h(at)g(some)505 3414 y(stage)25 b Fw(s)d FA(after)h(this)f(en)m(umeration)g(the)h(opp)s(onen)m(t)g(has)f (to)i(giv)m(e)f(a)g(short)f(description)505 3522 y(of)k Fw(A)672 3536 y Fx(s)734 3510 y Fz(\026)797 3522 y Fw(n)p FA(,)f(b)m(y)g(ensuring)e(that)j Fw(U)1636 3536 y Fx(s)1673 3522 y FA(\()p Fw(\033)s FA(\))g(=)f Fw(A)1988 3536 y Fx(s)2050 3510 y Fz(\026)2113 3522 y Fw(n)g FA(for)g(some)h Fw(\033)i FA(with)c Fr(j)p Fw(\033)s Fr(j)i Fz(6)f Fw(r)12 b FA(+)e Fw(d)g FA(+)g Fw(b)p FA(,)505 3630 y(where)21 b Fw(U)31 b FA(is)21 b(the)g(univ)m(ersal)f(pre\014x-free)g(mac)m(hine) h(that)h(is)f(b)s(eing)f(used)g(to)i(de\014ne)f Fw(K)7 b FA(.)505 3738 y(W)-8 b(e)27 b(use)e(the)h(T)-8 b(uring)24 b(completeness)h(of)h Fw(A)f FA(to)i(mak)m(e)f(the)g(appro)m(ximation)e (to)i Fw(A)3277 3726 y Fz(\026)3340 3738 y Fw(n)505 3846 y FA(c)m(hange)36 b(after)f(stage)h Fw(s)p FA(,)e(so)g(that)h(the)g (opp)s(onen)m(t)f(has)g(to)h(come)h(up)d(with)g(a)i(short)505 3954 y(description)29 b(of)i(the)f(new)h(appro)m(ximation)e(to)i Fw(A)2230 3942 y Fz(\026)2293 3954 y Fw(n)p FA(.)g(If)f(w)m(e)h(can)g (c)m(hange)g Fw(A)g FA(often)505 4062 y(enough,)37 b(then)f(the)h (measure)g(corresp)s(onding)d(to)k(the)e(opp)s(onen)m(t's)h (descriptions)505 4169 y(exceeds)32 b(1,)f(whic)m(h)e(is)g(imp)s (ossible.)588 4277 y(More)35 b(precisely)-8 b(,)33 b(b)m(y)h(the)g (Recursion)f(Theorem)h(w)m(e)g(can)g(assume)g(w)m(e)g(ha)m(v)m(e)i(an) 505 4385 y(index)c Fw(d)h FA(suc)m(h)g(that)g Fw(M)1328 4400 y Fx(d)1402 4385 y FA(is)e(a)j(pre\014x-free)e(mac)m(hine)h (corresp)s(onding)d(to)k Fw(L)f FA(in)e(the)505 4493 y(sense)45 b(of)g(the)h(Kraft-Chaitin)d(Theorem.)i(If)f(w)m(e)i(let)f Fw(c)k FA(=)h Fw(b)30 b FA(+)f Fw(d)p FA(,)46 b(then)e(as)i(an)505 4601 y(answ)m(er)30 b(to)g(our)g(en)m(umeration)f(of)h(the)g(request)f Fr(h)p Fw(r)m(;)15 b(n)p Fr(i)31 b FA(in)m(to)f Fw(L)p FA(,)g(the)f(opp)s(onen)m(t)h(has)505 4709 y(to)43 b(pro)m(vide)e(a)h Fw(U)10 b FA(-description)40 b(of)h Fw(A)1856 4697 y Fz(\026)1938 4709 y Fw(n)g FA(of)h(length)f Fw(r)30 b FA(+)d Fw(c)p FA(.)43 b(Let)f Fw(k)47 b FA(=)d(2)3142 4676 y Fx(c)p Fy(+1)3267 4709 y FA(.)e(If)505 4817 y(w)m(e)34 b(manage)g(to)f(put)f(requests)h(of)g(total)h(w)m(eigh)m(t)f(1)p Fw(=)p FA(2)h(in)m(to)f Fw(L)g FA(and)f(also)h(force)g(the)505 4925 y(appro)m(ximation)f(to)h Fw(A)1324 4913 y Fz(\026)1390 4925 y Fw(n)f FA(to)h(c)m(hange)h(at)f(least)g Fw(k)i FA(times)d(for)g(eac)m(h)i Fw(n)e FA(men)m(tioned)505 5033 y(in)41 b(our)h(requests,)g(then)g(the)g(total)h(measure)f(of)g (the)h(opp)s(onen)m(t's)f(descriptions)505 5141 y(will)g(exceed)j(1.)g (\(Here)g(w)m(e)g(only)e(coun)m(t)i(a)g(c)m(hange)g(in)e(the)i(appro)m (ximation)e(to)p eop %%Page: 33 33 33 32 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(33)505 541 y Fw(A)611 529 y Fz(\026)687 541 y Fw(n)37 b FA(if)g(the)h(opp)s(onen)m(t)f(has)h(pro)m(vided)e(a)j(short)e (description)f(of)i(the)g(previous)505 649 y(appro)m(ximation)30 b(to)h(this)e(initial)f(segmen)m(t.\))588 757 y(The)41 b(pro)s(of)f(is)g(m)m(uc)m(h)i(easier)f(if)f(w)m(e)h(assume)g(that)h Fr(;)2470 724 y Fq(0)2537 757 y Fz(6)2608 771 y Fy(wtt)2760 757 y Fw(A)p FA(.)g(In)e(this)g(case,)505 865 y(w)m(e)35 b(build)c(an)j(auxiliary)e(c.e.)k(set)e Fw(B)5 b FA(,)34 b(and)g(b)m(y)g(the)h(Recursion)e(Theorem)h(w)m(e)g(can)505 973 y(assume)d(w)m(e)g(are)f(giv)m(en)h(a)g(total)g(wtt-reduction)f (\000)g(suc)m(h)g(that)h Fw(B)f FA(=)25 b(\000)2948 940 y Fx(A)3005 973 y FA(,)31 b(with)e(use)505 1081 y(b)s(ounded)22 b(b)m(y)i(a)h(computable)f(increasing)e(function)h Fw(g)s FA(.)i(Let)g Fw(n)g FA(=)g Fw(g)s FA(\()p Fw(k)s FA(\).)h(W)-8 b(e)25 b(put)e(the)505 1189 y(single)29 b(request)h Fr(h)p FA(0)p Fw(;)15 b(n)p Fr(i)31 b FA(in)m(to)e Fw(L)p FA(.)h(\(The)g(w)m (eigh)m(t)g(of)g(this)f(single)f(request)i(is)f(1.\))i(Eac)m(h)505 1298 y(time)j(\000)773 1265 y Fx(A)826 1273 y Fl(s)863 1298 y FA(\()p Fw(j)5 b FA(\))36 b(con)m(v)m(erges)g(to)e Fw(B)1605 1312 y Fx(s)1642 1298 y FA(\()p Fw(j)5 b FA(\))35 b(for)f(all)f Fw(j)k(<)31 b(k)37 b FA(and)d(the)g(opp)s(onen)m(t)g(pro) m(vides)505 1406 y(a)j Fw(U)10 b FA(-description)35 b(of)h Fw(A)1338 1420 y Fx(s)1410 1394 y Fz(\026)1483 1406 y Fw(n)g FA(of)g(length)g Fz(6)f Fw(c)p FA(,)i(w)m(e)f(force)h(the)g (appro)m(ximation)e(to)505 1514 y Fw(A)599 1502 y Fz(\026)662 1514 y Fw(n)28 b FA(to)h(c)m(hange)g(b)m(y)f(putting)f(in)m(to)h Fw(B)33 b FA(the)28 b(largest)h(n)m(um)m(b)s(er)e Fw(<)e(k)31 b FA(that)e(is)e(not)h(y)m(et)505 1622 y(in)h Fw(B)5 b FA(.)30 b(Once)g(w)m(e)g(reac)m(h)h Fw(k)i FA(suc)m(h)d(c)m(hanges,)h (the)f(total)h(measure)f(of)g Fw(U)10 b FA(-descriptions)505 1730 y(is)30 b(at)h(least)g Fw(k)s FA(2)1016 1697 y Fq(\000)p Fx(c)1131 1730 y Fw(>)25 b FA(1,)31 b(whic)m(h)e(is)h(a)g(con)m (tradiction.)588 1838 y(F)-8 b(or)26 b(the)g(T)-8 b(uring)24 b(case,)i(w)m(e)g(still)d(build)f Fw(B)30 b FA(and)25 b(ha)m(v)m(e)h(a)g(total)g(reduction)e(\000)3143 1805 y Fx(A)3225 1838 y FA(=)h Fw(B)505 1946 y FA(giv)m(en)30 b(b)m(y)g(the)f(Recursion)g(Theorem,)h(but)f(no)m(w)g(there)h(is)f(no)g (computable)h(b)s(ound)505 2054 y(on)h(the)f(use)h Fw(\015)998 2021 y Fx(A)1085 2054 y FA(of)g(\000)1246 2021 y Fx(A)1303 2054 y FA(.)f(The)g(problem)f(no)m(w)h(is)g(that,)h(when)f(w)m(e)h(ha)m (v)m(e)g(an)g Fw(m)f FA(suc)m(h)505 2162 y(that)f Fw(\015)752 2129 y Fx(A)805 2137 y Fl(s)843 2162 y FA(\()p Fw(m)p FA(\))d(=)f Fw(n)j FA(and)f(w)m(e)i(put)f(a)h(request)f Fr(h)p Fw(r)m(;)15 b(n)p Fr(i)30 b FA(in)m(to)e Fw(L)p FA(,)h(the)f(opp)s(onen)m(t)h(migh)m(t,)505 2271 y(b)s(efore)39 b(pro)m(viding)d(a)j(short)g(description)d(of)j Fw(A)2178 2285 y Fx(s)2254 2259 y Fz(\026)2331 2271 y Fw(n)p FA(,)f(mo)m(v)m(e)j Fw(\015)2745 2238 y Fx(A)2802 2271 y FA(\()p Fw(m)p FA(\))e(b)s(ey)m (ond)f Fw(n)p FA(,)505 2379 y(thereb)m(y)d(depriving)d(us)i(of)h(the)g (p)s(ossibilit)m(y)c(of)k(causing)f(further)f(c)m(hanges)j(in)d(the)505 2487 y(appro)m(ximation)i(to)i Fw(A)1337 2475 y Fz(\026)1410 2487 y Fw(n)e FA(b)m(y)h(en)m(umerating)g(n)m(um)m(b)s(ers)f Fw(<)f(m)i FA(in)m(to)g Fw(B)5 b FA(.)36 b(Broadly)505 2595 y(sp)s(eaking,)c(the)h(solution)e(is)h(to)h(carry)g(out)g(man)m(y) f(attempts,)i(based)f(on)f(di\013eren)m(t)505 2703 y(computations)g (\000)1130 2670 y Fx(A)1186 2703 y FA(\()p Fw(m)p FA(\).)h(Eac)m(h)f (time)f(the)g(use)g(of)h(suc)m(h)f(a)h(computation)f(c)m(hanges,)505 2811 y(some)f(of)g(what)f(w)m(e)h(placed)e(in)g Fw(L)i FA(for)f(this)f(attempt)i(b)s(ecomes)g(\\garbage",)i(but)c(as)505 2919 y(the)34 b(reduction)d(\000)i(is)f(total,)i(this)e(only)g(happ)s (ens)f(\014nitely)g(often)j(for)e(eac)m(h)i Fw(m)p FA(.)f(W)-8 b(e)505 3027 y(ha)m(v)m(e)36 b(to)e(ensure)f(that)i(the)f(total)h(w)m (eigh)m(t)f(of)h(the)f(garbage)h(pro)s(duced)d(b)m(y)i(all)f(our)505 3135 y(attempts)f(is)d(limited,)g(since)g(otherwise)h Fw(L)g FA(will)e(not)j(b)s(e)e(a)i(Kraft-Chaitin)e(set.)588 3243 y(F)-8 b(or)42 b(eac)m(h)g Fw(s)p FA(,)e(w)m(e)i(can)f(e\013ectiv) m(ely)g(determine)f(a)h(stage)i Fw(f)10 b FA(\()p Fw(s)p FA(\))42 b Fw(>)g(s)e FA(suc)m(h)h(that)505 3351 y Fr(8)p Fw(n)36 b(<)g(s)15 b FA([)p Fw(K)914 3369 y Fx(f)7 b Fy(\()p Fx(s)p Fy(\))1047 3351 y FA(\()p Fw(A)1150 3369 y Fx(f)g Fy(\()p Fx(s)p Fy(\))1320 3339 y Fz(\026)1394 3351 y Fw(n)p FA(\))36 b Fz(6)g Fw(K)1704 3369 y Fx(f)7 b Fy(\()p Fx(s)p Fy(\))1837 3351 y FA(\()p Fw(n)p FA(\))25 b(+)f Fw(b)p FA(].)38 b(Let)f Fw(s)2421 3365 y Fy(0)2497 3351 y FA(=)f(0)h(and)f Fw(s)2912 3365 y Fx(i)p Fy(+1)3067 3351 y FA(=)g Fw(f)10 b FA(\()p Fw(s)3307 3365 y Fx(i)3334 3351 y FA(\).)505 3462 y(The)30 b(construction)f(is)g(restricted)g(to)h Fs(stages)38 b FA(in)28 b Fr(f)p Fw(s)2282 3476 y Fx(i)2336 3462 y FA(:)d Fw(i)h Fr(2)f Fw(!)s Fr(g)p FA(.)30 b(The)f(follo)m(wing) f(is)h(a)505 3570 y(w)m(a)m(y)36 b(to)f(k)m(eep)g(trac)m(k)g(of)f(the)h (n)m(um)m(b)s(er)e(of)h(times)g(the)h(opp)s(onen)m(t)e(has)h(had)g(to)h (giv)m(e)505 3678 y(new)d(descriptions)e(of)i(appro)m(ximations)f(to)i Fw(A)2155 3666 y Fz(\026)2221 3678 y Fw(n)p FA(.)f(W)-8 b(e)33 b(sa)m(y)g(that)g(the)f(n)m(um)m(b)s(er)f Fw(n)505 3786 y FA(is)i(in)f(a)h Fw(j)5 b FA(-set)35 b(if)d(this)g(has)h(happ)s (ened)f Fw(j)38 b FA(times.)33 b(More)h(precisely)-8 b(,)33 b(for)g(1)d Fz(6)g Fw(j)35 b Fz(6)30 b Fw(k)s FA(,)505 3894 y(w)m(e)37 b(sa)m(y)g(that)f(a)h(\014nite)e(set)h Fw(E)k Fr(\022)35 b Fw(!)k FA(is)c(a)h Fw(j)5 b Fs(-set)38 b(at)h(stage)f Fw(t)e FA(if,)f(for)h(all)f Fw(n)f Fr(2)g Fw(E)5 b FA(,)37 b(at)505 4002 y(some)i Fs(stage)45 b Fw(u)38 b(<)g(t)f FA(a)i(request)f Fr(h)p Fw(r)m(;)15 b(n)p Fr(i)39 b FA(w)m(en)m(t)g(in)m(to)f Fw(L)g FA(and)f(at)i(stage)g Fw(t)f FA(there)g(are)505 4110 y(at)e(least)f Fw(j)41 b FA(distinct)34 b(strings)f Fw(\033)39 b FA(of)c(length)f Fw(n)h FA(suc)m(h)f(that)i Fw(K)2607 4124 y Fx(t)2637 4110 y FA(\()p Fw(\033)s FA(\))e Fz(6)f Fw(r)26 b FA(+)d Fw(c)p FA(.)35 b(A)g(c.e.)505 4218 y(set)h Fw(E)k FA(with)33 b(an)i(en)m(umeration)f Fw(E)k FA(=)1836 4149 y Fp(S)1911 4244 y Fx(t)1956 4218 y Fw(E)2023 4232 y Fx(t)2088 4218 y FA(is)33 b(a)i Fw(j)5 b Fs(-set)44 b FA(if)34 b Fw(E)2646 4232 y Fx(t)2710 4218 y FA(is)g(a)h Fw(j)5 b FA(-set)36 b(at)f(eac)m(h)505 4328 y Fs(stage)42 b Fw(t)p FA(.)35 b(In)f(our)g(construction,)g(the)h(strings)e Fw(\033)38 b FA(will)32 b(ha)m(v)m(e)k(the)e(form)g Fw(A)3057 4342 y Fx(s)3127 4316 y Fz(\026)3197 4328 y Fw(n)g FA(for)505 4435 y(certain)g Fs(stages)42 b Fw(s)33 b FA(with)f Fw(u)f Fz(6)f Fw(s)g Fz(6)g Fw(t)p FA(.)k(Since)f(the)g(opp)s(onen)m(t)h(has)f (to)h(matc)m(h)h(ev)m(ery)505 4543 y(description)d(of)i Fw(n)f FA(w)m(e)h(pro)m(vide)f(via)g Fw(L)h FA(with)e(descriptions)g (of)i Fw(A)2802 4531 y Fz(\026)2870 4543 y Fw(n)f FA(that)i(are)f(at) 505 4651 y(most)e Fw(c)f FA(longer,)f(w)m(e)i(ha)m(v)m(e)g(the)e(follo) m(wing)g(straigh)m(tforw)m(ard)g(but)g(imp)s(ortan)m(t)g(fact.)505 4759 y(\(Recall)h(that)g Fw(k)d FA(=)d(2)1228 4726 y Fx(c)p Fy(+1)1353 4759 y FA(.\))1037 4896 y Fs(If)33 b(the)g(c.e.)f(set)g Fw(E)38 b Fs(is)33 b(a)g Fw(k)s Fs(-set,)f(then)h FA(wt\()p Fw(E)5 b FA(\))27 b Fz(6)d FA(1)p Fw(=)p FA(2)p Fs(.)505 5033 y FA(As)k(in)e(the)i(wtt)g(case,)h (our)e(construction)g(will)e(build)g(a)j Fw(k)s FA(-set)g Fw(C)2706 5048 y Fx(k)2776 5033 y FA(of)g(w)m(eigh)m(t)g Fw(>)d FA(1)p Fw(=)p FA(2)505 5141 y(to)32 b(reac)m(h)f(a)f(con)m (tradiction.)p eop %%Page: 34 34 34 33 bop 505 363 a FD(34)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FA(The)29 b(pro)s(cedure)f Fw(P)1253 555 y Fx(j)1318 541 y FA(\(2)e Fz(6)f Fw(j)31 b Fz(6)25 b Fw(k)s FA(\))30 b(en)m(umerates)f(a)g Fw(j)5 b FA(-set)31 b Fw(C)2628 555 y Fx(j)2664 541 y FA(.)f(The)e(construction)505 649 y(b)s(egins)33 b(b)m(y)h(calling)f Fw(P)1270 664 y Fx(k)1313 649 y FA(,)i(whic)m(h)e(calls)g Fw(P)1901 664 y Fx(k)r Fq(\000)p Fy(1)2069 649 y FA(sev)m(eral)h(times,)g(and)g(so)h(on)f(do)m (wn)g(to)505 757 y Fw(P)563 771 y Fy(2)603 757 y FA(,)d(whic)m(h)e(en)m (umerates)i Fw(L)f FA(\(and)g Fw(C)1766 771 y Fy(2)1806 757 y FA(\).)588 865 y(Eac)m(h)36 b(pro)s(cedure)d Fw(P)1303 879 y Fx(j)1375 865 y FA(has)h(rational)g(parameters)h Fw(q)s(;)15 b(\014)38 b Fr(2)32 b FA([0)p Fw(;)15 b FA(1].)37 b(The)d Fs(go)-5 b(al)45 b Fw(q)37 b FA(is)505 973 y(the)31 b(w)m(eigh)m(t)h(it)e(w)m(an)m(ts)i Fw(C)1366 987 y Fx(j)1433 973 y FA(to)f(reac)m(h,)h(and)e(the)h Fs(garb)-5 b(age)34 b(quota)39 b Fw(\014)c FA(is)30 b(ho)m(w)h(m)m(uc)m(h)g(it)505 1081 y(is)f(allo)m(w)m(ed)g(to)h(w)m(aste.)588 1246 y(W)-8 b(e)43 b(no)m(w)e(describ)s(e)e(the)i(pro)s(cedure)f Fw(P)1976 1260 y Fx(j)2013 1246 y FA(\()p Fw(q)s(;)15 b(\014)5 b FA(\),)42 b(where)f(1)i Fw(<)f(j)49 b Fz(6)43 b Fw(k)s FA(,)e(and)f(the)505 1354 y(parameters)31 b Fw(q)d FA(=)d(2)1186 1321 y Fq(\000)p Fx(x)1316 1354 y FA(and)k Fw(\014)i FA(=)25 b(2)1715 1321 y Fq(\000)p Fx(y)1842 1354 y FA(are)31 b(suc)m(h)f(that)h Fw(x)25 b Fz(6)g Fw(y)s FA(.)588 1488 y(1.)43 b(Cho)s(ose)30 b Fw(m)g FA(large.)588 1596 y(2.)43 b(W)-8 b(ait)31 b(un)m(til)e(\000) 1195 1563 y Fx(A)1251 1596 y FA(\()p Fw(m)p FA(\))10 b Fr(#)p FA(.)588 1704 y(3.)43 b(Let)31 b Fw(v)d Fz(>)d FA(1)31 b(b)s(e)f(the)g(n)m(um)m(b)s(er)f(of)i(times)f Fw(P)2124 1718 y Fx(j)2191 1704 y FA(has)g(gone)h(through)f(step)g(2.) 587 1812 y Fw(j)h FA(=)24 b(2:)43 b(Pic)m(k)33 b(a)h(large)f(n)m(um)m (b)s(er)f Fw(n)p FA(.)h(Put)g Fr(h)p Fw(r)2077 1826 y Fx(n)2125 1812 y Fw(;)15 b(n)p Fr(i)33 b FA(in)m(to)h Fw(L)p FA(,)f(where)g(2)2907 1779 y Fq(\000)p Fx(r)2994 1787 y Fl(n)3071 1812 y FA(=)d(2)3217 1779 y Fq(\000)p Fx(v)3313 1812 y Fw(\014)5 b FA(.)863 1920 y(W)-8 b(ait)31 b(for)f(a)g Fs(stage)38 b Fw(t)30 b FA(suc)m(h)g(that)g Fw(K)2075 1934 y Fx(t)2105 1920 y FA(\()p Fw(n)p FA(\))c Fz(6)f Fw(r)2393 1934 y Fx(n)2460 1920 y FA(+)19 b Fw(d)p FA(,)31 b(and)f(put)f Fw(n)h FA(in)m(to)g Fw(C)3330 1934 y Fy(1)3369 1920 y FA(.)863 2028 y(\(If)24 b Fw(M)1071 2043 y Fx(d)1135 2028 y FA(is)f(a)h(pre\014x-free)g(mac)m(hine)f (corresp)s(onding)f(to)j Fw(L)p FA(,)f(then)g Fw(t)f FA(exists.\))587 2138 y Fw(j)31 b(>)24 b FA(2:)43 b(Call)32 b Fw(P)1116 2152 y Fx(j)t Fq(\000)p Fy(1)1243 2138 y FA(\(2)1323 2105 y Fq(\000)p Fx(v)1419 2138 y Fw(\014)5 b(;)15 b(\014)1571 2105 y Fq(0)1596 2138 y FA(\),)34 b(where)f Fw(\014)2012 2105 y Fq(0)2066 2138 y FA(=)d(min)o(\()p Fw(\014)5 b(;)15 b FA(2)2495 2105 y Fx(j)t Fq(\000)p Fx(k)r Fq(\000)p Fx(w)r Fq(\000)p Fy(1)2824 2138 y FA(\))34 b(and)f Fw(w)j FA(is)c(the)863 2246 y(n)m(um)m(b)s(er)c(of)i Fw(P)1354 2260 y Fx(j)t Fq(\000)p Fy(1)1511 2246 y FA(pro)s(cedures)e (started)j(so)f(far.)g(\(The)f(most)h(imp)s(ortan)m(t)863 2354 y(p)s(oin)m(t)23 b(to)i(understand)e(here)h(is)g(that)h(the)f (goals)h(of)g(the)f Fw(P)2815 2368 y Fx(j)t Fq(\000)p Fy(1)2967 2354 y FA(pro)s(cedures)863 2462 y(called)j(b)m(y)i(a)g Fw(P)1375 2476 y Fx(j)1440 2462 y FA(pro)s(cedure)f(are)h(related)f(to) i(the)e(garbage)i(quota)g(of)e(the)863 2570 y Fw(P)921 2584 y Fx(j)986 2570 y FA(pro)s(cedure,)g(whic)m(h)f(ensures)g(that)j (ev)m(en)f(if)e(all)h(suc)m(h)g(pro)s(cedures)f(are)863 2678 y(canceled,)43 b(the)h(w)m(asted)g(measure)f(will)d(still)h(b)s(e) i(b)s(elo)m(w)f(this)g(garbage)863 2786 y(quota.)35 b(Another)f(imp)s (ortan)m(t)f(p)s(oin)m(t)g(is)g(that)h(the)g(garbage)i(quotas)e(are)863 2894 y(c)m(hosen)f(so)f(that)h(their)f(sum)f(o)m(v)m(er)j(all)d(pro)s (cedures)g(started)i(during)d(the)863 3002 y(construction)g(is)f(less)h (than)g(1)p Fw(=)p FA(2.\))701 3109 y(In)21 b(an)m(y)i(case,)h(if)d (wt\()p Fw(C)1456 3123 y Fx(j)t Fq(\000)p Fy(1)1583 3109 y FA(\))26 b Fw(<)f(q)g FA(then)d(rep)s(eat)h(step)g(3,)g(and)e (otherwise)h(return.)588 3217 y(4.)43 b(Put)37 b Fw(m)h FA(in)m(to)f Fw(B)5 b FA(.)38 b(This)e(forces)i Fw(A)g FA(to)g(c)m(hange)h(b)s(elo)m(w)e Fw(\015)5 b FA(\()p Fw(m)p FA(\))38 b Fw(<)f FA(min)o(\()p Fw(C)3207 3231 y Fx(j)t Fq(\000)p Fy(1)3334 3217 y FA(\),)701 3325 y(and)27 b(hence)g(mak)m(es)h Fw(C)1459 3339 y Fx(j)t Fq(\000)p Fy(1)1613 3325 y FA(a)g Fw(j)5 b FA(-set)29 b(\(if)e(w)m(e)h(assume)f (inductiv)m(ely)e(that)j Fw(C)3179 3339 y Fx(j)t Fq(\000)p Fy(1)3333 3325 y FA(is)701 3433 y(a)i(\()p Fw(j)d Fr(\000)20 b FA(1\)-set\).)32 b(So)e(put)g Fw(C)1637 3447 y Fx(j)t Fq(\000)p Fy(1)1794 3433 y FA(in)m(to)g Fw(C)2043 3447 y Fx(j)2080 3433 y FA(,)g(and)g(declare)h Fw(C)2686 3447 y Fx(j)t Fq(\000)p Fy(1)2837 3433 y FA(=)25 b Fr(;)p FA(.)505 3573 y(If)31 b Fw(\015)649 3540 y Fx(A)706 3573 y FA(\()p Fw(m)p FA(\))g(c)m(hanges)h(during)d(the)i(execution)g(of)g (the)g(lo)s(op)f(at)i(step)f(3,)h(then)e(cancel)505 3681 y(the)j(run)e(of)h(all)g(subpro)s(cedures,)e(and)h(go)j(to)f(step)f(2.) h(Despite)g(the)f(cancelations,)505 3789 y Fw(C)570 3803 y Fx(j)t Fq(\000)p Fy(1)735 3789 y FA(is)k(no)m(w)i(a)g Fw(j)5 b FA(-set)39 b(b)s(ecause)e(of)h(this)e(v)m(ery)i(c)m(hange.)h (\(This)d(is)h(an)g(imp)s(ortan)m(t)505 3897 y(p)s(oin)m(t,)29 b(as)h(it)g(ensures)f(that)h(the)g(measure)g(asso)s(ciated)g(with)e(n)m (um)m(b)s(ers)g(already)i(in)505 4005 y Fw(C)570 4019 y Fx(j)t Fq(\000)p Fy(1)727 4005 y FA(is)g(not)g(w)m(asted.\))i(So)e (put)g Fw(C)1700 4019 y Fx(j)t Fq(\000)p Fy(1)1857 4005 y FA(in)m(to)g Fw(C)2106 4019 y Fx(j)2143 4005 y FA(,)g(and)g(declare)g Fw(C)2748 4019 y Fx(j)t Fq(\000)p Fy(1)2900 4005 y FA(=)25 b Fr(;)p FA(.)588 4169 y(This)d(completes)i(the)g(description)e(of)h (the)h(pro)s(cedures.)f(The)g(construction)g(con-)505 4277 y(sists)40 b(of)g(calling)f Fw(P)1183 4292 y Fx(k)1226 4277 y FA(\(1)p Fw(;)15 b FA(1)p Fw(=)p FA(4\).)43 b(One)c(can)i(c)m (hec)m(k)h(that,)f(b)s(ecause)f(of)g(the)g(w)m(a)m(y)h(the)505 4385 y(garbage)28 b(quotas)e(are)h(c)m(hosen,)g Fw(L)f FA(is)f(a)h(Kraft-Chaitin)f(set.)i(The)e(set)i Fw(C)2950 4400 y Fx(k)3019 4385 y FA(is)e(a)h Fw(k)s FA(-set,)505 4493 y(and)f(therefore)h(should)e(ha)m(v)m(e)j(w)m(eigh)m(t)f(at)g (most)g(1)p Fw(=)p FA(2.)i(But,)e(since)f(\000)2814 4460 y Fx(A)2896 4493 y FA(is)g(total,)h(eac)m(h)505 4601 y(pro)s(cedure)36 b(returns)g(unless)f(canceled,)i(so)g(the)g(initial)d (pro)s(cedure)i Fw(P)2962 4616 y Fx(k)3005 4601 y FA(,)h(whic)m(h)e(is) 505 4709 y(nev)m(er)f(canceled,)f(ev)m(en)m(tually)g(ensures)g(that)g Fw(C)2166 4724 y Fx(k)2242 4709 y FA(has)g(w)m(eigh)m(t)g Fw(>)d FA(1)p Fw(=)p FA(2,)k(whic)m(h)e(is)g(a)505 4817 y(con)m(tradiction.)588 4925 y(W)-8 b(e)33 b(can)g(visualize)d(this)g (construction)i(b)m(y)f(thinking)f(of)i(a)g(mac)m(hine)g(similar)d(to) 505 5033 y(Lerman's)i(pin)m(ball)d(mac)m(hine)j(\(see)h([122)r(,)f (Chapter)g(VI)s(I)s(I.5]\).)g(Ho)m(w)m(ev)m(er,)j(since)c(w)m(e)505 5141 y(en)m(umerate)22 b(rational)d(quan)m(tities)h(instead)g(of)h (single)e(ob)5 b(jects,)21 b(w)m(e)g(replace)g(the)f(balls)p eop %%Page: 35 35 35 34 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(35)505 541 y FA(in)35 b(Lerman's)h(mac)m(hine)g(b)m(y)g(amoun)m (ts)g(of)h(a)f(precious)f(liquid,)e(sa)m(y)k(1955)h(Biondi-)505 649 y(San)m(ti)27 b(Brunello)e(wine.)h(Our)g(mac)m(hine)g(consists)h (of)g(decan)m(ters)h Fw(C)2767 664 y Fx(k)2810 649 y Fw(;)15 b(C)2915 664 y Fx(k)r Fq(\000)p Fy(1)3048 649 y Fw(;)g(:)g(:)g(:)32 b(;)15 b(C)3330 663 y Fy(0)3369 649 y FA(.)505 757 y(A)m(t)37 b(an)m(y)g(stage)g Fw(C)1123 771 y Fx(j)1195 757 y FA(is)e(a)i Fw(j)5 b FA(-set.)37 b(W)-8 b(e)37 b(put)f Fw(C)2019 771 y Fx(j)t Fq(\000)p Fy(1)2181 757 y FA(ab)s(o)m(v)m(e)h Fw(C)2509 771 y Fx(j)2582 757 y FA(so)f(that)g Fw(C)2966 771 y Fx(j)t Fq(\000)p Fy(1)3129 757 y FA(can)g(b)s(e)505 865 y(emptied)d(in)m(to)g Fw(C)1109 879 y Fx(j)1146 865 y FA(.)h(The)e(heigh)m(t)i(of)f(a)h (decan)m(ter)h(is)d(c)m(hangeable.)j(The)e(pro)s(cedure)505 973 y Fw(P)563 987 y Fx(j)600 973 y FA(\()p Fw(q)s(;)15 b(\014)5 b FA(\))39 b(w)m(an)m(ts)f(to)g(add)f(w)m(eigh)m(t)h Fw(q)i FA(to)e Fw(C)1975 987 y Fx(j)2012 973 y FA(,)f(b)m(y)h (\014lling)c Fw(C)2532 987 y Fx(j)t Fq(\000)p Fy(1)2696 973 y FA(up)j(to)h Fw(q)i FA(and)d(then)505 1081 y(empt)m(ying)g(it)f (in)m(to)h Fw(C)1264 1095 y Fx(j)1301 1081 y FA(.)g(The)f(empt)m(ying)g (corresp)s(onds)g(to)i(adding)d(one)i(more)g Fw(A)p FA(-)505 1189 y(c)m(hange.)588 1297 y(The)22 b(empt)m(ying)g(device)g(is)f(a)i (ho)s(ok)f(\(the)h Fw(\015)2026 1264 y Fx(A)2083 1297 y FA(\()p Fw(m)p FA(\)-mark)m(er\),)h(whic)m(h)c(b)s(esides)h(b)s(eing) 505 1405 y(used)27 b(on)h(purp)s(ose)d(ma)m(y)j(go)h(o\013)f (\014nitely)d(often)j(b)m(y)g(itself.)e(When)i Fw(C)2834 1419 y Fx(j)t Fq(\000)p Fy(1)2988 1405 y FA(is)e(emptied)505 1513 y(in)m(to)37 b Fw(C)761 1527 y Fx(j)834 1513 y FA(then)f Fw(C)1112 1527 y Fx(j)t Fq(\000)p Fy(2)1239 1513 y Fw(;)15 b(:)g(:)g(:)32 b(;)15 b(C)1521 1527 y Fy(0)1597 1513 y FA(are)37 b(spilled)d(on)j(the)f(\015o)s(or,)h(since)f(the)h(new)f (ho)s(oks)505 1626 y(empt)m(ying)21 b Fw(C)960 1640 y Fx(j)t Fq(\000)p Fy(1)1087 1626 y Fw(;)15 b(:)g(:)g(:)32 b(;)15 b(C)1369 1640 y Fy(0)1430 1626 y FA(ma)m(y)22 b(b)s(e)e(m)m(uc)m(h)i(longer)f(\(the)h Fw(\015)2465 1593 y Fx(A)2522 1626 y FA(\()p Fw(m)p FA(\)-mark)m(er)g(ma)m(y)g(mo)m (v)m(e)505 1734 y(to)j(a)g(m)m(uc)m(h)f(bigger)g(p)s(osition\),)e(and)i (so)g(w)m(e)h(cannot)g(use)e(them)i(an)m(y)f(more)g(to)h(empt)m(y)505 1842 y(those)31 b(decan)m(ters)h(in)d(their)g(old)h(p)s(ositions.)588 1950 y(W)-8 b(e)37 b(\014rst)e(p)s(our)f(wine)g(in)m(to)h(the)h (highest)e(decan)m(ter)j Fw(C)2486 1964 y Fy(0)2525 1950 y FA(,)f(represen)m(ting)e(the)i(left)505 2058 y(domain)24 b(of)h Fw(L)p FA(.)g(W)-8 b(e)27 b(w)m(an)m(t)e(to)h(ensure)e(that)i (at)g(least)f(half)f(the)h(wine)f(w)m(e)h(put)f(in)m(to)h Fw(C)3355 2072 y Fy(0)505 2166 y FA(reac)m(hes)h Fw(C)881 2181 y Fx(k)924 2166 y FA(.)f(Recall)f(that)i(the)f(parameter)g Fw(\014)30 b FA(is)24 b(the)h(amoun)m(t)g(of)g(garbage)i Fw(P)3147 2180 y Fx(j)3183 2166 y FA(\()p Fw(q)s(;)15 b(\014)5 b FA(\))505 2274 y(allo)m(ws.)35 b(If)g Fw(v)k FA(is)34 b(the)i(n)m(um)m(b)s(er)e(of)h(times)g(the)h(empt)m(ying)f (device)g(has)g(gone)h(o\013)g(b)m(y)505 2382 y(itself,)24 b(then)g Fw(P)1004 2396 y Fx(j)1065 2382 y FA(lets)h Fw(P)1285 2396 y Fx(j)t Fq(\000)p Fy(1)1436 2382 y FA(\014ll)e Fw(C)1627 2396 y Fx(j)t Fq(\000)p Fy(1)1778 2382 y FA(in)g(p)s(ortions) g(of)h(size)h(2)2538 2349 y Fq(\000)p Fx(v)2634 2382 y Fw(\014)5 b FA(.)25 b(Then)e(when)h Fw(C)3268 2396 y Fx(j)t Fq(\000)p Fy(1)505 2489 y FA(is)h(emptied)g(in)m(to)g Fw(C)1180 2503 y Fx(j)1217 2489 y FA(,)g(at)h(most)g(2)1636 2456 y Fq(\000)p Fx(v)1732 2489 y Fw(\014)31 b FA(m)m(uc)m(h)26 b(liquid)c(can)k(b)s(e)e(lost)i(b)s(ecause)f(of)h(b)s(eing)505 2597 y(in)36 b(higher)g(decan)m(ters)i Fw(C)1377 2611 y Fx(j)t Fq(\000)p Fy(2)1504 2597 y Fw(;)15 b(:)g(:)g(:)32 b(;)15 b(C)1786 2611 y Fy(0)1826 2597 y FA(.)37 b(The)g(pro)s(cedure)f Fw(P)2569 2611 y Fy(2)2609 2597 y FA(\()p Fw(q)s(;)15 b(\014)5 b FA(\))38 b(is)e(sp)s(ecial)g(but)505 2705 y(limits)29 b(the)h(garbage)i(in)d(the)i(same)g(w)m(a)m(y:)h(it)e(puts) g(requests)g Fr(h)p Fw(r)2677 2719 y Fx(n)2724 2705 y Fw(;)15 b(n)p Fr(i)31 b FA(in)m(to)g Fw(L)f FA(where)505 2813 y(2)550 2780 y Fq(\000)p Fx(r)637 2788 y Fl(n)709 2813 y FA(=)25 b(2)850 2780 y Fq(\000)p Fx(v)947 2813 y Fw(\014)5 b FA(.)26 b(Once)h(it)e(sees)i(the)f(corresp)s(onding)f Fw(A)2374 2801 y Fz(\026)2437 2813 y Fw(n)h FA(description,)e(it)i (empties)505 2921 y Fw(C)570 2935 y Fy(0)643 2921 y FA(in)m(to)34 b Fw(C)896 2935 y Fy(1)969 2921 y FA(\(but)f Fw(C)1239 2935 y Fy(0)1311 2921 y FA(ma)m(y)i(b)s(e)d(spilled)f(on)i(the)h(\015o) s(or)f(b)s(efore)g(that)h(b)s(ecause)f(of)h(a)505 3029 y(lo)m(w)m(er)d(decan)m(ter)g(b)s(eing)e(emptied\).)588 3299 y(W)-8 b(e)32 b(brie\015y)d(sk)m(etc)m(h)k(ho)m(w)d(to)i(sho)m(w)f (that)g(the)g(class)g(of)g Fw(K)7 b FA(-trivial)29 b(sets)i(is)f (closed)505 3407 y(do)m(wn)m(w)m(ard)f(under)f(T)-8 b(uring)27 b(reducibilit)m(y)-8 b(.)27 b(Let)j Fw(A)f FA(b)s(e)g Fw(K)7 b FA(-trivial)27 b(and)h(let)i(a)f(T)-8 b(uring)505 3515 y(reduction)30 b Fw(B)g FA(=)25 b(\000)1162 3482 y Fx(A)1249 3515 y FA(b)s(e)30 b(giv)m(en.)g(W)-8 b(e)32 b(cannot)f(c)m(hange)h Fw(B)j FA(at)c(will)d(an)m(y)i(more,)h(since)505 3623 y(w)m(e)i(do)g(not)f(directly)f(con)m(trol)i(it.)f(Ho)m(w)m(ev)m (er,)j(if)c Fw(B)37 b FA(do)s(es)32 b(not)h(c)m(hange)h(enough,)e(w)m (e)505 3731 y(can)40 b(build)c(a)k(Kraft-Chaitin)d(set)j Fw(W)51 b FA(sho)m(wing)39 b(that)h Fw(B)j FA(is)38 b Fw(K)7 b FA(-trivial.)38 b(W)-8 b(e)40 b(no)m(w)505 3839 y(ha)m(v)m(e)34 b(a)f(tree)g(of)f(runs)f(of)i(pro)s(cedures.)e(The)h (ro)s(ot)g(no)s(de)g(is)f(the)i(single)e(run)g(of)h Fw(P)3326 3854 y Fx(k)3369 3839 y FA(,)505 3947 y(whic)m(h)e(as)i(b)s(efore)f (tries)g(to)h(reac)m(h)g(a)f Fw(k)s FA(-set)i(of)e(w)m(eigh)m(t)h(1.)g (The)f(lea)m(v)m(es)h(b)s(eha)m(v)m(e)g(lik)m(e)505 4055 y(the)e Fw(P)719 4069 y Fy(2)787 4055 y FA(pro)s(cedure)e(ab)s(o)m(v)m (e.)j(A)e(no)s(de)f Fw(P)1863 4069 y Fx(j;\034)1984 4055 y FA(\(2)e Fw(<)f(j)31 b Fz(6)25 b Fw(k)s FA(\))k(calls)f(pro)s (cedures)g Fw(P)3180 4069 y Fx(j)t Fq(\000)p Fy(1)p Fx(;\033)3369 4055 y FA(,)505 4168 y(at)35 b(stages)h(where)e Fw(U)1222 4182 y Fx(s)1258 4168 y FA(\()p Fw(\033)s FA(\))f(=)e Fw(m)j FA(and)g Fw(B)1882 4182 y Fx(s)1950 4168 y FA(=)e(\000)2110 4135 y Fx(A)2163 4143 y Fl(s)2234 4168 y FA(con)m(v)m(erges)k(on)e(all) f(inputs)f Fw(<)g(m)p FA(.)505 4276 y(F)-8 b(or)39 b(in)d(this)h(case)i (w)m(e)f(w)m(an)m(t)g(to)h(en)m(umerate)f(a)h(request)e Fr(hj)p Fw(\033)s Fr(j)27 b FA(+)d Fw(d;)15 b(B)2937 4290 y Fx(s)3012 4264 y Fz(\026)3088 4276 y Fw(m)p Fr(i)38 b FA(in)m(to)505 4384 y(the)i(Kraft-Chaitin)d(set)j Fw(W)52 b FA(built)37 b(at)j(no)s(de)f Fw(P)2175 4398 y Fx(j;\034)2305 4384 y FA(\(where)g Fw(d)h FA(is)e(an)h(appropriate)505 4492 y(constan)m(t)32 b(dep)s(ending)c(on)i(the)h(no)s(de)e Fw(P)1869 4506 y Fx(j;\034)1961 4492 y FA(\).)588 4601 y(As)j(b)s(efore,)f(once)h Fw(P)1289 4615 y Fx(j)t Fq(\000)p Fy(1)p Fx(;\033)1510 4601 y FA(returns,)e Fw(P)1908 4615 y Fx(j;\034)2031 4601 y FA(needs)h(an)g Fw(A)2502 4589 y Fz(\026)2567 4601 y Fw(\015)5 b FA(\()p Fw(m)p FA(\))32 b(c)m(hange.)h(If)e(suc)m(h)505 4709 y(a)g(c)m(hange)h(happ)s(ens)c (su\016cien)m(tly)h(often)i(then)f Fw(P)2188 4723 y Fx(j;\034)2309 4709 y FA(reac)m(hes)i(its)d(goal.)j(Otherwise,)505 4817 y(the)24 b(cost)h(of)f(c)m(hanges)h(of)f Fw(B)5 b FA(,)24 b(in)e(the)i(sense)g(of)g(the)g(pro)s(of)f(of)h(Theorem)g(6.2,)h(is)e (small,)505 4925 y(so)j Fw(W)39 b FA(is)24 b(a)i(Kraft-Chaitin)e(set,)j (sho)m(wing)e(that)h Fw(B)k FA(is)25 b Fw(K)7 b FA(-trivial.)24 b(There)h(m)m(ust)h(b)s(e)f(a)505 5033 y(run)h(on)h(the)g(tree)g(where) g(this)f(cost)i(is)e(small)f(\(called)i(the)g(\\golden)g(run")e(in)h ([103)r(]\),)505 5141 y(as)40 b(otherwise)e(the)i(ro)s(ot)f(no)s(de)g (w)m(ould)f(reac)m(h)i(its)e(goal.)i(Ho)m(w)m(ev)m(er,)i(the)d(pro)s (of)f(is)p eop %%Page: 36 36 36 35 bop 505 363 a FD(36)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(non)m(uniform)24 b(since)g(one)i(cannot)g(iden)m(tify)e(the)i (golden)f(run)f(e\013ectiv)m(ely)-8 b(,)27 b(as)e(w)m(e)h(will)505 649 y(see)31 b(b)s(elo)m(w.)588 757 y(The)k(pro)s(of)g(of)h(Theorem)f (8.1)h(pro)s(ceeds)g(in)e(a)i(similar)c(w)m(a)m(y)-8 b(,)38 b(except)e(that)g Fw(P)3303 771 y Fx(j;\034)505 871 y FA(calls)30 b(pro)s(cedures)f Fw(P)1224 885 y Fx(j)t Fq(\000)p Fy(1)p Fx(;\033)1444 871 y FA(based)h(on)g(computations)g Fw(U)2461 838 y Fx(A)2518 871 y FA(\()p Fw(\033)s FA(\))c(=)f Fw(y)34 b FA(\(since)c(w)m(e)g(no)m(w)505 978 y(w)m(an)m(t)e(to)g(en)m (umerate)g(requests)f Fr(hj)p Fw(\033)s Fr(j)13 b FA(+)g Fw(d;)i(y)s Fr(i)p FA(\),)30 b(and)c(the)h(mark)m(er)g Fw(\015)5 b FA(\()p Fw(m)p FA(\))28 b(is)e(replaced)505 1086 y(b)m(y)f(the)g(use)g(of)g(this)e(computation.)i(The)g(details)e (can)i(b)s(e)g(found)e(in)h(Nies)g([103)r(].)91 b Fr(a)588 1212 y FA(W)-8 b(e)43 b(next)e(discuss)e(t)m(w)m(o)j(theorems)f(from)g (Nies)f([103)r(])h(that)h(can)f(b)s(e)f(obtained)505 1320 y(b)m(y)c(extending)e(the)h(metho)s(ds)g(in)f(the)h(pro)s(of)g(of) g(Theorem)g(8.2.)i(The)d(\014rst)h(sho)m(ws)505 1428 y(that)43 b(the)g(construction)e(in)g(the)h(pro)s(of)g(of)g(Theorem)g (6.2)h(actually)f(pro)m(vides)f(a)505 1536 y(c)m(haracterization)47 b(of)f(the)f Fw(K)7 b FA(-trivial)44 b(c.e.)j(sets.)f(That)f(is,)g(eac) m(h)i Fw(K)7 b FA(-trivial)43 b(c.e.)505 1643 y(set)f Fw(A)g FA(can)f(b)s(e)g(though)m(t)g(of)h(as)f(b)s(eing)f(built)f(b)m (y)i(suc)m(h)g(a)h(construction,)f(for)g(an)505 1751 y(appropriate)28 b(e\013ectiv)m(e)j(en)m(umeration.)d(The)h(pro)s(of)f (of)h(this)e(result)h(uses)g(the)h(pro)s(of)505 1859 y(of)j(do)m(wn)m(w)m(ard)g(closure)f(of)h Fr(K)q FA(,)h(for)f(the)g(sp) s(ecial)e(case)j(of)f(the)g(iden)m(tit)m(y)g(functional.)505 1967 y(W)-8 b(e)39 b(state)g(it)f(here)f(for)h(c.e.)h(sets,)f(but)f(a)h (v)m(ersion)f(for)h(\001)2524 1934 y Fy(0)2524 1992 y(2)2601 1967 y FA(sets)g(is)e(also)i(giv)m(en)g(in)505 2075 y([103)r(].)588 2237 y FB(Theorem)c FA(8.3)h(\(Nies)c([103)q(]\))p FB(.)46 b Fs(F)-7 b(or)31 b(any)f(c.e.)e(set)h Fw(A)p Fs(,)h(the)f(fol)5 b(lowing)30 b(ar)-5 b(e)31 b(e)-5 b(quiv-)505 2345 y(alent.)563 2470 y FA(\(i\))42 b Fw(A)32 b Fs(is)h Fw(K)7 b Fs(-trivial.)538 2578 y FA(\(ii\))41 b Fs(Ther)-5 b(e)33 b(is)g(a)g(c.e.)f(appr)-5 b(oximation)36 b Fr(f)p Fw(A)2022 2592 y Fx(s)2060 2578 y Fr(g)2105 2592 y Fx(s)p Fq(2)p Fx(!)2268 2578 y Fs(to)d Fw(A)f Fs(such)h(that)929 2656 y Fp(X)1060 2742 y Fr(f)o Fp(e)-50 b Fw(c)q FA(\()p Fw(x;)15 b(s)p FA(\))26 b(:)f Fw(x)33 b Fs(is)f(minimal)i(in)e Fw(A)2145 2756 y Fx(s)2182 2742 y FA(\()p Fw(x)p FA(\))21 b Fr(\000)f Fw(A)2484 2756 y Fx(s)p Fq(\000)p Fy(1)2611 2742 y FA(\()p Fw(x)p FA(\))p Fr(g)27 b Fw(<)e FA(1)p Fw(;)701 2921 y Fs(wher)-5 b(e)33 b Fp(e)-51 b Fw(c)q FA(\()p Fw(x;)15 b(s)p FA(\))26 b(=)1324 2853 y Fp(P)1420 2948 y Fx(x<y)r 36="" 37="" 363="" 505="" 541="" 555="" 649="" 757="" 771="" 862="" 865="" 973="" 1132="" 1228="" 1252="" 1349="" 1373="" 1448="" 1532="" 1606="" 1639="" 1653="" 1747="" 1855="" 1963="" 2089="" 2122="" 2146="" 2230="" 2338="" 2496="" 2604="" 2712="" 2820="" 2888="" 2896="" 2921="" 2928="" 3036="" 3091="" 3166="" 3194="" 3199="" 3208="" 3223="" 3302="" 3307="" 3461="" 3469="" 3569="" 3577="" 3591="" 3644="" 3677="" 3691="" 3701="" 3738="" 3802="" 3846="" 3910="" 3954="" 4062="" 4068="" 4176="" 4224="" 4332="" 4346="" 4440="" 4454="" 4547="" 4561="" 4709="" 4817="" 4925="" 5000="" 5033="" 5141="" fk(6)p="" fx(r)1659="" y="" fa(2)1704="" fq(\000)p="" fx(k)1819="" fl(s)1853="" fy(\()p="" fx(y)r="" fy(\))1949="" fs(.)505="" fa(note)32="" b(that)f="" fp(e)-51="" b="" fw(c)p="" fa(\()p="" fw(x;)15="" b(s)p="" fa(\))31="" b(is)f(the)g(cost)i(of)e(putting)g="" fw(x)g="" fa(in)m(to)g="" fw(a)h="" fa(at)g(stage)h="" fw(s)p="" fa(.)588="" y(as)j(an)g(application)e(of)i(this)f(c)m(haracterization)i(in)e(the)h="" (\001)2639="" fy(0)2639="" y(2)2713="" fa(case,)h(one)f="" (obtains)505="" y(the)c(fact)g(that)g="" fw(k)7="" fa(-trivialit)m(y)29="" b(is,)g(in)h(essence,)h(a)g(notion)e(ab)s(out)i(c.e.)g(sets.)588="" fb(theorem)j="" fa(8.4)h(\(nies)c([103)q(]\))p="" fb(.)46="" fs(f)-7="" b(or)51="" b(e)-5="" b(ach)49="" fs(-trivial)50="" b(set)f="" fw(a)p="" fs(,)g(ther)-5="" b(e)50="" b(is)f(a)g="" fs(-)505="" y(trivial)34="" b(c.e.)e(set)g="" fw(d)k="" fs(such)c(that)i="" fw(a)26="" fz(6)1755="" fi(tt)1831="" fw(d)s="" fs(.)588="" fa(a)e(further)f(application)f(of)h(the)h(metho)s(ds)f(in)g(the)="" g(pro)s(of)g(of)h(theorem)f(8.2)i(is)e(that)505="" y(there)31="" b(is)e(a)h(uniform)e(listing)f(of)j(the)h(c.e.)g(sets)f(in)f="" fr(k)i="" fa(that)g(includes)c(the)j(constan)m(ts)505="" y(via)e(whic)m(h)f="" fa(-trivialit)m(y)26="" b(holds.)h(\(this)f="" (result)h(can)i(b)s(e)e(extended)h(to)h(all)e(of)h="" fr(k)q="" fa(;)h(see)505="" y([38)r(].\))588="" fb(theorem)34="" fa(8.5)h(\(do)m(wney)-8="" b(,)33="" b(hirsc)m(hfeldt,)c(nies,)h(and)g="" (stephan)f([38)q(]\))p="" fb(.)47="" fs(ther)-5="" b(e)24="" b(is)505="" y(an)38="" b(e\013e)-5="" b(ctive)36="" b(listing)h="" fr(fhf)p="" fw(b)1465="" fx(e;s)1555="" fr(g)1600="" fx(s)p="" fq(2)p="" fx(!)1731="" fw(;)15="" b(d)1818="" fx(e)1855="" fr(ig)1935="" fx(e)p="" fx(!)2103="" fs(of)37="" b(c.e.)f(appr)-5="" b(oximations)41="" b(such)c(that)505="" y(every)g="" fs(-trivial)36="" b(set)h(o)-5="" b(c)g(curs)37="" b(as)g(a)g="" fw(b)1837="" fx(e)1906="" fa(=")32" b(lim)2136="" fx(s)2188="" fw(b)2257="" fx(e;s)2346="" fs(,)k(and)h(e)-5="" b(ach)38="" fw(b)2868="" fx(e)2941="" fs(is)f="" fs(-trivial)505="" y(via)33="" b(the)g(c)-5="" b(onstant)35="" fw(d)1215="" fx(e)1252="" fa(nies)29="" b([102)r(])g(pro)m(v)m(ed)g="" (that)h(theorem)e(8.5)i(fails)e(if)g(one)h(replaces)g(the)g(notion)f="" (\\)p="" fa(-)505="" b(via)i="" fw(d)p="" fa(")g(b)m(y)g(the)g(notion)f(\\lo)m(w)h(for)f="" fw(k)43="" fa(via)35="" fa(".)i(in)e(other)h(w)m(ords,)f(one)h(can)505="" y(not)i(list)d(the)j(c.e.)g(sets)f="" fw(b)42="" fa(in)35="" fr(k)k="" fa(while)c(also)i(pro)m(viding)e(constan)m(ts)j="" fw(d)f="" fa(suc)m(h)g(that)505="" fr(8)p="" fw(y)18="" fa([)p="" fw(y)s="" fa(\))26="" fz(6)f="" fw(k)1052="" fx(b)1112="" fa(\))c(+)f="" fa(].)31="" b(the)f(reason)h(is)e(that)i(from)f(suc)m(h)g(a)h(constan)m(t)="" h(one)e(can)505="" y(e\013ectiv)m(ely)e(obtain)e(an)h(index)e(for)i="" (the)g(lo)m(wness)f(of)h="" fw(b)k="" fa([103)q(,)c(prop)s(osition)e(2.8].)j="" (on)p="" eop="" %%page:="" bop="" a="" fc(calibra)-5="" b(ting)31="" b(randomness)888="" fd(37)505="" fa(the)36="" b(other)f(hand,)f(for)h(an)m(y)g(sequence)g="" fr(f)p="" fw(b)1981="" fx(e)2019="" fr(g)2064="" fx(!)2229="" fa(of)g(uniformly)d(lo)m(w)j(c.e.)h="" (sets,)g(an)505="" y(extension)h(of)g(the)g(construction)g(in)f="" (theorem)g(6.2)i(pro)m(vides)e(an)h="" fw(a)g="" fr(2)f(k)h="" fr(m)505="" fa(that)g(is)e(not)i(t)-8="" b(uring)34="" b(b)s(elo)m(w)g(an)m(y)h="" fw(b)1778="" fx(e)1815="" fa(,)h(and)e(hence)i(the)f(sequence)h="" fw(b)2972="" fx(e)3009="" fr(g)3054="" fx(!)3220="" fa(do)s(es)505="" y(not)j(exhaust)f="" fa(.)g(th)m(us)f(the)="" i(non)m(uniformit)m(y)c(in)i(the)h(pro)s(of)g(of)g(theorem)f(8.1)i(is)="" y(necessary)-8="" b(.)32="" b(details)e(can)h(b)s(e)f(found)f(in)g="" ([102)q(,)i(theorem)f(5.9].)588="" fb(cor)n(ollar)-6="" b(y)35="" fa(8.6)g(\(nies)c([102)q(]\))p="" b(e)34="" b(is)f(no)h(e\013e)-5="" b(ctive)32="" b(way)i(to)f(obtain)h(fr)-5="" b(om)505="" y(a)38="" b(p)-5="" b(air)39="" fw(a;)15="" b(d)p="" fa(\))p="" fs(,)39="" b(wher)-5="" b(e)38="" fw(a)f="" fs(is)h(a)f(c.e.)g(set)="" g(that)i(is)e="" fs(-trivial)38="" b(via)f="" fs(,)g(a)h(c)-5="" b(onstant)3361="" fp(e)3347="" fw(d)505="" fs(such)33="" b(that)h="" fs(is)f(low)i(for)f="" fw(k)39="" fs(via)1683="" fp(e)1669="" fa(w)-8="" b(e)41="" b(summarize)d(the)h(degree-theoretic)i="" (prop)s(erties)d(of)h="" fa(.)h(recall)f(that)g="" fa(is)505="" fw(!)s="" fa(-c.e.)34="" b(iff)d="" fw(a)e="" fz(6)1040="" fy(wtt)1178="" fr(;)1223="" fq(0)1246="" fa(.)k(it)f(follo)m(ws)g(from)f(theorem)h(8.4)i(that)f(ev)m(ery)="" g="" fa(-trivial)30="" b(set)505="" y(is)k="" fa(-c.e.)h(in)f(fact,)i(as)e(sho)m(wn)g(in)f(do)m(wney)-8="" b(,)36="" b(hirsc)m(hfeldt,)d(miller,)f(and)i(nies)g([36)q(],)505="" y(ev)m(ery)j="" fa(-trivial)34="" b(set)i(is)f(a)h="" fs(d.c.e.)i(r)-5="" b(e)g(al)p="" fa(,)37="" b(that)g(is,)e(the)h(di\013erence)f="" (of)h(t)m(w)m(o)h(left-c.e.)505="" y(reals.)588="" fb(theorem)d="" fa(8.7)h(\(nies)c([103)q(]\))p="" fs(the)33="" fs(-trivial)33="" b(sets)g(form)h(a)f(nonprincip)-5="" b(al)35="" fa(\006)3355="" fy(0)3355="" y(3)505="" fs(ide)-5="" b(al)31="" b(in)e(the)g(low)h="" fs(-c.e.)e(t)-7="" b(uring)29="" b(de)-5="" b(gr)g(e)g(es,)30="" b(which)g(is)f(gener)-5="" b(ate)g(d)31="" b(by)e(its)g(c.e.)g(mem-)505="" y(b)-5="" b(ers.)588="" fb(pr)n(oof.)41="" fa(that)25="" fa(is)c(an)i(ideal)f(follo)m(ws)g(from)g(theorem)h(8.2)h(and)e(the)h="" (closure)f(of)505="" fa(under)c(join)g(men)m(tioned)i(in)e="" (section)h(6.2.)j(by)e(corollary)e(7.9)j(and)e(theorem)g(8.4,)505="" y(this)31="" b(ideal)g(is)h(con)m(tained)g(in)f(the)h(lo)m(w)g="" fa(-c.e.)h(t)-8="" b(uring)31="" b(degrees,)i(and)e(is)g(generated)505="" y(b)m(y)44="" b(its)e(c.e.)j(mem)m(b)s(ers.)d(the)h(ideal)f(is)h="" (nonprincipal)c(b)s(ecause,)k(b)m(y)h(the)f(ab)s(o)m(v)m(e-)505="" y(men)m(tioned)27="" b(extension)h(of)f(the)h(construction)f(in)f="" (the)i(pro)s(of)f(of)g(theorem)g(6.2,)i(one)505="" y(can)i(build)c(a)k="" fa(-trivial)29="" b(set)i(not)f(t)-8="" b(b)s(elo)m(w)g(a)i(giv)m(en)g(lo)m(w)f(c.e.)h(set.)343="" fr(a)588="" fa(8.8)g(\(nies)c([103)q="" (]\))p="" b(e)25="" b(is)f(a)h(low)2342="" fy(2)2407="" fs(c.e.)e(set)h="" fw(e)30="" fs(such)24="" fz(6)3340="" fi(t)505="" fw(e)38="" fs(for)33="" b(every)g="" fa(by)h(theorem)e(8.4,)i(it)f="" (su\016ces)f(to)i(giv)m(e)f(suc)m(h)g(a)g(b)s(ound)e="" fw(e)46="" fa(for)40="" b(the)505="" fa(-trivial)28="" b(c.e.)j(sets.)f(by)g(w)m(ork)g(of)f(nies)g(to)i(b)s(e)e(published)c="" (in)k([33)q(])h(and)f([105)q(],)h(an)m(y)505="" y(prop)s(er)f(\006)="" fy(0)862="" y(3)932="" fa(ideal)g(in)g(the)i(c.e.)g="" (degrees)g(has)f(a)h(lo)m(w)2264="" fy(2)2334="" fa(c.e.)g(upp)s(er)e(b)s(ound.)306="" fa(another)34="" b(notion)f(of)h(computational)f(w)m(eakness)h(related)f="" (to)h(1-randomness)f(is)505="" y(that)e(of)g(bases)f(for)g="" (1-randomness.)588="" fb(definition)35="" fa(8.9)p="" fa(a)d(set)g="" fa(is)f(a)h="" fs(b)-5="" b(asis)46="" b(for)g="" fa(1)p="" fs(-r)-5="" b(andomness)54="" fa(if)43="" b(there)h(is)f(an)505="" fw(x)33="" fz(="">)684 4190 y Fy(T)764 4176 y Fw(A)e FA(suc)m(h)f(that)h Fw(X)37 b FA(is)30 b(1-random)g(relativ)m(e)h(to)g Fw(A)p FA(.)588 4335 y(Ku)m(\024)-43 b(cera)48 b([66)q(])f(and)f(G\023) -45 b(acs)47 b([46)q(])g(sho)m(w)m(ed)g(that)g(ev)m(ery)g(set)g(can)g (b)s(e)f(computed)505 4443 y(b)m(y)41 b(some)g(1-random)g(set)g(\(see)g (Theorem)g(12.1)h(b)s(elo)m(w\),)e(so)h(if)f Fw(A)g FA(is)g(lo)m(w)g (for)h(1-)505 4551 y(randomness)c(then)g Fw(A)h FA(is)e(a)i(basis)f (for)g(1-randomness.)g(In)g(the)h(other)g(direction,)505 4659 y(Ku)m(\024)-43 b(cera)38 b([67)q(])e(sho)m(w)m(ed)g(that)h(ev)m (ery)g(basis)e(for)h(1-randomness)f(is)h(GL)2947 4673 y Fy(1)2986 4659 y FA(.)g(More)h(re-)505 4766 y(cen)m(tly)-8 b(,)30 b(Hirsc)m(hfeldt,)d(Nies,)i(and)f(Stephan)f([50)q(])i(ga)m(v)m (e)i(an)d(exact)i(c)m(haracterization)505 4874 y(of)h(the)g(bases)f (for)g(1-randomness.)588 5033 y FB(Theorem)k FA(8.10)i(\(Hirsc)m (hfeldt,)29 b(Nies,)i(and)e(Stephan)h([50)q(]\))p FB(.)46 b Fs(A)36 b(set)g(is)g Fw(K)7 b Fs(-triv-)505 5141 y(ial)34 b(iff)e(it)g(is)h(a)g(b)-5 b(asis)33 b(for)g FA(1)p Fs(-r)-5 b(andomness.)p eop %%Page: 38 38 38 37 bop 505 363 a FD(38)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FB(Pr)n(oof)34 b(Sketch.)40 b FA(If)34 b(a)i(set)f(is)f Fw(K)7 b FA(-trivial)33 b(then)i(it)f(is)g(lo)m(w)h(for)g (1-randomness,)505 649 y(and)30 b(hence)h(is)e(a)i(basis)e(for)h (1-randomness.)588 757 y(The)f(con)m(v)m(erse)h(is)e(pro)m(v)m(ed)h(b)m (y)g(what)g(has)g(b)s(een)f(called)g(the)h(\\h)m(ungry)f(sets")i(con-) 505 865 y(struction.)k(Supp)s(ose)f(that)i Fw(A)f FA(is)f(a)i(basis)e (for)h(1-randomness,)g(and)g(let)g Fw(Z)41 b FA(and)34 b(\010)505 973 y(b)s(e)f(suc)m(h)f(that)i(\010)1105 940 y Fx(Z)1191 973 y FA(=)29 b Fw(A)k FA(and)g Fw(Z)39 b FA(is)32 b(1-random)h(relativ)m(e)g(to)h Fw(A)p FA(.)f(W)-8 b(e)34 b(en)m(umerate)g(a)505 1081 y(Kraft-Chaitin)29 b(set)j Fw(L)1284 1096 y Fx(d)1355 1081 y FA(for)e(eac)m(h)i Fw(d)26 b Fr(2)f Fw(!)s FA(.)31 b(W)-8 b(e)32 b(w)m(an)m(t)g(to)f (ensure)f(that)h(there)g(is)f(a)h Fw(d)505 1189 y FA(suc)m(h)j(that)g Fw(L)976 1204 y Fx(d)1049 1189 y FA(con)m(tains)g(a)g(request)f Fr(h)p Fw(K)7 b FA(\()p Fr(j)p Fw(\034)j Fr(j)p FA(\))24 b(+)e Fw(d)g FA(+)g(2)p Fw(;)15 b(\034)10 b Fr(i)35 b FA(for)e(eac)m(h)i Fw(\034)40 b Fr(\036)30 b Fw(A)p FA(.)k(The)505 1297 y(idea)h(is)f(to)h(build)d(sets)j Fw(C)1404 1264 y Fx(\034)1397 1325 y(d)1480 1297 y Fr(\022)d FA(2)1628 1264 y Fx(!)1714 1297 y FA(for)i Fw(d)g Fr(2)e Fw(!)38 b FA(and)c Fw(\034)43 b Fr(2)32 b FA(2)2528 1264 y Fx()g FA(2)1048 2277 y Fq(\000)p Fx(K)1163 2285 y Fl(s)1196 2277 y Fy(\()p Fq(j)p Fx(\034)8 b Fq(j)p Fy(\))p Fq(\000)p Fx(d)p Fq(\000)p Fy(1)1515 2310 y FA(.)43 b(Since)f(the)i Fw(C)2075 2277 y Fx(\034)2068 2338 y(d)2161 2310 y FA(are)f(pairwise)f(disjoin)m(t,)g(this)g(is)g(a) 505 2418 y(Kraft-Chaitin)28 b(set.)i(Since)f Fw(Z)35 b FA(is)29 b(1-random)g(relativ)m(e)h(to)g Fw(A)p FA(,)g(w)m(e)g(ha)m (v)m(e)g Fw(Z)42 b(=)-55 b Fr(2)25 b Fw(U)3216 2433 y Fx(d)3286 2418 y FA(for)505 2527 y(some)36 b Fw(d)f FA(and)f(hence)h Fw(\026)p FA(\()p Fw(C)1420 2494 y Fx(\034)1413 2554 y(d)1463 2527 y FA(\))e(=)f(2)1679 2494 y Fq(\000)p Fx(K)5 b Fy(\()p Fq(j)p Fx(\034)j Fq(j)p Fy(\))p Fq(\000)p Fx(d)2062 2527 y FA(for)34 b(all)g Fw(\034)43 b Fr(\036)32 b Fw(A)p FA(,)j(whic)m(h)f(implies)e(that)505 2635 y Fr(h)p Fw(K)7 b FA(\()p Fr(j)p Fw(\034)j Fr(j)p FA(\))22 b(+)e Fw(d)g FA(+)g(2)p Fw(;)15 b(\034)10 b Fr(i)27 b(2)e Fw(L)1410 2650 y Fx(d)1480 2635 y FA(for)30 b(all)g Fw(\034)35 b Fr(\036)25 b Fw(A)p FA(,)31 b(as)f(desired.)588 2748 y(T)-8 b(o)34 b(build)29 b(the)k Fw(C)1190 2715 y Fx(\034)1183 2776 y(d)1233 2748 y FA(,)g(as)f(long)h(as)g Fw(\026)p FA(\()p Fw(C)1880 2715 y Fx(\034)1873 2776 y(d)1922 2748 y FA(\))d Fw(<)f FA(2)2132 2715 y Fq(\000)p Fx(K)2247 2723 y Fl(s)2280 2715 y Fy(\()p Fq(j)p Fx(\034)8 b Fq(j)p Fy(\))p Fq(\000)p Fx(d)2508 2748 y FA(,)33 b(w)m(e)g(lo)s(ok)f(for)h (strings)e Fw(\033)505 2867 y FA(suc)m(h)j(that)g Fw(\034)41 b Fz(4)30 b FA(\010)1162 2834 y Fx(\033)1243 2867 y FA(and)j Fw(\026)p FA(\()p Fw(C)1585 2834 y Fx(\034)1578 2895 y(d)1627 2867 y FA(\))23 b(+)f(2)1823 2834 y Fq(\000j)p Fx(\033)r Fq(j)1996 2867 y Fz(6)30 b FA(2)2142 2834 y Fq(\000)p Fx(K)2257 2842 y Fl(s)2290 2834 y Fy(\()p Fq(j)p Fx(\034)8 b Fq(j)p Fy(\))p Fq(\000)p Fx(d)2519 2867 y FA(,)34 b(and)f(put)g([)p Fw(\033)s FA(])h(in)m(to)g Fw(C)3327 2834 y Fx(\034)3320 2895 y(d)3369 2867 y FA(.)505 2975 y(T)-8 b(o)33 b(k)m(eep)g(our)f(sets)h(pairwise)d(disjoin)m(t,)i (w)m(e)g(then)h(ensure)e(that)i(no)f([)p Fw(\033)2939 2942 y Fq(0)2963 2975 y FA(])h(suc)m(h)f(that)505 3083 y Fw(\033)560 3050 y Fq(0)618 3083 y FA(is)h(compatible)g(with)g Fw(\033)k FA(is)c(later)h(put)g(in)m(to)g(an)m(y)g Fw(C)2398 3050 y Fx(\027)2391 3111 y(d)2441 3083 y FA(.)g(If)g Fw(Z)48 b(=)-56 b Fr(2)31 b Fw(U)2848 3098 y Fx(d)2889 3083 y FA(,)j(then)g(no)g([)p Fw(\033)s FA(])505 3191 y(with)i Fw(\033)k Fr(\036)c Fw(Z)44 b FA(is)36 b(ev)m(er)i(put)f(in)m (to)g(an)m(y)g Fw(C)1936 3158 y Fx(\034)1929 3219 y(d)1979 3191 y FA(,)g(whic)m(h)f(means)h(that)h(the)f(measure)g(of)505 3310 y(eac)m(h)32 b Fw(C)782 3277 y Fx(\034)775 3337 y(d)855 3310 y FA(with)d Fw(\034)35 b Fr(\036)25 b Fw(A)h FA(=)f(\010)1489 3277 y Fx(Z)1575 3310 y FA(m)m(ust)30 b(ev)m(en)m(tually)h(exceed)g(2)2570 3277 y Fq(\000)p Fx(K)5 b Fy(\()p Fq(j)p Fx(\034)j Fq(j)p Fy(\))p Fq(\000)p Fx(d)p Fq(\000)p Fy(1)3008 3310 y FA(.)306 b Fr(a)588 3435 y FA(One)48 b(corollary)f(of)i(this)d(result)h(is)g(the)i(easier)f (direction)e(of)i(Theorem)g(7.4,)505 3543 y(namely)30 b(that)i(ev)m(ery)f(set)g(that)h(is)e(lo)m(w)g(for)g(1-randomness)h(is) f Fw(K)7 b FA(-trivial.)29 b(Another)505 3651 y(is)d(an)h(extension)f (of)h(Theorem)f(7.6:)i(if)d Fw(A)i FA(is)f(\001)2100 3618 y Fy(0)2100 3675 y(2)2165 3651 y FA(and)g(lo)m(w)h(for)f(\012,)h (then)f(it)g(is)g(a)h(basis)505 3759 y(for)k(1-randomness,)f(and)g (hence)g Fw(K)7 b FA(-trivial.)588 3867 y(One)36 b(w)m(a)m(y)h(to)f(lo) s(ok)g(at)g(Theorem)g(8.10)h(is)e(in)f(connection)i(with)f(the)h(follo) m(wing)505 3975 y(classical)30 b(theorem.)588 4126 y FB(Theorem)k FA(8.11)p FB(.)47 b FA(\(de)39 b(Leeu)m(w,)f(Mo)s(ore,)i (Shannon)d(and)g(Shapiro)g([74)q(],)i(Sac)m(ks)505 4234 y([126)r(]\).)33 b Fs(If)g Fw(A)g Fs(is)f(not)h(c)-5 b(omputable)35 b(then)e Fw(\026)p FA(\()p Fr(f)p Fw(X)g FA(:)26 b Fw(X)32 b Fz(>)2383 4248 y Fi(T)2463 4234 y Fw(A)p Fr(g)p FA(\))27 b(=)e(0)p Fs(.)588 4385 y FA(There)32 b(is)g(a)h(sense)f(in)f(whic)m(h)g(this)h(result)f(cannot)i(b)s(e)f (e\013ectivized,)h(since)f Fr(f)p Fw(X)k FA(:)505 4493 y Fw(X)d Fz(>)684 4507 y Fy(T)764 4493 y Fw(A)p Fr(g)27 b FA(is)f(nev)m(er)h(Martin-L\177)-45 b(of)27 b(n)m(ull,)e(as)i(b)m(y)f (Theorem)h(12.1)h(it)e(alw)m(a)m(ys)h(con)m(tains)505 4601 y(a)j(1-random)g(set.)g(Ho)m(w)m(ev)m(er,)i(if)d Fw(A)g FA(is)g(not)h(a)g(basis)e(for)h(1-randomness,)h(then)f Fr(f)p Fw(X)k FA(:)505 4709 y Fw(X)54 b Fz(>)705 4723 y Fy(T)805 4709 y Fw(A)p Fr(g)44 b FA(is)d(con)m(tained)i(in)f(the)h (univ)m(ersal)e(Martin-L\177)-45 b(of)42 b(test)i(relativ)m(e)f(to)g Fw(A)p FA(,)505 4817 y(and)38 b(hence)f(is)g(Martin-L\177)-45 b(of)38 b(n)m(ull)d Fs(r)-5 b(elative)40 b(to)g Fw(A)p FA(.)e(Th)m(us)f(w)m(e)h(ha)m(v)m(e)h(the)f(follo)m(wing)505 4925 y(consequence)43 b(of)f(Theorem)g(8.11,)h(whic)m(h)e(can)h(b)s(e)f (tak)m(en)i(to)g(sa)m(y)f(that)h(the)f Fw(K)7 b FA(-)505 5033 y(trivial)36 b(sets)h(are)h(exactly)g(those)g(relativ)m(e)f(to)h (whic)m(h)e(Theorem)h(8.11)i(cannot)f(b)s(e)505 5141 y(e\013ectivized.)p eop %%Page: 39 39 39 38 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(39)588 541 y FB(Cor)n(ollar)-6 b(y)35 b FA(8.12)h(\(Hirsc)m (hfeldt,)15 b(Nies,)g(and)29 b(Stephan)h([50)q(]\))p FB(.)46 b Fs(A)35 b(set)h Fw(A)g Fs(is)g(not)505 649 y Fw(K)7 b Fs(-trivial)33 b(iff)f Fr(f)p Fw(X)h FA(:)26 b Fw(X)33 b Fz(>)1389 663 y Fi(T)1469 649 y Fw(A)p Fr(g)g Fs(is)f(Martin-L\177)-46 b(of)33 b(nul)5 b(l)33 b(r)-5 b(elative)33 b(to)g Fw(A)p Fs(.)588 799 y FA(Theorem)d(8.10)i(has)e (found)f(a)h(surprising)d(application)h(to)j(computabilit)m(y)e(the-) 505 907 y(ory)-8 b(.)30 b(A)f Fs(Sc)-5 b(ott)32 b(set)37 b FA(is)28 b(a)h(T)-8 b(uring)27 b(ideal)g Fr(S)36 b FA(suc)m(h)28 b(that)h(for)g(eac)m(h)h(in\014nite)c(binary)h(tree)505 1014 y Fw(T)45 b Fr(2)32 b(S)7 b FA(,)35 b(there)g(is)f(an)g (in\014nite)e(path)j(of)g Fw(T)47 b FA(in)33 b Fr(S)7 b FA(.)35 b(Scott)g(sets)g(o)s(ccur)g(naturally)e(in)505 1122 y(v)-5 b(arious)36 b(con)m(texts,)i(suc)m(h)e(as)h(the)g(study)e (of)i(mo)s(dels)e(of)h(arithmetic)g(and)g(rev)m(erse)505 1230 y(mathematics.)43 b(H.)f(F)-8 b(riedman)41 b(and)g(A.)h (McAllister)e(indep)s(enden)m(tly)f(ask)m(ed)j(the)505 1338 y(follo)m(wing)29 b(question:)g(if)f Fr(S)37 b FA(is)29 b(a)h(Scott)g(set)h(and)e Fw(X)j Fr(2)25 b(S)37 b FA(is)28 b(not)i(computable,)g(do)s(es)505 1446 y(there)d(necessarily)f(exist)g (a)h Fw(Y)45 b Fr(2)25 b(S)34 b FA(suc)m(h)26 b(that)h Fw(X)33 b Fr(j)2267 1460 y Fy(T)2347 1446 y Fw(Y)20 b FA(?)27 b(Slaman)e(\(p)s(ersonal)h(com-)505 1554 y(m)m(unication\))37 b(has)g(recen)m(tly)g(giv)m(en)g(a)h(p)s(ositiv)m(e)e(answ)m(er)h(to)h (this)e(question)g(using)505 1662 y(Theorem)31 b(8.10.)588 1770 y(Most)41 b(of)e(the)h(topics)f(in)f(this)g(Section)h(are)h(surv)m (ey)m(ed)f(in)f([106)r(])i(and)e(the)i(cor-)505 1878 y(resp)s(onding)28 b(pro)s(ceedings)h(pap)s(er.)h(The)g(in)m(teraction) g(of)h Fw(K)7 b FA(-trivialit)m(y)28 b(and)i(1-ran-)505 1986 y(domness)24 b(via)g(T)-8 b(uring)23 b(reducibilit)m(y)e(is)i(in)g (the)i(fo)s(cus)e(of)i(curren)m(t)f(researc)m(h;)h(see)g([94)q(,)505 2094 y(Section)31 b(4].)588 2301 y Fu(x)p Ft(9.)53 b(Kummer)42 b(complex)i(c.e.)g(sets,)g(arra)m(y)g(noncomputabilit)m(y)-9 b(,)45 b(and)505 2409 y(c.e.-traceabilit)m(y.)h FA(As)21 b(w)m(e)g(sa)m(w)g(in)f(Theorem)g(7.3,)i(T)-8 b(erwijn)19 b(and)h(Zam)m(b)s(ella)g([134)q(])505 2517 y(sho)m(w)m(ed)32 b(that)g(the)f(sets)h(that)f(are)h(lo)m(w)f(for)g(1-randomness)g(are)g (c.e.-traceable.)j(In)505 2625 y(Theorem)26 b(11.6,)h(w)m(e)f(will)c (see)k(that)g(an)g(analogous)f(concept)i(to)f(c.e.-traceabilit)m(y)g (is)505 2733 y(relev)-5 b(an)m(t)22 b(to)h(the)f(c)m(haracterization)h (of)e(another)h(class)g(of)g(sets)g(satisfying)e(a)i(lo)m(wness)505 2841 y(notion.)32 b(In)f(the)h(case)h(of)f(c.e.)h(sets,)g(there)f(is)f (a)h(fascinating)f(connection)h(b)s(et)m(w)m(een)505 2949 y(Kolmogoro)m(v)26 b(complexit)m(y)e(and)g(the)h(notion)f(of)h (arra)m(y)g(computabilit)m(y)e(in)m(tro)s(duced)505 3057 y(b)m(y)28 b(Do)m(wney)-8 b(,)29 b(Jo)s(c)m(kusc)m(h,)f(and)f(Stob)h ([39)q(],)g(whic)m(h,)f(as)h(sho)m(wn)f(b)m(y)g(Ishm)m(ukhameto)m(v)505 3165 y([52)r(],)j(coincides)g(with)f(c.e.-traceabilit)m(y)i(on)g(the)f (c.e.)i(degrees.)588 3273 y(Again)22 b(w)m(e)h(are)g(concerned)f(with)f (initial)f(segmen)m(t)j(complexit)m(y)-8 b(,)23 b(this)e(time)h(of)g (c.e.)505 3381 y(sets.)31 b(W)-8 b(e)32 b(w)m(ork)e(with)g(plain)e (complexit)m(y)-8 b(.)31 b(The)e(follo)m(wing)g(result)h(is)f(w)m (ell-kno)m(wn.)588 3530 y FB(Theorem)34 b FA(9.1)h(\(Barzdins')30 b(Lemma)h([7]\))p FB(.)47 b Fs(L)-5 b(et)29 b Fw(A)h Fs(b)-5 b(e)29 b(a)h(c.e.)f(set.)h(Then)f Fw(C)7 b FA(\()p Fw(A)3357 3518 y Fz(\026)505 3638 y Fw(n)25 b Fr(j)h Fw(n)p FA(\))f Fz(6)g FA(log)16 b Fw(n)k FA(+)g Fw(O)s FA(\(1\))34 b Fs(and)f Fw(C)7 b FA(\()p Fw(A)1743 3626 y Fz(\026)1806 3638 y Fw(n)p FA(\))25 b Fz(6)g FA(2)15 b(log)i Fw(n)j FA(+)g Fw(O)s FA(\(1\))p Fs(.)588 3787 y FB(Pr)n(oof.)41 b FA(T)-8 b(o)35 b(compute)f Fw(A)1552 3775 y Fz(\026)1621 3787 y Fw(n)f FA(giv)m(en)h Fw(n)p FA(,)g(it)f(su\016ces)h(to)g(kno)m(w)g(the)g Fs(numb)-5 b(er)44 b FA(of)505 3895 y(elemen)m(ts)f Fz(6)i Fw(n)c FA(in)g Fw(A)p FA(,)i(since)e(w)m(e)i(can)g(run)d(the)j(en)m(umeration) e(of)i Fw(A)f FA(un)m(til)f(this)505 4003 y(man)m(y)34 b(elemen)m(ts)g(app)s(ear.)g(So)f Fw(A)1689 3991 y Fz(\026)1758 4003 y Fw(n)g FA(can)h(b)s(e)f(describ)s(ed)f(giv)m(en)h Fw(n)g FA(with)g(at)h(most)505 4111 y(log)17 b Fw(n)f FA(+)f Fw(O)s FA(\(1\))30 b(man)m(y)e(bits.)f(Similarly)-8 b(,)26 b(to)j(compute)f Fw(A)2429 4099 y Fz(\026)2492 4111 y Fw(n)p FA(,)g(it)g(su\016ces)g(to)h(kno)m(w)f Fw(n)505 4219 y FA(and)k(the)g(n)m(um)m(b)s(er)f(of)h(elemen)m(ts)h Fz(6)28 b Fw(n)j FA(in)g Fw(A)p FA(,)i(whic)m(h)e(can)h(b)s(e)f(enco)s (ded)h(in)f(a)i(string)505 4327 y(of)e(length)f(2)15 b(log)i Fw(n)p FA(.)2179 b Fr(a)588 4452 y FA(A)29 b(longstanding)d(op) s(en)i(question)f(w)m(as)i(whether)e(the)h(2)15 b(log)j Fw(n)27 b FA(is)g(optimal)h(in)e(the)505 4560 y(second)j(part)f(of)h (Theorem)f(9.1.)h(The)f(b)s(est)g(w)m(e)h(could)e(hop)s(e)h(for)g(is)g (to)h(ha)m(v)m(e)g Fw(C)7 b FA(\()p Fw(A)3357 4548 y Fz(\026)505 4668 y Fw(n)p FA(\))26 b Fz(>)f FA(2)15 b(log)i Fw(n)6 b Fr(\000)g Fw(O)s FA(\(1\))23 b(for)f(in\014nitely)e(man)m(y)k Fw(n)p FA(,)f(since)f(the)h(follo)m(wing)f(is)g(kno)m(wn)h(\(and)505 4776 y(implies)j(that)j(there)f(is)g(no)g(c.e.)h(set)g Fw(A)g FA(suc)m(h)f(that)g Fw(C)7 b FA(\()p Fw(A)2445 4764 y Fz(\026)2508 4776 y Fw(n)p FA(\))25 b Fz(>)g FA(2)15 b(log)i Fw(n)f Fr(\000)f Fw(O)s FA(\(1\))30 b(for)505 4884 y(ev)m(ery)i Fw(n)p FA(\).)588 5033 y FB(Theorem)i FA(9.2)h(\(Solo)m(v)-5 b(a)m(y)15 b(\(unpublished\)\))p FB(.)44 b Fs(Ther)-5 b(e)24 b(is)g(no)h(c.e.)e(set)h Fw(A)g Fs(such)g(that)505 5141 y Fw(C)7 b FA(\()p Fw(A)706 5129 y Fz(\026)769 5141 y Fw(n)25 b Fr(j)g Fw(n)p FA(\))g Fz(>)g FA(log)17 b Fw(n)j Fr(\000)g Fw(O)s FA(\(1\))33 b Fs(for)g(every)g Fw(n)p Fs(.)p eop %%Page: 40 40 40 39 bop 505 363 a FD(40)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FB(Pr)n(oof.)41 b FA(De\014ne)f Fw(g)s FA(\()p Fw(n)p FA(\))h(so)f(that)g Fw(A)1882 529 y Fz(\026)1961 541 y Fw(g)s FA(\()p Fw(n)p FA(\))h(has)e(exactly)i(2)2712 508 y Fx(n)2799 541 y FA(elemen)m(ts.)f(Note)505 649 y(that)26 b(log)17 b Fw(g)s FA(\()p Fw(n)p FA(\))26 b Fz(>)f Fw(n)p FA(.)g(W)-8 b(e)27 b(can)e(compute)h Fw(A)1998 637 y Fz(\026)2062 649 y Fw(g)s FA(\()p Fw(n)p FA(\))g(giv)m(en)f Fw(g)s FA(\()p Fw(n)p FA(\))h(and)f Fw(n)p FA(,)g(b)m(y)g(running)505 757 y(the)30 b(en)m(umeration)f(of)g Fw(A)g FA(un)m(til)f(2)1641 724 y Fx(n)1717 757 y FA(elemen)m(ts)i(en)m(ter)g Fw(A)f FA(b)s(elo)m(w)f Fw(g)s FA(\()p Fw(n)p FA(\).)i(Th)m(us)f(w)m(e)g(can) 505 865 y(describ)s(e)h Fw(A)954 853 y Fz(\026)1019 865 y Fw(g)s FA(\()p Fw(n)p FA(\))i(giv)m(en)g Fw(g)s FA(\()p Fw(n)p FA(\))g(using)f(only)f(log)17 b Fw(n)j FA(+)h Fw(O)s FA(\(1\))33 b(bits)d(of)i(information,)505 973 y(and)e(hence)h Fw(C)7 b FA(\()p Fw(A)1135 961 y Fz(\026)1198 973 y Fw(g)s FA(\()p Fw(n)p FA(\))26 b Fr(j)g Fw(g)s FA(\()p Fw(n)p FA(\)\))g Fo(\013)f FA(log)17 b Fw(g)s FA(\()p Fw(n)p FA(\))k Fr(\000)f Fw(O)s FA(\(1\).)937 b Fr(a)588 1101 y FA(Solo)m(v)-5 b(a)m(y)32 b(explicitly)d(ask)m(ed)j (whether)e(it)h(is)f(p)s(ossible)e(for)j Fw(C)7 b FA(\()p Fw(A)2778 1089 y Fz(\026)2842 1101 y Fw(n)p FA(\))26 b Fz(>)g FA(2)15 b(log)j Fw(n)i Fr(\000)505 1209 y Fw(O)s FA(\(1\))32 b(to)f(happ)s(en)e(in\014nitely)e(often)k(for)f(a)h(c.e.)g (set)g Fw(A)p FA(.)588 1359 y FB(Definition)k FA(9.3)p FB(.)47 b FA(W)-8 b(e)33 b(sa)m(y)g(that)g(a)g(c.e.)h(set)e Fw(A)h FA(is)e Fs(\(Kummer\))36 b(c)-5 b(omplex)44 b FA(if)32 b(for)505 1467 y(eac)m(h)g Fw(d)f FA(there)f(are)h (in\014nitely)c(man)m(y)k Fw(n)f FA(suc)m(h)g(that)h Fw(C)7 b FA(\()p Fw(A)2488 1455 y Fz(\026)2551 1467 y Fw(n)p FA(\))25 b Fz(>)g FA(2)15 b(log)i Fw(n)j Fr(\000)g Fw(d)p FA(.)588 1616 y(Kummer)33 b(pro)m(v)m(ed)i(that)f(suc)m(h)g (complex)g(sets)h(do)f(exist.)g(The)f(classi\014cation)g(of)505 1724 y(the)j(degrees)f(con)m(taining)g(Kummer)f(complex)g(sets)i(is)e (an)h(in)m(teresting)f(in)m(terpla)m(y)505 1832 y(b)s(et)m(w)m(een)i (computabilit)m(y)e(theory)i(and)f(algorithmic)f(complexit)m(y)-8 b(.)36 b(W)-8 b(e)36 b(need)f(the)505 1940 y(follo)m(wing)29 b(de\014nition.)588 2090 y FB(Definition)35 b FA(9.4)h(\(Do)m(wney)-8 b(,)32 b(Jo)s(c)m(kusc)m(h,)f(and)e(Stob)i([39)q(]\))p FB(.)563 2215 y FA(\(i\))42 b(Let)24 b Fw(D)932 2229 y Fy(0)972 2215 y Fw(;)15 b(D)1087 2229 y Fy(1)1127 2215 y Fw(;)g(:)g(:)g(:)41 b FA(b)s(e)24 b(a)g(standard)g(en)m(umeration)g (of)g(the)h(\014nite)e(sets.)i(A)f Fs(str)-5 b(ong)701 2323 y(arr)g(ay)39 b FA(is)29 b(a)h(set)g(of)g(the)f(form)g Fr(f)p Fw(D)1846 2341 y Fx(f)7 b Fy(\()p Fx(x)p Fy(\))2012 2323 y FA(:)26 b Fw(x)f Fr(2)g Fo(N)7 b Fr(g)36 b FA(for)29 b(a)h(computable)f(function)701 2431 y Fw(f)10 b FA(.)538 2539 y(\(ii\))41 b(A)k(strong)g(arra)m(y)g Fr(f)p Fw(D)1479 2557 y Fx(f)7 b Fy(\()p Fx(x)p Fy(\))1670 2539 y FA(:)50 b Fw(x)f Fr(2)g Fo(N)7 b Fr(g)51 b FA(is)44 b(called)h(a)g Fs(very)h(str)-5 b(ong)48 b(arr)-5 b(ay)55 b FA(if)701 2650 y Fr(j)p Fw(D)801 2669 y Fx(f)7 b Fy(\()p Fx(x)p Fy(\))941 2650 y Fr(j)26 b Fw(>)f Fr(j)p Fw(D)1188 2669 y Fx(f)7 b Fy(\()p Fx(y)r Fy(\))1326 2650 y Fr(j)30 b FA(for)g(all)g Fw(x)25 b(>)g(y)s FA(.)513 2762 y(\(iii\))40 b(F)-8 b(or)33 b(a)f(v)m(ery)h(strong)f(arra)m(y)h Fr(F)38 b FA(=)28 b Fr(f)p Fw(D)1987 2781 y Fx(f)7 b Fy(\()p Fx(x)p Fy(\))2156 2762 y FA(:)29 b Fw(x)g Fr(2)f Fo(N)6 b Fr(g)q FA(,)38 b(w)m(e)33 b(sa)m(y)g(that)g(a)f(c.e.)i(set)701 2870 y Fw(A)h FA(is)f Fr(F)9 b Fs(-arr)-5 b(ay)40 b(nonc)-5 b(omputable)44 b FA(if)34 b(for)h(eac)m(h)i(c.e.)g(set)e Fw(W)48 b FA(there)36 b(exists)f(a)g Fw(k)701 2978 y FA(suc)m(h)30 b(that)1478 3086 y Fw(W)j Fr(\\)20 b Fw(D)1753 3104 y Fx(f)7 b Fy(\()p Fx(k)r Fy(\))1917 3086 y FA(=)25 b Fw(A)c Fr(\\)e Fw(D)2257 3104 y Fx(f)7 b Fy(\()p Fx(k)r Fy(\))2396 3086 y Fw(:)515 3216 y FA(\(iv\))42 b(A)24 b(c.e.)h(set)f(is)f Fs(arr)-5 b(ay)29 b(nonc)-5 b(omputable)33 b FA(\()p Fs(a.n.c.)p FA(\))25 b(if)d(it)i(is)f Fr(F)9 b FA(-arra)m(y)25 b(noncomput-)701 3324 y(able)k(for)g(some)g(v)m(ery)h (strong)g(arra)m(y)g Fr(F)9 b FA(.)30 b(A)f(degree)h(is)e Fs(arr)-5 b(ay)34 b(nonc)-5 b(omputable)701 3432 y FA(if)27 b(it)h(con)m(tains)g(an)g(arra)m(y)h(noncomputable)e(set;)i(otherwise)f (it)f(is)g Fs(arr)-5 b(ay)33 b(c)-5 b(om-)701 3540 y(putable)p FA(.)588 3690 y(This)30 b(de\014nition)f(w)m(as)i(designed)f(to)i (capture)f(a)h(certain)f(kind)e(of)i(m)m(ultiple)e(p)s(er-)505 3798 y(mitting)i(construction.)g(The)g(in)m(tuition)e(is)h(that)i(for)g Fw(A)f FA(to)h(b)s(e)f Fr(F)9 b FA(-a.n.c.,)33 b Fw(A)f FA(needs)505 3906 y Fr(j)p Fw(D)605 3924 y Fx(f)7 b Fy(\()p Fx(k)r Fy(\))745 3906 y Fr(j)30 b FA(man)m(y)h(p)s(ermissions)c(in)i (order)h(to)h(agree)h(with)d Fw(W)42 b FA(on)31 b Fw(D)2767 3924 y Fx(f)7 b Fy(\()p Fx(k)r Fy(\))2906 3906 y FA(.)588 4017 y(In)29 b(Do)m(wney)-8 b(,)30 b(Jo)s(c)m(kusc)m(h,)g(and)e(Stob)h ([40)q(],)g(a)h(new)e(de\014nition)f(of)i(arra)m(y)g(noncom-)505 4125 y(putabilit)m(y)k(w)m(as)j(in)m(tro)s(duced,)e(based)g(on)h (domination)f(prop)s(erties)f(of)j(functions.)505 4233 y(W)-8 b(e)35 b(\014rst)e(recall)g(that)h Fw(f)40 b Fz(6)1458 4247 y Fy(wtt)1599 4233 y Fw(A)34 b FA(\(for)g(a)g(function)e Fw(f)43 b FA(and)33 b(a)h(set)g Fw(A)p FA(\))g(means)g(that)505 4342 y(there)44 b(are)f(an)g(index)e Fw(e)j FA(and)e(a)h(computable)g (function)e Fw(b)i FA(suc)m(h)g(that)g Fw(f)56 b FA(=)46 b(\010)3338 4309 y Fx(A)3338 4365 y(e)505 4452 y FA(and,)29 b(furthermore,)g(for)g(eac)m(h)h Fw(n)p FA(,)f(the)g(use)g(of)h(the)f (computation)g(\010)2848 4419 y Fx(A)2848 4474 y(e)2905 4452 y FA(\()p Fw(n)p FA(\))g(do)s(es)g(not)505 4559 y(exceed)h Fw(b)p FA(\()p Fw(n)p FA(\).)e(It)h(is)e(easily)h(seen)g (that)h Fw(f)34 b Fz(6)1986 4573 y Fy(wtt)2121 4559 y Fr(;)2166 4527 y Fq(0)2218 4559 y FA(iff)27 b(there)i(are)f(computable) g(func-)505 4667 y(tions)j Fw(h)p FA(\()p Fw(:;)15 b(:)p FA(\))33 b(and)d Fw(p)p FA(\()p Fw(:)p FA(\))i(suc)m(h)f(that,)h(for)f (all)e Fw(n)p FA(,)i(w)m(e)h(ha)m(v)m(e)g Fw(f)10 b FA(\()p Fw(n)p FA(\))26 b(=)g(lim)2905 4681 y Fx(s)2956 4667 y Fw(h)p FA(\()p Fw(n;)15 b(s)p FA(\))32 b(and)505 4775 y Fr(jf)p Fw(s)26 b FA(:)f Fw(h)p FA(\()p Fw(n;)15 b(s)p FA(\))26 b Fr(6)p FA(=)f Fw(h)p FA(\()p Fw(n;)15 b(s)21 b FA(+)f(1\))p Fr(gj)27 b Fz(6)e Fw(p)p FA(\()p Fw(n)p FA(\).)588 4925 y FB(Definition)35 b FA(9.5)p FB(.)47 b FA(A)30 b(degree)g Ft(a)g FA(is)f Fs(arr)-5 b(ay)34 b(nonc)-5 b(omputable)39 b FA(if)29 b(for)g(eac)m(h)i Fw(f)k Fz(6)3285 4939 y Fy(wtt)505 5033 y Fr(;)550 5000 y Fq(0)596 5033 y FA(there)21 b(is)g(a)h(function)e Fw(g)25 b FA(computable)c(in)f Ft(a)i FA(suc)m(h)f(that)h Fw(g)s FA(\()p Fw(n)p FA(\))k Fz(>)f Fw(f)10 b FA(\()p Fw(n)p FA(\))21 b(for)h(in\014nitely)505 5141 y(man)m(y)31 b Fw(n)p FA(.)f(Otherwise,)f Ft(a)h FA(is)g Fs(arr)-5 b(ay)34 b(c)-5 b(omputable)p FA(.)p eop %%Page: 41 41 41 40 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(41)588 541 y FB(Theorem)34 b FA(9.6)h(\(Do)m(wney)-8 b(,)33 b(Jo)s(c)m(kusc)m(h,)d(and)g(Stob)g([39)q(,)h(40)q(]\))p FB(.)46 b Fs(L)-5 b(et)44 b Ft(a)g Fs(b)-5 b(e)43 b(a)h(c.e.)505 649 y(de)-5 b(gr)g(e)g(e)44 b(and)f(let)f Fr(f)p Fw(F)1212 663 y Fx(n)1260 649 y Fr(g)1305 663 y Fx(n)p Fq(2)p Fg(N)1490 649 y Fs(b)-5 b(e)42 b(a)h(very)f(str)-5 b(ong)44 b(arr)-5 b(ay.)44 b(Then)f(the)f(fol)5 b(lowing)44 b(ar)-5 b(e)505 757 y(e)g(quivalent:)563 883 y FA(\(i\))42 b Fs(The)33 b(de)-5 b(gr)g(e)g(e)33 b Ft(a)g Fs(is)f(a.n.c.)h(in)f(the)h(sense)g (of)g(De\014nition)f(9.4.)538 991 y FA(\(ii\))41 b Fs(Ther)-5 b(e)31 b(is)f(a)h(c.e.)f(set)g Fw(A)h Fs(of)f(de)-5 b(gr)g(e)g(e)32 b Ft(a)e Fs(such)h(that)g Fr(8)p Fw(e)15 b Fr(9)p Fw(n)g FA([)p Fw(W)2722 1005 y Fx(e)2774 991 y Fr(\\)g Fw(F)2908 1005 y Fx(n)2980 991 y FA(=)25 b Fw(A)16 b Fr(\\)f Fw(F)3294 1005 y Fx(n)3341 991 y FA(])p Fs(.)513 1099 y FA(\(iii\))40 b Fs(The)33 b(de)-5 b(gr)g(e)g(e)33 b Ft(a)g Fs(is)f(a.n.c.)h(in)f(the) h(sense)g(of)g(De\014nition)f(9.5.)505 1224 y(Henc)-5 b(e)35 b(for)h(c.e.)f(de)-5 b(gr)g(e)g(es,)37 b(the)f(two)g (de\014nitions)h(of)f(arr)-5 b(ay)37 b(nonc)-5 b(omputability)38 b(c)-5 b(o-)505 1332 y(incide,)32 b(and)g(the)h(\014rst)f(de\014nition) h(is)e(indep)-5 b(endent)33 b(of)f(the)g(choic)-5 b(e)33 b(of)e(very)h(str)-5 b(ong)505 1440 y(arr)g(ay.)588 1593 y FA(It)34 b(is)f(w)m(ell)f(kno)m(wn)h(that)h(an)f(arbitrary)g(degree)h Ft(a)f FA(is)g(in)p 2557 1520 168 4 v 32 w(GL)2685 1607 y Fy(2)2758 1593 y FA(\(i.e.,)h(\()p Ft(a)22 b Fr([)g Ft(0)3211 1560 y Fq(0)3235 1593 y FA(\))3270 1560 y Fq(0)3324 1593 y Fw(<)505 1701 y Ft(a)556 1668 y Fq(00)599 1701 y FA(\))43 b(iff)e(for)i(eac)m(h)h(function)e Fw(f)52 b FA(computable)42 b(in)f Ft(a)29 b Fr([)f Ft(0)2476 1668 y Fq(0)2542 1701 y FA(there)43 b(is)f(a)h(function)f Fw(g)505 1809 y FA(computable)33 b(in)f Ft(a)i FA(suc)m(h)f(that)h Fw(g)s FA(\()p Fw(n)p FA(\))d Fz(>)f Fw(f)10 b FA(\()p Fw(n)p FA(\))33 b(for)g(in\014nitely)d(man)m(y)k Fw(n)p FA(.)f(F)-8 b(rom)34 b(this)505 1920 y(fact)j(it)e(immediately)e(follo) m(ws)i(that)h(if)e Ft(a)g Fr(2)p 2081 1847 V 33 w FA(GL)2209 1934 y Fy(2)2249 1920 y FA(,)i(then)f Ft(a)g FA(is)f(a.n.c.,)j(and)e (if)f Ft(a)h FA(is)505 2028 y(\001)581 1995 y Fy(0)581 2053 y(2)651 2028 y FA(and)30 b(arra)m(y)h(computable,)f(then)g Ft(a)g FA(is)f(lo)m(w)2094 2042 y Fy(2)2134 2028 y FA(.)588 2136 y(There)e(are)h(a)g(n)m(um)m(b)s(er)e(of)i(other)f(c)m (haracterizations)i(of)e(the)h(arra)m(y)g(noncomput-)505 2244 y(able)40 b(c.e.)g(degrees)g(\(see)h([39)q(,)f(40)q(]\).)g(F)-8 b(or)40 b(example,)g(the)g(a.n.c.)g(c.e.)h(degrees)f(are)505 2352 y(precisely)35 b(those)h(that)g(b)s(ound)e(c.e.)j(sets)f Fw(A)2032 2366 y Fy(1)2072 2352 y Fw(;)15 b(A)2180 2366 y Fy(2)2220 2352 y Fw(;)g(B)2329 2366 y Fy(1)2369 2352 y Fw(;)g(B)2478 2366 y Fy(2)2553 2352 y FA(suc)m(h)36 b(that)g Fw(A)3034 2366 y Fy(1)3098 2352 y Fr(\\)23 b Fw(A)3250 2366 y Fy(2)3324 2352 y FA(=)505 2460 y Fw(B)574 2474 y Fy(1)636 2460 y Fr(\\)e Fw(B)787 2474 y Fy(2)856 2460 y FA(=)29 b Fr(;)k FA(and)f(ev)m(ery)h(separating)g(set)g(for)g Fw(A)2252 2474 y Fy(1)2291 2460 y Fw(;)15 b(A)2399 2474 y Fy(2)2472 2460 y FA(is)32 b(T)-8 b(uring)31 b(incomparable)505 2568 y(with)37 b(ev)m(ery)i(separating)f(set)g(for)g Fw(B)1781 2582 y Fy(1)1820 2568 y Fw(;)15 b(B)1929 2582 y Fy(2)1969 2568 y FA(.)38 b(In)g(fact,)h(they)f(are)g(the)g(degrees)h (that)505 2676 y(b)s(ound)24 b(disjoin)m(t)h(c.e.)i(sets)g Fw(A;)15 b(B)31 b FA(that)26 b(ha)m(v)m(e)i(no)e(separating)g(set)g(of) g(degree)h Ft(0)3163 2643 y Fq(0)3187 2676 y FA(.)f(The)505 2784 y(a.n.c.)42 b(c.e.)f(degrees)g(also)f(form)g(an)g(in)m(v)-5 b(arian)m(t)40 b(class)g(for)g(the)h(p)s(erfect)f(thin)f(\005)3325 2751 y Fy(0)3325 2808 y(1)3364 2784 y FA(-)505 2892 y(classes,)32 b(whic)m(h)e(form)g(an)h(orbit)g(in)e(the)j(lattice)f(of)h(\005)2363 2859 y Fy(0)2363 2917 y(1)2402 2892 y FA(-classes,)g(in)e(the)h(same)h (w)m(a)m(y)505 3000 y(that)39 b(the)f(maximal)e(sets)i(realize)g(all)f (high)f(c.e.)j(degrees)f(and)g(are)g(an)f(in)m(v)-5 b(arian)m(t)505 3108 y(orbit)26 b(for)g(the)h(high)e(c.e.)j(degrees)f(\(see)h(Cholak,)e (Coles,)g(Do)m(wney)-8 b(,)29 b(and)d(Herrmann)505 3216 y([25)r(]\).)588 3324 y(Of)k(relev)-5 b(ance)31 b(here)f(is)g(the)g (follo)m(wing)f(result.)588 3478 y FB(Theorem)34 b FA(9.7)h(\(Ishm)m (ukhameto)m(v)c([52)q(]\))p FB(.)47 b Fs(A)22 b(c.e.)h(de)-5 b(gr)g(e)g(e)24 b(is)g(arr)-5 b(ay)25 b(c)-5 b(omputable)505 3585 y(iff)33 b(it)f(is)h(c.e.-tr)-5 b(ac)g(e)g(able.)588 3739 y FA(Using)42 b(this)f(c)m(haracterization,)i(Ishm)m(ukhameto)m(v) f(pro)m(v)m(ed)h(the)f(follo)m(wing)e(re-)505 3847 y(mark)-5 b(able)42 b(theorem.)h(A)g(degree)g Ft(m)f FA(is)f(a)i Fs(str)-5 b(ong)45 b(minimal)g(c)-5 b(over)53 b FA(of)43 b(a)g(degree)505 3955 y Ft(a)25 b Fw(<)g Ft(m)30 b FA(if)f(for)h(all)g (degrees)h Ft(d)25 b Fw(<)g Ft(m)p FA(,)30 b(w)m(e)g(ha)m(v)m(e)i Ft(d)25 b Fz(6)g Ft(a)p FA(.)588 4108 y FB(Theorem)34 b FA(9.8)h(\(Ishm)m(ukhameto)m(v)c([52)q(]\))p FB(.)47 b Fs(A)22 b(c.e.)h(de)-5 b(gr)g(e)g(e)24 b(is)g(arr)-5 b(ay)25 b(c)-5 b(omputable)505 4216 y(iff)33 b(it)f(has)i(a)f(str)-5 b(ong)34 b(minimal)g(c)-5 b(over.)588 4370 y FA(W)d(e)27 b(can)e(no)m(w)g(state)i(Kummer's)c(classi\014cation)h(of)i(the)f(c.e.) h(degrees)g(con)m(taining)505 4478 y(complex)k(c.e.)i(sets.)f(Again)f (there)h(is)e(a)i(deep)f(connection)h(with)e(traceabilit)m(y)-8 b(.)588 4631 y FB(Theorem)34 b FA(9.9)h(\(Kummer's)30 b(Gap)g(Theorem)h([70)q(]\))p FB(.)563 4757 y FA(\(i\))42 b Fs(A)32 b(c.e.)g(de)-5 b(gr)g(e)g(e)33 b(c)-5 b(ontains)34 b(a)f(c)-5 b(omplex)35 b(set)d(iff)g(it)h(is)g(arr)-5 b(ay)34 b(nonc)-5 b(omputable.)538 4864 y FA(\(ii\))41 b Fs(In)i(addition,)i(if)d Fw(A)i Fs(is)f(c.e.)f(and)i(of)f(arr)-5 b(ay)46 b(c)-5 b(omputable)44 b(de)-5 b(gr)g(e)g(e,)44 b(then)g(for)701 4972 y(every)32 b(unb)-5 b(ounde)g(d,)34 b(nonde)-5 b(cr)g(e)g(asing,)35 b(total)f(c)-5 b(omputable)34 b(function)f Fw(f)10 b Fs(,)1312 5118 y Fw(C)d FA(\()p Fw(A)1512 5106 y Fz(\026)1575 5118 y Fw(n)p FA(\))25 b Fz(6)g FA(log)17 b Fw(n)j FA(+)f Fw(f)10 b FA(\()p Fw(n)p FA(\))20 b(+)g Fw(O)s FA(\(1\))p Fw(:)p eop %%Page: 42 42 42 41 bop 505 363 a FD(42)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)513 541 y FA(\(iii\))40 b Fs(Henc)-5 b(e)24 b(the)h(c.e.)f(de)-5 b(gr)g(e)g(es)26 b(exhibit)f(the)g(fol)5 b(lowing)26 b(gap)g(phenomenon:)g(for)g(e)-5 b(ach)701 649 y(c.e.)32 b(de)-5 b(gr)g(e)g(e)33 b Ft(a)p Fs(,)f(either)700 757 y(\(a\))43 b(ther)-5 b(e)33 b(is)f(a)g(c.e.)g(set)g Fw(A)25 b Fr(2)g Ft(a)32 b Fs(such)g(that)i Fw(C)7 b FA(\()p Fw(A)2431 745 y Fz(\026)2494 757 y Fw(n)p FA(\))25 b Fz(>)g FA(2)15 b(log)i Fw(n)i Fr(\000)g Fw(O)s FA(\(1\))33 b Fs(for)863 865 y(in\014nitely)g(many)g Fw(n)p Fs(,)f(or)705 973 y(\(b\))42 b(ther)-5 b(e)41 b(ar)-5 b(e)42 b(no)f(c.e.)f(set)h Fw(A)f Fr(2)g Ft(a)g Fs(and)i Fw(")e(>)g FA(0)h Fs(such)g(that)h Fw(C)7 b FA(\()p Fw(A)3116 961 y Fz(\026)3194 973 y Fw(n)p FA(\))40 b Fz(>)863 1081 y FA(\(1)21 b(+)f Fw(")p FA(\))15 b(log)i Fw(n)j Fr(\000)g Fw(O)s FA(\(1\))33 b Fs(for)h(in\014nitely)f (many)g Fw(n)p Fs(.)588 1335 y FA(Th)m(us)d(w)m(e)h(ha)m(v)m(e)i(the)e (remark)-5 b(able)30 b(fact)h(that)h(a)f(c.e.)h(degree)g(con)m(tains)f (a)g(c.e.)h(set)505 1443 y(whose)d(initial)d(segmen)m(t)k(complexit)m (y)e(is)g(as)h(large)f(as)h(p)s(ossible)d(iff)i(it)g(has)g(a)h(strong) 505 1551 y(minimal)f(co)m(v)m(er!)588 1659 y(In)c(Theorem)g(10.30)i(w)m (e)f(will)d(see)j(that)g(the)f(degrees)h(con)m(taining)f(Kummer)f(com-) 505 1767 y(plex)33 b(c.e.)h(sets)g(are)f(the)g(same)h(as)f(those)h(con) m(taining)f(sets)g(that)h(are)g(random)e(rel-)505 1875 y(ativ)m(e)41 b(to)g(a)f(v)-5 b(ariation)39 b(of)h(Kurtz)f(randomness.) g(It)h(is)f(natural)g(to)h(ask)g(whether)505 1983 y(there)28 b(is)e(a)h(classi\014cation)f(of,)i(sa)m(y)-8 b(,)28 b(all)e(jump)g(classes)h(in)f(terms)h(of)g(initial)d(segmen)m(t)505 2091 y(complexit)m(y)-8 b(.)588 2404 y Fu(x)p Ft(10.)53 b(Other)39 b(notions)g(of)h(algorithmic)f(randomness.)45 b FA(W)-8 b(e)35 b(no)m(w)g(return)505 2512 y(to)d(our)f(consideration) f(of)i(the)f(basic)g(de\014nition)e(of)i(randomness.)g(W)-8 b(e)32 b(ha)m(v)m(e)g(seen)505 2620 y(that)c(the)f(three)g(approac)m (hes)h(\(through)e(measure)h(theory)-8 b(,)28 b(unpredictabilit)m(y)-8 b(,)24 b(and)505 2728 y(incompressibilit)m(y\))k(all)j(yield)f(the)i (same)g(notion)g(of)g(randomness.)f(But)h(consider)505 2836 y(the)23 b(c)m(haracterization)h(of)f(1-randomness)g(in)e(terms)i (of)g(martingales,)f(namely)g(that)505 2944 y(no)28 b Fs(c)-5 b(omputably)33 b(enumer)-5 b(able)35 b FA(martingale)28 b(succeeds)g(on)g(the)g(giv)m(en)g(set.)h(In)e([119)r(],)505 3052 y(Sc)m(hnorr)38 b(ga)m(v)m(e)k(this)c(c)m(haracterization,)i(then) f(analyzed)g(it.)g(He)g(argued)g(that)h(it)505 3160 y(demonstrates)29 b(a)f(clear)g Fs(failur)-5 b(e)35 b FA(of)28 b(the)g(in)m(tuition)e(b)s (ehind)f(the)j(notion)f(of)h(Martin-)505 3268 y(L\177)-45 b(of)38 b(randomness.)e(He)i(argued)g(that)g(randomness)e(should)f(b)s (e)i(concerned)h(with)505 3376 y(defeating)24 b Fs(c)-5 b(omputable)33 b FA(strategies)25 b(rather)f(than)g(computably)f(en)m (umerable)g(ones,)505 3484 y(since)38 b(the)f(latter)i(are)f(fundamen)m (tally)e(asymmetric,)h(in)g(the)h(same)g(w)m(a)m(y)h(that)f(a)505 3592 y(c.e.)h(set)e(is)f(semi-decidable)g(rather)g(than)h(decidable.)f (W)-8 b(e)38 b(can)g(mak)m(e)g(a)f(similar)505 3700 y(argumen)m(t)31 b(ab)s(out)f(Martin-L\177)-45 b(of)29 b(tests)i(b)s(eing)e(e\013ectiv)m (ely)i(n)m(ull)d(\(in)h(the)h(sense)g(that)505 3808 y(w)m(e)f(kno)m(w)e (ho)m(w)h(fast)g(they)g(con)m(v)m(erge)i(to)f(zero\),)g(but)e(not)h (e\013ectiv)m(ely)h(giv)m(en,)f(in)e(the)505 3915 y(sense)i(that)f(the) h(test)g(sets)f Fw(V)1484 3929 y Fx(n)1558 3915 y FA(themselv)m(es)h (are)f(not)h(computable,)f(but)f(rather)h(c.e.)505 4023 y(\(The)e(discussion)d(ma)m(y)j(ha)m(v)m(e)h(b)s(een)e(obscured)f(b)m (y)i(the)g(fact)g(that)g(for)g(a)g(Martin-L\177)-45 b(of)505 4131 y(test)29 b Fr(f)p Fw(V)778 4145 y Fx(n)826 4131 y Fr(g)871 4145 y Fx(n)p Fq(2)p Fx(!)1011 4131 y FA(,)g(the)f(sets)g Fw(V)1447 4145 y Fx(n)1522 4131 y FA(can)g(alw)m(a)m(ys)h(b)s(e)e(c)m (hosen)i(to)g(b)s(e)e(computable)h(\(as)g(sets)505 4239 y(of)42 b(\014nite)f(initial)e(segmen)m(ts\).)44 b(Ho)m(w)m(ev)m(er,)g (their)d Fs(me)-5 b(asur)g(es)50 b FA(are)43 b(not)f(necessarily)505 4347 y(computable.\))588 4455 y(Armed)34 b(with)e(this)h(fundamen)m (tal)g(insigh)m(t,)g(and)g(follo)m(wing)f(Sc)m(hnorr)h([119)r(],)h(w)m (e)505 4563 y(will)19 b(lo)s(ok)j(at)g(t)m(w)m(o)h(notions)e(of)h (randomness)f(that)h(re\014ne)f(the)h(notion)f(of)h(Martin-L\177)-45 b(of)505 4671 y(randomness.)23 b(Both)i(notions)e(are)h(natural,)f(one) h(b)s(eing)f(inspired)e(b)m(y)i(the)h(measure-)505 4779 y(theoretic)36 b(approac)m(h)g(and)f(the)g(other)g(b)m(y)h(the)f (martingale)g(approac)m(h.)h(The)e(\014rst)505 4887 y(notion)c(w)m(e)h (in)m(tro)s(duce)e(is)h(most)h(naturally)d(de\014ned)h(using)g(tests.) 588 5141 y FB(Definition)35 b FA(10.1)h(\(Sc)m(hnorr)30 b([119)q(]\))p FB(.)p eop %%Page: 43 43 43 42 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(43)563 541 y FA(\(i\))42 b(W)-8 b(e)32 b(sa)m(y)f(that)h(a)f (Martin-L\177)-45 b(of)30 b(test)i Fr(f)p Fw(V)2031 555 y Fx(n)2079 541 y Fr(g)2124 555 y Fx(n)p Fq(2)p Fx(!)2295 541 y FA(is)e(a)h Fs(Schnorr)k(test)40 b FA(if)29 b Fw(\026)p FA(\()p Fw(V)3215 555 y Fx(n)3263 541 y FA(\))d(=)701 649 y(2)746 616 y Fq(\000)p Fx(n)876 649 y FA(for)h(all)g Fw(n)p FA(.)h(In)f(this)f(case)j(w)m(e)f(call)g(an)m(y)g(subset)f(of) 2561 581 y Fp(T)2637 676 y Fx(n)2699 649 y Fw(V)2752 663 y Fx(n)2827 649 y FA(a)h Fs(Schnorr)k(nul)5 b(l)701 757 y(set)p FA(.)538 865 y(\(ii\))41 b(A)21 b(set)g Fw(A)g FA(is)f Fs(Schnorr)25 b(r)-5 b(andom)24 b FA(if)19 b Fw(A)36 b(=)-55 b Fr(2)2003 797 y Fp(T)2078 892 y Fx(n)2141 865 y Fw(V)2194 879 y Fx(n)2261 865 y FA(for)21 b(all)f(Sc)m(hnorr)g (tests)h Fr(f)p Fw(V)3136 879 y Fx(n)3184 865 y Fr(g)3229 879 y Fx(n)p Fq(2)p Fx(!)3369 865 y FA(.)588 1045 y(In)41 b(his)f(original)g(v)m(ersion)h(of)g(De\014nition)f(10.1,)j(Sc)m(hnorr) e(only)f(required)g(that)505 1153 y(the)27 b(n)m(um)m(b)s(ers)e Fw(\026)p FA(\()p Fw(V)1164 1167 y Fx(n)1211 1153 y FA(\))i(b)s(e)f (uniformly)d(computable,)j(whic)m(h)f(is)h(easily)f(seen)i(to)g(yield) 505 1261 y(the)32 b(same)f(notions)f(of)h(n)m(ull)e(set)j(and)e(random) g(set)i(as)f(the)g(de\014nition)d(giv)m(en)j(here.)505 1369 y(When)e(dealing)e(with)g(Sc)m(hnorr)g(randomness,)g(w)m(e)i(will) d(use)i(whic)m(hev)m(er)g(v)m(ersion)f(of)505 1477 y(the)k (de\014nition)d(is)h(most)i(con)m(v)m(enien)m(t.)588 1584 y(Our)f(next)g(notion)g(of)g(randomness)g(is)f(based)h(on)g (computable)g(martingales.)588 1764 y FB(Definition)35 b FA(10.2)h(\(Sc)m(hnorr)30 b([119)q(]\))p FB(.)563 1904 y FA(\(i\))42 b(A)26 b(martingale)g Fw(f)34 b FA(:)26 b(2)1421 1871 y Fx()25 b FA(1)p Fr(g)p Fw(:)588 3397 y FA(Sc)m(hnorr)31 b(p)s(oin)m(ted)g(out)g(that)i(the)f (rate)g(of)g(success)g(of)f(a)h(c.e.)h(martingale)e Fw(d)h FA(can)505 3505 y(b)s(e)44 b(so)g(slo)m(w)f(that)h(it)g(cannot)g(b)s(e) f(computably)g(detected.)i(Th)m(us,)e(rather)h(than)505 3613 y(w)m(orking)38 b(with)g(n)m(ull)e(sets)j(con)m(tained)g(in)e (sets)i(of)g(the)g(form)f Fw(S)5 b FA([)p Fw(d)p FA(])39 b(with)e Fw(d)j Fr(2)e FA(\006)3330 3580 y Fy(0)3330 3637 y(1)3369 3613 y FA(,)505 3721 y(he)c(w)m(ork)m(ed)h(with)d(n)m (ull)g(sets)j(con)m(tained)f(in)e(sets)j(of)f(the)g(form)g Fw(S)2768 3736 y Fx(h)2812 3721 y FA([)p Fw(d)p FA(])h(where)e(b)s(oth) 505 3829 y Fw(d)42 b FA(and)e Fw(h)i FA(are)f(computable.)g(He)h(sho)m (w)m(ed)f(that)h(these)g(n)m(ull)d(sets)i(are)h(the)f(same)505 3937 y(as)36 b(the)g(Sc)m(hnorr)e(n)m(ull)g(sets)i(from)f(De\014nition) f(10.1.)j(The)e(follo)m(wing)f(result)g(giv)m(es)505 4045 y(Sc)m(hnorr's)26 b(c)m(haracterization)i(of)e(Sc)m(hnorr)g (randomness)f(in)g(terms)h(of)h(computable)505 4153 y(martingales)33 b(and)f(the)h(\\sp)s(eed)f(of)h(success".)h(\(Note)g(that)g(it)e (implies)e(that)k(ev)m(ery)505 4261 y(computably)c(random)f(set)i(is)f (Sc)m(hnorr)f(random.\))588 4485 y FB(Theorem)34 b FA(10.5)i(\(Sc)m (hnorr)29 b([119)r(],)i(S\177)-45 b(atze)31 b(9.4,)h(9.5\))p FB(.)47 b Fr(A)25 b(\022)g FA(2)2721 4452 y Fx(!)2803 4485 y Fs(is)31 b(Schnorr)i(nul)5 b(l)505 4593 y(iff)33 b(ther)-5 b(e)33 b(ar)-5 b(e)34 b(c)-5 b(omputable)34 b(functions)f Fw(d)g Fs(and)g Fw(h)g Fs(such)g(that)h Fr(A)25 b(\022)g Fw(S)2851 4608 y Fx(h)2895 4593 y FA([)p Fw(d)p FA(])p Fs(.)588 4817 y FA(It)k(had)f(b)s(een)g(a)h(longstanding) d(op)s(en)i(problem)f(to)i(pro)m(vide)f(a)h(mac)m(hine)f(c)m(harac-)505 4925 y(terization)e(for)g(Sc)m(hnorr)f(randomness.)g(T)-8 b(erwijn)24 b([132)r(])i(had)f(made)h(some)h(progress)505 5033 y(in)j(this)f(area.)j(Do)m(wney)f(and)f(Gri\016ths)f(ga)m(v)m(e)k (the)d(follo)m(wing)f(mac)m(hine)i(c)m(haracter-)505 5141 y(ization)f(of)h(Sc)m(hnorr)e(randomness.)p eop %%Page: 45 45 45 44 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(45)588 541 y FB(Definition)35 b FA(10.6)h(\(Do)m(wney)c(and)d (Gri\016ths)g([30)q(]\))p FB(.)47 b FA(W)-8 b(e)21 b(sa)m(y)g(a)g (pre\014x-free)e(ma-)505 649 y(c)m(hine)30 b Fw(M)41 b FA(is)29 b Fs(c)-5 b(omputable)39 b FA(if)1379 809 y Fw(\026)p FA(\(dom\()p Fw(M)10 b FA(\)\))26 b(=)2077 722 y Fp(X)1966 924 y Fx(\033)r Fq(2)p Fy(dom\()p Fx(M)7 b Fy(\))2334 809 y FA(2)2379 771 y Fq(\000j)p Fx(\033)r Fq(j)505 1058 y FA(is)30 b(a)h(computable)e(real.)588 1213 y FB(Theorem)34 b FA(10.7)i(\(Do)m(wney)31 b(and)f(Gri\016ths)f ([30)q(]\))p FB(.)46 b Fs(A)c(set)g Fw(A)g Fs(is)f(Schnorr)j(r)-5 b(an-)505 1321 y(dom)40 b(iff)f Fw(K)911 1335 y Fx(M)990 1321 y FA(\()p Fw(A)1129 1309 y Fz(\026)1204 1321 y Fw(n)p FA(\))d Fz(>)g Fw(n)24 b Fr(\000)g Fw(O)s FA(\(1\))40 b Fs(for)f(al)5 b(l)39 b(c)-5 b(omputable)41 b(pr)-5 b(e\014x-fr)g(e)g(e)40 b(machines)505 1429 y Fw(M)10 b Fs(.)588 1584 y FA(The)35 b(pro)s(of)f(of)h(Theorem)g(10.7)i (\014ltered)d(through)g(a)h(Solo)m(v)-5 b(a)m(y)36 b(test)g(c)m (haracteri-)505 1692 y(zation)30 b(of)f(Sc)m(hnorr)f(randomness.)f (\(An)i(equiv)-5 b(alen)m(t)29 b(de\014nition)d(in)i(terms)g(of)h(mar-) 505 1800 y(tingales)h(is)g(giv)m(en)g(in)f(W)-8 b(ang)32 b([136)q(].\))588 1955 y FB(Definition)j FA(10.8)h(\(Do)m(wney)c(and)d (Gri\016ths)g([30)q(]\))p FB(.)47 b FA(A)24 b Fs(total)k(Solovay)h (test)k FA(is)23 b(a)505 2062 y(computable)i(collection)f(of)h(c.e.)h (op)s(en)e(sets)h Fr(f)p Fw(V)2126 2076 y Fx(i)2154 2062 y Fr(g)2199 2076 y Fx(i)p Fq(2)p Fx(!)2346 2062 y FA(suc)m(h)f(that) 2736 1994 y Fp(P)2832 2089 y Fx(i)2876 2062 y Fw(\026)p FA(\()p Fw(V)3019 2076 y Fx(i)3047 2062 y FA(\))h(is)f(\014nite)505 2170 y(and)36 b(computable.)f(A)h(set)h Fw(A)f FA(passes)g(this)e (total)j(Solo)m(v)-5 b(a)m(y)37 b(test)g(if)d Fw(A)h Fr(2)f Fw(V)3105 2184 y Fx(i)3169 2170 y FA(for)i(at)505 2278 y(most)31 b(\014nitely)e(man)m(y)h Fw(i)p FA(.)588 2433 y FB(Theorem)k FA(10.9)i(\(Do)m(wney)31 b(and)f(Gri\016ths)f([30)q (]\))p FB(.)46 b Fs(A)35 b(set)h(is)g(Schnorr)h(r)-5 b(andom)505 2541 y(iff)33 b(it)f(p)-5 b(asses)34 b(al)5 b(l)33 b(total)i(Solovay)f(tests.)588 2696 y FA(The)25 b(Ku)m(\024)-43 b(cera-Slaman)26 b(Theorem)f(4.6)i(sho)m(ws)e(that)h (all)e(1-random)h(left-c.e.)i(reals)505 2804 y(are)37 b(wtt-complete,)h(since)d(they)i(are)f(Solo)m(v)-5 b(a)m(y-complete.)39 b(\(As)d(w)m(e)h(ha)m(v)m(e)g(seen)g(in)505 2912 y(Theorem)j(4.1,)h(Ku) m(\024)-43 b(cera)40 b([66)q(])g(w)m(as)g(the)g(\014rst)e(to)j(pro)m(v) m(e)f(that)g(they)g(are)g(all)e(T)-8 b(ur-)505 3020 y(ing)36 b(complete.\))i(There)e(is)f(also)i(a)g(c)m(haracterization)h(of)f(the) f(T)-8 b(uring)35 b(degrees)i(of)505 3128 y(Sc)m(hnorr)29 b(random)g(left-c.e.)i(reals:)f(Do)m(wney)g(and)f(Gri\016ths)g([30)q(]) h(pro)m(v)m(ed)g(that)g(ev-)505 3235 y(ery)37 b(Sc)m(hnorr)e(random)h (left-c.e.)h(real)f(has)g(high)f(T)-8 b(uring)35 b(degree,)i(and)f (they)h(also)505 3343 y(pro)m(v)m(ed)i(that)g(there)f(is)g(a)g(T)-8 b(uring)37 b(incomplete)g(Sc)m(hnorr)h(random)f(left-c.e.)j(real.)505 3451 y(Later,)j(Do)m(wney)-8 b(,)43 b(Gri\016ths,)e(and)g(LaF)-8 b(orte)44 b([31)q(])e(pro)m(v)m(ed)g(that)g(ev)m(ery)h(high)d(c.e.)505 3559 y(T)-8 b(uring)32 b(degree)i(con)m(tains)f(a)g(Sc)m(hnorr)f (random)h(left-c.e.)h(real.)f(This)e(also)i(follo)m(ws)505 3667 y(from)i(Theorem)g(10.13)i(b)s(elo)m(w,)d(whic)m(h)g(is)g(due)h (to)g(Nies,)g(Stephan,)g(and)f(T)-8 b(erwijn)505 3775 y([107)r(].)588 3883 y(The)38 b(mac)m(hine)g(c)m(haracterization)i(of)f (Martin-L\177)-45 b(of)38 b(randomness)f(allo)m(ws)h(us)g(to)505 3991 y(calibrate)j(randomness)e(via)i Fz(6)1627 4005 y Fx(K)1695 3991 y FA(,)g(and)f(w)m(e)h(can)g(similarly)c(calibrate)k (the)g(com-)505 4099 y(plexit)m(y)30 b(of)g(sets)h(in)e(terms)i(of)f (their)g(Sc)m(hnorr)f(complexit)m(y)-8 b(.)588 4254 y FB(Definition)35 b FA(10.10)h(\(Do)m(wney)c(and)e(Gri\016ths)f([30)q (]\))p FB(.)46 b FA(W)-8 b(e)36 b(write)f Fw(A)e Fz(6)3086 4269 y Fi(Sch)3233 4254 y Fw(B)40 b FA(if)505 4362 y(for)30 b(eac)m(h)h(computable)f(pre\014x-free)f(mac)m(hine)h Fw(M)40 b FA(there)30 b(is)f(a)i(computable)e(pre\014x-)505 4479 y(free)i(mac)m(hine)1047 4456 y Fp(c)1036 4479 y Fw(M)40 b FA(suc)m(h)30 b(that)1286 4624 y Fw(K)1371 4643 y Ff(c)1363 4660 y Fx(M)1443 4624 y FA(\()p Fw(A)1571 4612 y Fz(\026)1635 4624 y Fw(n)p FA(\))25 b Fz(6)g Fw(K)1923 4638 y Fx(M)2002 4624 y FA(\()p Fw(B)2136 4612 y Fz(\026)2199 4624 y Fw(n)p FA(\))20 b(+)g Fw(O)s FA(\(1\))p Fw(;)505 4770 y FA(where)30 b(the)h(constan)m(t)h(dep)s(ends)c(on)i Fw(M)10 b FA(.)588 4925 y(Clearly)-8 b(,)33 b(if)g Fw(\013)e Fz(6)1172 4940 y Fy(Sc)n(h)1314 4925 y Fw(\014)38 b FA(for)c(all)e (left-c.e.)j(reals)e Fw(\013)p FA(,)h(then)f Fw(\014)39 b FA(is)33 b(Sc)m(hnorr)f(random.)505 5033 y(Virtually)22 b(nothing)i(is)f(kno)m(wn)h(ab)s(out)g Fz(6)1908 5048 y Fy(Sc)n(h)2019 5033 y FA(.)h(Do)m(wney)g(and)e(Gri\016ths)g (constructed)505 5141 y(a)31 b(\\Sc)m(hnorr)f(trivial")f(left-c.e.)j (real.)p eop %%Page: 46 46 46 45 bop 505 363 a FD(46)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FB(Theorem)34 b FA(10.11)i(\(Do)m(wney)c(and)e(Gri\016ths)f([30)q (]\))p FB(.)46 b Fs(Ther)-5 b(e)40 b(exist)f(nonc)-5 b(omput-)505 649 y(able)33 b(left-c.e.)f(r)-5 b(e)g(als)34 b Fw(\013)f Fs(such)g(that)h Fw(\013)25 b Fz(6)1867 664 y Fi(Sch)2006 649 y Fr(;)p Fs(.)588 797 y FA(Recen)m(tly)-8 b(,)29 b(Do)m(wney)-8 b(,)28 b(Gri\016ths,)e(and)h(LaF)-8 b(orte)29 b([31)q(])e(pro)m(v)m(ed)g(that)h(Sc)m(hnorr)e(triv-)505 905 y(ial)k(sets)g(are)h(quite)f(di\013eren)m(t)g(from)g Fw(K)7 b FA(-trivial)28 b(sets.)588 1053 y FB(Theorem)34 b FA(10.12)i(\(Do)m(wney)-8 b(,)32 b(Gri\016ths,)e(and)f(LaF)-8 b(orte)33 b([31)q(]\))p FB(.)563 1178 y FA(\(i\))42 b Fs(Ther)-5 b(e)33 b(exist)g(T)-7 b(uring)32 b(c)-5 b(omplete)34 b(c.e.)e(sets)h(that)h(ar)-5 b(e)34 b(Schnorr)g(trivial.)538 1286 y FA(\(ii\))41 b Fs(No)32 b(Schnorr)i(trivial)g(left-c.e.)d(r)-5 b(e)g(al)34 b(is)f(wtt-c)-5 b(omplete.)513 1394 y FA(\(iii\))40 b Fs(Ther)-5 b(e)33 b(exist)g(nonzer)-5 b(o)34 b(c.e.)e(de)-5 b(gr)g(e)g(es)34 b(c)-5 b(ontaining)34 b(no)f(Schnorr)h(trivial)f (sets.)588 1542 y Ft(10.2.)54 b(Computable)23 b(randomness.)46 b FA(It)22 b(is)f(not)h(hard)f(to)i(pro)m(v)m(e)g(that)f(there)g(is)505 1650 y(no)27 b(computable)g(en)m(umeration)g(of)g(all)f(computable)h (martingales.)f(Th)m(us,)g(as)i(with)505 1758 y(Sc)m(hnorr)37 b(randomness,)f(argumen)m(ts)i(ab)s(out)f(computable)g(randomness)g (need)g(to)505 1865 y(deal)26 b(with)f(\005)964 1832 y Fy(0)964 1890 y(2)1030 1865 y FA(b)s(eha)m(vior.)h(W)-8 b(e)28 b(ha)m(v)m(e)f(already)f(noted)h(that)g(there)f(is)g(a)g (computably)505 1973 y(random)h(set)h(that)g(is)f(not)h(Martin-L\177) -45 b(of)27 b(random.)g(W)-8 b(ang)29 b([137)r(])e(pro)m(v)m(ed)h(that) g(there)505 2081 y(is)38 b(also)g(a)h(Sc)m(hnorr)e(random)g(set)i(that) g(is)e(not)i(computably)e(random.)h(Do)m(wney)-8 b(,)505 2189 y(Gri\016ths,)23 b(and)g(LaF)-8 b(orte)26 b([31)q(],)e(and)f (indep)s(enden)m(tly)e(Nies,)i(Stephan,)g(and)g(T)-8 b(erwijn)505 2297 y([107)r(],)26 b(sho)m(w)m(ed)g(that)g(this)e(also)i (holds)e(for)h(left-c.e.)i(reals.)e(The)g(follo)m(wing)f(theorem)505 2405 y(sho)m(ws)31 b(precisely)d(ho)m(w)j(complex)f(it)g(is)f(to)i (separate)h(these)f(randomness)e(notions.)588 2553 y FB(Theorem)34 b FA(10.13)i(\(Nies,)31 b(Stephan,)f(and)f(T)-8 b(erwijn)29 b([107)r(]\))p FB(.)46 b Fs(F)-7 b(or)42 b(every)g(set)g Fw(A)p Fs(,)505 2661 y(the)33 b(fol)5 b(lowing)34 b(ar)-5 b(e)34 b(e)-5 b(quivalent.)563 2786 y FA(\(i\))42 b Fw(A)32 b Fs(is)h(high.)538 2894 y FA(\(ii\))41 b Fr(9)p Fw(B)29 b Fr(\021)921 2908 y Fi(T)1001 2894 y Fw(A)k Fs(s.t.)f Fw(B)37 b Fs(is)c(c)-5 b(omputably)35 b(r)-5 b(andom)35 b(but)d(not)h(Martin-L\177)-46 b(of)33 b(r)-5 b(andom.)513 3002 y FA(\(iii\))40 b Fr(9)p Fw(C)31 b Fr(\021)919 3016 y Fi(T)999 3002 y Fw(A)i Fs(s.t.)f Fw(C)39 b Fs(is)33 b(Schnorr)h(r)-5 b(andom)35 b(but)e(not)g(c)-5 b(omputably)34 b(r)-5 b(andom.)505 3127 y(F)e(urthermor)i(e,)38 b(if)d Fw(A)h Fs(is)f(a)h(left-c.e.)f(r)-5 b(e)g(al)36 b(then)g Fw(B)k Fs(and)d Fw(C)42 b Fs(c)-5 b(an)36 b(also)g(b)-5 b(e)36 b(chosen)g(to)505 3235 y(b)-5 b(e)33 b(left-c.e.)f(r)-5 b(e)g(als.)588 3383 y FB(Pr)n(oof.)41 b FA(W)-8 b(e)29 b(only)d(pro)m(v)m(e)j(that)f Fr(:)p FA(\(i\))f Fr(\))g(:)p FA(\(iii\))e Fr(^)i(:)p FA(\(iii\).)f(F)-8 b(or)28 b(the)g(other)f (impli-)505 3491 y(cations,)i(see)g([107)q(].)g(Let)f Fw(A)g FA(b)s(e)g(a)g(nonhigh)f(set)h(that)h(is)e(not)h(Martin-L\177) -45 b(of)28 b(random.)505 3599 y(Let)45 b Fr(f)p Fw(U)789 3613 y Fx(i)817 3599 y Fr(g)862 3613 y Fx(i)p Fq(2)p Fx(!)1028 3599 y FA(b)s(e)e(the)h(univ)m(ersal)e(Martin-L\177)-45 b(of)43 b(test.)i(W)-8 b(e)45 b(sho)m(w)e(that)i Fw(A)f FA(is)e(not)505 3707 y(Sc)m(hnorr)36 b(random,)g(and)g(hence)g(also)h (not)f(computably)g(random.)g(Let)h Fw(f)46 b FA(b)s(e)35 b(the)505 3815 y(function)h(that)h(tells)f(us)g(when)g Fw(A)h FA(is)f(co)m(v)m(ered)i(b)m(y)f(the)g Fw(U)2530 3829 y Fx(i)2558 3815 y FA(.)h(That)e(is,)g Fw(f)10 b FA(\()p Fw(i)p FA(\))37 b(is)f(the)505 3923 y(\014rst)29 b(stage)i(at)f(whic)m(h)e(an)i(initial)d(segmen)m(t)j(of)g Fw(A)f FA(en)m(ters)h Fw(U)2558 3937 y Fx(i)2587 3923 y FA(.)f(Notice)i(that)f Fw(f)39 b FA(is)28 b Fw(A)p FA(-)505 4031 y(computable.)f(Let)g Fw(g)k FA(b)s(e)26 b(a)h(computable)g(function)e(that)j(is)e(in\014nitely)e(often)j (larger)505 4138 y(than)g Fw(f)36 b FA(\(whic)m(h)26 b(exists)g(b)s(ecause)h Fw(A)g FA(is)f(not)h(high\).)f(W)-8 b(e)28 b(can)f(de\014ne)g(a)g(Sc)m(hnorr)f(test)505 4246 y Fr(f)p Fw(V)603 4260 y Fx(n)651 4246 y Fr(g)696 4260 y Fx(n)p Fq(2)p Fx(!)871 4246 y FA(b)m(y)34 b(stopping)f(the)h(en)m (umeration)g(of)h(eac)m(h)g Fw(U)2439 4260 y Fx(n)2520 4246 y FA(after)g Fw(g)s FA(\()p Fw(n)p FA(\))g(man)m(y)g(steps)505 4354 y(to)28 b(obtain)f Fw(V)946 4368 y Fx(n)993 4354 y FA(.)g(Then)f(ev)m(ery)i Fw(V)1569 4368 y Fx(n)1642 4354 y FA(is)e(\014nitely)g(presen)m(ted,)h(so)g Fr(f)p Fw(V)2668 4368 y Fx(n)2715 4354 y Fr(g)2760 4368 y Fx(n)p Fq(2)p Fx(!)2928 4354 y FA(is)f(a)i(Sc)m(hnorr)505 4462 y(test,)i(and)e Fw(A)g FA(is)f(in)g Fw(V)1223 4476 y Fx(n)1299 4462 y FA(for)h(in\014nitely)d(man)m(y)j Fw(n)p FA(,)g(whic)m(h)g(is)f(su\016cien)m(t)h(to)h(sho)m(w)f(that)505 4570 y Fw(A)i FA(is)e(not)i(Sc)m(hnorr)e(random)g(\(since)h(w)m(e)h (can)g(con)m(v)m(ert)h(the)e(Sc)m(hnorr)f(test)i Fr(f)p Fw(V)3161 4584 y Fx(n)3209 4570 y Fr(g)3254 4584 y Fx(n)p Fq(2)p Fx(!)505 4685 y FA(in)m(to)24 b(a)h(Sc)m(hnorr)e(test)i Fr(f)1313 4662 y FA(~)1299 4685 y Fw(V)1352 4699 y Fx(n)1399 4685 y Fr(g)1444 4699 y Fx(n)p Fq(2)p Fx(!)1609 4685 y FA(co)m(v)m(ering)g(just)f(as)g(m)m(uc)m(h)g(and)g(with)e(the)j (additional)505 4800 y(prop)s(ert)m(y)30 b(that)1088 4777 y(~)1074 4800 y Fw(V)1127 4814 y Fx(n)p Fy(+1)1290 4800 y Fr(\022)1399 4777 y FA(~)1385 4800 y Fw(V)1438 4814 y Fx(n)1485 4800 y FA(,)h(b)m(y)f(letting)1969 4777 y(~)1955 4800 y Fw(V)2008 4814 y Fx(n)2080 4800 y FA(=)2176 4731 y Fp(S)2252 4827 y Fx(m>n)2432 4800 y Fw(V)2485 4814 y Fx(m)2551 4800 y FA(\).)728 b Fr(a)588 4925 y FA(W)-8 b(e)35 b(\014nish)30 b(this)i(section)i(b)m(y)f(men)m(tioning)f (that)i(there)f(is)f(a)i(measure-theoretic)505 5033 y(c)m (haracterization)c(\(whic)m(h)d(could)h(b)s(e)g(turned)f(in)m(to)h(a)h (mac)m(hine)f(c)m(haracterization\))505 5141 y(of)j(computable)f (randomness.)p eop %%Page: 47 47 47 46 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(47)588 541 y FB(Definition)35 b FA(10.14)h(\(Do)m(wney)-8 b(,)33 b(Gri\016ths,)c(and)h(LaF)-8 b(orte)32 b([31)q(]\))p FB(.)46 b FA(W)-8 b(e)39 b(sa)m(y)g(that)505 649 y(a)j(Martin-L\177)-45 b(of)41 b(test)h Fr(f)p Fw(V)1355 663 y Fx(n)1403 649 y Fr(g)1448 663 y Fx(n)p Fq(2)p Fx(!)1630 649 y FA(is)e Fs(c)-5 b(omputably)45 b(gr)-5 b(ade)g(d)52 b FA(if)41 b(there)g(exists)g(a)h(com-)505 757 y(putable)35 b(map)h Fw(f)44 b FA(:)36 b(2)1242 724 y Fx()2944 3472 y Fy(T)3030 3458 y Fw(A)f FA(that)f(is)505 3566 y(computably)j(random)h(relativ)m(e)g (to)g Fw(A)g FA(\(cf.)h(Theorem)e(8.10\).)j(They)e(obtained)f(a)505 3674 y(partial)23 b(c)m(haracterization)h(in)f(terms)g(of)h(t)m(w)m(o)g (w)m(ell-kno)m(wn)f(classes)g(of)h(degrees.)g(The)505 3782 y Fs(P)-7 b(A-de)i(gr)g(e)g(es)42 b FA(are)35 b(the)g(degrees)g (of)f(complete)h(extensions)f(of)h(P)m(eano)g(Arithmetic.)505 3890 y(This)f(is)h(an)g(imp)s(ortan)m(t)g(class)g(of)h(degrees,)h(with) d(man)m(y)i(equiv)-5 b(alen)m(t)35 b(de\014nitions.)505 3998 y(F)-8 b(or)35 b(instance,)f(a)g(degree)g Ft(d)g FA(is)f(a)h(P)-8 b(A-degree)36 b(iff)c(ev)m(ery)j(computable)e (in\014nite)f(bi-)505 4106 y(nary)23 b(tree)h(has)f(a)h Ft(d)p FA(-computable)f(in\014nite)e(path.)i(\(More)i(generally)-8 b(,)23 b(w)m(e)h(sa)m(y)g(that)g Ft(d)505 4214 y FA(is)g(a)i(P)-8 b(A-degree)26 b(relativ)m(e)f(to)g Fw(A)g FA(if)f(ev)m(ery)i Fw(A)p FA(-computable)f(in\014nite)d(binary)h(tree)j(has)505 4322 y(a)f Ft(d)p FA(-computable)f(in\014nite)d(path.\))k(The)e(P)-8 b(A-degrees)26 b(are)e(also)g(those)h(that)f(con)m(tain)505 4430 y Fr(f)p FA(0)p Fw(;)15 b FA(1)p Fr(g)p FA(-v)-5 b(alued)33 b(total)g(functions)d Fw(f)40 b FA(that)32 b(are)g Fs(\014xe)-5 b(d-p)g(oint-fr)g(e)g(e)p FA(,)34 b(in)c(the)i(sense)f(that)505 4538 y Fr(8)p Fw(e)15 b FA([\010)704 4552 y Fx(e)741 4538 y FA(\()p Fw(e)p FA(\))26 b Fr(6)p FA(=)f Fw(f)10 b FA(\()p Fw(e)p FA(\)].)25 b(The)f(class)f(of) i(P)-8 b(A-degrees)25 b(is)e(strictly)g(con)m(tained)h(in)f(the)h (class)505 4646 y(of)i Fs(diagonal)5 b(ly)30 b(nonc)-5 b(omputable)30 b(\(DNC\))e(de)-5 b(gr)g(e)g(es)p FA(,)27 b(whic)m(h)d(are)i(those)g(that)g(con)m(tain)505 4754 y(\(not)31 b(necessarily)f Fr(f)p FA(0)p Fw(;)15 b FA(1)p Fr(g)p FA(-v)-5 b(alued\))32 b(\014xed-p)s(oin)m(t-free)d(functions.) 588 4908 y FB(Theorem)34 b FA(10.16)i(\(Hirsc)m(hfeldt,)30 b(Nies,)g(and)g(Stephan)f([50)q(]\))p FB(.)563 5033 y FA(\(i\))42 b Fs(If)25 b(a)g FA(\001)936 5000 y Fy(0)936 5057 y(2)1000 5033 y Fs(set)g(do)-5 b(es)27 b(not)e(have)h(DNC)e(de)-5 b(gr)g(e)g(e)26 b(then)g(it)f(is)g(a)g(b)-5 b(asis)26 b(for)g(c)-5 b(omputable)701 5141 y(r)g(andomness.)p eop %%Page: 48 48 48 47 bop 505 363 a FD(48)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)538 541 y FA(\(ii\))41 b Fs(No)32 b(set)h(of)g(P)-7 b(A-de)i(gr)g(e)g(e)32 b(is)h(a)g(b)-5 b(asis)33 b(for)g(c)-5 b(omputable)35 b(r)-5 b(andomness.)588 705 y FA(The)34 b(pro)s(of)f(of)h(the)g(second) f(part)h(of)g(Theorem)f(10.16)j(uses)d(a)h(lemma)g(of)g(inde-)505 813 y(p)s(enden)m(t)d(in)m(terest:)h(If)f Fw(A)h FA(has)f(P)-8 b(A-degree)34 b(relativ)m(e)d(to)i Fw(B)j FA(and)31 b Fw(X)39 b FA(is)30 b(computably)505 921 y(random)g(relativ)m(e)g(to)i Fw(A)p FA(,)e(then)g Fw(X)38 b FA(is)30 b(1-random)g(relativ)m(e)g(to)i Fw(B)5 b FA(.)588 1029 y(Let)46 b Fw(A)f FA(b)s(e)f(an)h Fw(n)p FA(-c.e.)h(set.)g(If)e Fw(A)h FA(is)f(T)-8 b(uring)43 b(incomplete)h(then)h Fw(A)g FA(do)s(es)f(not)505 1137 y(ha)m(v)m(e)33 b(diagonally)d(noncomputable)g(degree,)i(b)m(y)f(Jo)s (c)m(kusc)m(h,)h(Lerman,)f(Soare)g(and)505 1245 y(Solo)m(v)-5 b(a)m(y)33 b([55)q(])f(\(whic)m(h)f(extends)g(Arslano)m(v's)h (Completeness)f(Criterion\).)f(So)i Fw(A)f FA(is)505 1353 y(a)40 b(basis)e(for)g(computable)h(randomness.)f(On)g(the)h (other)g(hand,)f(if)g Fw(A)h FA(is)f(T)-8 b(uring)505 1460 y(complete)31 b(then)f Fw(A)g FA(has)g(P)-8 b(A-degree,)32 b(and)d(hence)h(is)f(not)i(a)f(basis)f(for)h(computable)505 1568 y(randomness.)g(Th)m(us)f(w)m(e)i(ha)m(v)m(e)h(the)e(follo)m(wing) f(result.)588 1732 y FB(Cor)n(ollar)-6 b(y)35 b FA(10.17)h(\(Hirsc)m (hfeldt,)30 b(Nies,)g(and)g(Stephan)f([50)q(]\))p FB(.)47 b Fs(A)n(n)25 b Fw(n)p Fs(-c.e.)g(set)505 1840 y(is)33 b(a)g(b)-5 b(asis)33 b(for)h(c)-5 b(omputable)34 b(r)-5 b(andomness)35 b(iff)d(it)h(is)g(T)-7 b(uring)32 b(inc)-5 b(omplete.)588 2003 y FA(It)39 b(w)m(ould)e(b)s(e)h(in)m(teresting)f (to)i(in)m(v)m(estigate)h(bases)e(for)g(other)h(notions)e(suc)m(h)h(as) 505 2111 y(Sc)m(hnorr)30 b(and)g(Kurtz)f(randomness.)588 2236 y Ft(10.3.)54 b(Kurtz)31 b(randomness.)45 b FA(In)27 b([71)q(],)h(Kurtz)f(in)m(tro)s(duced)e(a)j(new)f(notion)f(of)505 2344 y(randomness)37 b(whic)m(h)g(lo)s(oks)h(at)h(the)f(idea)g(from)g (another)g(p)s(ersp)s(ectiv)m(e.)g(Namely)-8 b(,)505 2452 y(instead)26 b(of)h(thinking)e(of)i(a)g(set)g(as)g(random)f(if)g (it)g Fs(avoids)36 b FA(all)25 b(e\013ectiv)m(ely)j(giv)m(en)f(n)m(ull) 505 2560 y(sets,)34 b(Kurtz)e(suggested)h(that)g(a)g(set)g(should)e(b)s (e)h(considered)f(random)h(if)g(it)g Fs(ob)-5 b(eys)505 2668 y FA(ev)m(ery)32 b(e\013ectiv)m(ely)f(giv)m(en)f(test)h(of)g (measure)f(1.)588 2831 y FB(Definition)35 b FA(10.18)h(\(Kurtz)31 b([71)q(]\))p FB(.)104 b FA(\(i\))41 b(A)i Fs(Kurtz)h(\(p)-5 b(ositive\))45 b(test)52 b FA(is)41 b(a)i(c.e.)701 2939 y(op)s(en)29 b(set)i Fw(U)41 b FA(suc)m(h)30 b(that)h Fw(\026)p FA(\()p Fw(U)10 b FA(\))25 b(=)g(1.)538 3047 y(\(ii\))41 b(A)32 b(set)h(is)e(called)h Fs(Kurtz)j(r)-5 b(andom)42 b FA(\(or)32 b Fs(we)-5 b(akly)36 b FA(1)p Fs(-r)-5 b(andom)7 b FA(\))35 b(if)c Fw(A)e Fr(2)f Fw(U)42 b FA(for)33 b(all)701 3155 y(Kurtz)d(tests)h Fw(U)10 b FA(.)588 3318 y(Kurtz)34 b(originally)e(called)i(this)f(notion)h Fs(we)-5 b(ak)37 b(r)-5 b(andomness)7 b FA(,)38 b(and)c(it)g(is)g(a)h (w)m(eak)505 3426 y(notion)e(in)f(that,)i(as)f(sho)m(wn)g(b)m(y)g(W)-8 b(ang)34 b([136)r(],)f(it)g(is)f(not)h Fs(sto)-5 b(chastic)40 b FA(in)32 b(the)i(sense)505 3534 y(of)45 b(Ch)m(urc)m(h.)937 3501 y Fy(5)1020 3534 y FA(It)f(is)g(nev)m(ertheless)g(a)h(v)m(ery)f (in)m(teresting)g(concept,)h(esp)s(ecially)e(in)505 3642 y(its)c(relativized)e(form.)i(As)g(w)m(e)g(will)d(see)j(in)f(Section)g (12,)i(Kurtz)f(2-randomness,)505 3750 y(whic)m(h)e(means)h(b)s(eing)f (in)f(ev)m(ery)j(\006)1739 3717 y Fy(0)1739 3774 y(2)1816 3750 y FA(op)s(en)f(set)g(of)h(measure)f(1,)g(is)f(equiv)-5 b(alen)m(t)38 b(to)505 3858 y(passing)27 b(ev)m(ery)i(\\generalized")f (Martin-L\177)-45 b(of)27 b(test)i Fr(f)p Fw(U)2363 3872 y Fx(n)2410 3858 y Fr(g)2455 3872 y Fx(n)p Fq(2)p Fx(!)2596 3858 y FA(,)f(where)f(w)m(e)i(still)c(ha)m(v)m(e)505 3966 y Fw(\026)p FA(\()p Fw(U)657 3980 y Fx(n)705 3966 y FA(\))g Fr(!)g FA(0,)31 b(but)f(there)g(ma)m(y)h(b)s(e)f(no)g (decreasing)g(computable)g(upp)s(er)e(b)s(ound)h(on)505 4074 y(the)i(measures.)588 4182 y(Most)36 b(of)e(the)g(de\014nitions)e (of)j(tests)g(so)f(far)g(ha)m(v)m(e)h(b)s(een)f(negativ)m(e.)i(There)d (is)h(an)505 4290 y(equiv)-5 b(alen)m(t)30 b(form)m(ulation)f(of)i (Kurtz)f(randomness)f(in)g(terms)h(of)h(a)g(negativ)m(e)g(test.)588 4453 y FB(Definition)k FA(10.19)h(\(W)-8 b(ang)32 b([136)r(]\))p FB(.)46 b FA(A)26 b Fs(Kurtz)j(nul)5 b(l)28 b(test)35 b FA(is)25 b(a)h(sequence)g(of)g(c.e.)505 4561 y(op)s(en)k(sets)h Fr(f)p Fw(V)1001 4575 y Fx(n)1048 4561 y Fr(g)1093 4575 y Fx(n)p Fq(2)p Fx(!)1264 4561 y FA(suc)m(h)f(that)563 4688 y(\(i\))42 b Fw(\026)p FA(\()p Fw(V)844 4702 y Fx(n)891 4688 y FA(\))25 b Fz(6)g FA(2)1092 4655 y Fq(\000)p Fx(n)1225 4688 y FA(and)p 505 4776 499 4 v 588 4835 a Fn(5)623 4867 y Fv(F)-6 b(or)31 b(more)g(on)g(sto)r(c)n(hasticit)n(y)-6 b(,)32 b(see)g(Am)n(b)r(os-Spies)d(and)i(Ku)n(\024)-36 b(cera)31 b([1)q(].)h(That)f(a)h(notion)f(is)h(not)505 4958 y(sto)r(c)n(hastic)27 b(migh)n(t)e(ev)n(en)g(disqualify)h(it)g (from)g(b)r(eing)g(called)h(a)f Fe(r)l(andomness)33 b Fv(notion.)27 b(Kurtz)e(ran-)505 5050 y(domness)18 b(is)g(called)h(so)g (mainly)e(b)r(ecause)i(of)g(the)e(analogy)i(with)g(other)f (de\014nitions)g(of)h(randomness,)505 5141 y(but)25 b(w)n(e)h(note)g (here)g(that)f(it)h(is)g(in)g(fact)g(more)f(lik)n(e)h(a)g(notion)g(of)g (genericit)n(y)-6 b(.)p eop %%Page: 49 49 49 48 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(49)538 541 y FA(\(ii\))41 b(there)f(is)f(a)h(computable)g (function)e Fw(f)51 b FA(:)41 b Fo(N)54 b Fr(!)41 b FA(\(2)2475 508 y Fx()g FA(1)22 b Fr(\000)f FA(2)1296 1379 y Fq(\000)p Fx(n)1399 1412 y FA(,)33 b(then)f(let)h Fw(V)1853 1426 y Fx(n)1929 1412 y FA(=)p 2029 1339 123 4 v 29 w Fw(W)2115 1426 y Fx(s)2151 1412 y FA(.)g(Note)h(that)f Fw(V)2683 1426 y Fx(n)2763 1412 y FA(is)f(of)g(the)h(correct)505 1520 y(form)26 b(to)h(b)s(e)f(able)g(to)h(de\014ne)f(a)g(function)f Fw(f)36 b FA(as)26 b(in)f(De\014nition)g(10.19,)k(and)3061 1451 y Fp(T)3137 1546 y Fx(n)3199 1520 y Fw(V)3252 1534 y Fx(n)3324 1520 y FA(=)p 505 1562 99 4 v 505 1635 a Fw(W)13 b FA(.)588 1743 y(F)-8 b(or)25 b(the)f(con)m(v)m(erse,)i(giv)m (en)e(a)g(Kurtz)f(n)m(ull)f(test)j Fr(f)p Fw(V)2274 1757 y Fx(n)2321 1743 y Fr(g)2366 1757 y Fx(n)p Fq(2)p Fx(!)2507 1743 y FA(,)f(let)g Fw(W)38 b FA(=)2901 1675 y Fp(S)2976 1770 y Fx(n)p 3038 1670 100 4 v 3038 1743 a Fw(V)3091 1757 y Fx(n)3138 1743 y FA(.)25 b(Then)505 1858 y Fw(W)43 b FA(is)30 b(a)g(c.e.)i(op)s(en)e(set)h(of)f(measure)g(1,)h(and)p 2060 1785 99 4 v 30 w Fw(W)38 b FA(=)2280 1790 y Fp(T)2356 1885 y Fx(n)2418 1858 y Fw(V)2471 1872 y Fx(n)2518 1858 y FA(.)796 b Fr(a)588 1984 y FA(There)22 b(is)f(a)h(martingale)f (de\014nition)f(of)i(Kurtz)f(randomness)g(\(cf.)h(the)g(martingale)505 2092 y(c)m(haracterization)32 b(of)f(Sc)m(hnorr)e(randomness)g(giv)m (en)i(b)m(y)f(Theorem)g(10.5\):)588 2249 y FB(Theorem)k FA(10.21)i(\(W)-8 b(ang)32 b([136)r(]\))p FB(.)46 b Fs(A)36 b(set)h Fw(A)g Fs(is)g(not)g(Kurtz)g(r)-5 b(andom)39 b(iff)e(ther)-5 b(e)505 2357 y(exist)47 b(a)f(c)-5 b(omputable)48 b(martingale)f Fw(d)g Fs(and)g(a)f(nonde)-5 b(cr)g(e)g(asing)49 b(unb)-5 b(ounde)g(d)47 b(c)-5 b(om-)505 2465 y(putable)34 b(function)f Fw(h)f Fs(such)h(that)h Fw(d)p FA(\()p Fw(A)1828 2453 y Fz(\026)1891 2465 y Fw(n)p FA(\))25 b Fw(>)g(h)p FA(\()p Fw(n)p FA(\))34 b Fs(for)f(al)5 b(l)33 b Fw(n)p Fs(.)588 2623 y FA(Because)41 b(of)e(this)f(result)f(w)m(e)j(easily)e (see)h(that)h(Sc)m(hnorr)e(randomness)f(implies)505 2731 y(Kurtz)30 b(randomness.)g(No)h(Kurtz)f(random)f(set)i(can)g(b)s(e)e(a) i(c.e.)h(set.)f(In)f(fact:)588 2888 y FB(Theorem)k FA(10.22)i(\(Jo)s(c) m(kusc)m(h,)31 b(see)g(Kurtz)f([71)q(]\))p FB(.)47 b Fs(If)25 b Fw(A)g Fs(is)g(Kurtz)g(r)-5 b(andom)28 b(then)505 3000 y(it)k(is)g(bi-immune;)f(that)h(is,)g(neither)g Fw(A)g Fs(nor)p 2057 2927 69 4 v 32 w Fw(A)g Fs(c)-5 b(ontains)33 b(an)f(in\014nite)g(c)-5 b(omputable)505 3108 y(subset.)45 b(Henc)-5 b(e,)45 b(by)f(Jo)-5 b(ckusch)46 b FA([53)q(])p Fs(,)f(ther)-5 b(e)46 b(ar)-5 b(e)46 b FA(2)2316 3075 y Fq(@)2359 3084 y Fn(0)2443 3108 y Fs(de)-5 b(gr)g(e)g(es)46 b(that)h(c)-5 b(ontain)46 b(no)505 3215 y(Kurtz)33 b(r)-5 b(andom)35 b(sets.)588 3373 y FA(While)23 b(no)g(c.e.)h(set)g(can)g(b)s(e)f(Kurtz)g(random,)g(as)g(with)f (Martin-L\177)-45 b(of)23 b(randomness,)505 3481 y(the)35 b(same)f(is)f(not)h(true)g(for)f(left-c.e.)j(reals.)d(Kurtz)h([71)q(,)g (Corollary)e(2.3a])k(pro)m(v)m(ed)505 3589 y(that)25 b(ev)m(ery)g(nonzero)g(c.e.)g(degree)g(con)m(tains)g(a)f(Kurtz)g (random)f(set.)i(The)f(follo)m(wing)505 3697 y(impro)m(v)m(es)30 b(this)g(result)f(to)i(left-c.e.)h(reals.)588 3854 y FB(Theorem)i FA(10.23)i(\(Do)m(wney)-8 b(,)32 b(Gri\016ths,)e(and)f (Reid)h([32)q(]\))p FB(.)46 b Fs(Each)26 b(nonzer)-5 b(o)27 b(c.e.)505 3962 y(de)-5 b(gr)g(e)g(e)34 b(c)-5 b(ontains)34 b(a)f(Kurtz)g(r)-5 b(andom)35 b(left-c.e.)d(r)-5 b(e)g(al.)588 4120 y FA(No)47 b(c)m(haracterization)g(of)g(the)f (degrees)h(con)m(taining)e(Kurtz)h(random)f(sets)h(is)505 4228 y(kno)m(wn.)d(Nies)g(and)f(Y)-8 b(u)43 b(\(unpublished\))38 b(ha)m(v)m(e)44 b(sho)m(wn)f(that)g(the)g(conclusion)e(of)505 4336 y(the)29 b(previous)f(theorem)h(can)g(b)s(e)f(strengthened:)h(Eac) m(h)g(nonzero)g(c.e.)h(degree)g(con-)505 4444 y(tains)k(a)h(w)m(eakly)f (1-generic)h(left-c.e.)h(real.)e(Here)h Fw(A)f FA(is)f(w)m(eakly)i (1-generic)g(if)e Fw(A)h FA(is)505 4552 y(in)29 b(eac)m(h)j(dense)e (c.e.)i(op)s(en)d(set.)588 4659 y(Do)m(wney)-8 b(,)46 b(Gri\016ths,)c(and)h(Reid)g([32)q(])h(ga)m(v)m(e)i(a)e(mac)m(hine)f(c) m(haracterization)i(of)505 4767 y(Kurtz)30 b(randomness)g(in)f(the)h (st)m(yle)h(of)f(Theorem)g(10.7.)588 4925 y FB(Definition)35 b FA(10.24)h(\(Do)m(wney)-8 b(,)33 b(Gri\016ths,)c(and)h(Reid)f([32)q (]\))p FB(.)46 b FA(W)-8 b(e)25 b(sa)m(y)f(a)f(pre\014x-)505 5033 y(free)36 b(mac)m(hine)f Fw(M)46 b FA(is)34 b Fs(c)-5 b(omputably)40 b(layer)-5 b(e)g(d)46 b FA(if)35 b(there)g(is)g(a)h (computable)f(function)505 5141 y Fw(f)g FA(:)25 b Fw(!)k Fr(!)c FA(\(2)917 5108 y Fx()f Fw(n)22 b Fr(\000)g Fw(O)s FA(\(1\))36 b Fs(for)g(e)-5 b(ach)35 b(c)-5 b(omputably)38 b(layer)-5 b(e)g(d)37 b(machine)505 1526 y Fw(M)10 b Fs(.)588 1694 y FA(In)m(terestingly)-8 b(,)42 b(there)h(is)e(y)m(et)i(another)g(mac)m(hine)e(c)m (haracterization)j(of)e(Kurtz)505 1802 y(randomness,)h(this)g(one)h(in) e(terms)i(of)g(computable)f(pre\014x-free)g(mac)m(hines)h(\(cf.)505 1910 y(Theorem)31 b(10.7\).)588 2078 y FB(Theorem)j FA(10.26)i(\(Do)m (wney)-8 b(,)58 b(Gri\016ths,)c(and)30 b(Reid)f([32)r(]\))p FB(.)46 b Fs(A)i(set)h Fw(A)g Fs(is)56 b FA(not)505 2186 y Fs(Kurtz)44 b(r)-5 b(andom)46 b(iff)d(ther)-5 b(e)44 b(ar)-5 b(e)45 b(a)e(c)-5 b(omputable)45 b(pr)-5 b(e\014x-fr)g(e)g(e)45 b(machine)f Fw(M)54 b Fs(and)44 b(a)505 2294 y(c)-5 b(omputable)35 b(function)d Fw(f)j FA(:)26 b Fo(N)37 b Fr(!)26 b Fo(N)45 b Fs(such)33 b(that)1352 2451 y Fr(8)p Fw(d)15 b FA([)p Fw(K)1567 2465 y Fx(M)1646 2451 y FA(\()p Fw(A)1775 2439 y Fz(\026)1838 2451 y Fw(f)10 b FA(\()p Fw(d)p FA(\)\))26 b Fw(<)f(f)10 b FA(\()p Fw(d)p FA(\))21 b Fr(\000)e Fw(d)p FA(])p Fw(:)588 2619 y FA(It)41 b(is)f(also)h(p)s(ossible)e(to)j(come)f (up)f(with)g(suitable)f(Solo)m(v)-5 b(a)m(y)42 b(t)m(yp)s(e)f(c)m (haracteri-)505 2727 y(zations)36 b(of)g(Kurtz)f(randomness,)f(as)i(p)s (er)e(W)-8 b(ang)37 b([136)q(])f(and)f(Do)m(wney)-8 b(,)37 b(Gri\016ths,)505 2835 y(and)27 b(Reid)f([32)q(].)h(Using)f(suc)m(h)h (a)g(c)m(haracterization,)i(Do)m(wney)-8 b(,)28 b(Gri\016ths,)e(and)h (Reid)505 2943 y([32)r(])g(pro)m(vided)f(a)i(c)m(haracterization)h(of)f (Sc)m(hnorr)e(randomness)h(in)f(terms)h(of)h(Kurtz)505 3051 y(randomness.)588 3159 y(Kurtz)42 b(p)s(ositiv)m(e)g(tests)h(can)g (also)f(b)s(e)g(used)g(to)h(de\014ne)f(1-randomness,)g(if)f(the)505 3267 y(sp)s(eed)30 b(of)g(en)m(umeration)g(is)g(also)g(tak)m(en)i(in)m (to)e(accoun)m(t.)588 3435 y FB(Theorem)k FA(10.27)i(\(Da)m(vie)c([28)q (]\))p FB(.)46 b Fs(A)37 b(set)g Fw(A)h Fs(is)f(1-r)-5 b(andom)40 b(iff)c(ther)-5 b(e)38 b(is)g(a)f(c)-5 b(on-)505 3543 y(stant)36 b Fw(c)g Fs(such)f(that)h(for)g(e)-5 b(ach)35 b Fw(p)p Fs(,)g(if)g(the)g Fw(p)p Fs(-th)g(c)-5 b(omputable)37 b(se)-5 b(quenc)g(e)34 b(of)i(intervals)505 3651 y Fw(I)545 3665 y Fy(1)585 3651 y Fw(;)15 b(I)665 3665 y Fy(2)705 3651 y Fw(;)g(:)g(:)g(:)52 b Fs(is)36 b(such)g(that)i Fw(\026)p FA(\()1493 3583 y Fp(S)1568 3678 y Fx(j)1620 3651 y Fw(I)1660 3665 y Fx(j)1697 3651 y FA(\))31 b(=)h(1)p Fs(,)k(then)g(ther)-5 b(e)37 b(is)f(an)h Fw(n)e Fs(with)i Fw(A)32 b Fr(2)f Fw(I)3168 3665 y Fx(n)3251 3651 y Fs(and)505 3782 y Fw(\026)p FA(\()595 3714 y Fp(S)671 3809 y Fx(j)t Fk(6)p Fx(n)821 3782 y Fw(I)861 3796 y Fx(j)897 3782 y FA(\))26 b Fw(<)f FA(1)20 b Fr(\000)g FA(2)1255 3749 y Fq(\000j)p Fx(p)p Fq(j\000)p Fx(c)1475 3782 y Fs(.)588 3954 y Ft(10.4.)54 b(Kolmogoro)m(v-Lo)m(v)m(eland)j (randomness.)45 b FA(The)j(computable)g(b)s(et-)505 4062 y(ting)36 b(strategies)h(\(martingales\))f(used)g(to)h(de\014ne)e (computable)h(randomness)f(are)505 4169 y(monotonic,)i(in)d(the)i (sense)g(that)g(they)g(b)s(et)g(on)g(the)g(bit)f(p)s(ositions)f(in)g (their)h(nat-)505 4277 y(ural)c(order.)g(Dropping)g(this)f(monotonicit) m(y)i(condition)e(yields)g(a)i(more)g(p)s(o)m(w)m(erful)505 4385 y(notion)e(of)h(b)s(etting)e(strategy)-8 b(.)588 4493 y(W)g(e)27 b(giv)m(e)f(an)f(informal)e(v)m(ersion)i(of)g(the)h (de\014nition)d(in)h(Muc)m(hnik,)g(Semeno)m(v,)i(and)505 4601 y(Usp)s(ensky)31 b([99)q(].)h(See)g(also)f(Merkle,)h(Miller,)e (Nies,)h(Reimann,)g(and)f(Stephan)h([91)q(])505 4709 y(for)38 b(a)g(more)g(formal)f(de\014nition.)f(A)i(nonmonotonic)f(b)s (etting)g(strategy)i(b)s(eha)m(v)m(es)505 4817 y(as)34 b(follo)m(ws.)g(Giv)m(en)f(a)h(set)h Fw(A)p FA(,)f(at)g(stage)h Fw(s)f FA(supp)s(ose)e(the)i(previously)d(scanned)i(bit)505 4925 y(p)s(ositions)h(are)h Fw(n)1104 4939 y Fy(0)1143 4925 y Fw(;)15 b(:)g(:)g(:)32 b(;)15 b(n)1415 4939 y Fx(s)p Fq(\000)p Fy(1)1577 4925 y FA(and)35 b(the)g(corresp)s(onding)e (v)-5 b(alues)35 b(are)g Fw(r)2982 4939 y Fx(i)3044 4925 y FA(=)e Fw(A)p FA(\()p Fw(n)3306 4939 y Fx(i)3334 4925 y FA(\).)505 5033 y(F)-8 b(rom)32 b Fw(n)800 5047 y Fy(0)839 5033 y Fw(;)15 b(:)g(:)g(:)32 b(;)15 b(n)1111 5047 y Fx(s)p Fq(\000)p Fy(1)1269 5033 y FA(and)31 b Fw(r)1488 5047 y Fy(0)1527 5033 y Fw(;)15 b(:)g(:)g(:)32 b(;)15 b(r)1785 5047 y Fx(s)p Fq(\000)p Fy(1)1912 5033 y FA(,)32 b(the)f(strategy)i(determines)d(a)i(new)f(p)s(osi-)505 5141 y(tion)f Fw(n)747 5155 y Fx(s)809 5141 y Fr(6)p FA(=)25 b Fw(n)960 5155 y Fy(0)999 5141 y Fw(;)15 b(:)g(:)g(:)31 b(;)15 b(n)1270 5155 y Fx(s)p Fq(\000)p Fy(1)1397 5141 y FA(,)31 b(or)f(ma)m(y)h(c)m(ho)s(ose)g(to)g(b)s(e)f(unde\014ned.)e (If)i(de\014ned,)f(it)h(also)p eop %%Page: 51 51 51 50 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(51)505 541 y FA(c)m(ho)s(oses)32 b(an)f Fw(i)c(<)f FA(2)32 b(and)e(mak)m(es)i(a)g(b)s(et)f Fw(v)f Fr(2)c Fo(Q)45 b FA(with)30 b(0)d Fz(6)f Fw(v)k Fz(6)c Fw(V)52 b FA(on)31 b Fw(A)p FA(\()p Fw(n)3077 555 y Fx(s)3114 541 y FA(\))g(b)s(eing)505 649 y(equal)37 b(to)h Fw(i)p FA(,)g(where)f Fw(V)58 b FA(is)36 b(the)i(curren)m(t)f(capital)g(\(if)f Fw(s)h FA(=)f(0,)i(then)f Fw(V)58 b(>)36 b FA(0)i(is)e(the)505 757 y(initial)g(capital\).)j(If)f(the)g(b)s(et)h(turns)e(out)h(righ)m (t,)h(then)f Fw(v)k FA(is)37 b(added)h(to)h(the)g(capi-)505 865 y(tal;)e(otherwise)e(it)g(is)g(subtracted.)i(A)f(set)g Fw(A)g FA(is)f Fs(Kolmo)-5 b(gor)g(ov-L)g(oveland)42 b(r)-5 b(andom)505 973 y FA(\()p Fs(KL-r)g(andom)7 b FA(\))38 b(if)c(no)g(computable)h(nonmonotonic)f(b)s(etting)g(strategy) i(succeeds)505 1081 y(on)30 b Fw(A)p FA(,)g(in)f(the)h(sense)f(that)i (the)f(limsup)c(of)k(the)g(capital)g(it)f(obtains)g(b)m(y)h(b)s(etting) f(on)505 1189 y Fw(A)g FA(is)f(in\014nit)m(y)-8 b(.)26 b(In)i([99)q(])h(suc)m(h)f(sets)h(are)g(called)f Fs(unpr)-5 b(e)g(dictable)p FA(,)30 b(and)e(they)h(ha)m(v)m(e)h(also)505 1297 y(b)s(een)e(called)f Fs(nonmonotonic)-5 b(al)5 b(ly)34 b(r)-5 b(andom)p FA(.)31 b(Clearly)-8 b(,)27 b(KL-random)h(sets)g(are)h (com-)505 1405 y(putably)37 b(random,)g(since)g(computable)g (martingales)g(are)h(a)g(particular)e(kind)g(of)505 1513 y(nonmonotonic)25 b(b)s(etting)g(strategy)-8 b(.)27 b(In)d(fact,)j(if)d Fw(A)h FA(is)f(KL-random)g(then)h(no)g Fs(p)-5 b(artial)505 1621 y FA(computable)35 b(martingale)g(succeeds)h(on)f Fw(A)p FA(.)h(Ho)m(w)m(ev)m(er,)i(b)m(y)d(results)f(in)g(Muc)m(hnik,) 505 1729 y(Semeno)m(v,)h(and)f(Usp)s(ensky)f([99)q(],)h(the)g(con)m(v)m (erse)i(is)d(not)i(true;)f(that)g(is,)g(there)g(are)505 1837 y(non-KL-random)21 b(sets)i(on)f(whic)m(h)e(no)i(partial)f (computable)h(martingale)f(succeeds.)588 1944 y(Muc)m(hnik,)27 b(Semeno)m(v,)g(and)g(Usp)s(ensky)f([99)q(])h(sho)m(w)m(ed)g(that)h(ev) m(ery)g(1-random)f(set)505 2052 y(is)j(KL-random.)f(Whether)i(the)g (con)m(v)m(erse)g(holds)e(is)h(a)g(ma)5 b(jor)31 b(op)s(en)f(problem.) 588 2238 y FB(Question)j FA(10.28)p FB(.)48 b FA(Is)30 b(ev)m(ery)h(KL-random)e(set)i(1-random?)588 2424 y(While)25 b(w)m(e)i(allo)m(w)e(partial)g(b)s(etting)h(strategies,)h(w)m(e)f (could)f(as)i(w)m(ell)e(require)g(them)505 2532 y(to)32 b(b)s(e)e(total:)i(Merkle)f(\(see)h([91)q(]\))g(pro)m(v)m(ed)f(that)h (for)f(eac)m(h)h(nonmonotonic)e(b)s(etting)505 2640 y(strategy)c Fw(M)35 b FA(there)25 b(exist)f(total)h(\(ev)m(en)h(primitiv)m(e)c (recursiv)m(e\))i(nonmonotonic)g(b)s(et-)505 2747 y(ting)34 b(strategies)h Fw(L)1167 2761 y Fy(0)1207 2747 y Fw(;)15 b(L)1309 2761 y Fy(1)1382 2747 y FA(suc)m(h)34 b(that,)h(if)e Fw(M)45 b FA(succeeds)34 b(on)g Fw(A)p FA(,)h(then)f(one)g(of)h Fw(L)3214 2761 y Fy(0)3253 2747 y Fw(;)15 b(L)3355 2761 y Fy(1)505 2855 y FA(succeeds)33 b(on)f Fw(A)p FA(.)h(Th)m(us)e(if)g (the)h(answ)m(er)g(to)h(Question)e(10.28)k(is)c(a\016rmativ)m(e,)i (then)505 2963 y(one)42 b(migh)m(t)f(argue)g(that)h(Sc)m(hnorr's)e (critique)g(of)h(1-randomness)g(ceases)h(to)g(ap-)505 3071 y(ply)-8 b(,)41 b(as)g(w)m(e)h(will)c(then)j(ha)m(v)m(e)i(a)e(c)m (haracterization)i(of)e(1-randomness)g(based)g(on)505 3179 y(computable)26 b(strategies.)1390 3146 y Fy(6)1457 3179 y FA(Ho)m(w)m(ev)m(er,)j(presen)m(tly)c(w)m(e)i(do)f(not)h(ev)m (en)g(kno)m(w)f(whether)505 3287 y(the)36 b(inclusion)c(holds)h(for)i (left-c.e.)i(reals.)d(Sev)m(eral)i(results)d(ha)m(v)m(e)k(b)s(een)d (obtained)505 3395 y(suggesting)d(that)h(KL-randomness)d(is)h(at)i (least)f(close)g(to)g(1-randomness.)g(Muc)m(h-)505 3503 y(nik)36 b(\(see)j([99)q(,)f(Theorem)f(9.1]\))i(sho)m(w)m(ed)e(that)h (if)f(there)g(is)g(an)g(un)m(b)s(ounded)e(com-)505 3611 y(putable)k(function)g Fw(d)h FA(suc)m(h)g(that)g Fw(K)7 b FA(\()p Fw(A)1946 3599 y Fz(\026)2025 3611 y Fw(n)p FA(\))41 b Fz(6)g Fw(n)26 b Fr(\000)g Fw(d)p FA(\()p Fw(n)p FA(\))41 b(for)e(all)g(but)g(\014nitely)505 3719 y(man)m(y)d Fw(n)p FA(,)e(then)h Fw(A)g FA(is)f(not)i(KL-random.)e (\(In)h(fact)h(the)f(w)m(eak)m(er)h(h)m(yp)s(othesis)e(that)505 3827 y Fr(8)p Fw(n)15 b FA([)p Fw(K)7 b FA(\()p Fw(A)864 3815 y Fz(\026)928 3827 y Fw(g)s FA(\()p Fw(n)p FA(\)\))27 b Fz(6)e Fw(g)s FA(\()p Fw(n)p FA(\))c Fr(\000)f Fw(n)p FA(],)31 b(where)f Fw(g)35 b FA(is)29 b(an)i(un)m(b)s(ounded)d (computable)i(func-)505 3935 y(tion,)h(su\016ces.\))588 4043 y(Merkle,)39 b(Miller,)d(Nies,)i(Reimann,)g(and)f(Stephan)g([91)r (])h(sho)m(w)m(ed)g(that)h(if)e Fw(A)i FA(=)505 4151 y Fw(A)573 4165 y Fy(0)619 4151 y Fr(\010)6 b Fw(A)764 4165 y Fy(1)827 4151 y FA(is)22 b(KL-random,)h(then)g(at)h(least)f(one) h(of)f Fw(A)2254 4165 y Fy(0)2294 4151 y Fw(;)15 b(A)2402 4165 y Fy(1)2465 4151 y FA(is)22 b(1-random,)i(and)f(in)f(fact)505 4258 y(b)s(oth)31 b(are)h(if)e Fw(A)i FA(is)e(\001)1226 4225 y Fy(0)1226 4283 y(2)1266 4258 y FA(.)h(Extending)g(this)f (argumen)m(t)i(sho)m(ws)f(that)h(lim)15 b(inf)3060 4272 y Fx(n)3107 4258 y FA(\()p Fw(K)7 b FA(\()p Fw(A)3357 4246 y Fz(\026)505 4367 y Fw(n)p FA(\))p Fw(=n)p FA(\))40 b(=)g(1)f(\(i.e.,)h(the)f(e\013ectiv)m(e)i(Hausdor\013)e(dimension)2532 4334 y Fy(7)2608 4367 y FA(of)g Fw(A)g FA(is)f(1\);)i(see)g([91)q(].) 505 4475 y(On)d(the)g(other)h(hand,)f(Merkle)g([88)q(])h(sho)m(w)m(ed)f (that)h(the)g(e\013ectiv)m(e)h(Hausdor\013)e(di-)505 4583 y(mension)c(of)h(a)g(set)g(on)g(whic)m(h)f(no)g(partial)g (computable)g(martingale)g(succeeds)h(is)505 4691 y(not)k(necessarily)e (1.)i(All)e(of)h(these)h(results)e(on)h(KL-randomness)f(are)h(pro)m(v)m (ed)h(b)m(y)p 505 4867 499 4 v 588 4926 a Fn(6)623 4958 y Fv(This)18 b(is)g(debatable,)g(ho)n(w)n(ev)n(er,)g(since)g(Sc)n (hnorr)f(b)r(eliev)n(ed)h(that)f(ev)n(en)g(computable)g(martingales)505 5050 y(are)27 b(not)e(e\013ectiv)n(e)h(enough,)g(b)r(ecause)g(their)f (rates)i(of)f(success)h(ma)n(y)d(not)i(b)r(e)f(computable.)588 5109 y Fn(7)623 5141 y Fv(See)h(Section)f(15)i(for)f(more)g(on)f (e\013ectiv)n(e)h(Hausdor\013)f(dimension.)p eop %%Page: 52 52 52 51 bop 505 363 a FD(52)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(constructing)25 b(sev)m(eral)g(strategies,)h(one)g(of)f(whic)m (h)e(succeeds.)j(The)e(problem)g(in)f(an-)505 649 y(sw)m(ering)30 b(Question)g(10.28)i(seems)f(to)g(b)s(e)f(to)i(understand)c(the)j(in)m (terpla)m(y)f(b)s(et)m(w)m(een)505 757 y(suc)m(h)g(strategies.)h(A)f(p) s(ossible)e(approac)m(h)i(to)g(answ)m(ering)g(this)e(question)h(migh)m (t)h(b)s(e)505 865 y(to)j(\014rst)d(in)m(v)m(estigate)j(\(apparen)m (tly\))f(w)m(eak)m(er)h(v)m(ersions)e(of)g(KL-randomness,)g(suc)m(h)505 973 y(as)d(p)s(erm)m(utation)e(randomness)f(and)i(injectiv)m(e)f (randomness.)g(F)-8 b(or)28 b(the)f(de\014nitions)505 1081 y(and)j(a)h(discussion)d(of)i(these)h(notions,)f(see)h(Miller)d (and)i(Nies)g([94)q(].)588 1189 y(It)g(follo)m(ws)e(from)h(Theorem)g (11.12)j(b)s(elo)m(w)c(that)i(ev)m(ery)g(set)g(that)g(is)f(lo)m(w)g (for)g(KL-)505 1297 y(randomness)h(is)f Fw(K)7 b FA(-trivial.)588 1421 y Ft(10.5.)54 b(Finite)25 b(randomness.)46 b FA(The)22 b(notions)g(ab)s(o)m(v)m(e)i(w)m(ould)e(seem)h(to)g(indicate)505 1529 y(that)38 b(all)d(of)i(the)f(randomness)g(notions)f(are)i (linearly)d(ordered)i(in)g(strength.)g(W)-8 b(e)505 1637 y(brie\015y)28 b(men)m(tion)i(a)g(further)f(randomness)g(notion)2266 1604 y Fy(8)2334 1637 y FA(whic)m(h)g(sho)m(ws)h(that)g(this)f(ma)m(y) 505 1745 y(not)j(alw)m(a)m(ys)g(b)s(e)f(the)g(case.)i(A)f Fs(\014nite)h(test)41 b Fr(f)p Fw(U)2083 1759 y Fx(n)2130 1745 y Fr(g)2175 1759 y Fx(n)p Fq(2)p Fx(!)2347 1745 y FA(is)31 b(a)g(Martin-L\177)-45 b(of)32 b(test)g(where)505 1853 y(eac)m(h)e Fw(U)770 1867 y Fx(n)846 1853 y FA(is)e(a)h (\014nitely)e(presen)m(ted)i(op)s(en)f(set.)h(F)-8 b(or)30 b(example,)f(a)g(Kurtz)f(n)m(ull)f(test)i(is)505 1961 y(\014nite.)c(W)-8 b(e)27 b(sa)m(y)g(that)f Fw(A)g FA(is)f Fs(\014nitely)k(r)-5 b(andom)35 b FA(if)25 b(it)g(passes)g(all)g (\014nite)g(tests.)i(It)e(is)g(not)505 2069 y(hard)33 b(to)h(see)f(that)h(a)g(\001)1339 2036 y Fy(0)1339 2093 y(2)1411 2069 y FA(set)g(is)e(Martin-L\177)-45 b(of)33 b(random)g(iff)f(it)g(is)h(\014nitely)e(random.)505 2177 y(W)-8 b(e)40 b(sa)m(y)f(that)h(a)e(\014nite)g(test)h Fr(f)p Fw(U)1657 2191 y Fx(n)1705 2177 y Fr(g)1750 2191 y Fx(n)p Fq(2)p Fx(!)1929 2177 y FA(is)f Fs(c)-5 b(omputably)42 b(b)-5 b(ounde)g(d)49 b FA(if)38 b(the)h Fw(U)3188 2191 y Fx(n)3273 2177 y FA(are)505 2285 y(presen)m(ted)27 b(b)m(y)f(sets)h Fw(P)1263 2299 y Fx(n)1337 2285 y FA(for)f(whic)m(h)f (there)i(is)e(a)i(computable)f(function)f Fw(g)30 b FA(suc)m(h)c(that) 505 2393 y Fr(j)p Fw(P)588 2407 y Fx(n)636 2393 y Fr(j)j Fw(<)f(g)s FA(\()p Fw(n)p FA(\).)34 b(W)-8 b(e)33 b(sa)m(y)g(that)g Fw(A)g FA(is)e Fs(c)-5 b(omputably)36 b(\014nitely)f(r)-5 b(andom)42 b FA(if)31 b(it)h(passes)h(all)505 2501 y(computably)d(b)s (ounded)e(tests.)588 2668 y FB(Theorem)34 b FA(10.29)i(\(Do)m(wney)-8 b(,)32 b(Miller,)d(and)h(Reimann)f([42)q(]\))p FB(.)563 2798 y FA(\(i\))42 b Fs(Martin-L\177)-46 b(of)28 b(r)-5 b(andomness)31 b(implies)e(\014nite)g(r)-5 b(andomness,)30 b(but)e(not)h(c)-5 b(onverse-)701 2906 y(ly.)45 b(However,)g(\014nite)g (r)-5 b(andomness)47 b(and)f(c)-5 b(omputably)47 b(\014nite)e(r)-5 b(andomness)701 3014 y(neither)35 b(imply)h(nor)f(ar)-5 b(e)36 b(implie)-5 b(d)36 b(by)f(either)g(Schnorr)i(or)e(c)-5 b(omputable)37 b(r)-5 b(an-)701 3122 y(domness.)538 3230 y FA(\(ii\))41 b Fs(On)29 b(the)g(left-c.e.)f(r)-5 b(e)g(als,)31 b(\(i\))f(r)-5 b(emains)30 b(true,)f(exc)-5 b(ept)30 b(that)h(\014nite)e(r)-5 b(andomness)701 3338 y(c)g(oincides)33 b(with)g(Martin-L\177)-46 b(of)33 b(r)-5 b(andomness.)588 3505 y FA(Once)39 b(again)f(w)m(e)h(see)g(a)g(connection)f(with)f (traceabilit)m(y)i(and)e(arra)m(y)i(noncom-)505 3613 y(putabilit)m(y:)588 3781 y FB(Theorem)34 b FA(10.30)i(\(Do)m(wney)-8 b(,)32 b(Miller,)d(and)h(Reimann)f([42)q(]\))p FB(.)46 b Fs(If)24 b(a)h(left-c.e.)e(r)-5 b(e)g(al)505 3889 y(is)33 b(c)-5 b(omputably)35 b(\014nitely)e(r)-5 b(andom)35 b(then)e(it)f(is)h(arr)-5 b(ay)35 b(nonc)-5 b(omputable.)588 4103 y Fu(x)p Ft(11.)53 b(Lo)m(wness)40 b(prop)s(erties)h(revisited.)k FA(Recall)35 b(that)g(in)f(Section)g(7)i(a)f(set)505 4211 y Fw(A)k FA(w)m(as)g(called)f(lo)m(w)g(for)g(a)h(class)f Fr(C)44 b FA(if)37 b Fr(C)44 b FA(=)39 b Fr(C)2123 4178 y Fx(A)2180 4211 y FA(.)g(When)f(discussing)e(lo)m(wness)i(for)505 4319 y(randomness)g(notions,)g(one)h(has)g(t)m(w)m(o)h(options.)e(F)-8 b(or)39 b(Sc)m(hnorr)f(randomness,)g(for)505 4427 y(instance,)h(one)f (can)h(lo)s(ok)f(at)h(sets)g Fw(A)f FA(that)h(are)g Fs(low)i(for)f(the) h(Schnorr)g(nul)5 b(l)40 b(sets)505 4535 y FA(\(called)27 b Fw(S)851 4549 y Fy(0)891 4535 y Fs(-low)37 b FA(in)27 b([1]\),)i(meaning)d(that)j(ev)m(ery)f(set)g(that)g(is)e(Sc)m(hnorr)h (n)m(ull)e(relativ)m(e)505 4643 y(to)41 b Fw(A)f FA(is)g(Sc)m(hnorr)f (n)m(ull,)f(or)i(one)h(can)f(lo)s(ok)g(at)h(the)f(p)s(oten)m(tially)f (larger)g(class)h(of)505 4751 y(sets)30 b Fw(A)e FA(that)i(are)f Fs(low)j(for)f(Schnorr)i(r)-5 b(andomness)p FA(,)32 b(meaning)c(that)h (ev)m(ery)h(Sc)m(hnorr)505 4858 y(random)d(set)g(is)g(Sc)m(hnorr)f (random)g(relativ)m(e)h(to)h Fw(A)p FA(.)g(In)e(the)h(case)h(of)g (1-randomness,)p 505 4958 499 4 v 588 5018 a Fn(8)623 5050 y Fv(The)18 b(notion)g(of)g(\014nite)f(randomness)g(discussed)h (here)g(is)g(not)f(sto)r(c)n(hastic,)i(so)g(the)e(same)g(remarks)505 5141 y(apply)26 b(as)g(in)f(the)h(case)g(of)h(Kurtz)e(randomness;)g (see)i(fo)r(otnote)f(5.)p eop %%Page: 53 53 53 52 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(53)505 541 y FA(there)35 b(is)e(no)g(di\013erence)h(b)s(et)m(w)m (een)g(these)h(notions)e(b)s(ecause)h(there)g(is)f(a)h(univ)m(ersal)505 649 y(Martin-L\177)-45 b(of)26 b(test.)i(Am)m(b)s(os-Spies)c(and)i(Ku)m (\024)-43 b(cera)27 b([1)q(,)f(Problem)f(4.5])i(ask)m(ed)g(whether)505 757 y(the)33 b(t)m(w)m(o)g(notions)e(are)h(di\013eren)m(t)g(for)g(Sc)m (hnorr)e(randomness.)h(It)i(will)c(follo)m(w)i(from)505 865 y(Theorem)44 b(11.10)i(b)s(elo)m(w)e(that)g(the)g(answ)m(er)g(is)g (no,)g(despite)f(the)h(absence)h(of)f(a)505 973 y(univ)m(ersal)36 b(Sc)m(hnorr)h(test.)i(F)-8 b(or)38 b(computable)f(randomness,)g(the)h (answ)m(er)f(is)g(ev)m(en)505 1081 y(easier:)46 b(the)f(only)f(sets)h (that)h(are)f(lo)m(w)g(for)g(computable)f(randomness)g(are)h(the)505 1189 y(computable)30 b(ones!)h(\(See)g(Theorem)f(11.14.\))588 1314 y Ft(11.1.)54 b(Lo)m(wness)33 b(for)g(Sc)m(hnorr)i(and)e(Kurtz)g (n)m(ull)h(sets.)45 b FA(One)29 b(nice)f(asp)s(ect)505 1421 y(of)i(Sc)m(hnorr)e(randomness)g(is)g(that)i(there)f(is)f(a)i (complete)g(c)m(haracterization)g(of)f(the)505 1529 y(sets)40 b(that)f(are)g(lo)m(w)g(for)f(Sc)m(hnorr)g(randomness.)g(As)h(usual,)e (let)i Fw(D)2881 1543 y Fx(n)2967 1529 y FA(denote)g(the)505 1637 y Fw(n)p FA(-th)30 b(canonical)g(\014nite)g(set.)588 1809 y FB(Definition)35 b FA(11.1)h(\(T)-8 b(erwijn)29 b(and)h(Zam)m(b)s(ella)54 b([134)q(];)j(c.f.)31 b(De\014nition)d(7.2\)) p FB(.)48 b FA(A)505 1917 y(set)35 b Fw(A)f FA(is)f Fs(c)-5 b(omputably)38 b(tr)-5 b(ac)g(e)g(able)43 b FA(if)33 b(there)h(is)f(a)h(computable)g(function)e Fw(p)i FA(\(called)505 2025 y(a)43 b Fs(b)-5 b(ound)9 b FA(\))45 b(suc)m(h)d(that,)h(for)g (eac)m(h)h(function)d Fw(g)49 b Fz(6)2262 2039 y Fy(T)2363 2025 y Fw(A)p FA(,)43 b(there)f(is)g(a)h(computable)505 2133 y(function)29 b Fw(h)i FA(satisfying,)e(for)i(all)e Fw(n)p FA(,)563 2266 y(\(i\))42 b Fr(j)p Fw(D)801 2285 y Fx(h)p Fy(\()p Fx(n)p Fy(\))944 2266 y Fr(j)25 b Fz(6)g Fw(p)p FA(\()p Fw(n)p FA(\))30 b(and)538 2378 y(\(ii\))41 b Fw(g)s FA(\()p Fw(n)p FA(\))26 b Fr(2)f Fw(D)1059 2396 y Fx(h)p Fy(\()p Fx(n)p Fy(\))1201 2378 y FA(.)588 2550 y(The)k(follo)m(wing)f(prop)s(osition)g(sho)m(ws)h(that)h(it)f(do)s(es) g(not)h(matter)g(what)f(b)s(ound)f Fw(p)505 2657 y FA(w)m(e)j(c)m(ho)s (ose.)588 2829 y FB(Pr)n(oposition)j FA(11.2)i(\(T)-8 b(erwijn)29 b(and)h(Zam)m(b)s(ella)f([134)q(]\))p FB(.)46 b Fs(L)-5 b(et)27 b Fw(A)f Fs(b)-5 b(e)26 b(c)-5 b(omputably)505 2937 y(tr)g(ac)g(e)g(able)43 b(and)e(let)g Fw(p)f Fs(b)-5 b(e)41 b(an)g(unb)-5 b(ounde)g(d)42 b(nonde)-5 b(cr)g(e)g(asing)43 b(c)-5 b(omputable)42 b(function)505 3045 y(such)33 b(that)h Fw(p)p FA(\(0\))26 b Fw(>)f FA(0)p Fs(.)33 b(Then)g Fw(A)g Fs(is)f(c)-5 b(omputably)35 b(tr)-5 b(ac)g(e)g(able)34 b(with)g(b)-5 b(ound)33 b Fw(p)p Fs(.)588 3217 y FA(A)42 b(degree)g(is)f Fs(hyp)-5 b(erimmune-fr)g(e)g(e)50 b FA(if)40 b(eac)m(h)j(function)d(of)h(that)h(degree)h(is)d(ma-)505 3325 y(jorized)33 b(b)m(y)f(some)h(computable)g(function.)e(If)i Fw(A)g FA(is)e(computably)h(traceable)i(then)505 3433 y(eac)m(h)41 b(function)c Fw(g)44 b Fz(6)1241 3447 y Fy(T)1335 3433 y Fw(A)39 b FA(is)f(ma)5 b(jorized)39 b(b)m(y)g(the)g(function)f Fw(f)10 b FA(\()p Fw(n)p FA(\))39 b(=)g(max)16 b Fw(D)3227 3451 y Fx(h)p Fy(\()p Fx(n)p Fy(\))3369 3433 y FA(,)505 3541 y(where)42 b Fw(h)g FA(is)f(as)h(in)e (De\014nition)h(11.1.)i(Th)m(us)e(ev)m(ery)i(computably)d(traceable)j (set)505 3649 y(has)d(h)m(yp)s(erimm)m(une-free)e(degree.)j(One)f(ma)m (y)g(think)f(of)h(computable)g(traceabil-)505 3756 y(it)m(y)49 b(as)f(a)h(uniform)d(v)m(ersion)i(of)g(h)m(yp)s(erimm)m(une-freeness.)e (T)-8 b(erwijn)47 b(and)h(Zam-)505 3864 y(b)s(ella)36 b([134)r(])h(sho)m(w)m(ed)h(that)g(a)g(simple)d(v)-5 b(ariation)37 b(of)h(the)f(standard)g(construction)505 3972 y(of)c(h)m(yp)s(erimm)m(une-free)d(sets)i(b)m(y)g(Miller)e(and)i (Martin)f([97)q(])i(pro)s(duces)d(con)m(tin)m(uum)505 4080 y(man)m(y)h(computably)e(traceable)j(sets.)588 4188 y(Kjos-Hanssen)37 b(and)g(Nies)g(\(unpublished\))c(ha)m(v)m(e)38 b(recen)m(tly)g(c)m(haracterized)g(the)505 4296 y(computably)g (traceable)h(sets)g(within)d(the)i(class)h(of)f(sets)h(of)f(h)m(yp)s (erimm)m(une-free)505 4404 y(degree)31 b(using)e(pre\014x-free)h (complexit)m(y)-8 b(.)588 4576 y FB(Definition)35 b FA(11.3)h (\(Kjos-Hanssen)30 b(and)g(Nies)g(\(unpublished\)\))p FB(.)563 4709 y FA(\(i\))42 b(A)34 b(set)h Fw(A)g FA(is)e Fs(we)-5 b(akly)38 b(c.e.)e(tr)-5 b(ac)g(e)g(able)43 b FA(if)33 b(De\014nition)g(7.2)j(holds)d(for)h(the)h(com-)701 4817 y(putably)22 b(b)s(ounded)g Fw(A)p FA(-computable)h(functions;)g (that)i(is,)e(if)f(there)i(is)f(a)i(b)s(ound)701 4925 y Fw(p)j FA(suc)m(h)h(that)g(for)g(ev)m(ery)g Fw(f)35 b Fz(6)1701 4939 y Fy(T)1781 4925 y Fw(A)29 b FA(that)h(is)d(ma)5 b(jorized)29 b(b)m(y)g(some)g(computable)701 5033 y(function,)21 b(there)j(is)d(a)j(c.e.)g(trace)g(for)e Fw(f)33 b FA(with)21 b(b)s(ound)g Fw(p)h FA(\(as)i(b)s(efore,)f(the)g(c)m(hoice)701 5141 y(of)30 b(b)s(ound)e(do)s(es)j(not)f(matter\).)p eop %%Page: 54 54 54 53 bop 505 363 a FD(54)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)538 541 y FA(\(ii\))41 b(A)33 b(set)g Fw(X)40 b FA(is)32 b Fs(facile)40 b FA(if)32 b(for)g(ev)m(ery)i(nondecreasing)e(un)m(b)s (ounded)e(computable)701 649 y(function)f Fw(h)p FA(,)i(for)f(almost)g (all)f Fw(n)h FA(w)m(e)h(ha)m(v)m(e)h Fw(K)7 b FA(\()p Fw(X)2379 637 y Fz(\026)2442 649 y Fw(n)25 b Fr(j)g Fw(n)p FA(\))g Fz(6)g Fw(h)p FA(\()p Fw(n)p FA(\).)588 806 y FB(Pr)n(oposition)34 b FA(11.4)i(\(Kjos-Hanssen)31 b(and)e(Nies)h (\(unpublished\)\))p FB(.)42 b Fs(A)25 b(set)g Fw(A)h Fs(is)505 914 y(we)-5 b(akly)34 b(c.e.)e(tr)-5 b(ac)g(e)g(able)35 b(iff)d(every)g(set)h Fw(X)g Fz(6)2011 928 y Fi(T)2091 914 y Fw(A)g Fs(is)f(facile.)588 1071 y FA(Th)m(us)26 b(ev)m(ery)i(computably)e(traceable)i(set)f(is)f(facile.)h(Con)m(v)m (ersely)-8 b(,)27 b(supp)s(ose)f(that)505 1179 y Fw(A)37 b FA(has)g(h)m(yp)s(erimm)m(une-free)e(degree)i(and)f(is)g(facile.)h (It)f(is)g(not)h(hard)f(to)i(see)f(that)505 1287 y(the)j(facile)e(sets) i(are)f(closed)g(do)m(wn)m(w)m(ards)f(under)g(wtt-reducibilit)m(y)-8 b(,)37 b(but)h(T)-8 b(uring)505 1394 y(reducibilit)m(y)20 b(implies)g(wtt-reducibilit)m(y)g(within)h(the)i(h)m(yp)s(erimm)m (une-free)e(degrees,)505 1502 y(so)k(ev)m(ery)h Fw(A)p FA(-computable)f(set)g(is)f(facile.)g(Th)m(us)g(b)m(y)g(Prop)s(osition) f(11.4,)k Fw(A)e FA(is)e(w)m(eakly)505 1610 y(c.e.)g(traceable,)g(and)e (hence)g(c.e.)i(traceable)g(\(since)e(ev)m(ery)h Fw(A)p FA(-computable)g(function)505 1718 y(is)37 b(computably)g(b)s (ounded\).)f(But)h(it)g(is)g(not)h(hard)e(to)j(use)e(the)h(fact)g(that) g Fw(A)g FA(has)505 1826 y(h)m(yp)s(erimm)m(une-free)28 b(degree)i(to)g(con)m(v)m(ert)h(a)f(c.e.)h(trace)g(in)m(to)e(a)h (computable)f(trace,)505 1934 y(so)i Fw(A)g FA(is)e(computably)g (traceable.)j(Th)m(us)d(w)m(e)i(ha)m(v)m(e)h(the)e(follo)m(wing)f (corollary)-8 b(.)588 2091 y FB(Cor)n(ollar)i(y)35 b FA(11.5)p FB(.)47 b Fs(Supp)-5 b(ose)35 b(that)g Fw(A)f Fs(has)g(hyp)-5 b(erimmune-fr)g(e)g(e)36 b(de)-5 b(gr)g(e)g(e.)34 b(Then)505 2199 y Fw(A)f Fs(is)g(c)-5 b(omputably)34 b(tr)-5 b(ac)g(e)g(able)35 b(iff)d Fw(A)h Fs(is)f(facile.)588 2355 y FA(Remark)-5 b(ably)d(,)44 b(the)g(class)f(of)h(sets)g(that)g (are)g(lo)m(w)g(for)f(the)h(Sc)m(hnorr)f(n)m(ull)e(sets)505 2463 y(is)c(c)m(haracterized)h(b)m(y)f(the)g(purely)e(computabilit)m (y-theoretic)i(prop)s(ert)m(y)f(of)i(com-)505 2571 y(putable)30 b(traceabilit)m(y)-8 b(.)588 2728 y FB(Theorem)34 b FA(11.6)i(\(T)-8 b(erwijn)54 b(and)h(Zam)m(b)s(ella)f([134)r(]\))p FB(.)88 b Fs(A)41 b(set)h(is)g(low)h(for)g(the)505 2836 y(Schnorr)35 b(nul)5 b(l)32 b(sets)h(iff)f(it)h(is)f(c)-5 b(omputably)35 b(tr)-5 b(ac)g(e)g(able.)588 2993 y FA(One)26 b(direction)e(of)i(the)h (pro)s(of)e(of)h(Theorem)f(11.6)j(relies)c(on)i(ideas)f(of)h (Raisonnier)505 3101 y([111)r(])38 b(on)h(rapid)d(\014lters)h(for)h (the)h(\\mathematical")g(pro)s(of)f(of)g(Shelah's)f(theorem)505 3209 y(that)45 b(the)g(inaccessible)e(cardinal)g(cannot)i(b)s(e)f(remo) m(v)m(ed)h(from)f(Solo)m(v)-5 b(a)m(y's)46 b([125)q(])505 3316 y(construction)28 b(of)h(a)f(mo)s(del)f(of)i(set)f(theory)h(where) f(ev)m(ery)h(set)f(of)h(reals)e(is)h(Leb)s(esgue)505 3424 y(measurable.)588 3532 y(In)m(terestingly)-8 b(,)32 b(w)m(e)g(ha)m(v)m(e)h(the)e(follo)m(wing)f(fact)j(ab)s(out)e(the)h (degrees)g(of)g(sets)g(that)505 3640 y(are)e(lo)m(w)g(for)f(Sc)m(hnorr) g(randomness)f(\(whic)m(h)h(w)m(as)h(pro)m(v)m(ed)g(b)m(y)g(T)-8 b(erwijn)27 b(and)j(Zam-)505 3748 y(b)s(ella)25 b([134)r(])i(for)f(the) h(sets)g(that)g(are)g(lo)m(w)f(for)h(the)f(Sc)m(hnorr)g(n)m(ull)f (sets,)i(b)s(efore)f(it)g(w)m(as)505 3856 y(kno)m(wn)k(that)h(these)g (t)m(w)m(o)h(classes)e(are)h(the)g(same\).)588 4013 y FB(Theorem)j FA(11.7)i(\(T)-8 b(erwijn)29 b(and)g(Zam)m(b)s(ella)g ([134)r(]\))p FB(.)46 b Fs(The)32 b(de)-5 b(gr)g(e)g(es)34 b(of)e(sets)g(that)505 4121 y(ar)-5 b(e)25 b(low)g(for)g(Schnorr)g(r)-5 b(andomness)27 b(ar)-5 b(e)25 b(a)f(pr)-5 b(op)g(er)27 b(sub)-5 b(class)25 b(of)f(the)g(hyp)-5 b(erimmune-)505 4229 y(fr)g(e)g(e)32 b(de)-5 b(gr)g(e)g(es,)32 b(and)g(henc)-5 b(e,)31 b(exc)-5 b(ept)32 b(for)g Ft(0)p Fs(,)f(none)g(of)g(them)h(ar) -5 b(e)32 b FA(\001)2768 4196 y Fy(0)2768 4253 y(2)2807 4229 y Fs(.)f(In)g(p)-5 b(articular,)505 4337 y(the)34 b(de)-5 b(gr)g(e)g(es)35 b(of)e(nonc)-5 b(omputable)36 b(sets)d(that)i(ar)-5 b(e)34 b(low)g(for)g FA(1)p Fs(-r)-5 b(andomness)36 b(and)e(the)505 4444 y(de)-5 b(gr)g(e)g(es)37 b(of)e(nonc)-5 b(omputable)38 b(sets)e(that)g(ar)-5 b(e)36 b(low)h(Schnorr)g(r)-5 b(andomness)38 b(ar)-5 b(e)43 b FA(dis-)505 4552 y(join)m(t)p Fs(.)588 4709 y FA(Recen)m(tly)-8 b(,)35 b(an)e(easier)g(result)f(relating)g(a)h(lo)m(wness)g(notion)f (to)i(the)f(computably)505 4817 y(traceable)i(and)f(h)m(yp)s(erimm)m (une-free)e(degrees)i(w)m(as)h(obtained)e(b)m(y)h(Do)m(wney)-8 b(,)35 b(Grif-)505 4925 y(\014ths,)c(and)g(Reid)g([32)q(].)h(W)-8 b(e)33 b(giv)m(e)f(the)g(pro)s(of)f(b)s(elo)m(w)g(since)g(it)g(is)g (represen)m(tativ)m(e)h(of)505 5033 y(the)k(m)m(uc)m(h)g(more)g (di\016cult)d(pro)s(of)i(of)h(Theorem)f(11.6.)i(A)f Fs(Kurtz)i(nul)5 b(l)37 b(set)45 b FA(is)34 b(an)m(y)505 5141 y(subset)k(of)g(the)g(in)m (tersection)g(of)g(a)h(Kurtz)e(n)m(ull)f(test)j(\(see)g(De\014nition)e (10.19\).)j(A)p eop %%Page: 55 55 55 54 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(55)505 541 y FA(set)35 b Fw(A)g FA(is)e Fs(low)38 b(for)f(the)g(Kurtz)f(nul)5 b(l)36 b(sets)43 b FA(if)33 b(ev)m(ery)i(Kurtz)f(n)m(ull)f(set)i(relativ)m(e)f(to)h Fw(A)505 649 y FA(is)c(Kurtz)g(n)m(ull.)f(It)h(is)g(curren)m(tly)g(not) g(kno)m(wn)h(whether)e(b)s(eing)h(lo)m(w)g(for)g(the)h(Kurtz)505 757 y(n)m(ull)d(sets)h(is)g(equiv)-5 b(alen)m(t)30 b(to)h(b)s(eing)e (lo)m(w)h(for)g(Kurtz)g(randomness.)588 946 y FB(Theorem)k FA(11.8)i(\(Do)m(wney)-8 b(,)32 b(Gri\016ths,)d(and)h(Reid)f([32)r(]\)) p FB(.)563 1094 y FA(\(i\))42 b Fs(If)30 b(a)h(set)g(is)g(c)-5 b(omputably)33 b(tr)-5 b(ac)g(e)g(able)32 b(then)g(it)f(is)f(low)i(for) f(the)h(Kurtz)f(nul)5 b(l)30 b(sets.)538 1202 y FA(\(ii\))41 b Fs(If)35 b(a)i(set)f(is)g(low)g(for)h(the)f(Kurtz)g(nul)5 b(l)36 b(sets)g(then)h(it)f(has)h(hyp)-5 b(erimmune-fr)g(e)g(e)701 1310 y(de)g(gr)g(e)g(e.)588 1499 y FB(Pr)n(oof.)41 b FA(\(i\))26 b(Let)h Fw(A)f FA(b)s(e)f(computably)g(traceable)i(and)e (let)h Fr(f)p Fw(V)2712 1513 y Fx(n)2760 1499 y Fr(g)2805 1513 y Fx(n)p Fq(2)p Fx(!)2971 1499 y FA(b)s(e)g(a)g(Kurtz)505 1607 y(n)m(ull)i(test)i(relativ)m(e)g(to)g Fw(A)p FA(.)g(W)-8 b(e)31 b(build)26 b(a)k(\(computable\))g(Kurtz)f(n)m(ull)e(test)j Fr(f)p Fw(U)3161 1621 y Fx(n)3209 1607 y Fr(g)3254 1621 y Fx(n)p Fq(2)p Fx(!)505 1714 y FA(suc)m(h)g(that)907 1646 y Fp(T)983 1741 y Fx(n)1045 1714 y Fw(U)1107 1728 y Fx(n)1180 1714 y Fr(\023)1276 1646 y Fp(T)1351 1741 y Fx(n)1413 1714 y Fw(V)1466 1728 y Fx(n)1513 1714 y FA(.)588 1822 y(Let)24 b Fw(E)811 1836 y Fy(0)851 1822 y Fw(;)15 b(E)958 1836 y Fy(1)998 1822 y Fw(;)g(:)g(:)g(:)40 b FA(b)s(e)23 b(an)g(e\013ectiv)m(e)i(listing)d(of)h(all)g(\014nite)f (subsets)h(of)h(2)2921 1789 y Fx()e(f)10 b FA(\(1\).)588 4346 y(Since)25 b Fw(\026)p FA(\()p Fw(V)964 4360 y Fy(2)1004 4346 y FA(\))g Fz(6)g FA(1)p Fw(=)p FA(4,)j(there)e(m)m(ust)f(b)s(e)g Fw(k)1962 4360 y Fy(1)2027 4346 y Fw(<)g(k)2170 4360 y Fy(2)2236 4346 y FA(suc)m(h)g(that)i Fw(V)2682 4360 y Fy(2)2747 4346 y FA(do)s(es)e(not)h(con)m(tain)505 4454 y(all)f(sets)h(with)e Fw(k)1049 4468 y Fy(1)1089 4454 y FA(-th)i(bit)f(0,)h(and)f(also)h(do)s (es)f(not)h(con)m(tain)g(all)e(sets)j(with)d Fw(k)3031 4468 y Fy(1)3071 4454 y FA(-th)h(bit)g(1)505 4562 y(and)i Fw(k)726 4576 y Fy(2)766 4562 y FA(-th)g(bit)f(0.)i(Again,)f Fw(k)1479 4576 y Fy(1)1546 4562 y FA(and)f Fw(k)1766 4576 y Fy(2)1833 4562 y FA(are)h(among)h Fw(f)10 b FA(\(1\))k(+)g(1)p Fw(;)h(f)10 b FA(\(1\))k(+)g Fw(f)c FA(\(2\))k(+)g(2)p Fw(;)h(:)g(:)g(:)h FA(,)505 4670 y(so)35 b(letting)e Fw(g)s FA(\(2\))g(=)e Fw(k)1255 4684 y Fy(2)1295 4670 y FA(,)j(w)m(e)g(ha)m(v)m(e)h Fw(g)s FA(\(2\))e Fw(>)e(f)10 b FA(\(2\).)35 b(W)-8 b(e)35 b(can)f(con)m(tin)m(ue)g(this)f(pro)s (cess)505 4778 y(to)f(de\014ne)d(a)i(computable)f(function)f Fw(g)34 b FA(dominating)29 b Fw(f)10 b FA(.)901 b Fr(a)588 4925 y Ft(11.2.)54 b(Lo)m(wness)24 b(for)h(pairs)g(of)g(randomness)g (notions.)46 b FA(A)21 b(more)h(complete)505 5033 y(view)e(of)h(lo)m (wness)f(arises)f(when)h(w)m(e)h(consider)e(lo)m(wness)h(for)g(an)m(y)h (pair)e(of)i(randomness)505 5141 y(notions)33 b Fr(C)i(\022)30 b(D)s FA(.)j(Since)g(relativizing)e Fr(D)36 b FA(usually)31 b(mak)m(es)j(it)f(smaller,)f(one)i(w)m(ould)p eop %%Page: 56 56 56 55 bop 505 363 a FD(56)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(exp)s(ect)31 b(that)g(in)e(general)i Fr(C)f(6\022)25 b(D)1657 508 y Fx(A)1714 541 y FA(.)31 b(The)e(follo)m(wing)g(class)i (consists)e(of)i(the)g(sets)f Fw(A)505 649 y FA(for)h(whic)m(h)e(the)h (inclusion)e(still)g(holds.)588 819 y FB(Definition)35 b FA(11.9)p FB(.)47 b FA(A)31 b(set)g Fw(A)f FA(is)f(in)g(Lo)m(w)r(\()p Fr(C)5 b Fw(;)15 b Fr(D)s FA(\))31 b(if)f Fr(C)g(\022)25 b(D)2640 786 y Fx(A)2697 819 y FA(.)588 1000 y(Clearly)-8 b(,)32 b(if)g Fr(C)j(\022)1207 977 y Fp(e)1193 1000 y Fr(C)f(\022)1394 977 y Fp(e)1375 1000 y Fr(D)e(\022)d(D)35 b FA(are)f(randomness)d(notions,)i(then)f(Lo)m(w)r(\()3107 977 y Fp(e)3093 1000 y Fr(C)5 b Fw(;)3205 977 y Fp(e)3186 1000 y Fr(D)s FA(\))30 b Fr(\022)505 1108 y FA(Lo)m(w)r(\()p Fr(C)5 b Fw(;)15 b Fr(D)s FA(\).)35 b(That)e(is,)g(w)m(e)h(mak)m(e)g (the)g(class)f(Lo)m(w)q(\()p Fr(C)5 b Fw(;)15 b Fr(D)s FA(\))35 b(larger)e(b)m(y)g(decreasing)g Fr(C)505 1215 y FA(or)e(increasing)e Fr(D)s FA(.)588 1323 y(Let)39 b Fd(MLRand)o Fw(;)15 b Fd(CRand)38 b FA(and)g Fd(SRand)f FA(denote)h(the)h(classes)f(of)g(1-random,)h(com-)505 1431 y(putably)k(random,)h(and)f(Sc)m(hnorr)g(random)g(sets,)i(resp)s (ectiv)m(ely)-8 b(.)44 b(Then,)g(for)g(in-)505 1539 y(stance,)h(Lo)m(w) q(\()p Fd(MLRand)p Fw(;)15 b Fd(CRand)p FA(\))43 b(is)f(the)h(class)g (of)g(sets)h Fw(A)f FA(suc)m(h)f(that)i(ev)m(ery)g(1-)505 1647 y(random)38 b(set)h(is)f(computably)f(random)h(relativ)m(e)h Fw(A)p FA(.)g(W)-8 b(e)39 b(w)m(an)m(t)h(to)f(c)m(haracterize)505 1755 y(lo)m(wness)30 b(for)g(an)m(y)h(pair)e(of)i(randomness)e (notions.)588 1863 y(Recall)41 b(from)g(De\014nition)f(7.2)j(that)f(a)g (set)g Fw(A)f FA(is)g Fs(c.e.-tr)-5 b(ac)g(e)g(able)49 b FA(if)40 b(there)i(is)e(a)505 1971 y(computable)28 b Fw(p)f FA(suc)m(h)h(that)g(for)g(ev)m(ery)g(function)f Fw(f)35 b Fz(6)2342 1985 y Fy(T)2422 1971 y Fw(A)p FA(,)28 b(there)g(is)f(a)h(computable)505 2079 y(function)i Fw(h)h FA(suc)m(h)f(that)i(for)e(all)g Fw(n)p FA(,)g(w)m(e)i(ha)m(v)m(e)g Fr(j)p Fw(W)2180 2097 y Fx(h)p Fy(\()p Fx(n)p Fy(\))2322 2079 y Fr(j)26 b Fz(6)g Fw(p)p FA(\()p Fw(n)p FA(\))31 b(and)f Fw(f)10 b FA(\()p Fw(n)p FA(\))25 b Fr(2)h Fw(W)3227 2097 y Fx(h)p Fy(\()p Fx(n)p Fy(\))3369 2079 y FA(.)505 2187 y(W)-8 b(e)30 b(ha)m(v)m(e)h(seen)e(that)g(b)s(oth)f (c.e.-traceabilit)m(y)i(and)f(computable)f(traceabilit)m(y)-8 b(,)29 b(de-)505 2295 y(\014ned)38 b(in)g(the)h(previous)e(subsection,) h(are)h(deeply)f(related)h(to)h(lo)m(wness)e(notions.)505 2403 y(The)30 b(follo)m(wing)f(result)g(expands)h(on)g(this)f (relationship.)588 2572 y FB(Theorem)34 b FA(11.10)i(\(Kjos-Hanssen,)31 b(Nies,)f(and)g(Stephan)g([62)q(]\))p FB(.)563 2703 y FA(\(i\))42 b Fs(A)32 b(set)g(is)h(in)g FA(Lo)m(w)q(\()p Fd(MLRand)p Fw(;)15 b Fd(SRand)o FA(\))33 b Fs(iff)f(it)h(is)f(c.e.-tr) -5 b(ac)g(e)g(able.)538 2811 y FA(\(ii\))41 b Fs(A)32 b(set)g(is)h(in)g FA(Lo)m(w)q(\()p Fd(CRand)p Fw(;)15 b Fd(SRand)o FA(\))33 b Fs(iff)f(it)h(is)f(c)-5 b(omputably)35 b(tr)-5 b(ac)g(e)g(able.)588 2980 y FB(Pr)n(oof)34 b(Sketch.)40 b FA(\(i\))i(Notice)i(that)f Fw(A)i Fr(2)g FA(Lo)m(w)r(\()p Fd(MLRand)p Fw(;)15 b Fd(SRand)o FA(\))43 b(iff)e(ev)m(ery)505 3088 y(Sc)m(hnorr)32 b(n)m(ull)e(set)j(relativ)m(e)g(to)g Fw(A)g FA(is)f(con)m(tained)g(in)g(the)g(in)m(tersection)h(of)f(the)h (uni-)505 3196 y(v)m(ersal)c(Martin-L\177)-45 b(of)29 b(test.)i(This)c(already)i(lo)s(oks)f(quite)h(similar)d(to)k(\\there)g (is)e(a)h(c.e.)505 3304 y(trace)g(for)e(the)h(functions)e(computable)g (in)g Fw(A)p FA(".)j(W)-8 b(e)28 b(obtain)f(\(i\))g(b)m(y)h(mo)s (difying)c(the)505 3412 y(metho)s(ds)32 b(in)f(T)-8 b(erwijn)30 b(and)i(Zam)m(b)s(ella)f([134)r(])h(to)h(the)g(case)g(of)f(c.e.)i (traces)f(instead)505 3520 y(of)e(computable)f(ones.)588 3628 y(\(ii\))35 b(Because)h(of)f(Theorem)g(11.6,)h(it)f(only)f (remains)g(to)h(b)s(e)g(sho)m(wn)f(that)i(ev)m(ery)505 3736 y(set)31 b(in)e(Lo)m(w)r(\()p Fd(CRand)p Fw(;)15 b Fd(SRand)o FA(\))31 b(is)e(computably)h(traceable.)588 3844 y(1.)35 b(The)d(\014rst)h(step)h(w)m(as)f(made)h(b)m(y)f(Bedregal) h(and)f(Nies)g([11)q(,)g(62)q(],)h(who)f(pro)m(v)m(ed)505 3952 y(that)23 b(ev)m(ery)h(set)f(in)e(Lo)m(w)q(\()p Fd(CRand)p Fw(;)15 b Fd(SRand)o FA(\))23 b(has)f(h)m(yp)s(erimm)m (une-free)f(degree.)i(T)-8 b(o)23 b(see)505 4060 y(this,)30 b(assume)g(that)h Fw(A)g FA(has)f(h)m(yp)s(erimm)m(une)e(degree,)k(so)f (that)g(there)f(is)g(a)h(function)505 4168 y Fw(g)37 b Fz(6)656 4182 y Fy(T)744 4168 y Fw(A)e FA(not)g(dominated)f(b)m(y)h (an)m(y)g(computable)g(function.)f(Use)h Fw(g)k FA(to)c(de\014ne)g(an) 505 4276 y Fw(A)p FA(-computable)30 b(martingale)f(that)h(succeeds)f (in)f(the)i(sense)f(of)h(Sc)m(hnorr,)e(with)g(the)505 4383 y(computable)35 b(lo)m(w)m(er)g(b)s(ound)e Fw(n=)p FA(4,)i(on)g(some)g Fw(Z)40 b Fr(2)32 b Fd(CRand)o FA(.)k(The)e (construction)g(of)505 4491 y(this)e(martingale)g(uses)h(the)g(fact)h (that)f Fw(g)j FA(is)c(in\014nitely)e(often)j(ab)s(o)m(v)m(e)h(the)f (running)505 4599 y(time)38 b(of)g(eac)m(h)g(computable)g(martingale.)f (Sp)s(ecial)f(care)i(has)g(to)g(b)s(e)f(tak)m(en)i(with)505 4707 y(partial)30 b(martingales,)g(whic)m(h)f(results)g(in)g(a)i(set)f Fw(Z)37 b FA(that)31 b(is)f(only)f(\001)2850 4674 y Fy(0)2850 4732 y(3)2889 4707 y FA(.)588 4815 y(2.)35 b(Next)g(w)m(e)g(use)f(the)g (fact)h(that)g(if)e Fw(A)h FA(has)g(h)m(yp)s(erimm)m(une-free)e(degree) j(and)e(is)505 4923 y(c.e.-traceable,)26 b(then)c Fw(A)h FA(is)e(computably)h(traceable.)h(T)-8 b(o)23 b(see)h(this,)d(let)i Fw(f)34 b Fz(6)3077 4937 y Fy(T)3157 4923 y Fw(A)23 b FA(and)505 5031 y(let)h Fw(h)h FA(b)s(e)e(as)h(in)f(the)h(de\014nition) e(of)i(c.e.-traceabilit)m(y)-8 b(.)26 b(Let)f Fw(g)s FA(\()p Fw(n)p FA(\))f(b)s(e)g(the)g(least)g Fw(s)g FA(suc)m(h)505 5139 y(that)34 b Fw(f)10 b FA(\()p Fw(n)p FA(\))29 b Fr(2)g Fw(W)1090 5157 y Fx(h)p Fy(\()p Fx(n)p Fy(\))p Fx(;s)1285 5139 y FA(.)k(Then)f Fw(g)h Fz(6)1730 5153 y Fy(T)1814 5139 y Fw(A)g FA(and)g(so,)g(since)f Fw(A)h FA(is)f(h)m(yp)s(erimm)m(une-free,)p eop %%Page: 57 57 57 56 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(57)505 541 y Fw(g)33 b FA(is)28 b(ma)5 b(jorized)29 b(b)m(y)g(a)h(computable)f(function)e Fw(r)s FA(.)i(So)g(if)g(w)m(e)g (c)m(ho)s(ose)h(a)g(computable)505 657 y(function)862 633 y Fp(e)864 657 y Fw(h)i FA(suc)m(h)h(that)g Fw(D)1429 674 y Ff(e)1430 691 y Fx(h)p Fy(\()p Fx(n)p Fy(\))1601 657 y FA(=)c Fw(W)1787 676 y Fx(h)p Fy(\()p Fx(n)p Fy(\))p Fx(;r)r Fy(\()p Fx(n)p Fy(\))2113 657 y FA(for)j(all)f Fw(n)p FA(,)i(then)f(w)m(e)h(obtain)f(a)h(com-)505 779 y(putable)d(trace)h(for)f Fw(A)p FA(.)588 887 y(3.)25 b(By)g(part)f(\(i\),)g(if)g Fw(A)h Fr(2)g FA(Lo)m(w)r(\()p Fd(CRand)o Fw(;)15 b Fd(SRand)p FA(\))24 b(then)g Fw(A)h FA(is)e(c.e.-traceable,)k(whic)m(h)505 995 y(b)m(y)k(1)f(and)g(2)h (imply)d(that)j Fw(A)f FA(is)g(computably)f(traceable.)857 b Fr(a)588 1120 y FA(If)43 b(a)g(set)h(is)e(lo)m(w)g(for)h(the)g(Sc)m (hnorr)f(n)m(ull)f(sets)i(then)g(it)f(is)g(ob)m(viously)g(lo)m(w)h(for) 505 1228 y(Sc)m(hnorr)27 b(randomness.)f(Con)m(v)m(ersely)-8 b(,)28 b(if)e Fw(A)h FA(is)f(lo)m(w)h(for)g(Sc)m(hnorr)f(randomness)g (then)505 1336 y Fw(A)33 b Fr(2)g FA(Lo)m(w)r(\()p Fd(CRand)o Fw(;)15 b Fd(SRand)p FA(\),)35 b(and)g(hence)g(b)m(y)f(part)h(\(ii\))f (of)i(Theorem)e(11.10,)j Fw(A)e FA(is)505 1444 y(computably)22 b(traceable.)i(But)e(then)h(b)m(y)f(Theorem)g(11.6,)j Fw(A)d FA(is)g(lo)m(w)g(for)h(the)g(Sc)m(hnorr)505 1552 y(n)m(ull)29 b(sets.)i(Th)m(us)e(w)m(e)i(ha)m(v)m(e)g(the)g(follo)m (wing)e(result.)588 1709 y FB(Cor)n(ollar)-6 b(y)35 b FA(11.11)h(\(Kjos-Hanssen,)15 b(Nies,)g(and)31 b(Stephan)15 b([62]\))p FB(.)46 b Fs(A)25 b(set)g(is)f(low)505 1817 y(for)34 b(Schnorr)g(r)-5 b(andomness)35 b(iff)d(it)h(is)f(low)i(for)f (the)g(Schnorr)h(nul)5 b(l)33 b(sets.)588 1974 y FA(W)-8 b(e)37 b(ha)m(v)m(e)f(seen)f(that)g Fw(K)7 b FA(-trivialit)m(y)33 b(is)h(equiv)-5 b(alen)m(t)35 b(to)g(lo)m(wness)g(for)f(1-random-)505 2082 y(ness.)c(Nies)f([103)q(])h(sho)m(w)m(ed)g(that)g Fw(K)7 b FA(-trivialit)m(y)28 b(already)h(follo)m(ws)f(from)h(the)h(w)m (eak)m(er)505 2190 y(h)m(yp)s(othesis)40 b(of)h(b)s(eing)e(in)g(Lo)m(w) r(\()p Fd(MLRand)o Fw(;)15 b Fd(CRand)p FA(\).)42 b(\(One)e (consequence)i(of)f(this)505 2298 y(result)31 b(is)f(that)i(ev)m(ery)g (set)g(that)g(is)e(lo)m(w)i(for)f(Kolmogoro)m(v-Lo)m(v)m(eland)i (randomness)505 2406 y(is)d Fw(K)7 b FA(-trivial.)29 b(Nothing)h(ab)s(out)g(this)f(class)h(of)h(sets)f(is)g(kno)m(wn)g(b)s (ey)m(ond)f(that.\))588 2563 y FB(Theorem)34 b FA(11.12)i(\(Nies)31 b([103)q(]\))p FB(.)46 b Fs(A)24 b(set)g(is)g FA(Lo)m(w)q(\()p Fd(MLRand)p Fw(;)15 b Fd(CRand)p FA(\))24 b Fs(iff)f(it)h(is)g Fw(K)7 b Fs(-)505 2671 y(trivial.)588 2828 y FB(Pr)n(oof)34 b(Sketch.)40 b FA(The)20 b(\\if)7 b(")21 b(direction)e(has)h(b)s(een)g (discussed)f(in)g(Section)h(8.)i(F)-8 b(or)505 2936 y(the)38 b(remaining)e(direction,)g(supp)s(ose)g(that)i Fw(A)f Fr(2)g FA(Lo)m(w)q(\()p Fd(MLRand)p Fw(;)15 b Fd(CRand)p FA(\).)38 b(Then)505 3044 y(for)31 b(ev)m(ery)g(set)h Fw(Z)7 b FA(,)30 b(if)g Fw(N)41 b FA(is)29 b(an)i Fw(A)p FA(-computable)g(martingale)f(that)h(succeeds)g(on)g Fw(Z)7 b FA(,)505 3152 y(then)25 b Fw(Z)42 b(=)-55 b Fr(2)25 b Fd(MLRand)o FA(.)g(W)-8 b(e)27 b(\014rst)d(sho)m(w)h(that)h (this)e(fact)i(implies)c(a)k(certain)f(condition)505 3260 y(on)34 b(\014nite)f(strings.)g(Let)i Fw(R)f FA(b)s(e)g(an)m(y)g (c.e.)h(op)s(en)f(set)g(suc)m(h)g(that)g Fw(\026)p FA(\()p Fw(R)q FA(\))e Fw(<)f FA(1)j(and)g(all)505 3368 y(sets)41 b(in)e(the)h(complemen)m(t)g(of)g Fw(R)h FA(are)f(1-random)g(\(for)g (instance,)g(let)g Fw(R)i FA(=)g Fr(f)p Fw(\034)51 b FA(:)505 3476 y Fr(9)p Fw(\033)34 b Fz(4)d Fw(\034)25 b FA([)p Fw(K)7 b FA(\()p Fw(\033)s FA(\))32 b Fz(6)e Fr(j)p Fw(\033)s Fr(j)23 b(\000)f FA(1])p Fr(g)p FA(\).)36 b(The)d(follo)m(wing)f(condition)h(expresses)g(the)h(failure)505 3583 y(to)e(build)27 b(a)k(set)41 b Fw(=)-55 b Fr(2)24 b Fw(R)31 b FA(on)g(whic)m(h)e Fw(N)40 b FA(succeeds.)588 3740 y FB(Lemma)34 b FA(11.13)p FB(.)48 b Fs(L)-5 b(et)33 b Fw(N)43 b Fs(b)-5 b(e)32 b(any)i(martingale)g(that)g(suc)-5 b(c)g(e)g(e)g(ds)35 b(only)e(on)h(sets)f(not)505 3848 y(in)g Fd(MLRand)o Fs(.)f(Then)h(ther)-5 b(e)34 b(ar)-5 b(e)33 b Fw(\033)c Fr(2)c FA(2)1832 3815 y Fx()d FA(2)2031 3957 y Fx(d)2098 3994 y Fr(\))h FA([)p Fw(\034)10 b FA(])26 b Fr(\022)f Fw(R)q FA(])p Fw(:)-2051 b FA(\(1\))588 4151 y FB(Pr)n(oof.)41 b FA(Otherwise)g(one)h(could)f(inductiv)m(ely)f (build)f(a)j(sequence)h(of)f(strings)505 4259 y Fw(\033)557 4273 y Fy(0)634 4259 y Fr(\036)36 b Fw(\033)793 4273 y Fy(1)869 4259 y Fr(\036)h(\001)15 b(\001)g(\001)53 b FA(suc)m(h)37 b(that)h([)p Fw(\033)1628 4273 y Fx(i)1656 4259 y FA(])f Fo(*)g Fw(R)h FA(and)f Fw(N)10 b FA(\()p Fw(\033)2287 4273 y Fx(i)2315 4259 y FA(\))37 b Fz(>)g FA(2)2540 4226 y Fx(i)2568 4259 y FA(,)h(whic)m(h)e(w)m(ould)g(imply) 505 4367 y(that)42 b Fw(N)50 b FA(succeeds)41 b(on)g(the)g(set)g Fw(Z)49 b FA(=)1891 4299 y Fp(S)1967 4394 y Fx(n)2029 4367 y Fw(\033)2081 4381 y Fx(n)2128 4367 y FA(,)41 b(whic)m(h)e(is)h (not)h(in)e Fw(R)q FA(,)i(since)f Fw(R)h FA(is)505 4475 y(op)s(en.)2619 b Fr(a)588 4601 y FA(Note)31 b(that)g Fw(\033)38 b(=)-55 b Fr(2)25 b Fw(R)30 b FA(implies)d(that)j(the)g (relativ)m(e)g(measure)g Fw(\026)2661 4615 y Fx(\033)2707 4601 y FA(\()p Fw(R)q FA(\))c(=)f(2)3014 4568 y Fq(\000j)p Fx(\033)r Fq(j)3155 4601 y Fw(\026)p FA(\()p Fw(R)20 b Fr(\\)505 4709 y FA([)p Fw(\033)s FA(]\))31 b(is)d(less)g(than)h(1)h (\(otherwise)e(let)i Fw(X)42 b(=)-55 b Fr(2)25 b Fw(R)30 b FA(b)s(e)e(a)i(set)g(extending)e Fw(\033)s FA(;)i(then)e Fw(X)37 b FA(is)28 b(a)505 4817 y(1-random)j(set)g(in)e(a)h(\005)1306 4784 y Fy(0)1306 4841 y(1)1346 4817 y FA(-class)h(of)f(measure)g(0,)h (whic)m(h)e(is)h(imp)s(ossible\).)588 4925 y(As)f(in)f(the)h(pro)s(of)g (of)g(Theorem)g(7.6,)h(w)m(e)f(w)m(an)m(t)h(to)g(en)m(umerate)g(a)f (Kraft-Chaitin)505 5033 y(set)h Fw(W)42 b FA(sho)m(wing)29 b(that)h Fw(A)f FA(is)g Fw(K)7 b FA(-trivial,)27 b(in)h(the)i(sense)f (that)h(for)g(eac)m(h)g Fw(n)f FA(there)h(is)e(a)505 5141 y(request)k Fr(h)p Fw(r)m(;)15 b(A)1034 5129 y Fz(\026)1099 5141 y Fw(n)p Fr(i)28 b(2)g Fw(W)44 b FA(with)31 b Fw(r)f Fz(6)d Fw(K)7 b FA(\()p Fw(n)p FA(\))22 b(+)f Fw(O)s FA(\(1\).)33 b(Recall)e(that)i(in)d(that)j(pro)s(of,)p eop %%Page: 58 58 58 57 bop 505 363 a FD(58)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(the)25 b(Kraft-Chaitin)e(set)i(\(whic)m(h)e(w)m(as)i(called)f Fw(L)2132 556 y Fx(d)2172 541 y FA(\))h(dep)s(ended)e(on)h(a)h(n)m(um)m (b)s(er)e Fw(d)i FA(suc)m(h)505 651 y(that)37 b(\012)43 b Fw(=)-55 b Fr(2)34 b Fw(R)972 618 y Fx(A)971 679 y(d)1029 651 y FA(.)h(This)f(time)i(the)f(Kraft-Chaitin)f(set)i Fw(W)48 b FA(dep)s(ends)34 b(on)i(a)g(witness)505 760 y Fr(h)p Fw(\033)n(;)15 b(d;)g(u)p Fr(i)26 b FA(for)d(Lemma)h(11.13,)h (where)e Fw(u)g FA(is)g(a)h(n)m(um)m(b)s(er)e(suc)m(h)h(that)h(2)2777 727 y Fq(\000)p Fx(u)2903 760 y Fz(6)h FA(1)6 b Fr(\000)g Fw(\026)3182 774 y Fx(\033)3229 760 y FA(\()p Fw(R)q FA(\).)505 869 y(W)-8 b(e)30 b(build)25 b(a)k(T)-8 b(uring)26 b(functional)h Fw(L)h FA(suc)m(h)g(that)h Fw(L)2232 836 y Fx(X)2328 869 y FA(is)e(a)i(martingale)e(for)h(eac)m(h)i(set)505 977 y Fw(X)7 b FA(.)40 b(W)-8 b(e)41 b(\014rst)e(pretend)f(that)i(w)m (e)g(kno)m(w)g(a)f(witness)g Fr(h)p Fw(\033)n(;)15 b(d;)g(u)p Fr(i)41 b FA(for)e(Lemma)h(11.13)505 1087 y(with)29 b Fw(N)36 b FA(=)25 b Fw(L)979 1054 y Fx(A)1036 1087 y FA(.)588 1195 y(W)-8 b(e)29 b(ha)m(v)m(e)g(to)f(appro)m(ximate)f(the)h (p)s(ossible)c(initial)h(segmen)m(ts)j Fw(A)2809 1183 y Fz(\026)2872 1195 y Fw(n)f FA(to)h(mak)m(e)g Fw(W)505 1303 y FA(c.e.,)j(and)e(need)g(to)h(b)s(e)e(careful)h(not)g(to)h(mak)m (e)g(to)s(o)g(man)m(y)g(errors,)f(since)f(otherwise)505 1410 y Fw(W)53 b FA(will)37 b(not)k(b)s(e)e(a)h(Kraft-Chaitin)f(set.)h (Roughly)g(sp)s(eaking,)f(w)m(e)h(w)m(ork)g(with)f(a)505 1518 y(computable)24 b(sequence)g(of)g(\014nite)f(trees)i Fr(f)p Fw(T)1992 1532 y Fx(s)2029 1518 y Fr(g)2074 1532 y Fx(s)p Fq(2)p Fx(!)2204 1518 y FA(,)g(where)e(at)i(eac)m(h)g(stage)g Fw(s)p FA(,)f(strings)505 1626 y Fw(\015)42 b FA(on)36 b Fw(T)779 1640 y Fx(s)852 1626 y FA(represen)m(t)g(the)g(p)s(ossible)e (initial)g(segmen)m(ts)j(of)f Fw(A)g FA(of)h(length)e Fz(6)g Fw(s)p FA(.)h(The)505 1734 y(tree)i Fw(T)747 1748 y Fx(s)822 1734 y FA(c)m(hec)m(ks)g(whether)f(condition)f(\(1\))j (holds)c(at)k(stage)f Fw(s)p FA(:)g(if)e(for)h(some)h Fw(\034)47 b FA(w)m(e)505 1844 y(ha)m(v)m(e)35 b(de\014ned)d Fw(L)1096 1811 y Fx(\015)1140 1844 y FA(\()p Fw(\034)10 b FA(\))31 b Fz(>)f FA(2)1437 1811 y Fx(d)1511 1844 y FA(for)j(a)h(string)f Fw(\015)38 b FA(on)33 b(a)h(previous)e(tree)i Fw(T)2890 1858 y Fx(t)2953 1844 y FA(with)e Fw(t)e(<)g(s)p FA(,)505 1952 y(then)g Fw(\015)36 b FA(is)29 b(only)h(allo)m(w)m(ed)g (to)h(b)s(e)f(on)g(the)h(presen)m(t)f(tree)h Fw(T)2478 1966 y Fx(s)2545 1952 y FA(if)f([)p Fw(\034)10 b FA(])26 b Fr(\022)f Fw(R)2920 1966 y Fx(s)2956 1952 y FA(.)588 2060 y(A)k Fs(pr)-5 b(o)g(c)g(e)g(dur)g(e)38 b Fw(\013)28 b FA(is)g(a)g(pair)f Fr(h)p Fw(\032;)15 b(\015)5 b Fr(i)p FA(,)30 b(where)e Fw(\032;)15 b(\015)30 b Fr(2)25 b FA(2)2341 2027 y Fx()e FA(2)3210 2789 y Fx(d)3286 2822 y FA(for)505 2930 y(eac)m(h)i Fw(X)i Fz(<)27 b Fw(\015)37 b FA(and)31 b(eac)m(h)h(string)f Fw(\034)37 b Fr(2)27 b Fw(C)38 b FA(of)32 b(minimal)d(length.)i(If)g (at)i(a)f(stage)h Fw(t)27 b(>)g(s)505 3038 y FA(w)m(e)e(once)g(again)g (ha)m(v)m(e)h Fw(\015)k Fr(2)25 b Fw(T)1491 3052 y Fx(t)1521 3038 y FA(,)f(then)g Fw(C)32 b Fr(\022)25 b Fw(R)2033 3052 y Fx(t)2063 3038 y FA(,)f(and)g Fw(\013)h FA(no)m(w)g(has)f(p)s (ermission)d(to)k(put)505 3145 y Fr(h)p Fw(r)14 b FA(+)d Fw(c;)k(\015)5 b Fr(i)27 b FA(in)m(to)f Fw(W)13 b FA(.)26 b(In)f(short,)h(the)g(w)m(eigh)m(t)g(of)g(requests)g(put)f(in)m(to)h Fw(W)38 b FA(is)25 b(accoun)m(ted)505 3253 y(against)30 b(the)g(measure)f(of)g(new)g(en)m(umerations)g(in)m(to)h Fw(R)q FA(.)f(If)g(the)g(sets)h(b)s(elonging)e(to)505 3361 y(di\013eren)m(t)i(pro)s(cedures)f(are)i(disjoin)m(t,)e(then)h Fw(W)43 b FA(is)29 b(a)i(Kraft-Chaitin)e(set.)588 3469 y(W)-8 b(e)45 b(discuss)d(ho)m(w)i(to)h(guaran)m(tee)g(this)e(disjoin)m (tness.)f(Supp)s(ose)g Fw(\014)53 b Fr(6)p FA(=)47 b Fw(\013)d FA(is)f(a)505 3577 y(pro)s(cedure)33 b(that)i(w)m(an)m(ts)g (to)g(c)m(ho)s(ose)g(its)f(set)g Fw(C)7 b FA(\()p Fw(\014)e FA(\))35 b(at)g(a)f(stage)i Fw(s)2784 3544 y Fq(0)2838 3577 y Fw(>)c(s)p FA(.)i(If)f Fw(\015)40 b FA(is)33 b(in)505 3685 y(some)e Fw(T)786 3699 y Fx(q)855 3685 y FA(with)f Fw(s)25 b(<)g(q)k(<)c(s)1436 3652 y Fq(0)1490 3685 y FA(,)31 b(then)f Fw(C)7 b FA(\()p Fw(\013)p FA(\))26 b Fr(\022)g Fw(R)2145 3700 y Fx(s)2178 3681 y Fj(0)2204 3685 y FA(,)31 b(so)g(there)g(is)e(no)i(problem,)e(since)505 3793 y Fw(\014)34 b FA(c)m(ho)s(oses)28 b(its)g(set)g(disjoin)m(t)f (from)g Fw(R)1778 3808 y Fx(s)1811 3789 y Fj(0)1838 3793 y FA(.)h(Ho)m(w)m(ev)m(er,)i(if)d Fw(\015)33 b FA(has)28 b(not)g(app)s(eared)f(in)g(an)m(y)505 3901 y(suc)m(h)36 b(tree)h(\(and)f(it)f(p)s(ossibly)f(nev)m(er)i(will\),)e(then)i Fw(\013)g FA(w)m(an)m(ts)h(to)g(k)m(eep)g Fw(C)7 b FA(\()p Fw(\013)p FA(\))37 b(a)m(w)m(a)m(y)505 4009 y(from)27 b(p)s(ossible)e(assignmen)m(t)i(to)h(other)f(pro)s(cedures,)g(whic)m(h) f(ma)m(y)h(cause)h(a)g(con\015ict)505 4117 y(b)s(ecause)j Fw(C)7 b FA(\()p Fw(\013)p FA(\))32 b(is)e(relativ)m(ely)h(large.)g (The)g(solution)e(to)j(this)e(problem)g(is)g(to)i(build)505 4225 y(up)k(the)g(set)h Fw(C)7 b FA(\()p Fw(\013)p FA(\))38 b(in)d(small)g(p)s(ortions)g Fw(D)s FA(,)h(whose)g(measure)h(is)e(a)i (\014xed)f(fraction)505 4339 y(of)k(2)663 4306 y Fq(\000)p Fy(\()p Fx(r)r Fy(+)p Fx(c)p Fy(\))896 4339 y FA(,)f(and)g(only)f (assign)g(a)h(new)g(set)g Fw(D)j FA(once)e(the)f(old)f(one)h(is)f(in)g Fw(R)q FA(.)h(If)f Fw(\015)505 4446 y FA(alw)m(a)m(ys)g(reapp)s(ears)e (on)g(a)h(tree)h(after)f(suc)m(h)g(a)g(set)g(is)f(assigned,)g(then)h (ev)m(en)m(tually)505 4555 y Fw(C)7 b FA(\()p Fw(\013)p FA(\))31 b(reac)m(hes)g(the)g(required)d(measure)i(2)1964 4522 y Fq(\000)p Fy(\()p Fx(r)r Fy(+)p Fx(c)p Fy(\))2198 4555 y FA(,)g(in)f(whic)m(h)g(case)i Fw(\013)g FA(is)e(allo)m(w)m(ed)h (to)505 4663 y(en)m(umerate)36 b(the)g(request)f Fr(h)p Fw(r)26 b FA(+)d Fw(c;)15 b(y)s Fr(i)36 b FA(in)m(to)f Fw(W)13 b FA(.)35 b(Otherwise,)f Fw(\013)h FA(k)m(eeps)h(a)m(w)m(a)m(y) h(from)505 4771 y(assignmen)m(t)32 b(to)g(other)g(pro)s(cedures)f(only) f(a)i(single)f(set)h Fw(D)s FA(,)g(whose)f(measure)h(is)e(so)505 4879 y(small)37 b(that)i(the)f(union)e(\(o)m(v)m(er)k(all)d(pro)s (cedures\))g(of)h(the)g(measures)g(of)g(sets)h(k)m(ept)505 4987 y(a)m(w)m(a)m(y)33 b(in)c(this)g(fashion)g(is)h(at)h(most)f(the)h (small)e(quan)m(tit)m(y)h(2)2555 4954 y Fq(\000)p Fx(u)p Fq(\000)p Fy(2)2746 4987 y FA(.)p eop %%Page: 59 59 59 58 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(59)588 541 y FA(T)-8 b(o)36 b(ensure)d(that)j Fw(L)1278 508 y Fx(X)1345 541 y FA(\()p Fw(\034)10 b FA(\))33 b Fz(>)f FA(2)1646 508 y Fx(d)1721 541 y FA(for)j(eac)m(h)h Fw(X)j Fz(<)32 b Fw(\015)5 b FA(,)35 b(the)g(pro)s(cedure)f Fw(\013)e FA(=)g Fr(h)p Fw(\032;)15 b(\015)5 b Fr(i)505 649 y FA(acts)34 b(as)g(follo)m(ws.)e(Once)h Fw(U)1433 663 y Fx(s)1470 649 y FA(\()p Fw(\032)p FA(\))d(=)f Fr(j)p Fw(\015)5 b Fr(j)p FA(,)34 b(it)f(claims)f Fw(")e FA(=)f(2)2470 616 y Fq(\000)p Fx(r)2596 649 y FA(of)k(the)h(initial)c(capital)505 757 y(1)37 b(of)g Fw(L)f FA(at)i(the)e(ro)s(ot)h(no)s(de)f Fw(\025)h FA(\(recall)f(that)h Fw(r)h FA(=)d Fr(j)p Fw(\032)p Fr(j)p FA(\),)j(and)e(generally)f(preserv)m(es)505 865 y(it)g(along)h(b)s(oth)f(extensions)g(of)g(a)h(string.)f(It)h(c)m(ho)s (oses)g(its)f(strings)f Fw(\034)46 b FA(of)35 b(the)h(form)505 973 y Fw(\027)6 b FA(0)601 940 y Fx(r)r Fy(+)p Fx(d)p Fy(+1)821 973 y FA(,)31 b(where)f Fw(\027)36 b FA(is)29 b(a)i(string)f(where)g(the)g(capital)h(claimed)e(b)m(y)i Fw(\013)f FA(is)g(still)e(a)m(v)-5 b(ail-)505 1081 y(able.)38 b(A)m(t)g Fw(\027)43 b FA(it)36 b(\\withdra)m(ws")h(this)f(capital,)h (b)m(y)g(de\014ning)f Fw(L)2649 1048 y Fx(\015)2693 1081 y FA(\()p Fw(\027)6 b FA(0\))38 b(=)e Fw(L)3066 1048 y Fx(\015)3110 1081 y FA(\()p Fw(\027)6 b FA(\))26 b(+)e Fw(")505 1189 y FA(and,)33 b(to)g(main)m(tain)e(the)i(martingale)f (prop)s(ert)m(y)-8 b(,)33 b Fw(L)2276 1156 y Fx(\015)2320 1189 y FA(\()p Fw(\027)6 b FA(1\))30 b(=)e Fw(L)2677 1156 y Fx(\015)2722 1189 y FA(\()p Fw(\027)6 b FA(\))22 b Fr(\000)f Fw(")p FA(.)33 b(F)-8 b(rom)33 b Fw(\027)6 b FA(0)505 1297 y(on,)26 b(it)f(doubles)f(the)h(capital)h(along)f Fw(\034)10 b FA(,)26 b(alw)m(a)m(ys)g(b)s(etting)e(all)h(the)g(capital) g(on)h(0,)g(th)m(us)505 1405 y(ev)m(en)m(tually)31 b(reac)m(hing)f(an)g (increase)h(of)f(2)1920 1372 y Fx(d)1991 1405 y FA(at)h Fw(\034)10 b FA(.)588 1513 y(Di\013eren)m(t)27 b(pro)s(cedures)e Fr(h)p Fw(\032;)15 b(\015)5 b Fr(i)27 b FA(and)f Fr(h)p Fw(\032)1907 1480 y Fq(0)1930 1513 y Fw(;)15 b(\015)2022 1480 y Fq(0)2046 1513 y Fr(i)26 b FA(ha)m(v)m(e)i(to)f(c)m(ho)s(ose)g (their)e(sets)h Fw(D)j FA(to)e(b)s(e)505 1621 y(disjoin)m(t,)i(but)g (there)i(is)e(no)h(con\015ict)g(as)g(far)g(as)g(the)g(capital)g(is)f (concerned:)i(if)e Fw(\015)5 b(;)15 b(\015)3371 1588 y Fq(0)505 1729 y FA(are)32 b(incompatible,)d(then)i(they)g(refer)f(to) i(di\013eren)m(t)e(martingales)g Fw(L)2871 1696 y Fx(X)2939 1729 y FA(.)h(Otherwise)505 1837 y(they)h(claim)e(di\013eren)m(t)g (amoun)m(ts)h(of)h(the)f(initial)d(capital.)j(F)-8 b(or)32 b(an)m(y)f Fw(L)2927 1804 y Fx(X)2994 1837 y FA(,)h(the)f(total)505 1944 y(capital)38 b(claimed)f(is)h(at)h(most)f(\012,)g(as)h(a)f(pro)s (cedure)f Fr(h)p Fw(\032;)15 b(\015)5 b Fr(i)40 b FA(with)d Fw(\015)43 b Fr(\036)38 b Fw(X)46 b FA(claims)505 2058 y(2)550 2025 y Fq(\000j)p Fx(\032)p Fq(j)717 2058 y FA(m)m(uc)m(h)31 b(of)h(the)g(capital)f(only)g(once)h Fw(U)10 b FA(\()p Fw(\032)p FA(\))32 b(con)m(v)m(erges,)i(and)d(there)g(is)g(at)h(most) 505 2166 y(one)f(suc)m(h)f(pro)s(cedure)f(for)h(eac)m(h)i Fw(\032)p FA(.)588 2274 y(Finally)-8 b(,)40 b(as)i(a)f(witness)f(for)h (Lemma)g(11.13)j(is)c(not)h(actually)g(kno)m(wn,)g(w)m(e)h(do)505 2382 y(the)d(ab)s(o)m(v)m(e)h(for)e(eac)m(h)i(p)s(ossible)c(witness)h Fr(h)p Fw(\033)n(;)15 b(d;)g(u)p Fr(i)p FA(.)41 b(Let)e Fr(fh)p Fw(\033)2641 2396 y Fx(m)2709 2382 y Fw(;)15 b(d)2796 2396 y Fx(m)2863 2382 y Fw(;)g(u)2955 2396 y Fx(m)3022 2382 y Fr(ig)3102 2396 y Fx(m)p Fq(2)p Fx(!)3301 2382 y FA(b)s(e)505 2490 y(an)36 b(e\013ectiv)m(e)i(listing)33 b(of)j(suc)m(h)g(witnesses.)f(F)-8 b(or)36 b(eac)m(h)h Fw(m)f FA(w)m(e)g(build)d(a)j(martingale)505 2598 y(functional)g Fw(L)1001 2565 y Fx(X)1001 2620 y(m)1106 2598 y FA(as)i(ab)s(o)m(v)m (e,)h(but)e(no)m(w)h(with)e(initial)f(capital)i(2)2720 2565 y Fq(\000)p Fx(m)2842 2598 y FA(,)h(and)f(mak)m(e)i(it)505 2706 y(ev)m(en)m(tually)23 b(constan)m(t)g(if)f(it)f(turns)h(out)g (that)h Fw(\026)2076 2720 y Fx(\033)2116 2728 y Fl(m)2179 2706 y FA(\()p Fw(R)2283 2720 y Fx(s)2320 2706 y FA(\))j Fw(>)f FA(1)t Fr(\000)t FA(2)2646 2673 y Fq(\000)p Fx(u)2742 2681 y Fl(m)2827 2706 y FA(for)d(some)h Fw(s)p FA(.)f(W)-8 b(e)505 2816 y(no)m(w)32 b(ha)m(v)m(e)g(to)g(c)m(ho)s(ose)h Fw(\034)41 b FA(of)31 b(the)h(form)f Fw(\027)6 b FA(0)1963 2783 y Fx(m)p Fy(+)p Fx(r)r Fy(+)p Fx(d)p Fy(+1)2331 2816 y FA(to)32 b(mak)m(e)g(up)e(for)h(the)h(smaller)505 2924 y(capital.)38 b(No)m(w)g(simply)e(let)h Fw(L)h FA(=)1695 2856 y Fp(P)1791 2951 y Fx(m)1872 2924 y Fw(L)1934 2938 y Fx(m)2001 2924 y FA(.)g(Since)e Fw(A)i Fr(2)f FA(Lo)m(w)r(\()p Fd(MLRand)o Fw(;)15 b Fd(CRand)p FA(\),)505 3038 y(Lemma)42 b(11.13)h(holds)d(for)h Fw(N)54 b FA(=)43 b Fw(L)1791 3005 y Fx(A)1848 3038 y FA(,)e(via)g(a)h(witness)e Fr(h)p Fw(\033)2578 3052 y Fx(m)2645 3038 y Fw(;)15 b(d)2732 3052 y Fx(m)2799 3038 y Fw(;)g(u)2891 3052 y Fx(m)2958 3038 y Fr(i)p FA(.)42 b(No)m(w)g(\(1\))505 3147 y(holds)30 b(for)g Fw(N)36 b FA(=)26 b Fw(L)1151 3114 y Fx(A)1151 3169 y(m)1217 3147 y FA(,)31 b(since)f Fw(L)1558 3114 y Fx(A)1641 3147 y Fz(>)25 b Fw(L)1799 3114 y Fx(A)1799 3169 y(m)1866 3147 y FA(.)31 b(Th)m(us)e(eac)m(h)j Fw(\015)f Fr(\036)26 b Fw(A)31 b FA(reapp)s(ears)f(in\014nitely)505 3255 y(often)i(on)g(then)f(trees)h Fw(T)1344 3269 y Fx(s)1381 3255 y FA(,)g(and)f(so)h(the)f(\\accoun)m(ting)i(against")g(tric)m(k)e (outlined)f(in)505 3363 y(the)d(pro)s(of)e(of)h(Theorem)g(7.6)i(allo)m (ws)d(us)g(to)i(de\014ne)f(the)g(desired)f(Kraft-Chaitin)f(set)505 3471 y Fw(W)43 b FA(\(where)30 b(the)h(constan)m(t)h Fw(c)e FA(is)g Fw(d)1664 3485 y Fx(m)1751 3471 y FA(+)20 b Fw(m)g FA(+)g Fw(u)2085 3485 y Fx(m)2171 3471 y FA(+)g(3\).)972 b Fr(a)588 3668 y FA(W)-8 b(e)23 b(ha)m(v)m(e)f(seen)g(c)m (haracterizations)g(of)f(lo)m(wness)g(for)g(1-randomness)g(and)g(Sc)m (hnorr)505 3776 y(randomness.)26 b(These)f(results)g(raise)h(the)g (question)f(of)i(c)m(haracterizing)f(lo)m(wness)g(for)505 3884 y(computable)40 b(randomness.)f(In)g([1)q(,)h(Problem)f(4.8],)j (Am)m(b)s(os-Spies)c(and)i(Ku)m(\024)-43 b(cera)505 3992 y(ask)m(ed)37 b(whether)f(there)g(is)f(a)i(noncomputable)e(set)h(that)h (is)e(lo)m(w)h(for)g(computable)505 4100 y(randomness.)j(Do)m(wney)h (conjectured)f(that)h(the)f(answ)m(er)g(is)f(negativ)m(e,)j(and)e(this) 505 4208 y(conjecture)32 b(w)m(as)e(con\014rmed)g(b)m(y)g(Nies)g([103)r (].)588 4458 y FB(Theorem)k FA(11.14)i(\(Nies)31 b([103)q(]\))p FB(.)46 b Fs(A)34 b(set)f(is)h(low)h(for)f(c)-5 b(omputable)36 b(r)-5 b(andomness)505 4566 y(iff)33 b(it)f(is)h(c)-5 b(omputable.)588 4817 y FA(The)40 b(pro)s(of)f(in)g([103)q(])h(is)f(a)i (direct)e(argumen)m(t)h(similar)e(to)i(but)f(preceding)g(the)505 4925 y(pro)s(of)e(of)g(Theorem)g(11.12)i(discussed)c(ab)s(o)m(v)m(e.)k (But)e(w)m(e)h(can)f(also)g(use)g(Theorem)505 5033 y(11.12:)48 b(Supp)s(ose)c(that)i Fw(A)g FA(is)e(lo)m(w)i(for)f(computable)g (randomness.)g(Then)f Fw(A)51 b Fr(2)505 5141 y FA(Lo)m(w)r(\()p Fd(MLRand)p Fw(;)15 b Fd(CRand)o FA(\),)44 b(and)d(hence)i Fw(A)g FA(is)e Fw(K)7 b FA(-trivial,)41 b(and)h(th)m(us)h(\001)2998 5108 y Fy(0)2998 5165 y(2)3037 5141 y FA(.)g(On)e(the)p eop %%Page: 60 60 60 59 bop 505 363 a FD(60)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(other)36 b(hand,)e(b)m(y)h(a)h(result)e(of)h(Bedregal)g(and)g (Nies)g([11)q(],)h Fw(A)f FA(has)g(h)m(yp)s(erimm)m(une-)505 649 y(free)43 b(degree.)h(But)f(the)g(only)f(\001)1663 616 y Fy(0)1663 674 y(2)1745 649 y FA(sets)h(of)g(h)m(yp)s(erimm)m (une-free)e(degree)i(are)g(the)505 757 y(computable)30 b(ones,)h(b)m(y)f(Miller)f(and)h(Martin)f([97)q(].)588 865 y(The)24 b(original)e(pro)s(of)h(has)g(the)h(adv)-5 b(an)m(tage)26 b(of)e(b)s(eing)e(extendible)h(to)h(the)g(resource)505 973 y(b)s(ounded)g(setting,)i(and)e(also)i(to)g(sho)m(w)f(that)h(eac)m (h)h(set)f(in)e(Lo)m(w)r(\()p Fd(PrecRand)p Fw(;)15 b Fd(CRand)p FA(\))505 1081 y(is)43 b(computable.)g(Here)g Fd(PrecRand)h FA(is)e(the)h(class)g(of)g(sets)h(on)f(whic)m(h)f(no)h (partial)505 1189 y(computable)24 b(martingale)g(succeeds)g(\(i.e.,)h (no)f(martingale)g(whose)g(v)-5 b(alues)23 b(are)i(uni-)505 1297 y(formly)36 b(computable,)g(but)g(that)h(ma)m(y)h(c)m(ho)s(ose)f (to)h(b)s(e)e(unde\014ned)e(on)j(strings)e(o\013)505 1405 y(the)c(giv)m(en)f(set\).)588 1629 y Fu(x)p Ft(12.)53 b(Relativized)25 b(randomness.)46 b FA(W)-8 b(e)22 b(ha)m(v)m(e)h(so)f (far)f(fo)s(cused)g(on)g(1-random-)505 1737 y(ness)28 b(and)g(w)m(eak)m(er)h(notions.)f(W)-8 b(e)29 b(can)g(also)f(obtain)g (stronger)g(notions)g(of)g(random-)505 1845 y(ness)40 b(b)m(y)h(increasing)e(the)h(complexit)m(y)g(of)h(tests)g(in)e(terms)h (of)h(the)g(arithmetical)505 1953 y(hierarc)m(h)m(y)-8 b(.)32 b(These)e(notions)h(are)g(of)g(particular)e(in)m(terest)j(when)e (w)m(e)h(study)f(the)h(re-)505 2061 y(lationship)25 b(b)s(et)m(w)m(een) j(randomness)e(and)g(T)-8 b(uring)26 b(degrees.)i(The)e(basic)h(result) f(con-)505 2169 y(necting)38 b(1-randomness)f(to)h(T)-8 b(uring)36 b(reducibilit)m(y)e(is)i(the)i(celebrated)f(one)h(often)505 2277 y(attributed)30 b(only)g(to)h(G\023)-45 b(acs,)31 b(but)f(actually)g(\014rst)f(pro)m(v)m(ed)i(b)m(y)f(Ku)m(\024)-43 b(cera.)588 2453 y FB(Theorem)34 b FA(12.1)i(\(Ku)m(\024)-43 b(cera)32 b([66)q(],)e(G\023)-45 b(acs)32 b([46)q(]\))p FB(.)46 b Fs(Every)d(set)g(is)g(wtt-r)-5 b(e)g(ducible)44 b(to)505 2561 y(a)33 b FA(1)p Fs(-r)-5 b(andom)35 b(set.)505 2737 y FA(The)i(easiest)h(pro)s(of)e(of)i(Theorem)f(12.1)h(is)f(the)g (recen)m(t)h(one)g(of)f(Merkle)g(and)g(Mi-)505 2845 y(hailo)m(vi)m (\023)-43 b(c)32 b([89)q(].)f(That)g(pro)s(of)g(sho)m(ws)g(that)h(the)f (b)s(ound)e(on)i(the)g(wtt-reduction)g(can)505 2953 y(b)s(e)42 b(tak)m(en)h(to)g(b)s(e)e Fw(n)28 b FA(+)f Fw(o)p FA(\()p Fw(n)p FA(\).)43 b(Ho)m(w)m(ev)m(er,)h(this)d(b)s(ound)f(cannot)j(b)s (e)e(impro)m(v)m(ed)h(to)505 3061 y Fw(n)21 b FA(+)f Fw(O)s FA(\(1\);)32 b(that)f(is,)g(there)g(are)g(sets)g(that)h(are)f (not)g(sw-reducible)e(to)i(a)h(1-random)505 3169 y(set)44 b(\(see)h([33)q(])f(for)f(a)h(pro)s(of)7 b(\).)43 b(It)h(is)e(an)i(op)s (en)f(question)f(whether)h(ev)m(ery)h(set)g(is)505 3277 y(rK-reducible)28 b(\(or)j(ev)m(en)g Fw(K)7 b FA(-reducible\))28 b(to)k(a)e(1-random)h(set.)588 3385 y(Other)g(results)f(on)h(the)h(T)-8 b(uring)29 b(degrees)j(of)g(1-random)f(sets)g(include)e(Ku)m(\024)-43 b(cera's)505 3493 y(theorem)28 b([66)q(])g(that)f(all)g(degrees)g(ab)s (o)m(v)m(e)i Ft(0)1964 3460 y Fq(0)2015 3493 y FA(con)m(tain)e (1-random)g(sets.)h(As)f(noted)h(in)505 3601 y(Section)g(4,)g(for)g (eac)m(h)h Fw(c)p FA(,)f(the)g(collection)g(of)g(sets)g(with)e(initial) f(segmen)m(t)k(pre\014x-free)505 3708 y(complexit)m(y)40 b Fz(>)i Fw(n)26 b Fr(\000)h Fw(c)41 b FA(for)f(all)f Fw(n)h FA(is)f(a)i(\005)1984 3675 y Fy(0)1984 3733 y(1)2023 3708 y FA(-class,)g(so)g(there)f(are)h(1-random)f(sets)505 3817 y(of)32 b(lo)m(w)e(T)-8 b(uring)30 b(degree.)i(Ku)m(\024)-43 b(cera)32 b([66)q(])f(pro)m(v)m(ed)g(that)h(the)f(\001)2593 3784 y Fy(0)2593 3841 y(2)2663 3817 y FA(degrees)h(con)m(taining)505 3925 y(1-random)38 b(sets)g(are)g(not)g(closed)f(up)m(w)m(ards.)g(On)g (the)h(other)g(hand,)f(their)f(jumps)505 4033 y(are)44 b(b)s(etter)f(b)s(eha)m(v)m(ed:)h(using)e(a)h(new)g(basis)f(theorem)i (for)f(\005)2707 4000 y Fy(0)2707 4057 y(1)2746 4033 y FA(-classes)h(with)e(no)505 4141 y(computable)e(mem)m(b)s(ers,)f(Do)m (wney)i(and)e(Miller)f([41)q(])i(pro)m(v)m(ed)h(that)f(for)g(ev)m(ery)h Fw(S)505 4249 y FA(that)31 b(is)f(c.e.)h(in)e(and)h(ab)s(o)m(v)m(e)i Fr(;)1541 4216 y Fq(0)1565 4249 y FA(,)e(there)h(is)e(a)i(\001)2096 4216 y Fy(0)2096 4273 y(2)2165 4249 y FA(1-random)g(set)g Fw(A)f FA(with)f Fw(A)3090 4216 y Fq(0)3139 4249 y Fr(\021)3210 4263 y Fy(T)3290 4249 y Fw(S)5 b FA(.)588 4357 y(Ku)m(\024)-43 b(cera)23 b(also)e(observ)m(ed)g(that)h(1-randomness)e(is)h(connected)h (to)g(the)f(P)-8 b(A-degrees,)505 4464 y(whic)m(h)29 b(w)m(ere)i(discussed)e(in)g(Section)h(10.2.)i(This)c(connection)j(w)m (as)g(recen)m(tly)f(clari-)505 4572 y(\014ed)g(b)m(y)g(Stephan)g([130)r (],)g(who)g(pro)m(v)m(ed)h(the)g(follo)m(wing.)588 4749 y FB(Theorem)j FA(12.2)i(\(Stephan)30 b([130)q(]\))p FB(.)46 b Fs(If)34 b Fw(X)40 b Fs(is)33 b FA(1)p Fs(-r)-5 b(andom)36 b(and)e(has)h(P)-7 b(A-de)i(gr)g(e)g(e,)505 4857 y(then)34 b Fr(;)753 4824 y Fq(0)802 4857 y Fz(6)873 4871 y Fi(T)953 4857 y Fw(X)7 b Fs(.)588 5033 y FA(The)27 b(follo)m(wing)e(is)h(another)g(result)g(demonstrating)g(the)h (computational)f(w)m(eak-)505 5141 y(ness)k(of)h(the)f(1-random)h(sets) g(that)g(cannot)g(compute)f Fr(;)2454 5108 y Fq(0)2478 5141 y FA(.)p eop %%Page: 61 61 61 60 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(61)588 541 y FB(Theorem)34 b FA(12.3)i(\(Hirsc)m(hfeldt,)29 b(Nies,)i(and)e(Stephan)h([50)q(]\))p FB(.)46 b Fs(Supp)-5 b(ose)49 b Fw(A)f Fs(is)g(a)505 649 y(c.e.)41 b(set)h(and)h Fw(X)49 b Fz(>)1218 663 y Fi(T)1314 649 y Fw(A)42 b Fs(is)f FA(1)p Fs(-r)-5 b(andom)44 b(and)f(such)e(that)i Fr(;)2589 616 y Fq(0)2655 649 y Fo(\012)2726 663 y Fi(T)2822 649 y Fw(X)7 b Fs(.)42 b(Then)g Fw(A)g Fs(is)505 757 y Fw(K)7 b Fs(-trivial.)588 913 y FA(Theorems)28 b(12.2)i(and)d(12.3)j (establish)c(that)j(there)f(are)h(t)m(w)m(o)g(kinds)d(of)j(1-random)505 1021 y(sets.)35 b(The)e(\014rst)g(are)h(those)h(that)f(are)g (computationally)f(ric)m(h)g(and)g(can)h(compute)505 1129 y(the)24 b(halting)e(problem.)f(The)i(second)g(are)h(those)g(that) f(are)h(computationally)e(feeble)505 1237 y(and)39 b(cannot)h(ev)m(en)g (compute)g(a)f Fr(f)p FA(0)p Fw(;)15 b FA(1)p Fr(g)p FA(-v)-5 b(alued)41 b(\014xed-p)s(oin)m(t-free)e(function)f(or)h(a)505 1345 y(non-)p Fw(K)7 b FA(-trivial)26 b(c.e.)j(set.)f(As)f(w)m(e)h(see) g(b)s(elo)m(w,)f(this)g(means)g(that)h(all)f(2-random)g(sets)505 1453 y(\(de\014ned)j(b)s(elo)m(w\))g(are)h(computationally)e(w)m(eak.) 588 1561 y(The)40 b(basic)f(de\014nition)e(of)j(1-randomness)f(can)h(b) s(e)f(generalized)h(quite)f(easily)-8 b(.)505 1669 y(W)g(e)38 b(will)33 b(use)j(the)g(follo)m(wing)f(de\014nitions,)e(noted)k(b)m(y)f (sev)m(eral)g(researc)m(hers,)h(suc)m(h)505 1777 y(as)31 b(Solo)m(v)-5 b(a)m(y)31 b([126)r(])f(and)g(Kurtz)g([71)q(].)588 1933 y FB(Definition)35 b FA(12.4)p FB(.)105 b FA(\(i\))41 b(A)e(\006)1707 1900 y Fy(0)1707 1956 y Fx(n)1753 1933 y Fs(-test)47 b FA(is)37 b(a)h(sequence)h Fr(f)p Fw(V)2637 1948 y Fx(k)2680 1933 y Fr(g)2725 1948 y Fx(k)r Fq(2)p Fx(!)2899 1933 y FA(of)f(uniformly)701 2043 y(\006)767 2010 y Fy(0)767 2066 y Fx(n)813 2043 y FA(-classes)g(suc)m(h)f(that)h Fw(\026)p FA(\()p Fw(V)1699 2058 y Fx(k)1742 2043 y FA(\))g Fz(6)e FA(2)1967 2010 y Fq(\000)p Fx(k)2065 2043 y FA(.)i(A)g(set)g Fw(A)f FA(passes)h(this)e(test)i(if)f Fw(A)47 b(=)-55 b Fr(2)701 2083 y Fp(T)776 2178 y Fx(k)834 2151 y Fw(V)887 2166 y Fx(k)930 2151 y FA(.)538 2262 y(\(ii\))41 b(A)30 b(set)h(is)e(\006)1098 2229 y Fy(0)1098 2285 y Fx(n)1145 2262 y Fs(-r)-5 b(andom)39 b FA(or)31 b Fw(n)p Fs(-r)-5 b(andom)39 b FA(if)29 b(it)h(passes)g(all)f(\006)2688 2229 y Fy(0)2688 2285 y Fx(n)2765 2262 y FA(tests.)513 2370 y(\(iii\))40 b(One)30 b(can)g(similarly)d(de\014ne)j(\005)1762 2337 y Fy(0)1762 2393 y Fx(n)1809 2370 y FA(,)h(\001)1941 2337 y Fy(0)1941 2393 y Fx(n)1987 2370 y FA(,)g(etc.)h(tests)f(and)e (randomness.)515 2478 y(\(iv\))42 b(A)30 b(set)h(is)e Fs(arithmetic)-5 b(al)5 b(ly)35 b(r)-5 b(andom)40 b FA(if)30 b(it)g(is)f Fw(n)p FA(-random)g(for)i(all)e Fw(n)p FA(.)588 2634 y(These)j(de\014nitions)e(can)i(b)s(e)g(relativized)f(in)f(the)j (same)f(w)m(a)m(y)h(as)g(1-randomness,)505 2742 y(to)f(yield)c(notions) i(suc)m(h)g(as)h Fw(n)p FA(-randomness)e(relativ)m(e)h(to)h(a)g(set)g Fw(X)7 b FA(.)588 2850 y(W)-8 b(e)27 b(ha)m(v)m(e)f(iden)m(ti\014ed)d (\006)1401 2817 y Fy(0)1401 2874 y(1)1440 2850 y FA(-classes)i(of)g (sets)h(with)e(c.e.)i(sets)f(of)g(strings,)f(since)h(ev)m(ery)505 2959 y(\006)571 2926 y Fy(0)571 2983 y(1)611 2959 y FA(-class)38 b(is)f(equiv)-5 b(alen)m(t)38 b(to)1520 2890 y Fp(S)1595 2959 y Fr(f)p FA([)p Fw(\033)s FA(])i(:)e Fw(\033)k Fr(2)c Fw(W)13 b Fr(g)38 b FA(for)g(some)g(\(pre\014x-free\))h(c.e.)g(set)505 3066 y(of)j(strings)e Fw(W)13 b FA(.)41 b(Ho)m(w)m(ev)m(er,)i(w)m(e)f (cannot)g(do)f(the)h(same)f(at)h(higher)e(lev)m(els)h(of)g(the)505 3174 y(arithmetical)d(hierarc)m(h)m(y)-8 b(.)39 b(F)-8 b(or)40 b(example,)e(consider)g(the)h(\006)2602 3141 y Fy(0)2602 3199 y(2)2641 3174 y FA(-class)g(consisting)e(of)505 3283 y(those)i(sets)g(that)g(are)g(zero)h(from)e(some)h(p)s(oin)m(t)e (on)m(w)m(ards.)i(This)e(\006)2877 3250 y Fy(0)2877 3307 y(2)2916 3283 y FA(-class)h(is)g Fs(not)505 3391 y FA(equiv)-5 b(alen)m(t)33 b(to)h(one)f(of)g(the)g(form)1706 3323 y Fp(S)1782 3391 y Fr(f)p FA([)p Fw(\033)s FA(])e(:)f Fw(\033)i Fr(2)e Fw(W)13 b Fr(g)33 b FA(for)f(some)i(\006)2808 3358 y Fy(0)2808 3416 y(2)2880 3391 y FA(set)f(of)h(strings)505 3499 y Fw(W)13 b FA(.)588 3607 y(The)20 b(use)g(of)h(op)s(en)f(sets)g (is)g(basic)g(in)f(man)m(y)h(argumen)m(ts)h(in)m(v)m(olving)e (1-randomness.)505 3715 y(F)-8 b(ortunately)g(,)37 b(this)e(tec)m (hnique)g(can)h(b)s(e)f(resurrected)g(for)g(higher-order)f(random-)505 3824 y(ness,)d(as)f(w)m(e)h(no)m(w)f(see.)i(W)-8 b(e)31 b(denote)g(the)g Fw(n)p FA(-th)f(jump)f(of)h Fr(;)h FA(b)m(y)f Fr(;)2723 3791 y Fy(\()p Fx(n)p Fy(\))2825 3824 y FA(.)588 3980 y FB(Theorem)k FA(12.5)i(\(Kurtz)30 b([71)q(],)h(Kautz)g([60)q (]\))p FB(.)563 4105 y FA(\(i\))42 b Fs(F)-7 b(r)i(om)29 b(the)f(index)g(of)g(a)g FA(\006)1560 4072 y Fy(0)1560 4128 y Fx(n)1607 4105 y Fs(-class)g Fw(S)k Fs(and)d Fw(q)f Fr(2)d Fo(Q)9 b Fs(,)33 b(we)28 b(c)-5 b(an)28 b(c)-5 b(ompute)29 b(the)f(index)701 4232 y(of)j(a)h FA(\006)950 4197 y Fq(;)985 4174 y Fn(\()p Fl(n)p Fj(\000)p Fn(1\))950 4258 y Fy(1)1158 4232 y Fs(-class)f Fw(U)k Fr(\023)25 b Fw(S)36 b Fs(that)d(is)e(also)i(an)e(op)-5 b(en)33 b FA(\006)2565 4199 y Fy(0)2565 4255 y Fx(n)2611 4232 y Fs(-class)f(and)g(such)f(that)701 4340 y Fw(\026)p FA(\()p Fw(U)10 b FA(\))20 b Fr(\000)g Fw(\026)p FA(\()p Fw(S)5 b FA(\))26 b Fw(<)f(q)s Fs(.)538 4448 y FA(\(ii\))41 b Fs(F)-7 b(r)i(om)29 b(the)e(index)h(of)f(a)g FA(\005)1559 4415 y Fy(0)1559 4471 y Fx(n)1607 4448 y Fs(-class)g Fw(T)40 b Fs(and)28 b Fw(q)g Fr(2)d Fo(Q)8 b Fs(,)33 b(we)28 b(c)-5 b(an)27 b(c)-5 b(ompute)29 b(the)e(index)701 4575 y(of)j(a)h FA(\005)950 4540 y Fq(;)985 4516 y Fn(\()p Fl(n)p Fj(\000)p Fn(1\))950 4601 y Fy(1)1158 4575 y Fs(-class)g Fw(V)45 b Fr(\022)25 b Fw(T)43 b Fs(that)32 b(is)e(also)i(a)e(close)-5 b(d)32 b FA(\005)2567 4542 y Fy(0)2567 4597 y Fx(n)2614 4575 y Fs(-class)f(and)g(such)f(that)701 4683 y Fw(\026)p FA(\()p Fw(T)13 b FA(\))20 b Fr(\000)g Fw(\026)p FA(\()p Fw(V)g FA(\))26 b Fw(<)f(q)s Fs(.)513 4797 y FA(\(iii\))40 b Fs(F)-7 b(r)i(om)44 b(the)f(index)h(of)f(a)g FA(\006)1636 4764 y Fy(0)1636 4819 y Fx(n)1682 4797 y Fs(-class)h Fw(S)j Fs(and)d Fw(q)j Fr(2)c Fo(Q)9 b Fs(,)48 b(we)43 b(c)-5 b(an)43 b Fr(;)2934 4764 y Fy(\()p Fx(n)p Fy(\))3037 4797 y Fs(-c)-5 b(ompute)701 4905 y(the)34 b(index)g(of)h(a)f(close)-5 b(d)35 b FA(\005)1619 4872 y Fy(0)1619 4929 y Fx(n)p Fq(\000)p Fy(1)1757 4905 y Fs(-class)f Fw(V)48 b Fr(\022)27 b Fw(S)39 b Fs(such)34 b(that)h Fw(\026)p FA(\()p Fw(S)5 b FA(\))22 b Fr(\000)f Fw(\026)p FA(\()p Fw(V)f FA(\))28 b Fw(<)g(q)s Fs(.)701 5027 y(Mor)-5 b(e)g(over,)30 b(if)g Fw(\026)p FA(\()p Fw(S)5 b FA(\))30 b Fs(is)f(c)-5 b(omputable)32 b(fr)-5 b(om)31 b Fr(;)2245 4994 y Fy(\()p Fx(n)p Fq(\000)p Fy(1\))2467 5027 y Fs(then)f(the)g(index)g(of)g Fw(V)50 b Fs(c)-5 b(an)701 5141 y(b)g(e)32 b(found)h(c)-5 b(omputably)35 b(fr)-5 b(om)34 b Fr(;)1797 5108 y Fy(\()p Fx(n)p Fq(\000)p Fy(1\))1989 5141 y Fs(.)p eop %%Page: 62 62 62 61 bop 505 363 a FD(62)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)515 541 y FA(\(iv\))42 b Fs(F)-7 b(r)i(om)44 b(the)e(index)h(of)g(a)f FA(\005)1635 508 y Fy(0)1635 564 y Fx(n)1682 541 y Fs(-class)h Fw(T)55 b Fs(and)43 b Fw(q)j Fr(2)c Fo(Q)9 b Fs(,)48 b(we)43 b(c)-5 b(an)42 b Fr(;)2934 508 y Fy(\()p Fx(n)p Fy(\))3037 541 y Fs(-c)-5 b(ompute)701 649 y(the)34 b(index)g(of)g(an)g (op)-5 b(en)35 b FA(\006)1616 616 y Fy(0)1616 674 y Fx(n)p Fq(\000)p Fy(1)1753 649 y Fs(-class)f Fw(U)j Fr(\023)27 b Fw(T)46 b Fs(such)34 b(that)h Fw(\026)p FA(\()p Fw(U)10 b FA(\))22 b Fr(\000)e Fw(\026)p FA(\()p Fw(T)13 b FA(\))28 b Fw(<)f(q)s Fs(.)701 772 y(Mor)-5 b(e)g(over,)30 b(if)g Fw(\026)p FA(\()p Fw(S)5 b FA(\))30 b Fs(is)g(c)-5 b(omputable)31 b(fr)-5 b(om)31 b Fr(;)2245 739 y Fy(\()p Fx(n)p Fq(\000)p Fy(1\))2468 772 y Fs(then)f(the)g(index)g(of)g Fw(U)40 b Fs(c)-5 b(an)701 886 y(b)g(e)32 b(found)h(c)-5 b(omputably)35 b(fr)-5 b(om)34 b Fr(;)1797 853 y Fy(\()p Fx(n)p Fq(\000)p Fy(1\))1989 886 y Fs(.)588 1051 y FA(Using)44 b(the)g(ab)s(o)m(v)m(e)h (result,)f(w)m(e)g(can)h(easily)e(sho)m(w,)h(for)g(instance,)g(that)h (\()p Fw(n)29 b FA(+)505 1165 y(1\)-randomness)40 b(coincides)f(with)f (1-randomness)h(relativ)m(e)h(to)h Fr(;)2812 1132 y Fy(\()p Fx(n)p Fy(\))2914 1165 y FA(,)f(whic)m(h)e(is)h(a)505 1273 y(theorem)31 b(of)g(Kurtz)f([71)q(].)588 1383 y(Let)c Fw(U)35 b FA(b)s(e)25 b(the)g(standard)f(univ)m(ersal)g(pre\014x-free)g (mac)m(hine.)h(Then)g Fw(U)2972 1350 y Fx(X)3064 1383 y FA(will)e(b)s(e)h(a)505 1490 y(univ)m(ersal)j(mac)m(hine)i(relativ)m (e)g(to)g(an)m(y)g Fw(X)7 b FA(,)30 b(and)e(w)m(e)h(obtain)f(the)h (follo)m(wing)e(natural)505 1598 y(\()p Fw(n)21 b FA(+)e(1\)-random)31 b(sets.)1528 1732 y(\012)1594 1694 y Fy(\()p Fx(n)p Fy(\))1721 1732 y FA(:=)1928 1645 y Fp(X)1842 1871 y Fx(U)1897 1852 y Fj(;)1929 1834 y Fn(\()p Fl(n)p Fn(\))2024 1871 y Fy(\()p Fx(\033)r Fy(\))-12 b Fq(#)2160 1732 y FA(2)2205 1694 y Fq(\000j)p Fx(\033)r Fq(j)2346 1732 y Fw(:)505 1994 y FA(See)40 b(Section)g(14)g(for)g(more)g(on)f(relativizing)f(\012.)i (There)f(are)h(other)g(natural)f(ex-)505 2102 y(amples)h(of)g Fw(n)p FA(-random)f(sets,)i(de\014ned)d(without)i(the)g(use)f(of)i (relativization;)e(see)505 2210 y(for)34 b(instance)f(Bec)m(her,)i (Daicz,)g(and)e(Chaitin)e([9)q(];)j(Bec)m(her)h(and)e(Chaitin)e([8)q (];)j(and)505 2317 y(Bec)m(her)e(and)e(Grigorie\013)f([10)r(].)588 2425 y(There)24 b(is)g(a)g(v)m(ery)h(in)m(teresting)f(in)m(tert)m (wining)e(of)i(plain)e(Kolmogoro)m(v)k(complexit)m(y)505 2533 y(and)k(relativized)f(randomness.)588 2699 y FB(Definition)35 b FA(12.6)p FB(.)47 b FA(A)31 b(set)g Fw(A)f FA(is)f Fs(Kolmo)-5 b(gor)g(ov)36 b(r)-5 b(andom)40 b FA(if)29 b(for)h(some)h Fw(c)p FA(,)1475 2852 y Fr(9)1526 2814 y Fq(1)1600 2852 y Fw(n)15 b FA([)p Fw(C)7 b FA(\()p Fw(A)1895 2840 y Fz(\026)1958 2852 y Fw(n)p FA(\))26 b Fz(>)f Fw(n)19 b Fr(\000)h Fw(c)p FA(])p Fw(:)-1919 b FA(\(2\))505 3004 y(W)-8 b(e)39 b(sa)m(y)f(that)g Fw(A)f FA(is)f Fs(time-b)-5 b(ounde)g(d)40 b(Kolmo)-5 b(gor)g(ov)42 b(r)-5 b(andom)47 b FA(with)36 b(time)h(b)s(ound)e Fw(t)505 3112 y FA(if)h(\(2\))i(holds)d(with)h Fw(C)1278 3079 y Fx(t)1344 3112 y FA(instead)g(of)h Fw(C)7 b FA(,)36 b(where)g Fw(C)2248 3079 y Fx(t)2314 3112 y FA(is)g(the)h(time-)p Fw(t)p FA(-b)s(ounded)e(Kol-)505 3220 y(mogoro)m(v)g(complexit)m(y)-8 b(.)34 b(\(F)-8 b(or)34 b(more)f(on)g(time-b)s(ounded)e(complexit)m(y) -8 b(,)33 b(see)h(Li)e(and)505 3328 y(Vit\023)-45 b(an)m(yi)30 b([79)q(].\))588 3494 y(While)39 b(w)m(e)h(ha)m(v)m(e)h(seen)e(in)g (Section)g(3.3.1)i(that)g(no)e(set)h(can)g(satisfy)f(\(2\))i(with)505 3602 y Fr(8)d FA(in)f(place)i(of)f Fr(9)1111 3569 y Fq(1)1185 3602 y FA(,)h(the)f(class)g(of)h(Kolmogoro)m(v)g(random)f(sets)h(has)f (measure)g(1.)505 3709 y(The)28 b(next)h(theorem)g(sho)m(ws)f(that)h Fs(Kolmo)-5 b(gor)g(ov)34 b(r)-5 b(andomness)33 b(is)e(e)-5 b(quivalent)31 b(to)h FA(2)p Fs(-)505 3817 y(r)-5 b(andomness)p FA(.)30 b(Y)-8 b(u,)26 b(Ding,)h(and)e(Do)m(wney)j([141)q(])f(pro)m(v)m (ed)g(that)f(ev)m(ery)i(3-random)e(set)505 3925 y(is)34 b(Kolmogoro)m(v)i(random.)f(They)g(also)g(observ)m(ed)g(that)g(there)h (is)e(no)h(\001)3028 3892 y Fy(0)3028 3950 y(2)3102 3925 y FA(Kolmo-)505 4033 y(goro)m(v)f(random)d(set.)i(This)d(fact)j(is)e (also)h(implied)d(b)m(y)i(the)i(follo)m(wing)d(result,)h(since)505 4141 y(2-random)g(sets)g(cannot)g(b)s(e)e(\001)1590 4108 y Fy(0)1590 4166 y(2)1629 4141 y FA(.)588 4307 y FB(Theorem)34 b FA(12.7)i(\(Nies,)30 b(Stephan,)g(and)g(T)-8 b(erwijn)29 b([107)q(]\))p FB(.)46 b Fs(L)-5 b(et)28 b Fw(g)i Fs(b)-5 b(e)27 b(a)h(c)-5 b(omput-)505 4415 y(able)32 b(time)f(b)-5 b(ound)32 b(such)f(that)h Fw(g)s FA(\()p Fw(n)p FA(\))26 b Fz(>)f Fw(n)1893 4382 y Fy(2)1949 4415 y Fr(\000)16 b Fw(O)s FA(\(1\))p Fs(.)32 b(The)f(fol)5 b(lowing)32 b(ar)-5 b(e)32 b(e)-5 b(quivalent)505 4523 y(for)34 b(any)f(set)g Fw(Z)7 b Fs(:)579 4651 y FA(\(i\))41 b Fw(Z)e Fs(is)33 b FA(2)p Fs(-r)-5 b(andom.)553 4759 y FA(\(ii\))41 b Fw(Z)e Fs(is)33 b(Kolmo)-5 b(gor)g(ov)35 b(r)-5 b(andom.)528 4867 y FA(\(iii\))40 b Fw(Z)f Fs(is)33 b(Kolmo)-5 b(gor)g(ov)35 b(r)-5 b(andom)35 b(with)f(time)f(b)-5 b(ound)33 b Fw(g)s Fs(.)588 5033 y FA(The)h(implication)d(\(i\))j Fr(\))g FA(\(ii\))f(in)f(Theorem)i(12.7)h(w)m(as)f(pro)m(v)m(ed)h(indep)s (enden)m(tly)505 5141 y(and)30 b(earlier)f(b)m(y)i(Miller)d([92)q(].)p eop %%Page: 63 63 63 62 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(63)588 541 y FA(A)27 b(set)g Fw(A)g FA(is)e Fs(str)-5 b(ongly)31 b(Chaitin)f(r)-5 b(andom)36 b FA(if)26 b(there)g(is)g(a)h Fw(c)g FA(suc)m(h)f(that)h Fr(9)2975 508 y Fq(1)3049 541 y Fw(n)15 b FA([)p Fw(K)7 b FA(\()p Fw(A)3357 529 y Fz(\026)505 649 y Fw(n)p FA(\))26 b Fz(>)f Fw(n)8 b FA(+)g Fw(K)f FA(\()p Fw(n)p FA(\))h Fr(\000)g Fw(c)p FA(].)25 b(Solo)m(v)-5 b(a)m(y)25 b([126)r(])g(sho)m(w)m(ed)f(that)h (\(up)f(to)h(additiv)m(e)f(constan)m(ts\))i(if)505 757 y(a)h(string)e(has)i(maximal)e(pre\014x-free)h(Kolmogoro)m(v)h (complexit)m(y)f(then)g(it)g(has)g(max-)505 865 y(imal)35 b(plain)g(Kolmogoro)m(v)i(complexit)m(y)-8 b(,)37 b(so)g(b)m(y)f (Theorem)g(12.7,)i(strong)f(Chaitin)505 973 y(randomness)k(implies)d (2-randomness.)k(It)f(is)g(also)g(kno)m(wn)g(that)h(3-randomness)505 1081 y(implies)31 b(strong)j(Chaitin)e(randomness)h(\(see)i(Theorem)e (13.11)j(b)s(elo)m(w\).)e(It)f(is)g(not)505 1189 y(kno)m(wn)f(whether)g (strong)h(Chaitin)e(randomness)g(is)h(equiv)-5 b(alen)m(t)32 b(to)h(either)f(2-ran-)505 1297 y(domness)e(or)g(3-randomness.)588 1405 y(Another)i(c)m(haracterization)h(of)f(2-randomness)f(can)h(b)s(e) f(giv)m(en)h(b)m(y)f(considering)505 1513 y(sets)g(that)g(are)g(lo)m(w) f(for)g(\012)g(\(see)h(De\014nition)e(7.5\).)588 1677 y FB(Theorem)34 b FA(12.8)i(\(Nies,)30 b(Stephan,)g(and)g(T)-8 b(erwijn)29 b([107)q(]\))p FB(.)46 b Fs(A)34 b(set)g(is)g FA(2)p Fs(-r)-5 b(andom)505 1785 y(iff)33 b(it)f(is)h FA(1)p Fs(-r)-5 b(andom)35 b(and)e(low)h(for)f FA(\012)p Fs(.)588 1950 y FB(Pr)n(oof.)41 b FA(By)c(Corollary)e(12.18)k(b)s(elo)m (w,)c(for)i(an)m(y)f(t)m(w)m(o)i(sets)f Fw(A)g FA(and)e Fw(B)5 b FA(,)37 b(if)e Fw(A)h FA(is)505 2058 y(1-random)25 b(and)g Fw(B)k FA(is)24 b(1-random)h(relativ)m(e)g(to)h Fw(A)p FA(,)f(then)g Fw(A)g FA(is)f(1-random)h(relativ)m(e)g(to)505 2166 y Fw(B)5 b FA(.)31 b(Th)m(us)e(if)h Fw(A)g FA(is)g(1-random,)g (then)h Fw(A)f FA(is)g(2-random)g Fr(,)h Fw(A)f FA(is)g(1-random)g (relativ)m(e)505 2274 y(to)c(\012)f Fr(,)f FA(\012)h(is)f(1-random)h (relativ)m(e)g(to)g Fw(A)g Fr(,)g Fw(A)g FA(is)f(lo)m(w)h(for)g(\012.)f (Since)g(an)m(y)i(2-random)505 2381 y(set)31 b(is)f(1-random,)g(the)h (equiv)-5 b(alence)30 b(follo)m(ws.)1235 b Fr(a)588 2509 y FA(Th)m(us)32 b(ev)m(ery)h(2-random)f(set)h(is)e(lo)m(w)h(for)g (\012,)g(whic)m(h)f(b)m(y)h(Corollary)f(7.8)j(giv)m(es)e(us)505 2617 y(the)f(follo)m(wing)e(result.)588 2781 y FB(Cor)n(ollar)-6 b(y)35 b FA(12.9)h(\(Sac)m(ks)31 b(and)f(Stillw)m(ell,)d(see)k(Kautz)g ([60)q(,)g(Thm.)e(IV.2.4]\))p FB(.)505 2889 y Fs(Every)k FA(2)p Fs(-r)-5 b(andom)35 b(set)d(is)h(GL)1541 2903 y Fy(1)1581 2889 y Fs(.)588 3054 y FA(It)h(is)f(straigh)m(tforw)m(ard)h (to)h(de\014ne)e(Kurtz,)g(Sc)m(hnorr,)h(and)f(computably)g Fw(n)p FA(-ran-)505 3162 y(dom)27 b(sets)g(for)f(all)g Fw(n)g FA(b)m(y)h(analogy)g(with)e(the)i(ab)s(o)m(v)m(e.)h(It)f(is)f (not)h(di\016cult)d(to)k(see)f(that)505 3270 y(b)s(eing)33 b(Kurtz)h(2-random)g(coincides)f(with)f(passing)h(all)g(generalized)h (Martin-L\177)-45 b(of)505 3378 y(tests)43 b Fr(f)p Fw(U)837 3392 y Fx(n)884 3378 y Fr(g)929 3392 y Fx(n)p Fq(2)p Fx(!)1070 3378 y FA(,)f(where)e(w)m(e)i(ha)m(v)m(e)h Fw(\026)p FA(\()p Fw(U)1928 3392 y Fx(n)1975 3378 y FA(\))h Fr(!)g FA(0,)e(but)f(there)g(ma)m(y)i(b)s(e)d(no)i(com-)505 3486 y(putable)c(decreasing)g(b)s(ound)e(on)i Fw(\026)p FA(\()p Fw(U)1864 3500 y Fx(n)1911 3486 y FA(\).)h(Th)m(us)e(ev)m(ery)i (Kurtz)f(2-random)g(set)g(is)505 3593 y(1-random.)27 b(Relativizing)d(this)h(observ)-5 b(ation)26 b(and)g(the)g(fact)h(that) g(ev)m(ery)g(1-random)505 3701 y(set)43 b(is)d(Kurtz)i(1-random)f (\(see)i(Section)e(10.3\),)j(w)m(e)e(see)h(that)f(ev)m(ery)g Fw(n)p FA(-random)505 3809 y(set)30 b(is)d(Kurtz)i Fw(n)p FA(-random,)f(and)g(ev)m(ery)h(Kurtz)g(\()p Fw(n)16 b FA(+)h(1\)-random)29 b(set)g(is)f Fw(n)p FA(-random.)505 3917 y(Neither)i(implication)e(can)j(b)s(e)f(rev)m(ersed.)588 4082 y FB(Theorem)k FA(12.10)i(\(Kurtz)30 b([71)r(]\))p FB(.)46 b Fs(F)-7 b(or)38 b(every)e Fw(n)d Fz(>)f FA(1)p Fs(,)37 b(ther)-5 b(e)38 b(is)f(an)g Fw(n)p Fs(-r)-5 b(andom)505 4190 y(set)33 b(that)h(c)-5 b(annot)34 b(b)-5 b(e)33 b(c)-5 b(ompute)g(d)34 b(by)f(any)g(Kurtz)g FA(\()p Fw(n)20 b FA(+)g(1\))p Fs(-r)-5 b(andom)35 b(set.)588 4354 y FB(Pr)n(oof.)41 b FA(By)36 b(relativizing)e(the)h(pro)s(of)g (that)h(there)g(are)g(1-random)f(sets)h(b)s(elo)m(w)505 4463 y Fr(;)550 4430 y Fq(0)574 4463 y FA(,)31 b(w)m(e)g(see)g(that)f (there)h(is)e(an)i Fw(n)p FA(-random)e(set)i Fw(A)26 b Fz(6)2283 4477 y Fy(T)2363 4463 y Fr(;)2408 4430 y Fy(\()p Fx(n)p Fy(\))2510 4463 y FA(.)31 b(F)-8 b(or)31 b(eac)m(h)g Fw(e)p FA(,)g(let)713 4614 y Fw(P)771 4628 y Fx(e)833 4614 y FA(=)25 b Fr(f)p Fw(B)31 b FA(:)25 b(\010)1190 4577 y Fx(B)1190 4637 y(e)1276 4614 y FA(=)g Fw(A)p Fr(g)h FA(=)f Fr(f)p Fw(B)30 b FA(:)25 b Fr(8)p Fw(x)15 b Fr(8)p Fw(s)g FA([\010)2119 4577 y Fx(B)2119 4637 y(e;s)2207 4614 y FA(\()p Fw(x)p FA(\))10 b Fr(#!)27 b FA(\010)2568 4577 y Fx(B)2568 4637 y(e;s)2656 4614 y FA(\()p Fw(x)p FA(\))f(=)f Fw(A)p FA(\()p Fw(x)p FA(\)])p Fr(g)p Fw(:)505 4769 y FA(By)32 b(Theorem)g(8.11,)h(if)e Fw(A)h FA(is)e(not)i(computable)g(then)f Fw(\026)p FA(\()p Fr(f)p Fw(B)h FA(:)c Fw(A)g Fz(6)2849 4783 y Fy(T)2931 4769 y Fw(B)5 b Fr(g)p FA(\))28 b(=)f(0,)32 b(so)505 4895 y(eac)m(h)j Fw(P)771 4909 y Fx(e)843 4895 y FA(is)d(a)j(\005)1085 4860 y Fq(;)1120 4837 y Fn(\()p Fl(n)p Fn(\))1085 4921 y Fy(1)1215 4895 y FA(-class)f(of)g(measure)g(0,)g(and)f(hence)h(eac)m (h)p 2675 4822 96 4 v 35 w Fw(P)2733 4909 y Fx(e)2804 4895 y FA(is)f(a)i(\006)3045 4862 y Fy(0)3045 4920 y Fx(n)p Fy(+1)3181 4895 y FA(-class)505 5015 y(of)28 b(measure)g(1.)h (So)e(ev)m(ery)i(Kurtz)e(\()p Fw(n)15 b FA(+)g(1\)-random)28 b(set)h(m)m(ust)e(b)s(e)g(in)g(eac)m(h)p 3100 4942 V 29 w Fw(P)3158 5029 y Fx(e)3195 5015 y FA(,)h(and)505 5123 y(hence)j(cannot)g(compute)g Fw(A)p FA(.)1821 b Fr(a)p eop %%Page: 64 64 64 63 bop 505 363 a FD(64)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FB(Cor)n(ollar)h(y)35 b FA(12.11)h(\(to)c(the)e(pro)s(of)g(of)g (Theorem)h(12.10\))p FB(.)563 675 y FA(\(i\))42 b Fs(No)32 b(Kurtz)h FA(\()p Fw(n)20 b FA(+)g(1\))p Fs(-r)-5 b(andom)36 b(set)c(is)h(c)-5 b(omputable)34 b(fr)-5 b(om)34 b Fr(;)2718 642 y Fy(\()p Fx(n)p Fy(\))2820 675 y Fs(.)538 789 y FA(\(ii\))41 b Fs(Ther)-5 b(e)33 b(is)g(an)g Fw(n)p Fs(-r)-5 b(andom)34 b(set)f(c)-5 b(omputable)34 b(fr)-5 b(om)34 b Fr(;)2479 756 y Fy(\()p Fx(n)p Fy(\))2581 789 y Fs(.)588 954 y FA(The)d(follo)m(wing)f(result)h(w)m(as)h(\014rst)e(pro)m(v)m(ed) i(b)m(y)g(Kautz,)g(although)f(it)g(w)m(as)h(stated)505 1062 y(without)27 b(pro)s(of)g(b)m(y)h(Gaifman)f(and)h(Snir)d([47)r(].) j(Kautz's)g(pro)s(of)f(w)m(as)i(fairly)d(compli-)505 1170 y(cated,)37 b(but)d(w)m(e)h(can)g(obtain)g(a)g(simpler)d(pro)s(of) j(using)e(relativizations)h(of)h(results)505 1278 y(men)m(tioned)30 b(ab)s(o)m(v)m(e.)588 1442 y FB(Theorem)k FA(12.12)i(\(Kautz)31 b([60)q(],)g(Kurtz)f([71)q(])h(for)f Fw(n)25 b FA(=)g(1\))p FB(.)46 b Fs(L)-5 b(et)41 b Fw(n)e Fz(>)g FA(1)p Fs(.)i(Ther)-5 b(e)505 1550 y(is)33 b(a)g(Kurtz)g Fw(n)p Fs(-r)-5 b(andom)34 b(set)f(that)h(is)e(not)h Fw(n)p Fs(-r)-5 b(andom.)588 1715 y FB(Pr)n(oof.)41 b FA(Let)f Fw(X)47 b FA(b)s(e)39 b(a)g(\006)1520 1680 y Fq(;)1555 1657 y Fn(\()p Fl(n)p Fj(\000)p Fn(1\))1520 1741 y Fy(1)1767 1715 y FA(set)h(suc)m(h)f(that)h Fr(;)2383 1682 y Fy(\()p Fx(n)p Fq(\000)p Fy(1\))2616 1715 y Fw(<)2687 1729 y Fy(T)2782 1715 y Fw(X)47 b(<)2975 1729 y Fy(T)3070 1715 y Fr(;)3115 1682 y Fy(\()p Fx(n)p Fy(\))3218 1715 y FA(.)39 b(By)505 1832 y(Theorem)24 b(10.23)i(relativized)d(to)i Fr(;)1700 1799 y Fy(\()p Fx(n)p Fq(\000)p Fy(1\))1893 1832 y FA(,)f(there)g(is)g(a)g(Kurtz)g Fw(n)p FA(-random)f(set)i Fw(A)f FA(suc)m(h)505 1941 y(that)30 b Fw(A)18 b Fr(\010)g(;)921 1908 y Fy(\()p Fx(n)p Fq(\000)p Fy(1\))1138 1941 y Fr(\021)1209 1955 y Fy(T)1289 1941 y Fw(X)7 b FA(.)30 b(On)f(the)g(other)g(hand,)g(b)m(y) g(Theorem)g(4.1)h(relativized)e(to)505 2050 y Fr(;)550 2017 y Fy(\()p Fx(n)p Fq(\000)p Fy(1\))743 2050 y FA(,)22 b Fw(A)g FA(cannot)h(b)s(e)f Fw(n)p FA(-random,)f(since)h(that)g(w)m (ould)f(imply)f(that)j Fw(X)32 b Fr(\021)2995 2064 y Fy(T)3076 2050 y Fr(;)3121 2017 y Fy(\()p Fx(n)p Fy(\))3223 2050 y FA(.)91 b Fr(a)588 2178 y FA(Kurtz)30 b([71)q(],)h(Kautz)g([60)q (],)g(and)e(v)-5 b(an)30 b(Lam)m(balgen)h([72)q(])f(examined)g(the)g (relation-)505 2285 y(ship)21 b(b)s(et)m(w)m(een)j(relativized)e (randomness)g(and)g(the)h(T)-8 b(uring)22 b(degrees.)i(They)e(pro)m(v)m (ed)505 2393 y(a)h(n)m(um)m(b)s(er)e(of)h(classic)g(results.)f(W)-8 b(e)23 b(giv)m(e)g(a)f(sample,)g(along)g(with)f(some)h(more)g(recen)m (t)505 2501 y(related)31 b(results,)e(and)h(include)e(a)j(few)f(pro)s (ofs)f(as)i(examples.)588 2666 y FB(Theorem)j FA(12.13)i(\(v)-5 b(an)31 b(Lam)m(balgen)f([72)q(],)h(Kautz)g([60)q(]\))p FB(.)563 2794 y FA(\(i\))42 b Fs(If)32 b Fw(A)20 b Fr(\010)g Fw(B)37 b Fs(is)c Fw(n)p Fs(-r)-5 b(andom,)34 b(then)f(so)g(ar)-5 b(e)34 b Fw(A)f Fs(and)g Fw(B)5 b Fs(.)538 2908 y FA(\(ii\))41 b Fs(If)32 b Fw(A)h Fs(is)f Fw(n)p Fs(-r)-5 b(andom,)34 b(then)g(so)f(is)f Fw(A)1924 2875 y Fy([)p Fx(n)p Fy(])2011 2908 y Fs(,)g(the)h Fw(n)p Fs(-th)f(c)-5 b(olumn)34 b(of)f Fw(A)p Fs(.)588 3073 y FB(Theorem)h FA(12.14)i(\(v)-5 b(an)31 b(Lam)m(balgen)f([72)q(]\))p FB(.)46 b Fs(If)34 b Fw(A)21 b Fr(\010)g Fw(B)38 b Fs(is)c Fw(n)p Fs(-r)-5 b(andom,)35 b(then)f Fw(A)505 3180 y Fs(is)f Fw(n)p Fs(-r)-5 b(andom)34 b(r)-5 b(elative)34 b(to)f Fw(B)5 b Fs(.)588 3345 y FB(Cor)n(ollar)-6 b(y)35 b FA(12.15)h(\(Ku)m(\024)-43 b(cera)32 b(\(see)f([60)r(]\),)g(v)-5 b(an)30 b(Lam)m(balgen)g([72)q (]\))p FB(.)47 b Fs(If)23 b Fw(A)p Fr(\010)p Fw(B)29 b Fs(is)505 3453 y FA(1)p Fs(-r)-5 b(andom,)35 b(then)e Fw(A)26 b Fr(j)1263 3467 y Fi(T)1344 3453 y Fw(B)5 b Fs(.)588 3618 y FA(Th)m(us)25 b(ev)m(ery)h(1-random)g(set)g Fw(X)33 b FA(splits)24 b(in)m(to)h(t)m(w)m(o)i(T)-8 b(uring)24 b(incomparable)g(halv)m(es,)505 3726 y(b)s(oth)30 b(of)h(whic)m(h)e (are)h(computable)g(in)f Fw(X)7 b FA(.)31 b(So)g(w)m(e)g(ha)m(v)m(e)g (the)g(follo)m(wing)e(result.)588 3891 y FB(Cor)n(ollar)-6 b(y)35 b FA(12.16)h(\(Kurtz)31 b([71)q(]\))p FB(.)46 b Fs(No)32 b FA(1)p Fs(-r)-5 b(andom)35 b(set)e(has)h(minimal)g(de)-5 b(gr)g(e)g(e.)588 4056 y FA(Using)28 b(Theorem)f(10.13,)j(Y)-8 b(u)28 b([138)r(])g(has)g(recen)m(tly)g(sho)m(wn)f(that)i(Theorem)f (12.14)505 4164 y(fails)k(for)h(Sc)m(hnorr)f(and)g(computable)h (randomness,)f(ev)m(en)i(for)e Fw(n)e FA(=)f(1.)34 b(Th)m(us)e(this)505 4272 y(imp)s(ortan)m(t)25 b(to)s(ol)h(in)e(the)i(theory)f(of)h (1-randomness)f(is)g(not)h(a)m(v)-5 b(ailable)24 b(in)h(the)g(study)505 4379 y(of)31 b(these)g(notions.)588 4487 y(The)c(follo)m(wing)e(is)g(a) j(con)m(v)m(erse)g(to)f(Theorem)g(12.14)h(\(whic)m(h,)e(as)h(p)s(oin)m (ted)f(out)h(b)m(y)505 4595 y(Y)-8 b(u)31 b([138)q(],)g(do)s(es)f(also) h(hold)e(for)h(Sc)m(hnorr)f(and)h(computable)g(randomness\).)588 4760 y FB(Theorem)k FA(12.17)i(\(v)-5 b(an)31 b(Lam)m(balgen)f([72)q (]\))p FB(.)46 b Fs(If)36 b Fw(B)k Fs(is)c Fw(n)p Fs(-r)-5 b(andom)38 b(and)f Fw(A)f Fs(is)f Fw(n)p Fs(-)505 4868 y(r)-5 b(andom)35 b(r)-5 b(elative)34 b(to)f Fw(B)5 b Fs(,)32 b(then)h Fw(A)21 b Fr(\010)e Fw(B)37 b Fs(is)c Fw(n)p Fs(-r)-5 b(andom.)588 5033 y FB(Pr)n(oof.)41 b FA(W)-8 b(e)31 b(giv)m(e)e(a)h(pro)s(of)e(due)g(to)i(Nies.)f(Supp)s (ose)e Fw(A)18 b Fr(\010)f Fw(B)34 b FA(is)28 b(not)h Fw(n)p FA(-random.)505 5141 y(W)-8 b(e)31 b(sho)m(w)f(that)g(either)g Fw(B)k FA(is)29 b(not)h Fw(n)p FA(-random)f(or)g Fw(A)h FA(is)f(not)h Fw(n)p FA(-random)f(relativ)m(e)h(to)p eop %%Page: 65 65 65 64 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(65)505 541 y Fw(B)5 b FA(.)32 b(By)h(Theorem)e(12.5,)j(w)m(e)f (can)f(c)m(ho)s(ose)h(a)g(sequence)f Fr(f)p Fw(V)2533 555 y Fx(i)2562 541 y Fr(g)2607 555 y Fx(i)p Fq(2)p Fx(!)2761 541 y FA(of)g(uniformly)d(\006)3348 508 y Fy(0)3348 564 y Fx(n)505 649 y FA(op)s(en)h(sets)h(suc)m(h)f(that)h Fw(A)20 b Fr(\010)g Fw(B)30 b Fr(2)1669 581 y Fp(T)1745 676 y Fx(i)1788 649 y Fw(V)1841 663 y Fx(i)1899 649 y FA(and)g Fw(\026)p FA(\()p Fw(V)2219 663 y Fx(i)2248 649 y FA(\))25 b Fz(6)g FA(2)2449 616 y Fq(\000)p Fy(2)p Fx(i)2568 649 y FA(.)588 757 y(Let)37 b Fw(\025)f FA(b)s(e)f(the)i (empt)m(y)f(string.)f(W)-8 b(e)38 b(write)d([)p Fw(\033)27 b Fr(\010)d Fw(\034)10 b FA(])36 b(for)g(the)g(collection)g(of)g(sets) 505 865 y Fw(X)d FA(=)25 b Fw(X)784 879 y Fy(0)844 865 y Fr(\010)20 b Fw(X)1010 879 y Fy(1)1080 865 y FA(suc)m(h)30 b(that)h Fw(\033)d Fr(\036)d Fw(X)1733 879 y Fy(0)1803 865 y FA(and)30 b Fw(\034)36 b Fr(\036)25 b Fw(X)2227 879 y Fy(1)2266 865 y FA(.)588 973 y(Let)1170 1087 y Fw(S)1226 1101 y Fx(i)1279 1087 y FA(=)1375 1000 y Fp([)1476 1087 y Fr(f)p FA([)p Fw(\033)s FA(])h(:)g Fw(\026)p FA(\()p Fw(V)1846 1101 y Fx(i)1894 1087 y Fr(\\)20 b FA([)p Fr(;)h(\010)f Fw(\033)s FA(]\))26 b Fz(>)f FA(2)2439 1049 y Fq(\000)p Fx(i)p Fq(\000j)p Fx(\033)r Fq(j)2659 1087 y Fr(g)p Fw(:)505 1245 y FA(Clearly)-8 b(,)29 b Fw(S)894 1259 y Fx(i)p Fy(+1)1038 1245 y Fr(\022)c Fw(S)1190 1259 y Fx(i)1218 1245 y FA(,)30 b(and)f(the)h Fw(S)1661 1259 y Fx(i)1718 1245 y FA(are)h(uniformly)26 b(\006)2349 1212 y Fy(0)2349 1268 y Fx(n)2426 1245 y FA(op)s(en)j(sets.)h(W)-8 b(e)31 b(sho)m(w)f(that)505 1353 y Fw(\026)p FA(\()p Fw(S)651 1367 y Fx(i)679 1353 y FA(\))d Fz(6)g FA(2)884 1320 y Fq(\000)p Fx(i)967 1353 y FA(.)32 b(Let)f Fw(\033)1239 1367 y Fy(0)1279 1353 y Fw(;)15 b(\033)1371 1367 y Fy(1)1410 1353 y Fw(;)g(:)g(:)g(:)48 b FA(b)s(e)31 b(a)g(listing)e(of)i(the)h (strings)e Fw(\033)k FA(that)e(are)f(minimal)505 1467 y(\(under)j(the)g(substring)f(relation\))h(suc)m(h)g(that)h Fw(\026)p FA(\()p Fw(V)2289 1481 y Fx(i)2341 1467 y Fr(\\)22 b FA([)p Fr(;)i(\010)f Fw(\033)s FA(]\))33 b Fz(>)e FA(2)2907 1434 y Fq(\000)p Fx(i)p Fq(\000j)p Fx(\033)r Fq(j)3128 1467 y FA(.)k(Then)505 1575 y Fw(S)561 1589 y Fx(i)616 1575 y FA(=)713 1507 y Fp(S)789 1602 y Fx(i)817 1575 y FA([)p Fw(\033)894 1589 y Fx(i)923 1575 y FA(].)c(Since)f(the)i(sets) f Fw(V)1631 1589 y Fx(i)1680 1575 y Fr(\\)21 b FA([)p Fr(;)g(\010)f Fw(\033)1996 1589 y Fx(i)2025 1575 y FA(])31 b(are)g(pairwise)f(disjoin)m(t)f(and)i Fw(\026)p FA(\()p Fw(V)3234 1589 y Fx(i)3262 1575 y FA(\))c Fz(6)505 1693 y FA(2)550 1660 y Fq(\000)p Fy(2)p Fx(i)669 1693 y FA(,)k(w)m(e)g(see)g (that)1203 1625 y Fp(P)1299 1720 y Fx(i)1342 1693 y FA(2)1387 1660 y Fq(\000)p Fx(i)p Fq(\000j)p Fx(\033)1581 1670 y Fl(i)1607 1660 y Fq(j)1656 1693 y Fz(6)25 b FA(2)1797 1660 y Fq(\000)p Fy(2)p Fx(i)1916 1693 y FA(,)31 b(and)e(hence)i Fw(\026)p FA(\()p Fw(S)2547 1707 y Fx(i)2575 1693 y FA(\))26 b Fz(6)2732 1625 y Fp(P)2828 1720 y Fx(i)2871 1693 y FA(2)2916 1660 y Fq(\000j)p Fx(\033)3031 1670 y Fl(i)3058 1660 y Fq(j)3107 1693 y Fz(6)e FA(2)3247 1660 y Fq(\000)p Fx(i)3331 1693 y FA(.)588 1801 y(If)k Fw(B)h Fr(2)861 1733 y Fp(T)937 1828 y Fx(i)980 1801 y Fw(S)1036 1815 y Fx(i)1064 1801 y FA(,)f(then)f Fw(B)33 b FA(is)26 b(not)i Fw(n)p FA(-random.)f(Otherwise,)g(there)h(is)e(a)i Fw(j)34 b FA(suc)m(h)27 b(that)505 1909 y Fw(B)40 b(=)-55 b Fr(2)25 b Fw(S)746 1923 y Fx(i)804 1909 y FA(for)30 b(all)g Fw(i)25 b(>)g(j)5 b FA(.)31 b(F)-8 b(or)32 b(suc)m(h)e Fw(i)p FA(,)g(let)1104 2074 y Fw(R)1174 2036 y Fx(k)1173 2096 y(i)1242 2074 y FA(=)1338 1988 y Fp([)1439 2074 y Fr(f)p FA([)p Fw(\033)s FA(])c(:)f Fr(j)p Fw(\033)s Fr(j)h FA(=)f Fw(k)49 b Fr(^)c FA([)p Fw(\033)24 b Fr(\010)c Fw(B)2384 2062 y Fz(\026)2448 2074 y Fw(k)s FA(])25 b Fr(\022)g Fw(V)2697 2088 y Fx(i)2725 2074 y Fr(g)p Fw(:)505 2260 y FA(Then)30 b Fw(\026)p FA(\()p Fw(R)903 2227 y Fx(k)902 2285 y(i)946 2260 y FA(\))c Fz(6)f FA(2)1148 2227 y Fq(\000)p Fx(i)1232 2260 y FA(,)30 b(since)g Fw(B)41 b(=)-55 b Fr(2)25 b Fw(S)1752 2274 y Fx(i)1780 2260 y FA(.)31 b(Moreo)m(v)m(er,)i (since)d Fw(V)2537 2274 y Fx(i)2595 2260 y FA(is)g(op)s(en,)g Fw(R)3002 2227 y Fx(k)3001 2285 y(i)3070 2260 y Fr(\022)c Fw(R)3237 2222 y Fx(k)r Fy(+1)3236 2287 y Fx(i)3369 2260 y FA(.)505 2375 y(Let)36 b Fw(R)742 2389 y Fx(i)804 2375 y FA(=)908 2306 y Fp(S)984 2401 y Fx(k)1042 2375 y Fw(R)1112 2342 y Fx(k)1111 2400 y(i)1154 2375 y FA(.)g(The)f Fw(R)1476 2389 y Fx(i)1539 2375 y FA(are)h(op)s(en)f(and)f(uniformly)f(\006)2588 2342 y Fy(0)2588 2397 y Fx(n)2670 2375 y FA(relativ)m(e)i(to)h Fw(B)5 b FA(,)35 b(and)505 2489 y Fw(\026)p FA(\()p Fw(R)664 2503 y Fx(i)693 2489 y FA(\))g(=)g(sup)n Fr(f)p Fw(\026)p FA(\()p Fw(R)1210 2456 y Fx(k)1209 2514 y(i)1253 2489 y FA(\))h(:)f Fw(k)j Fr(2)c Fw(!)s Fr(g)i Fz(6)e FA(2)1855 2456 y Fq(\000)p Fx(i)1975 2489 y FA(for)i Fw(i)f(>)g(j)5 b FA(.)37 b(F)-8 b(urthermore,)36 b Fw(A)f Fr(2)f Fw(R)3221 2503 y Fx(i)3286 2489 y FA(for)505 2597 y(eac)m(h)e Fw(i)26 b(>)e(j)5 b FA(,)32 b(so)e Fw(A)h FA(is)e(not)i Fw(n)p FA(-random)e(relativ)m(e)i(to)g Fw(B)5 b FA(.)963 b Fr(a)588 2725 y FA(Com)m(bining)28 b(Theorems)i(12.14)j(and)c(12.17,)k(w)m(e)e (ha)m(v)m(e)g(the)g(follo)m(wing)d(corollary)-8 b(,)505 2833 y(whic)m(h)30 b(will)d(b)s(e)j(useful)f(sev)m(eral)h(times)g(b)s (elo)m(w.)588 2998 y FB(Cor)n(ollar)-6 b(y)35 b FA(12.18)h(\(v)-5 b(an)31 b(Lam)m(balgen)f([72)q(]\))p FB(.)46 b Fs(If)c Fw(B)47 b Fs(is)42 b Fw(n)p Fs(-r)-5 b(andom)43 b(and)g Fw(A)g Fs(is)505 3106 y Fw(n)p Fs(-r)-5 b(andom)35 b(r)-5 b(elative)33 b(to)g Fw(B)5 b Fs(,)32 b(then)h Fw(B)k Fs(is)c Fw(n)p Fs(-r)-5 b(andom)34 b(r)-5 b(elative)33 b(to)h Fw(A)p Fs(.)588 3271 y FA(The)c(follo)m(wing)f(is)h(an)g (application)f(of)h(this)f(result.)588 3436 y FB(Theorem)34 b FA(12.19)i(\(Miller)29 b(and)h(Y)-8 b(u)30 b([95)q(]\))p FB(.)46 b Fs(If)38 b Fw(A)g Fs(is)g Fw(n)p Fs(-r)-5 b(andom)39 b(and)g Fw(B)g Fz(6)3237 3450 y Fi(T)3326 3436 y Fw(A)505 3544 y Fs(is)33 b FA(1)p Fs(-r)-5 b(andom,)35 b(then)e Fw(B)k Fs(is)c Fw(n)p Fs(-r)-5 b(andom.)588 3710 y FB(Pr)n(oof.)41 b FA(Let)c Fw(X)42 b Fr(\021)1300 3724 y Fy(T)1389 3710 y Fr(;)1434 3677 y Fy(\()p Fx(n)p Fq(\000)p Fy(1\))1663 3710 y FA(b)s(e)35 b(1-random.)h(Since)f Fw(A)h FA(is)f Fw(n)p FA(-random,)g(it)h(is)f(1-)505 3818 y(random)43 b(relativ)m(e)h(to)g Fw(X)7 b FA(.)44 b(So)f Fw(X)51 b FA(is)42 b(1-random)i(relativ)m(e)f(to)h Fw(A)p FA(,)g(and)f (therefore)505 3926 y(relativ)m(e)31 b(to)f Fw(B)5 b FA(.)30 b(Hence)h Fw(B)k FA(is)29 b(1-random)h(relativ)m(e)g(to)h Fw(X)7 b FA(,)31 b(and)e(th)m(us)h(is)f Fw(n)p FA(-random.)3339 4034 y Fr(a)588 4162 y FA(By)k(di\013eren)m(t)e(means,)i(Miller)d(and)h (Y)-8 b(u)32 b([95)r(])g(pro)m(v)m(ed)g(the)h(stronger)f(result)f(that) 505 4270 y(for)e(an)m(y)h Fw(X)7 b FA(,)30 b(if)e Fw(A)h FA(is)f(1-random)h(relativ)m(e)g(to)h Fw(X)37 b FA(and)28 b Fw(B)i Fz(6)2517 4284 y Fy(T)2597 4270 y Fw(A)f FA(is)f(1-random,)i (then)505 4378 y Fw(B)35 b FA(is)30 b(1-random)g(relativ)m(e)g(to)i Fw(X)7 b FA(.)588 4543 y FB(Theorem)34 b FA(12.20)i(\(Kautz)31 b([60)q(]\))p FB(.)47 b Fs(L)-5 b(et)42 b Fw(n)h Fz(>)g FA(2)p Fs(.)g(If)f Fw(A)g Fs(and)i Fw(B)j Fs(ar)-5 b(e)43 b Fw(n)p Fs(-r)-5 b(andom)505 4651 y(r)g(elative)34 b(to)f(e)-5 b(ach)33 b(other,)h(then)f(their)h(de)-5 b(gr)g(e)g(es)33 b(form)h(a)f(minimal)h(p)-5 b(air.)588 4816 y FB(Pr)n(oof.)41 b FA(Supp)s(ose)28 b(that)i(0)25 b Fw(<)1636 4830 y Fy(T)1716 4816 y Fw(C)32 b Fz(6)1884 4830 y Fy(T)1964 4816 y Fw(A;)15 b(B)5 b FA(.)30 b(Let)g Fw(e)f FA(b)s(e)g(suc)m(h)g(that)h(\010)3023 4783 y Fx(A)3023 4838 y(e)3105 4816 y FA(=)24 b Fw(C)7 b FA(.)29 b(It)505 4925 y(is)h(easy)h(to)g(c)m(hec)m(k)h(that)g Fr(f)p Fw(X)h FA(:)26 b(\010)1621 4892 y Fx(X)1621 4947 y(e)1713 4925 y FA(=)f Fw(C)7 b Fr(g)31 b FA(is)e(a)i(\005)2192 4892 y Fx(C)2192 4949 y Fy(2)2251 4925 y FA(-class,)g(and)f(b)m(y)g (Theorem)h(8.11,)505 5033 y(it)k(has)f(measure)h(0.)g(So)f Fw(A)h FA(is)f(2-random)h(relativ)m(e)f(to)i Fw(C)7 b FA(,)34 b(and)g(hence)h(relativ)m(e)g(to)505 5141 y Fw(B)5 b FA(,)30 b(con)m(trary)i(to)f(h)m(yp)s(othesis.)1785 b Fr(a)p eop %%Page: 66 66 66 65 bop 505 363 a FD(66)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(Th)m(us,)24 b(b)m(y)h(Theorem)f(12.14,)k(ev)m(ery)d(2-random)g (set)g(is)f(the)h(join)f(of)g(a)i(minimal)c(pair.)588 649 y(Theorem)33 b(12.20)i(cannot)f(b)s(e)e(extended)h(to)h(the)g Fw(n)29 b FA(=)g(1)34 b(case.)g(Indeed,)f(Ku)m(\024)-43 b(cera)505 757 y([66)r(])27 b(sho)m(w)m(ed)g(that)g(no)g(t)m(w)m(o)h (\001)1544 724 y Fy(0)1544 782 y(2)1610 757 y FA(1-random)f(sets)h (form)e(a)h(minimal)d(pair.)i(Ho)m(w)m(ev)m(er,)505 865 y(w)m(e)31 b(ha)m(v)m(e)h(the)e(follo)m(wing)f(consequence)i(of)g (Theorem)f(8.10.)588 1034 y FB(Theorem)k FA(12.21)i(\(Hirsc)m(hfeldt,) 30 b(Nies,)g(and)g(Stephan)f([50)q(]\))p FB(.)47 b Fs(If)30 b Fw(A)g Fs(and)h Fw(B)k Fs(ar)-5 b(e)505 1141 y FA(\001)581 1108 y Fy(0)581 1166 y(2)647 1141 y Fs(and)27 b FA(1)p Fs(-r)-5 b(andom)29 b(r)-5 b(elative)27 b(to)g(e)-5 b(ach)28 b(other,)f(then)g(any)g Fw(X)33 b Fz(6)2635 1155 y Fi(T)2715 1141 y Fw(A;)15 b(B)31 b Fs(is)c Fw(K)7 b Fs(-trivial.)588 1310 y FB(Pr)n(oof.)41 b FA(Since)35 b Fw(A)g FA(is)g(1-random)g (relativ)m(e)g(to)i Fw(B)5 b FA(,)35 b(it)g(is)f(1-random)h(relativ)m (e)h(to)505 1418 y Fw(X)7 b FA(.)31 b(So)g Fw(X)37 b FA(is)30 b(a)h(basis)e(for)h(1-randomness,)g(and)g(th)m(us)g(is)f Fw(K)7 b FA(-trivial.)481 b Fr(a)588 1548 y FA(Randomness)29 b(is)e(link)m(ed)h(to)h(prop)s(erties)f(of)h(\\almost)g(all")f (degrees.)i(Classically)-8 b(,)505 1656 y(Kolmogoro)m(v's)46 b(0-1)f(la)m(w)f(states)i(that)e(an)m(y)h(class)f(of)g(sets)h(closed)f (under)f(\014nite)505 1764 y(translations)32 b(has)g(measure)h(0)g(or)f (1)h(\(see)h(e.g.)f(Oxtob)m(y)g([110)r(]\).)g(There)f(is)g(also)h(an) 505 1872 y(e\013ectiv)m(e)f(0-1)g(la)m(w.)588 2040 y FB(Lemma)i FA(12.22)j(\(Kurtz)30 b([71)q(],)h(Kautz)f([60)r(],)g(Ku)m (\024)-43 b(cera)32 b(for)e Fw(n)25 b FA(=)g(1\))p FB(.)46 b Fs(L)-5 b(et)49 b Fw(X)56 b Fs(b)-5 b(e)48 b(a)505 2150 y(set,)40 b(let)g Fw(n)e Fz(>)g FA(1)p Fs(,)i(and)h(let)f Fw(T)52 b Fs(b)-5 b(e)40 b(a)g FA(\005)1829 2117 y Fx(X)1829 2172 y(n)1897 2150 y Fs(-class)g(of)g(p)-5 b(ositive)41 b(me)-5 b(asur)g(e.)41 b(If)e Fw(A)g Fr(2)f FA(2)3344 2117 y Fx(!)505 2258 y Fs(is)c Fw(n)p Fs(-r)-5 b(andom)35 b(r)-5 b(elative)35 b(to)f Fw(X)7 b Fs(,)34 b(then)g(ther)-5 b(e)34 b(ar)-5 b(e)35 b Fw(\033)30 b Fr(2)d FA(2)2399 2225 y Fx()f Fw(k)c Fr(\000)d Fw(O)s FA(\(1\).)505 660 y(Ho)m(w)m(ev)m(er,)32 b(one)d(w)m(ould)f(naturally)f(exp)s(ect)i Fw(n)p FA(-)g(and)g(\()p Fw(n)17 b FA(+)g(1\)-random)29 b(sets)h(to)f(ha)m(v)m(e)505 774 y(di\013eren)m(t)h(unrelativized)e(initial)g(segmen)m(t)k (complexities.)d(This)g(is)g(true)h(for)g(\012)3261 741 y Fy(\()p Fx(n)p Fy(\))3363 774 y FA(.)588 926 y FB(Theorem)k FA(12.24)i(\(Y)-8 b(u,)31 b(Ding,)f(and)g(Do)m(wney)h([141)r(]\))p FB(.)46 b Fs(F)-7 b(or)34 b(al)5 b(l)33 b Fw(c)g Fs(and)g Fw(n)25 b(<)g(m)p Fs(,)1211 1082 y Fr(9)1262 1044 y Fq(1)1336 1082 y Fw(k)18 b FA([)p Fw(K)7 b FA(\(\012)1611 1044 y Fy(\()p Fx(n)p Fy(\))1738 1070 y Fz(\026)1801 1082 y Fw(k)s FA(\))26 b Fw(<)f(K)7 b FA(\(\012)2193 1044 y Fy(\()p Fx(m)p Fy(\))2340 1070 y Fz(\026)2403 1082 y Fw(k)s FA(\))21 b Fr(\000)e Fw(c)p FA(])p Fw(:)505 1234 y FA(F)-8 b(or)31 b Fw(n)25 b FA(=)g(0)31 b(and)f Fw(m)25 b FA(=)g(1,)31 b(Theorem)f(12.24)i(w)m(as)f(pro)m(v)m(ed)f(b)m (y)g(Solo)m(v)-5 b(a)m(y)32 b([126)q(],)f(using)505 1342 y(totally)g(di\013eren)m(t)f(metho)s(ds.)588 1450 y(In)c(con)m(trast)i (to)g(this)d(result,)h(Miller)e(and)i(Y)-8 b(u)27 b([95)q(])g(sho)m(w)m (ed)g(that)g(the)g(relativiza-)505 1558 y(tions)21 b(of)g(\012)g(ha)m (v)m(e)h(incomparable)e Fw(K)7 b FA(-degrees.)22 b(Indeed,)e(they)h (pro)m(v)m(ed)h(the)f(follo)m(wing)505 1666 y(stronger)31 b(result,)e(whic)m(h)h(will)d(b)s(e)j(further)f(discussed)g(in)g(the)h (next)h(section.)588 1817 y FB(Theorem)j FA(12.25)i(\(Miller)29 b(and)h(Y)-8 b(u)30 b([95)q(]\))p FB(.)46 b Fs(F)-7 b(or)31 b(al)5 b(l)31 b Fw(m)25 b Fr(6)p FA(=)g Fw(n)p Fs(,)k(the)h Fw(K)7 b Fs(-de)-5 b(gr)g(e)g(es)30 b(of)505 1931 y FA(\012)571 1898 y Fy(\()p Fx(m)p Fy(\))725 1931 y Fs(and)k FA(\012)968 1898 y Fy(\()p Fx(n)p Fy(\))1102 1931 y Fs(have)f(no)g(upp)-5 b(er)34 b(b)-5 b(ound.)588 2139 y Fu(x)p Ft(13.)53 b(Results)24 b(of)g(Miller)g(and)g(Y)-9 b(u,)24 b(and)g(v)-6 b(an)24 b(Lam)m(balgen)e(reducibilit)m(y)-9 b(.)505 2247 y FA(Recen)m(tly)h(,) 31 b(Jo)s(e)e(Miller)e(and)i(Liang)f(Y)-8 b(u)29 b([93)q(,)h(95)q(,)f (96)q(])g(ha)m(v)m(e)i(pro)m(v)m(ed)e(some)g(remark-)505 2355 y(able)43 b(results)e(on)i(the)g(initial)c(segmen)m(t)44 b(complexities)e(of)h(random)e(sets,)j(whic)m(h)505 2463 y(highligh)m(t)24 b(b)s(oth)i(the)g(strengths)f(and)h(limitations)d(of) j(initial)e(segmen)m(t)j(complexit)m(y)505 2571 y(as)g(a)f(measure)g (of)h(relativ)m(e)f(randomness.)f(Motiv)-5 b(ated)27 b(b)m(y)f(v)-5 b(an)26 b(Lam)m(balgen's)h(The-)505 2679 y(orems)k(12.14)h(and)d(12.17,)k(they)d(in)m(tro)s(duced)f(the)h(follo) m(wing)f(measure)h(of)g(relativ)m(e)505 2786 y(randomness.)588 2938 y FB(Definition)35 b FA(13.1)h(\(Miller)29 b(and)g(Y)-8 b(u)31 b([95)q(]\))p FB(.)46 b FA(W)-8 b(e)39 b(sa)m(y)e(that)h Fw(A)g FA(is)e Fs(van)j(L)-5 b(amb)g(al-)505 3046 y(gen)41 b(r)-5 b(e)g(ducible)46 b FA(to)40 b Fw(B)5 b FA(,)38 b(and)h(write)f Fw(A)i Fz(6)1932 3060 y Fy(vL)2057 3046 y Fw(B)5 b FA(,)38 b(if)g(for)h(all)f Fw(C)46 b Fr(2)39 b FA(2)2825 3013 y Fx(!)2876 3046 y FA(,)g(if)f Fw(A)26 b Fr(\010)g Fw(C)45 b FA(is)505 3154 y(1-random)31 b(then)f Fw(B)25 b Fr(\010)19 b Fw(C)37 b FA(is)30 b(1-random.)588 3306 y(This)23 b(notion)h(is)f(closely)h(related)g(to)h(one)g(in)m(tro) s(duced)e(b)m(y)h(Nies.)g(In)g([103)q(,)h(Section)505 3414 y(8],)37 b(Nies)f(de\014ned)f Fw(A)g Fz(6)1336 3428 y Fy(LR)1471 3414 y Fw(B)40 b FA(if)35 b(ev)m(ery)i(set)g(that)g(is)e (1-random)h(relativ)m(e)g(to)h Fw(B)j FA(is)505 3522 y(1-random)h(relativ)m(e)g(to)h Fw(A)p FA(.)f(\(So,)g(for)g(instance,)g Fw(A)i Fz(6)2434 3536 y Fy(LR)2576 3522 y Fr(;)e FA(iff)f Fw(A)h FA(is)f(lo)m(w)g(for)h(1-)505 3630 y(randomness.\))34 b(Notice)h(that)g(this)e(relation)h(is)f(implied)e(b)m(y)k(T)-8 b(uring)32 b(reducibilit)m(y)-8 b(.)505 3738 y(Nies)32 b([103)q(])g(studied)e(the)i(monotone)g(\006)1889 3705 y Fy(0)1889 3762 y(3)1960 3738 y FA(op)s(erator)g Fr(LR)p FA(\()p Fw(B)5 b FA(\))27 b(:=)h Fr(f)p Fw(A)g FA(:)g Fw(A)f Fz(6)3123 3752 y Fy(LR)3250 3738 y Fw(B)5 b Fr(g)p FA(.)505 3846 y(He)28 b(also)e(sho)m(w)m(ed)i(that)f(if)f Fw(A)h FA(and)f Fw(B)31 b FA(are)c(c.e.,)i(then)d Fw(A)f Fz(6)2470 3860 y Fy(LR)2596 3846 y Fw(B)31 b FA(implies)24 b Fw(A)3069 3813 y Fq(0)3118 3846 y Fz(6)3189 3860 y Fy(tt)3273 3846 y Fw(B)3347 3813 y Fq(0)3369 3846 y FA(.)505 3954 y(Moreo)m(v)m(er,)40 b(applying)34 b(the)j(tec)m(hnique)g(of)g (pseudo-jumps)d(from)j([56)q(])g(to)g(the)g(c.e.)505 4062 y(op)s(erator)22 b(giv)m(en)f(b)m(y)g(the)g(construction)g(of)h(a) f(set)h(that)g(is)e(lo)m(w)h(for)g(1-randomness,)g(he)505 4169 y(sho)m(w)m(ed)k(that)h(there)f(is)e(a)i(c.e.)h(set)g(that)f(is)f (T)-8 b(uring)23 b(incomplete)h(but)g(LR-complete.)588 4277 y(If)35 b Fw(A)g FA(and)g Fw(B)k FA(are)d(b)s(oth)e(1-random)h (then)g(Theorems)g(12.14)i(and)e(12.17)i(imply)505 4385 y(that)43 b Fw(A)j Fz(6)899 4399 y Fy(LR)1043 4385 y Fw(B)h FA(iff)41 b Fw(B)50 b Fz(6)1472 4399 y Fy(vL)1602 4385 y Fw(A)p FA(.)43 b(\(Notice)g(the)g(in)m(v)m(erse)f (relationship.\))e(If)i Fw(A)g FA(is)505 4493 y(not)37 b(1-random,)g(then)g Fw(A)24 b Fr(\010)g Fw(C)44 b FA(is)35 b(nev)m(er)i(1-random,)g(no)g(matter)h(what)e Fw(C)43 b FA(is,)36 b(so)505 4601 y(the)f(least)g(vL-degree)g(consists)f(of)h (all)e(sets)i(that)g(are)g(not)f(1-random.)h(Th)m(us)e(vL-)505 4709 y(reducibilit)m(y)c(is)i(in)m(teresting)g(only)g(on)g(the)h (1-random)g(sets.)g(It)g(migh)m(t)f(b)s(e)h(fruitful)505 4817 y(to)39 b(explore)e(extensions)g(of)h(vL-reducibilit)m(y)c(that)39 b(b)s(eha)m(v)m(e)f(non)m(trivially)d(on)i(the)505 4925 y(non-1-random)31 b(sets.)588 5033 y(The)41 b(follo)m(wing)f(result)g (summarizes)g(some)i(of)f(the)h(basic)e(prop)s(erties)g(of)h(vL-)505 5141 y(reducibilit)m(y)27 b(and)j(the)h(resulting)d(vL-degrees.)p eop %%Page: 68 68 68 67 bop 505 363 a FD(68)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FB(Theorem)34 b FA(13.2)i(\(Miller)29 b(and)g(Y)-8 b(u)31 b([95)q(]\))p FB(.)563 668 y FA(\(i\))42 b Fs(If)32 b Fw(A)h Fs(is)f Fw(n)p Fs(-r)-5 b(andom)35 b(and)e Fw(A)26 b Fz(6)1754 682 y Fi(vL)1860 668 y Fw(B)5 b Fs(,)32 b(then)h Fw(B)k Fs(is)c Fw(n)p Fs(-r)-5 b(andom.)538 776 y FA(\(ii\))41 b Fs(If)30 b Fw(A)17 b Fr(\010)e Fw(B)36 b Fs(is)30 b FA(1)p Fs(-r)-5 b(andom)33 b(then)f(the)f(vL-de)-5 b(gr)g(e)g(es)31 b(of)g Fw(A)g Fs(and)h Fw(B)j Fs(have)c(no)g(upp)-5 b(er)701 884 y(b)g(ound.)33 b(Thus)g(ther)-5 b(e)34 b(is)e(no)h(join)g(op)-5 b(er)g(ator)36 b(on)d(the)g(vL-de)-5 b(gr)g(e)g(es.)513 992 y FA(\(iii\))40 b Fs(If)32 b Fw(A)26 b Fz(6)961 1006 y Fi(T)1041 992 y Fw(B)37 b Fs(and)c Fw(A)g Fs(is)g FA(1)p Fs(-r)-5 b(andom,)34 b(then)g Fw(B)29 b Fz(6)2330 1006 y Fi(vL)2437 992 y Fw(A)p Fs(.)515 1100 y FA(\(iv\))42 b Fs(Ther)-5 b(e)33 b(ar)-5 b(e)34 b FA(1)p Fs(-r)-5 b(andom)35 b(sets)d Fw(A)26 b Fr(\021)1869 1114 y Fi(vL)1975 1100 y Fw(B)37 b Fs(such)c(that)h Fw(A)26 b(<)2639 1114 y Fi(T)2719 1100 y Fw(B)5 b Fs(.)540 1208 y FA(\(v\))43 b Fs(Ther)-5 b(e)33 b(ar)-5 b(e)34 b(no)f(maximal)h(or)f(minimal)h (vL-de)-5 b(gr)g(e)g(es)33 b(of)g FA(1)p Fs(-r)-5 b(andom)35 b(sets.)515 1316 y FA(\(vi\))42 b Fs(If)32 b Fw(A)20 b Fr(\010)g Fw(B)37 b Fs(is)c FA(1)p Fs(-r)-5 b(andom)35 b(then)e Fw(A)21 b Fr(\010)e Fw(B)30 b(<)2139 1330 y Fi(vL)2246 1316 y Fw(A;)15 b(B)5 b Fs(.)490 1423 y FA(\(vii\))41 b Fs(Every)36 b(\014nite)h(p)-5 b(artial)40 b(or)-5 b(der)38 b(c)-5 b(an)38 b(b)-5 b(e)37 b(emb)-5 b(e)g(dde)g(d)39 b(into)f(the)f(vL-de)-5 b(gr)g(e)g(es,)38 b(and)701 1531 y(henc)-5 b(e)33 b(the)g FA(\006)1169 1498 y Fy(0)1169 1556 y(1)1208 1531 y Fs(-the)-5 b(ory)33 b(of)g(the)g(vL-de)-5 b(gr)g(e)g(es)34 b(is)e(de)-5 b(cidable.)588 1695 y FA(One)33 b(of)g(the)h(attractiv)m(e)h(features)e(of)g(vL-reducibilit)m(y)d(is)i (that)i(it)f(can)g(b)s(e)f(used)505 1803 y(to)g(pro)m(v)m(e)g(results)d (ab)s(out)i Fw(K)7 b FA(-)31 b(and)f Fw(C)7 b FA(-reducibilit)m(y)-8 b(,)28 b(in)i(w)m(a)m(ys)h(that)h(are)f(often)h(eas-)505 1911 y(ier)42 b(than)g(dealing)f(directly)g(with)f(these)j (reducibilities.)38 b(The)k(follo)m(wing)f(result)505 2019 y(is)33 b(what)h(allo)m(ws)f(for)h(the)g(transfer)g(of)g(results)e (from)i(vL-reducibilit)m(y)d(to)j Fw(K)7 b FA(-)34 b(and)505 2126 y Fw(C)7 b FA(-reducibilit)m(y)-8 b(.)588 2290 y FB(Theorem)34 b FA(13.3)i(\(Miller)29 b(and)g(Y)-8 b(u)31 b([95)q(]\))p FB(.)46 b Fs(F)-7 b(or)34 b(any)f(sets)g Fw(A)g Fs(and)g Fw(B)5 b Fs(,)563 2416 y FA(\(i\))42 b Fw(A)25 b Fz(6)865 2430 y Fx(K)958 2416 y Fw(B)30 b Fr(\))25 b Fw(A)h Fz(6)1338 2430 y Fi(vL)1444 2416 y Fw(B)37 b Fs(and)538 2524 y FA(\(ii\))k Fw(A)25 b Fz(6)865 2538 y Fx(C)949 2524 y Fw(B)30 b Fr(\))25 b Fw(A)h Fz(6)1329 2538 y Fi(vL)1435 2524 y Fw(B)5 b Fs(.)588 2688 y FA(W)-8 b(e)39 b(state)g(the)e(follo)m(wing)f(consequences)i(for)f Fw(K)7 b FA(-reducibilit)m(y)-8 b(,)35 b(but)i(they)h(also)505 2796 y(hold)29 b(for)i Fw(C)7 b FA(-reducibilit)m(y)-8 b(.)588 2959 y FB(Cor)n(ollar)i(y)35 b FA(13.4)h(\(Miller)29 b(and)g(Y)-8 b(u)31 b([95)q(]\))p FB(.)563 3086 y FA(\(i\))42 b Fs(If)32 b Fw(A)26 b Fz(6)961 3100 y Fx(K)1054 3086 y Fw(B)37 b Fs(and)c Fw(A)g Fs(is)g Fw(n)p Fs(-r)-5 b(andom,)34 b(then)f Fw(B)k Fs(is)c Fw(n)p Fs(-r)-5 b(andom.)538 3193 y FA(\(ii\))41 b Fs(If)f Fw(A)26 b Fr(\010)g Fw(B)45 b Fs(is)40 b FA(1)p Fs(-r)-5 b(andom,)43 b(then)e Fw(A)f Fr(j)2002 3207 y Fx(K)2110 3193 y Fw(B)5 b Fs(,)40 b(and)h(the)g Fw(K)7 b Fs(-de)-5 b(gr)g(e)g(es)41 b(of)g Fw(A)g Fs(and)701 3301 y Fw(B)i Fs(have)d(no)f(upp)-5 b(er)40 b(b)-5 b(ound.)40 b(Thus)g(ther)-5 b(e)39 b(is)g(no)h(join)f(op)-5 b(er)g(ator)42 b(on)e(the)f Fw(K)7 b Fs(-)701 3409 y(de)-5 b(gr)g(e)g(es.)505 3573 y FA(According)33 b(to)h(Miller)d(and)i(Y)-8 b(u)33 b([95)q(],)h(R.)f(Rettinger)g(indep)s(enden)m(tly)d(announced)505 3681 y(that)h(if)f Fw(A)20 b Fr(\010)g Fw(B)35 b FA(is)29 b(1-random)i(then)f Fw(A)25 b Fr(j)1895 3695 y Fx(K)1989 3681 y Fw(B)5 b FA(.)588 3789 y(Theorem)34 b(12.25)h(follo)m(ws)e(from) g(the)h(second)g(part)f(of)h(the)g(ab)s(o)m(v)m(e)g(result,)f(since)505 3897 y(if)k Fw(m)g Fr(6)p FA(=)g Fw(n)g FA(then)h(\012)1194 3864 y Fy(\()p Fx(m)p Fy(\))1352 3897 y FA(and)f(\012)1602 3864 y Fy(\()p Fx(n)p Fy(\))1741 3897 y FA(are)h(1-random)g(relativ)m (e)g(to)g(eac)m(h)h(other,)f(and)505 4006 y(hence)31 b(\012)824 3973 y Fy(\()p Fx(m)p Fy(\))965 4006 y Fr(\010)20 b FA(\012)1122 3973 y Fy(\()p Fx(n)p Fy(\))1254 4006 y FA(is)29 b(1-random.)588 4114 y(Th)m(us)23 b(w)m(e)h(see)h(that,)f (while)e(there)i(is)e(no)i(direct)f(correlation)h(b)s(et)m(w)m(een)g (increasing)505 4222 y(lev)m(els)43 b(of)f(randomness)g(and)g (increasing)f Fw(K)7 b FA(-degrees,)43 b(there)g Fs(is)50 b FA(a)43 b(relationship)505 4330 y(b)s(et)m(w)m(een)31 b(lev)m(els)f(of)h(randomness)e(and)h(initial)d(segmen)m(t)32 b(complexit)m(y)-8 b(.)588 4438 y(Although)44 b(certain)h(results)e(on) h(vL-reducibilit)m(y)e(can)i(b)s(e)g(transfered)g(to)i Fw(K)7 b FA(-)505 4546 y(reducibilit)m(y)-8 b(,)29 b(there)j(are)g (some)g(imp)s(ortan)m(t)f(di\013erences)g(b)s(et)m(w)m(een)h(these)g (notions.)505 4654 y(F)-8 b(or)44 b(instance,)e(b)m(y)g(part)g(\(iii\)) f(of)h(Theorem)h(13.2,)h(ev)m(ery)f(\001)2677 4621 y Fy(0)2677 4678 y(2)2758 4654 y FA(1-random)g(set)f(is)505 4762 y Fz(>)576 4776 y Fy(vL)705 4762 y FA(\012.)f(On)f(the)h(other)g (hand,)f(w)m(e)i(ha)m(v)m(e)g(the)f(follo)m(wing)f(result,)g(whic)m(h)g (sho)m(ws)505 4870 y(that)31 b(if)f(\012)25 b(=)g(\012)1039 4884 y Fy(0)1098 4870 y Fr(\010)20 b FA(\012)1255 4884 y Fy(1)1324 4870 y FA(then)30 b(\012)1597 4884 y Fy(0)1667 4870 y FA(is)f(an)h(example)g(of)h(a)g(\001)2496 4837 y Fy(0)2496 4894 y(2)2565 4870 y FA(1-random)g(set)f Fo(\013)3187 4884 y Fx(K)3281 4870 y FA(\012.)588 5033 y FB(Theorem)k FA(13.5)i(\(Miller)29 b(and)g(Y)-8 b(u)31 b([95)q(]\))p FB(.)46 b Fs(If)35 b Fw(A)22 b Fr(\010)g Fw(B)39 b Fs(is)c FA(1)p Fs(-r)-5 b(andom)38 b(then)d Fw(A)30 b Fr(j)3326 5047 y Fx(K)505 5141 y Fw(A)21 b Fr(\010)f Fw(B)5 b Fs(.)p eop %%Page: 69 69 69 68 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(69)588 541 y FA(An)44 b(ev)m(en)g(more)g(signi\014can)m(t)e (di\013erence)h(b)s(et)m(w)m(een)h(the)g(vL-degrees)g(and)f(the)505 649 y Fw(K)7 b FA(-degrees)40 b(is)e(that)h(the)g(former)f(are)h(in)m (v)-5 b(arian)m(t)38 b(under)f(computable)i(p)s(erm)m(uta-)505 757 y(tions)e(\(b)m(y)g(part)g(\(iii\))e(of)i(Theorem)g(13.2)h(and)f (the)g(closure)f(of)h(the)g(notion)g(of)g(1-)505 865 y(randomness)30 b(under)e(computable)i(p)s(erm)m(utations\),)g(but)g (the)h(latter)f(are)h(not.)588 1022 y FB(Theorem)j FA(13.6)i(\(Miller) 29 b(and)g(Y)-8 b(u)31 b([95)q(]\))p FB(.)46 b Fs(Ther)-5 b(e)42 b(is)f(a)h(c)-5 b(omputable)43 b(p)-5 b(ermuta-)505 1130 y(tion)37 b Fw(f)k FA(:)32 b Fw(!)j Fr(!)d Fw(!)39 b Fs(such)d(that)i(for)e(every)g FA(1)p Fs(-r)-5 b(andom)39 b(set)d Fw(A)p Fs(,)h(the)f Fw(K)7 b Fs(-de)-5 b(gr)g(e)g(es)37 b(of)f Fw(A)505 1238 y Fs(and)e Fw(f)10 b FA(\()p Fw(A)p FA(\))33 b Fs(have)g(no)g(upp)-5 b(er)34 b(b)-5 b(ound.)588 1394 y FA(This)23 b(result)f(suggests)j(that)g Fw(K)7 b FA(-reducibilit)m(y)20 b(ma)m(y)25 b(b)s(e)e(to)s(o)i(strong)f(as)g (a)h(measure)505 1502 y(of)d(relativ)m(e)g(randomness)f(on)h(the)g (1-random)g(sets,)g(and)g(that)g(vL-reducibilit)m(y)d(ma)m(y)505 1610 y(b)s(e)30 b(a)h(b)s(etter)f(measure)h(in)e(this)g(con)m(text.)588 1718 y(As)21 b(p)s(oin)m(ted)f(out)h(b)m(y)f(Miller)f(and)h(Y)-8 b(u)21 b([95)q(],)g(it)f(follo)m(ws)g(from)h(part)f(\(ii\))g(of)h (Theorem)505 1826 y(13.2)37 b(that)f(if)e(w)m(e)i(let)f(\012)1333 1840 y Fx(n)1415 1826 y FA(b)s(e)g(the)g Fw(n)p FA(-th)g(column)f(of)h (\012,)g(then)g Fr(f)p FA(\012)2791 1840 y Fx(n)2872 1826 y FA(:)e Fw(n)g Fr(2)g Fw(!)s Fr(g)j FA(is)e(a)505 1934 y(vL-an)m(tic)m(hain,)27 b(and)g(hence)g(a)g Fw(K)7 b FA(-an)m(tic)m(hain.)27 b(So)g(w)m(e)h(ha)m(v)m(e)g(a)f(concrete)i (example)d(of)505 2042 y(in\014nitely)d(man)m(y)i(pairwise)e (incomparable)h Fw(K)7 b FA(-degrees)25 b(of)h(1-random)f(sets.)h(It)f (w)m(as)505 2150 y(a)38 b(v)m(exing)f(op)s(en)f(question)g(whether)g (there)h(are)h(an)m(y)f Fs(c)-5 b(omp)g(ar)g(able)47 b Fw(K)7 b FA(-degrees)37 b(of)505 2258 y(1-random)24 b(sets.)g(Miller)e(and)g(Y)-8 b(u)24 b([96)q(])g(recen)m(tly)g(answ)m (ered)f(this)f(question,)h(making)505 2366 y(use)g(of)g(a)g(con)m(v)m (erse)h(to)f(part)f(\(ii\))g(of)h(Theorem)f(3.10,)j(whic)m(h)c(sho)m (ws)h(that)i(the)e(Ample)505 2473 y(Excess)31 b(Theorem)f(\(part)h (\(i\))f(of)h(Theorem)f(3.10\))i(is)d(in)g(a)i(sense)f(tigh)m(t.)588 2630 y FB(Theorem)k FA(13.7)i(\(Miller)29 b(and)g(Y)-8 b(u)31 b([96)q(]\))p FB(.)46 b Fs(L)-5 b(et)41 b Fw(f)50 b Fs(b)-5 b(e)41 b(any)g(function)g(such)g(that)505 2676 y Fp(P)601 2771 y Fx(n)663 2744 y FA(2)708 2711 y Fq(\000)p Fx(f)7 b Fy(\()p Fx(n)p Fy(\))932 2744 y Fw(<)25 b Fr(1)p Fs(.)32 b(Ther)-5 b(e)34 b(is)e(a)h FA(1)p Fs(-r)-5 b(andom)35 b(set)e Fw(A)g Fs(such)g(that)1372 2892 y Fw(K)7 b FA(\()p Fw(A)1584 2880 y Fz(\026)1647 2892 y Fw(n)p FA(\))26 b Fz(6)f Fw(n)19 b FA(+)h Fw(f)10 b FA(\()p Fw(n)p FA(\))20 b(+)g Fw(O)s FA(\(1\))p Fw(:)588 3048 y FB(Cor)n(ollar)-6 b(y)35 b FA(13.8)h(\(Miller)29 b(and)g(Y)-8 b(u)31 b([96)q(]\))p FB(.)46 b Fs(L)-5 b(et)32 b Fw(B)k Fs(b)-5 b(e)31 b FA(1)p Fs(-r)-5 b(andom.)34 b(Ther)-5 b(e)33 b(is)e(a)505 3156 y FA(1)p Fs(-r)-5 b(andom)48 b(set)c Fw(A)k(<)1267 3170 y Fx(K)1382 3156 y Fw(B)5 b Fs(.)44 b(In)h(fact,)f Fw(A)h Fs(c)-5 b(an)45 b(b)-5 b(e)45 b(chosen)g(so)h(that)g FA(lim)3055 3170 y Fx(n)3117 3156 y Fw(K)7 b FA(\()p Fw(B)3357 3144 y Fz(\026)505 3264 y Fw(n)p FA(\))21 b Fr(\000)e Fw(K)7 b FA(\()p Fw(A)919 3252 y Fz(\026)982 3264 y Fw(n)p FA(\))25 b(=)g Fr(1)p Fs(.)588 3420 y FB(Pr)n(oof.)41 b FA(Let)g Fw(g)s FA(\()p Fw(n)p FA(\))i(=)f Fw(K)7 b FA(\()p Fw(B)1678 3408 y Fz(\026)1758 3420 y Fw(n)p FA(\))27 b Fr(\000)f Fw(n)p FA(.)41 b(By)g(the)f(Ample)g(Excess)h(Theorem,)505 3466 y Fp(P)601 3561 y Fx(n)663 3534 y FA(2)708 3501 y Fq(\000)p Fx(g)r Fy(\()p Fx(n)p Fy(\))928 3534 y Fw(<)25 b Fr(1)p FA(,)31 b(so)h(there)f(is)e(a)j(function)d Fw(f)40 b FA(suc)m(h)31 b(that)g(lim)2656 3548 y Fx(n)2718 3534 y Fw(g)s FA(\()p Fw(n)p FA(\))21 b Fr(\000)f Fw(f)10 b FA(\()p Fw(n)p FA(\))26 b(=)g Fr(1)505 3652 y FA(and)693 3584 y Fp(P)789 3679 y Fx(n)851 3652 y FA(2)896 3619 y Fq(\000)p Fx(f)7 b Fy(\()p Fx(n)p Fy(\))1137 3652 y Fw(<)42 b Fr(1)p FA(.)f(Let)h Fw(A)f FA(b)s(e)f(as)h(in)f(Theorem)g (13.7.)j(Then)d(lim)3059 3666 y Fx(n)3121 3652 y Fw(K)7 b FA(\()p Fw(B)3357 3640 y Fz(\026)505 3762 y Fw(n)p FA(\))21 b Fr(\000)e Fw(K)7 b FA(\()p Fw(A)919 3750 y Fz(\026)982 3762 y Fw(n)p FA(\))25 b(=)g Fr(1)p FA(.)2030 b Fr(a)505 3888 y FA(Miller)24 b(and)g(Y)-8 b(u)25 b([96)q(])h(sho)m(w) m(ed)f(that)h(it)e(is)g(also)h(p)s(ossible)e(to)j(ensure)e(that)i Fw(A)10 b Fr(\010)g Fw(C)31 b(<)3327 3902 y Fx(K)505 3996 y Fw(B)k FA(for)c(all)e Fw(C)7 b FA(.)30 b(As)h(they)f(p)s(oin)m (ted)g(out)g(in)g([95)q(],)h(if)e(w)m(e)i(tak)m(e)h(a)f Fw(C)37 b FA(that)31 b(is)f(1-random)505 4103 y(relativ)m(e)d(to)h Fw(A)p FA(,)f(then)f Fw(A)13 b Fr(\010)g Fw(C)32 b FA(is)26 b(1-random,)h(and)f(the)h Fw(K)7 b FA(-degree)27 b(of)g Fw(B)k FA(b)s(ounds)24 b(the)505 4211 y Fw(K)7 b FA(-degrees)27 b(of)g Fw(A)f FA(and)g Fw(A)12 b Fr(\010)g Fw(C)7 b FA(,)26 b(whic)m(h)f(do)s(es)h(a)m(w)m(a)m(y)i(with)d(a)h(p)s(ossible)e(impro)m (v)m(emen)m(t)505 4319 y(of)31 b(Theorem)f(13.5.)588 4427 y(Using)48 b(new)f(tec)m(hniques)h(and)f(extensions)h(of)g(the)h (ab)s(o)m(v)m(e)g(metho)s(ds,)e(Miller)505 4535 y(pro)m(v)m(ed)31 b(the)g(follo)m(wing.)588 4692 y FB(Theorem)j FA(13.9)i(\(Miller)29 b([93)q(]\))p FB(.)563 4817 y FA(\(i\))42 b Fs(If)33 b Fw(A)h Fs(and)h Fw(B)j Fs(ar)-5 b(e)35 b FA(1)p Fs(-r)-5 b(andom)36 b(and)f Fw(A)28 b Fr(\021)2095 4831 y Fx(K)2190 4817 y Fw(B)5 b Fs(,)33 b(then)h Fw(A)2596 4784 y Fq(0)2647 4817 y Fr(\021)2718 4831 y Fi(tt)2797 4817 y Fw(B)2871 4784 y Fq(0)2894 4817 y Fs(.)f(Thus)i(every)701 4925 y Fw(K)7 b Fs(-de)-5 b(gr)g(e)g(e)33 b(of)f FA(1)p Fs(-r)-5 b(andom)35 b(sets)e(is)g(c)-5 b(ountable.)538 5033 y FA(\(ii\))41 b Fs(If)i Fw(A)h Fs(and)h Fw(B)j Fs(ar)-5 b(e)44 b FA(3)p Fs(-r)-5 b(andom)46 b(and)f Fw(A)g Fz(6)2181 5047 y Fx(K)2295 5033 y Fw(B)5 b Fs(,)43 b(then)h Fw(B)50 b Fz(6)2843 5047 y Fi(T)2943 5033 y Fw(A)28 b Fr(\010)g(;)3183 5000 y Fq(0)3251 5033 y Fs(and)701 5141 y Fw(B)775 5108 y Fq(0)823 5141 y Fz(6)894 5155 y Fi(T)974 5141 y Fw(A)1042 5108 y Fq(0)1065 5141 y Fs(.)p eop %%Page: 70 70 70 69 bop 505 363 a FD(70)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)588 541 y FA(As)37 b(noted)h(b)m(y)f(Miller)e([93)q(],)i(it)g(follo)m(ws)f (from)h(part)g(\(ii\))f(that)i(the)f(upp)s(er)e(cone)505 649 y(ab)s(o)m(v)m(e)e(the)f Fw(K)7 b FA(-degree)32 b(of)f(a)h (3-random)g(set)g(is)e(alw)m(a)m(ys)i(coun)m(table.)g(On)f(the)g(other) 505 757 y(hand,)i(Miller)e(and)i(Y)-8 b(u)33 b([96)q(])h(sho)m(w)m(ed)f (that)h(there)g(is)e(a)h(1-random)h(set)f(whose)h Fw(K)7 b FA(-)505 867 y(degree)31 b(is)f(b)s(elo)m(w)f(an)i(an)m(tic)m(hain)f (of)g Fw(K)7 b FA(-degrees)31 b(of)g(size)f(2)2515 834 y Fq(@)2558 843 y Fn(0)2597 867 y FA(.)588 975 y(As)j(w)m(e)h(ha)m(v)m (e)g(seen)f(in)e(Theorem)i(3.5,)h(Miller)d(and)h(Y)-8 b(u)33 b([95)q(])g(ga)m(v)m(e)i(a)e(plain)e(Kol-)505 1083 y(mogoro)m(v)25 b(complexit)m(y)f(c)m(haracterization)g(of)g (1-randomness.)f(They)g(also)h(ga)m(v)m(e)h(the)505 1191 y(follo)m(wing)k(\\mixed")h(c)m(haracterization.)588 1343 y FB(Theorem)k FA(13.10)i(\(Miller)29 b(and)h(Y)-8 b(u)30 b([95)q(]\))p FB(.)46 b Fs(A)33 b(set)f Fw(A)h Fs(is)g FA(1)p Fs(-r)-5 b(andom)35 b(iff)1363 1488 y Fw(C)7 b FA(\()p Fw(A)1563 1476 y Fz(\026)1627 1488 y Fw(n)p FA(\))25 b Fz(>)g Fw(n)20 b Fr(\000)f Fw(K)7 b FA(\()p Fw(n)p FA(\))21 b Fr(\000)f Fw(O)s FA(\(1\))p Fw(:)588 1640 y FA(Sev)m(eral)i(results)e(w)m(e)i(ha)m(v)m(e)h(seen)e (p)s(oin)m(t)f(to)j(the)e(computational)g(w)m(eakness)h(of)g(ran-)505 1747 y(dom)31 b(sets.)i(Indeed,)d(there)i(are)g(w)m(a)m(ys)g(in)e(whic) m(h)g(su\016cien)m(tly)g(random)h(sets)h(b)s(egin)505 1855 y(to)h(resem)m(ble)f(highly)e(nonrandom)h(sets)h(suc)m(h)g(as)g Fw(K)7 b FA(-trivial)31 b(sets.)2809 1822 y Fy(9)2881 1855 y FA(F)-8 b(or)33 b(instance,)505 1963 y(Miller)19 b([93)q(])i(has)f(sho)m(wn)g(that)h(if)f Fw(A)g FA(is)g(3-random,)g (then)h(it)f(is)f(often)i(useless)f(in)f(lo)m(w)m(er-)505 2071 y(ing)26 b(the)h(pre\014x-free)f(complexit)m(y)g(of)h(strings,)f (so)h(that)g Fw(A)g FA(resem)m(bles)f(sets)h(that)g(are)505 2179 y(lo)m(w)21 b(for)g Fw(K)7 b FA(.)21 b(W)-8 b(e)22 b(sa)m(y)g(that)g Fw(A)f FA(is)f Fs(we)-5 b(akly)25 b(low)g(for)g Fw(K)i FA(if)20 b Fr(9)2371 2146 y Fq(1)2461 2179 y FA([)p Fw(K)7 b FA(\()p Fw(n)p FA(\))25 b Fz(6)g Fw(K)2900 2146 y Fx(A)2957 2179 y FA(\()p Fw(n)p FA(\))r(+)r Fw(O)s FA(\(1\)].)505 2287 y(That)35 b(is,)f(for)g(in\014nitely)d(man)m(y)k Fw(n)p FA(,)f(the)h(information)e(in)g Fw(A)i FA(is)e(so)i(useless)f (that)h(it)505 2395 y(cannot)c(help)e(to)i(compress)g Fw(n)p FA(.)588 2546 y FB(Theorem)j FA(13.11)i(\(Miller)29 b([93)q(]\))p FB(.)563 2672 y FA(\(i\))42 b Fs(If)32 b Fw(A)h Fs(is)f FA(3)p Fs(-r)-5 b(andom,)35 b(then)e(it)g(is)g(we)-5 b(akly)33 b(low)h(for)f Fw(K)7 b Fs(.)538 2780 y FA(\(ii\))41 b Fs(If)36 b Fw(A)h Fs(is)f(we)-5 b(akly)38 b(low)f(for)h Fw(K)43 b Fs(and)37 b FA(1)p Fs(-r)-5 b(andom,)39 b(then)e(it)g(is)f (str)-5 b(ongly)39 b(Chaitin)701 2888 y(r)-5 b(andom)35 b(\(as)e(de\014ne)-5 b(d)34 b(in)e(Se)-5 b(ction)34 b(12\).)588 3095 y Fu(x)p Ft(14.)53 b(Relativizing)39 b FA(\012)p Ft(.)46 b FA(So)33 b(far)g(when)f(w)m(e)i(ha)m(v)m(e)h(lo)s(ok)m(ed)e (at)h(relativizations,)505 3203 y(w)m(e)22 b(ha)m(v)m(e)h(alw)m(a)m(ys) f(lo)s(ok)m(ed)f(at)h(the)g(\\standard)f(w)m(a)m(y")h(of)g(forming)e(a) i(univ)m(ersal)d(pre\014x-)505 3312 y(free)j(mac)m(hine)g(\(as)g(in)e (the)i(de\014nition)e(of)h(\012)1958 3279 y Fy(\()p Fx(n)p Fy(\))2082 3312 y FA(in)f(Section)h(12\).)j(Relativization)d(acts)505 3420 y(strangely)k(on)f(randomness)g(notions,)g(since)g(w)m(e)h(are)g (dealing)f(with)f(c.e.)j(op)s(erators,)505 3528 y(but)j(not)g(CEA)g (\(computably)f(en)m(umerable)h(in)e Fs(and)33 b(ab)-5 b(ove)7 b FA(\))30 b(op)s(erators.)f(W)-8 b(e)30 b(ha)m(v)m(e)505 3636 y(already)24 b(seen)h(this)e(\(in)h(Section)g(8\))h(since,)f(for)g (instance,)g(if)g Fw(A)g FA(is)g(not)g(GL)3023 3650 y Fy(1)3063 3636 y FA(,)g(or)h(ev)m(en)505 3744 y(just)36 b(not)g Fw(K)7 b FA(-trivial,)35 b(then)h(no)g(set)g(that)h(is)e (1-random)i(relativ)m(e)f(to)h Fw(A)f FA(is)f(T)-8 b(uring)505 3851 y(ab)s(o)m(v)m(e)30 b Fw(A)f FA(\(Ku)m(\024)-43 b(cera)30 b([67)q(];)f(Hirsc)m(hfeldt,)f(Nies,)g(and)h(Stephan)e([50)r (]\).)i(In)f(particular,)505 3961 y(whatev)m(er)k(\012)961 3928 y Fy(\012)1046 3961 y FA(is,)d(\012)c Fo(\012)1324 3975 y Fy(T)1404 3961 y FA(\012)1470 3928 y Fy(\012)1525 3961 y FA(.)588 4069 y(Notice)45 b(that)g(this)e(means)g(that)i(if)e(w) m(e)h(could)f(construe)h(\012)f(as)i(an)e(in)m(v)-5 b(arian)m(t)505 4177 y(op)s(erator,)35 b(meaning)f(that)h Fw(A)d Fr(\021)1641 4191 y Fy(T)1728 4177 y Fw(B)39 b FA(implies)31 b(\012)2214 4144 y Fx(A)2303 4177 y Fr(\021)2374 4191 y Fy(T)2460 4177 y FA(\012)2526 4144 y Fx(B)2587 4177 y FA(,)j(then)g(it)g(w)m (ould)f(b)s(e)h(a)505 4285 y(degree)e(in)m(v)-5 b(arian)m(t)29 b(op)s(erator)i(that)g(is)e(not)i(the)g(jump)e(or)h(an)g(iterate)i(of)e (the)h(jump,)505 4393 y(thereb)m(y)k(resolving)e(a)h(longstanding)f (conjecture)i(of)f(Martin)g(\(see)h([61)q(,)f(p.)g(279]\).)505 4501 y(As)k(w)m(e)f(will)e(see)j(b)s(elo)m(w,)f(this)f(is)g(not)i(the)f (case,)i(but)d(giv)m(en)h(the)h(cen)m(tral)f(role)g(of)505 4608 y(\012)31 b(in)f(this)g(area,)i(it)f(is)f(natural)g(to)i(try)f(to) h(understand)d(it)i(as)g(an)g(op)s(erator)h(in)d(the)p 505 4684 499 4 v 588 4744 a Fn(9)623 4776 y Fv(Miller)39 b(has)e(suggested)h(that)e(w)n(e)i(should)f(\014nd)f(w)n(a)n(ys)h(to)g (quan)n(tify)f(the)h(idea)h(that,)f(while)505 4867 y(highly)30 b(random)f(sets)i(ha)n(v)n(e)e(a)h(great)h(deal)f(of)h(information,)g (it)f(is)g Fe(useless)j(information)p Fv(.)d(There)505 4958 y(are)j(existing)g(concepts)g(of)g(useful)g(information)g(\(see)g (An)n(tunes)e(and)h(F)-6 b(ortno)n(w)33 b([3)q(])g(for)g(sev)n(eral)505 5050 y(references\),)39 b(but)e(none)g(seem)g(to)g(ha)n(v)n(e)g(b)r (een)g(successfully)h(applied)g(y)n(et)f(to)g(the)g(con)n(text)g(of)505 5141 y(higher)26 b(order)g(randomness.)p eop %%Page: 71 71 71 70 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(71)505 541 y FA(same)28 b(w)m(a)m(y)h(that)f(w)m(e)g(seek)g(to)g (understand)e(the)h(halting)g(problem,)f(and)h(hence)g(the)505 649 y(jump)i(op)s(erator,)i(in)e(classical)h(computabilit)m(y)f(theory) -8 b(.)588 757 y(The)21 b(\014rst)f(thing)h(w)m(e)g(need)g(to)h (understand)d(is)h(what)h(w)m(e)h(actually)f(mean)g(when)f(w)m(e)505 865 y(talk)33 b(ab)s(out)f(relativizing)e(\012.)j(Clearly)-8 b(,)31 b(an)m(y)i(reasonable)f(de\014nition)e(m)m(ust)i(ensure)505 973 y(that)25 b(\012)762 940 y Fx(X)854 973 y FA(is)e(an)i Fw(X)7 b FA(-left-c.e.)26 b(real)e(and)g(is)f(1-random)i(relativ)m(e)f (to)h Fw(X)7 b FA(.)25 b(F)-8 b(urthermore,)505 1081 y(the)37 b(de\014nition)d(should)g(b)s(e)i(relativ)m(ely)g (oracle-indep)s(enden)m(t,)f(in)g(the)i(sense)f(that)505 1189 y(it)29 b(should)f(only)g(in)m(v)m(olv)m(e)i(mac)m(hines)f Fw(U)39 b FA(suc)m(h)29 b(that)h Fw(U)2350 1156 y Fx(X)2446 1189 y FA(is)f(a)h(univ)m(ersal)d(pre\014x-free)505 1297 y(mac)m(hine)39 b(relativ)m(e)h(to)g Fw(X)46 b FA(for)39 b(ev)m(ery)h(oracle)g Fw(X)7 b FA(,)40 b(and)e(the)i(co)s(ding)e (constan)m(ts)i(of)505 1405 y(pre\014x-free)34 b(oracle)g(mac)m(hines)g (do)g(not)g(dep)s(end)e(on)i(the)g(oracle.)h(More)f(precisely)-8 b(,)505 1513 y(w)m(e)41 b(ha)m(v)m(e)h(the)e(follo)m(wing)f (de\014nitions)f(from)i([36)q(].)h(An)f(oracle)h(mac)m(hine)f Fw(M)51 b FA(is)39 b(a)505 1622 y Fs(pr)-5 b(e\014x-fr)g(e)g(e)33 b(or)-5 b(acle)32 b(machine)k FA(if)28 b Fw(M)1734 1589 y Fx(A)1820 1622 y FA(is)g(pre\014x-free)g(for)g(ev)m(ery)h Fw(A)d Fr(2)f FA(2)2938 1589 y Fx(!)2989 1622 y FA(.)k(A)f(pre\014x-) 505 1730 y(free)33 b(oracle)g(mac)m(hine)g Fw(U)42 b FA(is)32 b Fs(universal)42 b FA(if)32 b(for)g(ev)m(ery)i(pre\014x-free) e(oracle)h(mac)m(hine)505 1838 y Fw(M)41 b FA(there)30 b(is)g(a)h Fw(\034)k Fr(2)25 b FA(2)1240 1805 y Fx()e FA(0)33 b Fs(then)g Fw(X)40 b Fs(is)33 b(a)g(left-c.e.)e(r)-5 b(e)g(al.)588 4274 y FA(It)30 b(is)e(sho)m(wn)h(in)f([36)q(])h(that)h Fw(A)f FA(is)g(lo)m(w)g(for)g(\012)f(iff)g(there)i(is)e(a)i(univ)m (ersal)d(pre\014x-free)505 4384 y(oracle)i(mac)m(hine)e Fw(U)38 b FA(suc)m(h)28 b(that)g(\012)1682 4351 y Fx(A)1682 4411 y(U)1768 4384 y FA(is)f(a)h(left-c.e.)i(real.)d(By)i(Theorem)e (12.8,)j(ev)m(ery)505 4492 y(2-random)k(set)f(is)f(lo)m(w)h(for)g (\012,)g(so)h(almost)f(ev)m(ery)h(set)g(is)e(tak)m(en)i(to)g(a)f (left-c.e.)i(real)505 4600 y(b)m(y)c Fs(some)38 b FA(Omega)31 b(op)s(erator.)588 4708 y(Another)23 b(result)f(in)f([36)q(])i(is)f (that)i(for)e(an)m(y)h(univ)m(ersal)e(pre\014x-free)i(oracle)g(mac)m (hine)505 4817 y Fw(U)45 b FA(and)33 b(an)m(y)i(set)g Fw(X)7 b FA(,)35 b(the)f(set)h(of)f(all)g Fw(B)k FA(suc)m(h)c(that)h (\012)2383 4784 y Fx(B)2383 4844 y(U)2478 4817 y FA(is)e(1-random)h (relativ)m(e)h(to)505 4925 y Fw(X)47 b FA(has)38 b(p)s(ositiv)m(e)g (measure.)h(T)-8 b(ogether)40 b(with)e(the)h(\014rst)f(part)h(of)g (Theorem)g(14.3,)505 5033 y(this)30 b(giv)m(es)h(a)g(resoundingly)e (negativ)m(e)j(solution)d(to)i(the)g(question)g(of)f(the)i(degree-)505 5141 y(in)m(v)-5 b(ariance)30 b(of)g(Omega)i(op)s(erators.)p eop %%Page: 73 73 73 72 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(73)588 541 y FB(Theorem)34 b FA(14.4)i(\(Do)m(wney)-8 b(,)32 b(Hirsc)m(hfeldt,)d(Miller,)g(and)h(Nies)g([36)q(]\))p FB(.)563 670 y FA(\(i\))42 b Fs(F)-7 b(or)34 b(al)5 b(l)34 b Fw(X)g Fr(2)26 b FA(2)1243 637 y Fx(!)1294 670 y Fs(,)33 b(ther)-5 b(e)34 b(ar)-5 b(e)34 b Fw(A;)15 b(B)32 b Fr(2)26 b FA(2)2078 637 y Fx(!)2162 670 y Fs(with)34 b Fw(A)27 b FA(=)2526 637 y Fq(\003)2591 670 y Fw(B)38 b Fs(such)c(that)g FA(\012)3158 637 y Fx(A)3158 697 y(U)3250 670 y Fs(is)f(a)701 784 y(left-c.e.)e(r)-5 b(e)g(al)34 b(and)g FA(\012)1444 751 y Fx(B)1444 811 y(U)1537 784 y Fs(is)e FA(1)p Fs(-r)-5 b(andom)35 b(r)-5 b(elative)34 b(to)f Fw(X)7 b Fs(.)538 898 y FA(\(ii\))41 b Fs(Ther)-5 b(e)35 b(ar)-5 b(e)35 b Fw(A;)15 b(B)34 b Fr(2)28 b FA(2)1465 865 y Fx(!)1550 898 y Fs(such)35 b(that)h Fw(A)28 b FA(=)2114 865 y Fq(\003)2182 898 y Fw(B)39 b Fs(and)c FA(\012)2534 865 y Fx(A)2534 925 y(U)2622 898 y Fr(j)2647 912 y Fi(T)2731 898 y FA(\012)2797 865 y Fx(B)2797 925 y(U)2891 898 y Fs(\(and)h(in)e(fact,)701 1011 y FA(\012)767 978 y Fx(A)767 1038 y(U)858 1011 y Fs(and)f FA(\012)1100 978 y Fx(B)1100 1038 y(U)1193 1011 y Fs(ar)-5 b(e)34 b FA(1)p Fs(-r)-5 b(andom)35 b(r)-5 b(elative)33 b(to)g(e)-5 b(ach)34 b(other\).)588 1181 y FB(Pr)n(oof.)41 b FA(\(i\))d(Let)g Fr(S)45 b FA(=)37 b Fr(f)p Fw(A)i FA(:)f(\012)1736 1148 y Fx(A)1736 1208 y(U)1825 1181 y FA(is)29 b(a)i(left-c.e.)g(real)p Fr(g)38 b FA(and)g Fr(R)f FA(=)h Fr(f)p Fw(B)k FA(:)c(\012)3235 1148 y Fx(B)3235 1208 y(U)3333 1181 y FA(is)505 1305 y(1-random)32 b(relativ)m(e)f(to)h Fw(X)7 b Fr(g)p FA(.)33 b(Let)1722 1282 y Fp(b)1701 1305 y Fr(R)27 b FA(=)f Fr(f)p Fw(A)i FA(:)f Fr(9)p Fw(B)k Fr(2)c(R)g FA(\()p Fw(A)g FA(=)2639 1272 y Fq(\003)2705 1305 y Fw(B)5 b FA(\))p Fr(g)p FA(.)32 b(Since)e Fr(R)i FA(has)505 1425 y(p)s(ositiv)m(e)i (measure,)i(Kolmogoro)m(v's)g(0-1)g(la)m(w)e(implies)f(that)i Fw(\026)p FA(\()2757 1402 y Fp(b)2736 1425 y Fr(R)p FA(\))f(=)e(1.)k (Since)e Fr(S)505 1545 y FA(has)d(p)s(ositiv)m(e)e(measure,)h(there)h (is)e(an)i Fw(A)25 b Fr(2)g(S)i(\\)2197 1522 y Fp(b)2176 1545 y Fr(R)p FA(.)588 1653 y(\(ii\))i(By)h(part)f(1,)h(there)g(are)g Fw(A;)15 b(B)30 b Fr(2)25 b FA(2)1898 1620 y Fx(!)1978 1653 y FA(with)j Fw(A)e FA(=)2349 1620 y Fq(\003)2413 1653 y Fw(B)34 b FA(suc)m(h)29 b(that)h(\012)2982 1620 y Fx(A)2982 1680 y(U)3070 1653 y FA(is)f(a)g(left-)505 1766 y(c.e.)i(real)d(and)h(\012)1083 1733 y Fx(B)1083 1793 y(U)1172 1766 y FA(is)f(2-random.)i(Hence)g(\012)2031 1733 y Fx(B)2031 1793 y(U)2120 1766 y FA(is)e(\012)2276 1733 y Fx(A)2276 1793 y(U)2335 1766 y FA(-random)g(and,)h(b)m(y)g (Corollary)505 1880 y(12.18,)k(\012)834 1847 y Fx(A)834 1907 y(U)923 1880 y FA(is)c(\012)1080 1847 y Fx(B)1080 1907 y(U)1141 1880 y FA(-random.)h(This)e(implies)g(that)j(\012)2309 1847 y Fx(A)2309 1907 y(U)2393 1880 y Fr(j)2418 1894 y Fy(T)2498 1880 y FA(\012)2564 1847 y Fx(B)2564 1907 y(U)2625 1880 y FA(.)689 b Fr(a)588 2007 y FA(In)42 b(ligh)m(t)g(of)h (this)e(result,)h(one)h(migh)m(t)f(w)m(onder)g(whether)g(there)h(are)g Fs(any)51 b FA(de-)505 2115 y(grees)30 b(on)f(whic)m(h)f(Omega)i(op)s (erators)f(are)h(in)m(v)-5 b(arian)m(t.)29 b(Once)g(again,)g(the)g (answ)m(er)g(is)505 2223 y(connected)j(with)d Fw(K)7 b FA(-trivialit)m(y)-8 b(.)588 2388 y FB(Theorem)34 b FA(14.5)i(\(Do)m(wney)-8 b(,)32 b(Hirsc)m(hfeldt,)d(Miller,)g(and)h (Nies)g([36)q(]\))p FB(.)46 b Fs(L)-5 b(et)50 b Fw(A)57 b Fr(2)505 2496 y FA(2)550 2463 y Fx(!)601 2496 y Fs(.)33 b(the)g(fol)5 b(lowing)33 b(ar)-5 b(e)34 b(e)-5 b(quivalent.)563 2623 y FA(\(i\))42 b Fw(A)32 b Fs(is)h Fw(K)7 b Fs(-trivial.)538 2731 y FA(\(ii\))41 b Fs(Every)32 b(Ome)-5 b(ga)33 b(op)-5 b(er)g(ator)36 b(takes)d Fw(A)f Fs(to)i(a)f(left-c.e.)e(r)-5 b(e)g(al.)513 2839 y FA(\(iii\))40 b Fs(Every)32 b(Ome)-5 b(ga)33 b(op)-5 b(er)g(ator)36 b(is)c(de)-5 b(gr)g(e)g(e)34 b(invariant)g(on)f(the)g(de)-5 b(gr)g(e)g(e)34 b(of)e Fw(A)p Fs(.)505 3004 y FA(There)i(is)f(an)h(example)f(in)g([36)q(])h (of)g(an)g(Omega)h(op)s(erator)f(that)g(is)f(degree)i(in)m(v)-5 b(ari-)505 3111 y(an)m(t)33 b(only)d(on)i(the)f Fw(K)7 b FA(-trivial)30 b(degrees.)i(It)g(is)e(not)i(kno)m(wn)f(whether)g(ev)m (ery)h(Omega)505 3219 y(op)s(erator)f(has)f(this)f(prop)s(ert)m(y)-8 b(.)588 3327 y(F)g(or)37 b(further)d(prop)s(erties)g(of)h(Omega)i(op)s (erators,)f(including)c(their)i(in)m(teresting)505 3435 y(analytic)c(b)s(eha)m(vior,)g(see)h([36)q(].)588 3646 y Fu(x)p Ft(15.)53 b(Hausdor\013)37 b(dimension)f(and)g(partial)f (randomness.)46 b FA(In)30 b(this)h(sec-)505 3754 y(tion,)26 b(w)m(e)h(lo)s(ok)e(at)i(the)f(follo)m(wing)e(in)m(tuitiv)m(e)h (notion,)h(whic)m(h)e(pro)m(vides)h(y)m(et)i(another)505 3862 y(w)m(a)m(y)g(to)g(calibrate)e(randomness.)g(Supp)s(ose)f(that)j (\012)e(=)g Fw(:a)2483 3876 y Fy(1)2523 3862 y Fw(a)2571 3876 y Fy(2)2625 3862 y Fw(:)15 b(:)g(:)h FA(.)27 b(Then)d(w)m(e)j(w)m (ould)505 3970 y(exp)s(ect)j Fw(:a)865 3984 y Fy(1)905 3970 y FA(0)p Fw(a)998 3984 y Fy(2)1037 3970 y FA(0)15 b Fw(:)g(:)g(:)46 b FA(to)30 b(b)s(e)e(\\)1535 3934 y Fy(1)p 1535 3949 36 4 v 1535 4001 a(2)1581 3970 y FA(-random".)h(T)-8 b(o)29 b(mak)m(e)h(this)e(notion)g(of)h(partial)f(ran-)505 4078 y(domness)38 b(precise,)h(and)f(describ)s(e)f(some)i(results)e (that)j(are)f(v)m(ery)g(in)m(teresting)f(in)505 4186 y(their)f(o)m(wn)g(righ)m(t,)g(w)m(e)g(need)g(to)h(detour)f(through)g (the)g(theory)g(of)h(Hausdor\013)f(di-)505 4293 y(mension.)588 4418 y Ft(15.1.)54 b(Classical)28 b(Hausdor\013)h(dimension.)46 b FA(First)24 b(w)m(e)h(recall)g(the)g(de\014nition)505 4526 y(of)h(classical)e(Hausdor\013)g(dimension)e([48)r(].)j(F)-8 b(or)26 b(commen)m(ts)g(and)e(discussion)e(see)k(for)505 4634 y(instance)k(F)-8 b(alconer)32 b([44)q(].)588 4798 y FB(Definition)j FA(15.1)p FB(.)105 b FA(\(i\))41 b Fw(C)32 b Fr(\022)25 b FA(2)1772 4765 y Fx()f Fw(n)p FA(.)538 4906 y(\(ii\))41 b Fw(C)36 b Fs(c)-5 b(overs)39 b Fr(A)25 b(\022)g FA(2)1322 4873 y Fx(!)1403 4906 y FA(if)k Fr(A)c(\022)1680 4838 y Fp(S)1756 4933 y Fx(\033)r Fq(2)p Fx(C)1905 4906 y FA([)p Fw(\033)s FA(].)513 5050 y(\(iii\))40 b(De\014ne)30 b Fw(H)1065 5013 y Fx(")1058 5073 y(n)1105 5050 y FA(\()p Fr(A)p FA(\))25 b(:=)h(inf)1520 4949 y Fp(n)1603 4964 y(X)1596 5161 y Fx(\033)r Fq(2)p Fx(C)1756 5050 y FA(2)1801 5013 y Fq(\000)p Fx(")p Fq(j)p Fx(\033)r Fq(j)2000 5050 y FA(:)g Fw(C)36 b FA(is)30 b(an)g Fw(n)p FA(-co)m(v)m(er)i(of)f Fr(A)2867 4949 y Fp(o)2927 5050 y FA(.)p eop %%Page: 74 74 74 73 bop 505 363 a FD(74)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)515 541 y FA(\(iv\))42 b(De\014ne)27 b Fw(H)1062 504 y Fx(")1098 541 y FA(\()p Fr(A)p FA(\))f(:=)54 b(lim)1388 596 y Fx(n)p Fq(!1)1587 541 y Fw(H)1670 504 y Fx(")1663 564 y(n)1710 541 y FA(\()p Fr(A)p FA(\).)27 b(This)e(is)g(the)i Fw(")p Fs(-dimensional)k(outer)f(Haus-)701 676 y(dor\013)j(me)-5 b(asur)g(e)39 b FA(of)30 b Fr(A)p FA(.)588 840 y FB(Lemma)k FA(15.2)p FB(.)47 b Fs(L)-5 b(et)33 b Fr(A)25 b(\022)g FA(2)1559 807 y Fx(!)1610 840 y Fs(.)32 b(Ther)-5 b(e)34 b(exists)f Fw(")25 b Fr(2)g FA([0)p Fw(;)15 b FA(1])34 b Fs(such)f(that)563 973 y FA(\(i\))42 b Fw(H)784 940 y Fx(")817 917 y Fj(0)843 973 y FA(\()p Fr(A)p FA(\))25 b(=)g(0)33 b Fs(for)g Fw(")1372 940 y Fq(0)1421 973 y Fw(>)25 b(")33 b Fs(and)538 1089 y FA(\(ii\))41 b Fw(H)784 1056 y Fx(")817 1032 y Fj(0)843 1089 y FA(\()p Fr(A)p FA(\))25 b(=)g Fr(1)33 b Fs(for)g FA(0)26 b Fz(6)f Fw(")1585 1056 y Fq(0)1633 1089 y Fw(<)g(")p Fs(.)588 1252 y FA(The)30 b Fw(")h FA(in)e(Lemma)i(15.2)g(is)f(called)f(the)i Fs(Hausdor\013)j (dimension)k FA(of)31 b Fr(A)p FA(:)588 1415 y FB(Definition)k FA(15.3)p FB(.)47 b FA(dim)o(\()p Fr(A)p FA(\))25 b(:=)g(inf)6 b Fr(f)p Fw(")26 b FA(:)g Fw(H)2138 1382 y Fx(")2174 1415 y FA(\()p Fr(A)p FA(\))g(=)f(0)p Fr(g)p FA(.)588 1578 y(Hausdor\013)30 b(dimension)e(has)i(a)h(n)m(um)m(b)s(er)e(of)i (basic)e(prop)s(erties:)563 1705 y(\(i\))42 b(It)25 b(giv)m(es)g(a)g (re\014nemen)m(t)g(of)g(the)g(notion)f(of)h(measure)g(zero:)h(If)e Fw(\026)p FA(\()p Fw(X)7 b FA(\))26 b Fr(6)p FA(=)f(0,)h(then)701 1813 y(dim)n(\()p Fw(X)7 b FA(\))27 b(=)e(1.)538 1920 y(\(ii\))41 b(\(monotonicit)m(y\))31 b(If)f Fw(X)i Fr(\022)25 b Fw(Y)51 b FA(then)30 b(dim)n(\()p Fw(X)7 b FA(\))27 b Fz(6)e FA(dim)n(\()p Fw(Y)c FA(\).)513 2028 y(\(iii\))40 b(\(coun)m(table)31 b(stabilit)m(y\))e(If)h Fw(I)37 b FA(is)30 b(coun)m(table,)h(then)1342 2204 y(dim)1509 2103 y Fp(\020)1581 2118 y([)1578 2315 y Fx(i)p Fq(2)p Fx(I)1700 2204 y Fw(Y)1753 2218 y Fx(i)1781 2103 y Fp(\021)1860 2204 y FA(=)25 b(sup)1971 2282 y Fx(i)p Fq(2)p Fx(I)2108 2130 y Fp(\010)2177 2204 y FA(dim)n(\()p Fw(Y)2416 2218 y Fx(i)2444 2204 y FA(\))2479 2130 y Fp(\011)2533 2204 y Fw(:)701 2445 y FA(In)k(particular,)g(dim)o(\()p Fw(X)f Fr([)20 b Fw(Y)g FA(\))30 b(is)g(max)p Fr(f)p FA(dim)o(\()p Fw(X)7 b FA(\))p Fw(;)15 b FA(dim\()p Fw(Y)20 b FA(\))p Fr(g)p FA(.)588 2571 y Ft(15.2.)54 b(E\013ectiv)m(e)38 b(Hausdor\013)i(dimension.)45 b FA(W)-8 b(e)35 b(no)m(w)f(discuss)e (e\013ectiviza-)505 2679 y(tions)e(of)h(Hausdor\013)e(dimension.)f (There)i(has)g(b)s(een)g(a)g(large)h(amoun)m(t)f(of)h(researc)m(h)505 2787 y(in)24 b(e\013ectiv)m(e)j(dimension,)c(and)h(w)m(e)i(only)e (scratc)m(h)i(the)f(surface)g(here.)h(In)e(particular,)505 2895 y(w)m(e)35 b(do)f(not)h(discuss)d(w)m(ork)i(in)f(e\013ectivizing)h (other)h(t)m(yp)s(es)f(of)g(fractal)h(dimension,)505 3003 y(suc)m(h)42 b(as)h(pac)m(king)f(dimension)e(\(see)j(A)m(threy)m (a,)h(Hitc)m(hco)s(c)m(k,)g(Lutz,)f(and)f(Ma)m(y)m(or-)505 3111 y(domo)33 b([4)q(]\).)g(F)-8 b(or)33 b(more)g(on)f(e\013ectiv)m(e) j(dimension)29 b(theory)-8 b(,)34 b(see)f(Reimann)e([113)r(])i(or)505 3219 y(Lutz)g([85)q(];)h(for)e(results)g(on)g(the)h(complexit)m(y)g(of) g(these)g(and)f(related)h(notions,)f(see)505 3326 y(Hitc)m(hco)s(c)m (k,)g(Lutz,)f(and)f(T)-8 b(erwijn)28 b([51)r(].)588 3434 y(Recall)44 b(from)g(De\014nition)f(10.4)j(the)f(n)m(ull)d(sets)j(of)f (the)h(form)f Fw(S)2877 3449 y Fx(h)2921 3434 y FA([)p Fw(d)p FA(].)i(Sc)m(hnorr)505 3542 y(also)35 b(addressed)e(n)m(ull)g (sets)h(of)h Fs(exp)-5 b(onential)46 b FA(order,)34 b(whic)m(h)f(ha)m (v)m(e)j(the)e(form)g Fw(S)3252 3557 y Fx(h)3297 3542 y FA([)p Fw(d)p FA(])505 3650 y(for)42 b Fw(h)p FA(\()p Fw(n)p FA(\))j(=)g(2)1039 3617 y Fx("n)1160 3650 y FA(with)c Fw(")k Fr(2)f FA(\(0)p Fw(;)15 b FA(1].)44 b(Although)d(he)h(did)e(not) i(mak)m(e)h(an)f(explicit)505 3758 y(reference)23 b(to)g(Hausdor\013)e (dimension,)f(it)i(turns)e(out)j(that)f(the)h(theory)f(of)g (Hausdor\013)505 3866 y(dimension)30 b(can)j(b)s(e)f(cast)h(precisely)e (in)g(terms)h(of)h(suc)m(h)f(n)m(ull)e(sets)j(of)g(exp)s(onen)m(tial) 505 3974 y(order.)588 4082 y(Lutz)43 b(constructivized)f(Hausdor\013)h (dimension)d(in)i([83)q(,)h(84)q(],)g(using)f(what)h(he)505 4190 y(called)29 b Fw(s)p Fs(-gales)37 b FA(\(a)31 b(generalization)e (of)h(martingales\).)g(Let)g Fw(s)25 b Fr(2)f FA([0)p Fw(;)15 b Fr(1)p FA(\).)32 b(An)d Fw(s)p Fs(-gale)505 4298 y FA(is)h(a)h(function)e Fw(d)c FA(:)h(2)1197 4265 y Fx()h FA(instead)g(of)h(equalit)m (y)-8 b(.)21 b(The)g(success)505 4817 y(set)31 b Fw(S)5 b FA([)p Fw(d)p FA(])31 b(is)f(de\014ned)f(exactly)i(as)g(w)m(as)f (done)h(for)f(martingales)f(in)h(Section)g(2.2.)588 4925 y(Although)21 b(the)h(follo)m(wing)e(theorem)i(sho)m(ws)f(that)h(w)m(e) g(do)g(not)g(really)e(need)h Fw(s)p FA(-gales)505 5033 y(for)35 b(the)g(treatmen)m(t)i(of)e(Hausdor\013)f(dimension,)f(it)h (is)g(sometimes)h(con)m(v)m(enien)m(t)h(to)505 5141 y(use)30 b(them.)p eop %%Page: 75 75 75 74 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(75)588 541 y FB(Theorem)34 b FA(15.4)p FB(.)47 b FA(\(Lutz)38 b([83)q(],)g(Am)m(b)s(os-Spies,)f(Merkle,)h(Reimann,)f (and)g(Ste-)505 649 y(phan)h([2],)h(Calude,)f(Staiger,)h(and)e(T)-8 b(erwijn)37 b([19)q(]\).)k Fs(F)-7 b(or)41 b(any)g Fr(A)d(\022)h FA(2)2977 616 y Fx(!)3068 649 y Fs(and)i Fw(r)g Fr(2)505 757 y FA([0)p Fw(;)15 b FA(1])p Fs(,)35 b(the)e(fol)5 b(lowing)33 b(ar)-5 b(e)34 b(e)-5 b(quivalent:)579 883 y FA(\(i\))41 b Fr(A)32 b Fs(has)i(Hausdor\013)f(dimension)h Fw(r)s Fs(,)553 991 y FA(\(ii\))41 b Fw(r)28 b FA(=)d(inf)5 b Fr(f)p Fw(s)26 b Fr(2)f Fo(Q)40 b FA(:)58 b Fs(ther)-5 b(e)33 b(is)g(an)g Fw(s)p Fs(-\(sup)-5 b(er\)gale)33 b Fw(d)g Fs(s.t.)g Fr(A)24 b(\022)h Fw(S)5 b FA([)p Fw(d)p FA(])p Fr(g)p Fs(.)528 1099 y FA(\(iii\))40 b Fw(r)28 b FA(=)d(inf)5 b Fr(f)p Fw(s)26 b Fr(2)f Fo(Q)40 b FA(:)58 b Fs(ther)-5 b(e)33 b(is)g(a)g(\(sup)-5 b(er\)martingale)35 b Fw(d)e Fs(s.t.)g Fr(A)25 b(\022)f Fw(S)2994 1124 y Fy(2)3029 1105 y Fn(\(1)p Fj(\000)p Fl(s)p Fn(\))p Fl(n)3232 1099 y FA([)p Fw(d)p FA(])p Fr(g)p Fw(:)588 1255 y FA(So)j(w)m(e)g(see) h(that)f(the)g(theory)g(of)g(\(e\013ectiv)m(e\))i(Hausdor\013)e (dimension)d(falls)h(out)i(as)505 1363 y(a)k(sp)s(ecial)e(case)j(of)e (Sc)m(hnorr's)g(treatmen)m(t)i(of)e(e\013ectiv)m(e)i(measure)e(theory) -8 b(.)588 1471 y(Theorem)31 b(15.4)g(motiv)-5 b(ates)32 b(the)e(follo)m(wing)f(de\014nition:)588 1627 y FB(Definition)35 b FA(15.5)p FB(.)47 b FA(Let)c Fr(C)k FA(b)s(e)42 b(a)g(complexit)m(y)g (class.)g(A)h(class)f Fr(A)i(\022)h FA(2)3170 1594 y Fx(!)3263 1627 y FA(has)505 1735 y Fr(C)5 b Fs(-dimension)34 b Fw(r)f FA(if)594 1881 y Fw(r)28 b FA(=)d(inf)6 b Fr(f)p Fw(s)25 b Fr(2)g Fo(Q)40 b FA(:)25 b Fr(9)p Fw(d)g Fr(2)g(C)c FA([)p Fw(d)31 b FA(is)e(a)i(sup)s(ermartingale)d(and)i Fr(A)25 b(\022)f Fw(S)2859 1906 y Fy(2)2894 1887 y Fn(\(1)p Fj(\000)p Fl(s)p Fn(\))p Fl(n)3097 1881 y FA([)p Fw(d)p FA(])15 b(])p Fr(g)p Fw(:)588 2026 y FA(The)30 b Fr(C)5 b FA(-dimension)29 b(of)h(a)h(set)g Fw(A)f FA(is)g(the)g Fr(C)5 b FA(-dimension)29 b(of)h(the)h(singleton)e Fr(f)p Fw(A)p Fr(g)p FA(.)588 2183 y(There)d(is)f(an)h(imp)s(ortan)m(t)f (connection)h(b)s(et)m(w)m(een)h(\006)2344 2150 y Fy(0)2344 2207 y(1)2383 2183 y FA(-dimension)c(and)j(Kolmogo-)505 2291 y(ro)m(v)36 b(complexit)m(y)-8 b(,)35 b(whic)m(h)e(w)m(as)j (established)d(in)g(the)i(form)g(giv)m(en)f(b)s(elo)m(w)g(b)m(y)h(Ma)m (y-)505 2399 y(ordomo)c([87)q(])g(and)f(pre\014gured)f(b)m(y)i(Ry)m (abk)m(o)h([116)q(,)f(117)r(],)g(Staiger)f([127)r(,)h(128)q(],)g(and) 505 2507 y(Cai)38 b(and)g(Hartmanis)f([13)q(])i(\(see)g(Staiger)f([129) r(])h(for)f(a)g(discussion)e(of)i(these)h(and)505 2614 y(other)31 b(related)f(pap)s(ers\).)588 2771 y FB(Theorem)k FA(15.6)i(\(Ma)m(y)m(ordomo)c([87)r(]\))p FB(.)46 b Fs(F)-7 b(or)28 b(any)f(set)g Fw(A)p Fs(,)g(the)h FA(\006)2818 2738 y Fy(0)2818 2795 y(1)2857 2771 y Fs(-dimension)f(of)505 2879 y Fw(A)33 b Fs(is)g(e)-5 b(qual)33 b(to)1575 3041 y FA(lim)15 b(inf)1842 3055 y Fx(n)1924 2979 y Fw(K)7 b FA(\()p Fw(A)2137 2967 y Fz(\026)2200 2979 y Fw(n)p FA(\))p 1924 3020 366 4 v 2080 3103 a Fw(n)2300 3041 y(:)588 3223 y FA(Since)38 b(plain)f(and)h(pre\014x-free)g(Kolmogoro)m (v)h(complexit)m(y)g(are)g(equal)f(up)g(to)h(a)505 3331 y(log)31 b(factor,)g(this)f(theorem)h(also)f(holds)f(with)g Fw(C)37 b FA(in)29 b(place)h(of)h Fw(K)7 b FA(.)588 3456 y Ft(15.3.)54 b(The)25 b(picture)h(of)g(implications.)45 b FA(The)22 b(follo)m(wing)f(relationships)f(hold)505 3564 y(b)s(et)m(w)m(een)i(v)-5 b(arious)20 b(notions)h(of)g(e\013ectiv) m(e)i(randomness)d(and)h(dimension,)d(where)j(\001)3325 3531 y Fy(0)3325 3588 y(2)3364 3564 y FA(-)505 3672 y(randomness)36 b(and)g(Sc)m(hnorr)g(\001)1610 3639 y Fy(0)1610 3696 y(2)1649 3672 y FA(-randomness)g(are)h(the)g(relativizations)e(to)j Fr(;)3261 3639 y Fq(0)3321 3672 y FA(of)505 3780 y(computable)30 b(randomness)g(and)f(Sc)m(hnorr)h(randomness,)f(resp)s(ectiv)m(ely)-8 b(.)1087 3919 y(\001)1163 3886 y Fy(0)1163 3944 y(2)1202 3919 y FA(-random)1283 4027 y Fr(+)919 4136 y FA(Sc)m(hnorr)29 b(\001)1331 4103 y Fy(0)1331 4160 y(2)1370 4136 y FA(-random)91 b(=)-15 b Fr(\))254 b FA(\001)2272 4103 y Fy(0)2272 4160 y(2)2311 4136 y FA(-dimension)29 b(1)1283 4244 y Fr(+)1140 b(+)1122 4352 y FA(1-random)294 b(=)-15 b Fr(\))259 b FA(\006)2267 4319 y Fy(0)2267 4376 y(1)2306 4352 y FA(-dimension)29 b(1)1283 4460 y Fr(+)911 4568 y FA(computably)g(random)2507 4568 y gsave currentpoint currentpoint translate -90 neg rotate neg exch neg exch translate 2507 4568 a 2426 4591 a FA(=)-15 b Fr(\))2507 4568 y currentpoint grestore moveto 2507 4568 a 1283 4675 a Fr(+)991 4783 y FA(Sc)m(hnorr)30 b(random)163 b(=)-15 b Fr(\))82 b FA(computable)30 b(dimension)e(1)505 4925 y(No)34 b(other)f(implications)d(hold)i(than)h(the)g(ones)g (indicated.)f(That)h(these)g(implica-)505 5033 y(tions)d(hold)f(follo)m (ws)g(easily)h(from)f(the)i(de\014nitions.)d(The)i(strictness)f(of)i (the)f(impli-)505 5141 y(cations)c(in)d(the)j(\014rst)e(column)g(w)m (as)h(discussed)f(in)f(Section)i(10)h(\(except)g(for)f(the)h(fact)p eop %%Page: 76 76 76 75 bop 505 363 a FD(76)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y FA(that)25 b(there)f(is)f(a)h(Sc)m(hnorr)f(\001)1482 508 y Fy(0)1482 566 y(2)1521 541 y FA(-random)h(set)g(that)h(is)e(not)h (1-random,)g(whic)m(h)e(follo)m(ws)505 650 y(immediately)36 b(from)g(the)i(fact)g(that)f(no)g(\001)2013 617 y Fy(0)2013 674 y(2)2090 650 y FA(set)g(can)h(b)s(e)e(Sc)m(hnorr)g(\001)2961 617 y Fy(0)2961 674 y(2)3000 650 y FA(-random\).)505 758 y(That)42 b(there)f(are)h(no)f(more)h(implications)d(b)s(et)m(w)m (een)j(the)f(\014rst)g(and)g(the)g(second)505 866 y(column)33 b(follo)m(ws)h(from)g(the)g(next)g(prop)s(osition.)f(The)g(strictness)h (of)g(the)h(t)m(w)m(o)g(im-)505 974 y(plications)g(in)g(the)i(second)f (column)f(follo)m(ws)h(b)m(y)g(similar)e(means.)i(\(It)h(is)f(easy)h (to)505 1082 y(sho)m(w)28 b(\(see)h([84)q(]\))g(that)g(the)f(class)g (of)g(computable)f(sets)i(has)f(\006)2663 1049 y Fy(0)2663 1106 y(1)2702 1082 y FA(-dimension)e(0,)i(but)505 1190 y(is)k(not)g(computably)g(n)m(ull,)e(so)j(in)e(particular)g(the)h (class)h(of)f(computable)g(sets)h(has)505 1297 y(computable)26 b(dimension)d(1.)k(Also,)f(Lutz)g([84)q(])g(has)g(sho)m(wn)f(that)h (for)g(ev)m(ery)h(\001)3183 1264 y Fy(0)3183 1322 y(2)3248 1297 y FA(real)505 1406 y Fw(r)h Fr(2)d FA([0)p Fw(;)15 b FA(1],)29 b(there)f(is)e(a)i(\001)1361 1373 y Fy(0)1361 1430 y(2)1428 1406 y FA(set)f(of)h(\006)1733 1373 y Fy(0)1733 1430 y(1)1772 1406 y FA(-dimension)d Fw(r)s FA(,)i(but)g(it)g(is)f(ob)m (vious)h(that)h(ev)m(ery)505 1514 y(\001)581 1481 y Fy(0)581 1539 y(2)651 1514 y FA(set)j(has)f(\001)1031 1481 y Fy(0)1031 1539 y(2)1070 1514 y FA(-dimension)e(0.\))588 1673 y FB(Pr)n(oposition)34 b FA(15.7)i(\(T)-8 b(erwijn)29 b([133)r(]\))p FB(.)46 b Fs(Ther)-5 b(e)25 b(ar)-5 b(e)25 b(sets)g(that)h(ar)-5 b(e)25 b(not)g(Schnorr)505 1780 y(r)-5 b(andom)35 b(but)e(have)g FA(\001)1275 1747 y Fy(0)1275 1805 y(2)1314 1780 y Fs(-dimension)g(1.) 588 1939 y FA(An)28 b(imp)s(ortan)m(t)f(related)h(op)s(en)f(question,)h (form)m(ulated)f(indep)s(enden)m(tly)e(b)m(y)j(Rei-)505 2047 y(mann)46 b(and)f(b)m(y)h(T)-8 b(erwijn)44 b(\(see)j([94)q(]\),)g (is)e(whether)h(ev)m(ery)g(set)h(of)f(p)s(ositiv)m(e)f(\006)3325 2014 y Fy(0)3325 2071 y(1)3364 2047 y FA(-)505 2154 y(dimension)21 b(computes)j(a)g(1-random)g(set.)g(Ev)m(en)g(if)f(w)m(e)h(strengthen)f (the)h(h)m(yp)s(othesis)505 2262 y(to)32 b(\006)683 2229 y Fy(0)683 2287 y(1)722 2262 y FA(-dimension)c(1,)j(the)f(question)g (is)f(still)g(op)s(en.)588 2387 y Ft(15.4.)54 b(P)m(artial)37 b(randomness.)45 b FA(There)32 b(are)g(at)h(least)g(t)m(w)m(o)g(p)s (ossible)d(v)m(ersions)505 2495 y(of)d(partial)d(Martin-L\177)-45 b(of)26 b(randomness,)f(as)i(w)m(e)f(no)m(w)g(discuss.)f(First)g(w)m(e) i(migh)m(t)e(base)505 2603 y(the)36 b(de\014nition)c(up)s(on)i(a)h (straigh)m(tforw)m(ard)g(generalization)f(of)i(the)f(original)e (de\014-)505 2711 y(nition.)c(A)i(natural)e(v)-5 b(ariation)30 b(on)g Fw(s)p FA(-gales)h(is)e Fw(s)p Fs(-me)-5 b(asur)g(e)7 b FA(:)1569 2870 y Fw(\026)1624 2884 y Fx(s)1660 2870 y FA(\()p Fw(V)21 b FA(\))k(:=)1958 2784 y Fp(X)1951 2980 y Fx(\033)r Fq(2)p Fx(V)2112 2870 y FA(2)2157 2832 y Fq(\000)p Fx(s)p Fq(j)p Fx(\033)r Fq(j)505 3103 y FA(for)41 b(an)g(op)s(en)g(set)h Fw(V)20 b FA(.)41 b(Here)h(it)e(is)h(imp)s (ortan)m(t)f(to)i(think)e(of)h Fw(V)61 b FA(as)42 b(a)f(pre\014x-free) 505 3211 y(collection)i(of)g(strings,)f(since)g(if)g Fw(s)k Fr(6)p FA(=)f(1)f(then)e(di\013eren)m(t)h(presen)m(tations)f(of) h(the)505 3319 y(same)31 b(op)s(en)f(set)h(can)f(ha)m(v)m(e)i (di\013eren)m(t)e Fw(s)p FA(-measures.)588 3427 y(W)-8 b(e)32 b(can)e(use)g(this)f(notion)g(to)i(de\014ne)e(partial)g (Martin-L\177)-45 b(of)30 b(randomness)f(as)h(fol-)505 3535 y(lo)m(ws.)588 3693 y FB(Definition)35 b FA(15.8)h(\(T)-8 b(adaki)30 b([131)r(]\))p FB(.)104 b FA(\(i\))41 b(A)49 b Fs(we)-5 b(ak)49 b(Martin-L\177)-46 b(of)50 b Fw(s)p Fs(-test)56 b FA(is)47 b(a)701 3801 y(computable)32 b(collection)h(of)g (c.e.)h(op)s(en)f(sets)g Fr(f)p Fw(V)2371 3816 y Fx(k)2414 3801 y Fr(g)2459 3816 y Fx(k)r Fq(2)p Fx(!)2629 3801 y FA(suc)m(h)g(that)g Fw(\026)3091 3815 y Fx(s)3128 3801 y FA(\()p Fw(V)3216 3816 y Fx(k)3259 3801 y FA(\))d Fz(6)701 3911 y FA(2)746 3878 y Fq(\000)p Fx(k)874 3911 y FA(for)g(all)f Fw(k)s FA(.)538 4019 y(\(ii\))41 b(W)-8 b(e)27 b(sa)m(y)g(that)f Fw(A)h FA(is)e Fs(we)-5 b(akly)30 b(Martin-L\177)-46 b(of)29 b Fw(s)p Fs(-r)-5 b(andom)34 b FA(if)25 b Fw(A)36 b(=)-55 b Fr(2)2795 3951 y Fp(T)2870 4046 y Fx(k)2928 4019 y Fw(V)2981 4034 y Fx(k)3050 4019 y FA(for)26 b(ev)m(ery)701 4127 y(w)m(eak)31 b(Martin-L\177)-45 b(of)30 b Fw(s)p FA(-test)h Fr(f)p Fw(V)1742 4142 y Fx(k)1785 4127 y Fr(g)1830 4142 y Fx(k)r Fq(2)p Fx(!)1967 4127 y FA(.)588 4285 y(W)-8 b(e)44 b(can)f(also)g(de\014ne)f(a)h(set)g Fw(A)g FA(to)g(b)s(e)f(w)m (eakly)h(Levin-Chaitin)d Fw(s)p FA(-random)i(if)505 4393 y Fw(K)7 b FA(\()p Fw(A)730 4381 y Fz(\026)805 4393 y Fw(n)p FA(\))38 b Fz(>)f Fw(sn)24 b Fr(\000)h Fw(O)s FA(\(1\))39 b(for)e(all)g Fw(n)p FA(.)g(The)g(analog)h(of)g(Sc)m (hnorr's)f(Theorem)g(3.8)505 4501 y(that)45 b(Levin-Chaitin)40 b(random)j(is)g(the)g(same)h(as)g(Martin-L\177)-45 b(of)44 b(random)e(can)i(b)s(e)505 4609 y(established)29 b(with)g(a)i(similar)d (pro)s(of.)588 4767 y FB(Theorem)34 b FA(15.9)i(\(T)-8 b(adaki)30 b([131)r(]\))p FB(.)46 b Fs(A)25 b(set)h Fw(A)h Fs(is)e(we)-5 b(akly)28 b(Martin-L\177)-46 b(of)26 b Fw(s)p Fs(-r)-5 b(andom)505 4875 y(iff)33 b Fw(A)f Fs(is)h(we)-5 b(akly)34 b(L)-5 b(evin-Chaitin)33 b Fw(s)p Fs(-r)-5 b(andom.)588 5033 y FA(Armed)44 b(with)g(this)f(result,)h(w)m(e)h(can)g (em)m(ulate)g(the)g(pro)s(of)f(that)h(\012)g(is)e(Levin-)505 5141 y(Chaitin)29 b(random)g(to)i(sho)m(w)g(the)f(follo)m(wing:)p eop %%Page: 77 77 77 76 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(77)588 541 y FB(Theorem)34 b FA(15.10)i(\(T)-8 b(adaki)31 b([131)q(]\))p FB(.)46 b Fs(L)-5 b(et)43 b FA(0)h Fw(<)e(s)h Fz(6)g FA(1)g Fs(b)-5 b(e)42 b(a)h(c)-5 b(omputable)44 b(r)-5 b(e)g(al)505 649 y(and)34 b(de\014ne)1618 800 y FA(\012)1684 762 y Fx(s)1745 800 y FA(:=)1889 714 y Fp(X)1866 915 y Fx(U)7 b Fy(\()p Fx(\033)r Fy(\))-12 b Fq(#)2058 800 y FA(2)2103 762 y Fq(\000)2168 729 y Fj(j)p Fl(\033)r Fj(j)p 2168 747 76 3 v 2191 788 a Fl(s)2257 800 y Fw(;)505 1047 y Fs(wher)-5 b(e)30 b Fw(U)39 b Fs(is)28 b(a)h(universal)h(pr)-5 b(e\014x-fr)g(e)g(e)29 b(machine.)h(Then)f FA(\012)2515 1014 y Fx(s)2580 1047 y Fs(is)f(we)-5 b(akly)30 b(Martin-L\177)-46 b(of)505 1155 y Fw(s)p Fs(-r)-5 b(andom.)588 1326 y FA(Similarly)d(,)42 b(w)m(e)j(can)f(construct)h(a)g(univ)m (ersal)d(w)m(eak)k(Martin-L\177)-45 b(of)44 b Fw(s)p FA(-test,)h(and)505 1434 y(establish)29 b(similar)f(analogues)j(to)g (results)e(on)h(Martin-L\177)-45 b(of)30 b(randomness.)588 1542 y(This)j(notion)g(squares)h(with)e(our)i(in)m(tuition)d(that)k(if) e Fw(A)e FA(=)g Fw(a)2699 1556 y Fy(1)2739 1542 y Fw(a)2787 1556 y Fy(2)2841 1542 y Fw(:)15 b(:)g(:)50 b FA(is)33 b(random)505 1650 y(then)22 b Fw(B)29 b FA(=)c Fw(a)946 1664 y Fy(1)986 1650 y FA(0)p Fw(a)1079 1664 y Fy(2)1119 1650 y FA(0)p Fw(a)1212 1664 y Fy(2)1267 1650 y Fw(:)15 b(:)g(:)37 b FA(should)20 b(b)s(e)h(\\somewhat")h(random.)f(Indeed)g Fw(B)26 b FA(is)20 b(w)m(eakly)505 1758 y(Martin-L\177)-45 b(of)987 1722 y Fy(1)p 987 1737 36 4 v 987 1789 a(2)1033 1758 y FA(-random.)36 b(T)-8 b(o)36 b(see)h(this)e(supp)s(ose)g(that)i (for)f(eac)m(h)h Fw(d)g FA(there)f(are)h(in-)505 1876 y(\014nitely)e(man)m(y)i Fw(n)f FA(suc)m(h)g(that)h Fw(K)7 b FA(\()p Fw(B)1804 1864 y Fz(\026)1877 1876 y Fw(n)p FA(\))36 b Fw(<)2119 1840 y Fy(1)p 2119 1855 V 2119 1907 a(2)2164 1876 y Fw(n)24 b Fr(\000)g Fw(d)p FA(.)37 b(Consider)d(the)j (pre\014x-free)505 1984 y(mac)m(hine)28 b Fw(M)37 b FA(that)28 b(sim)m(ulates)f(the)h(univ)m(ersal)e(pre\014x-free)h(mac)m(hine)g Fw(U)37 b FA(and,)28 b(when)505 2092 y(it)35 b(\014nds)f(that)i Fw(U)10 b FA(\()p Fw(\033)s FA(\))20 b Fr(#)36 b FA(is)e(of)i(the)g (form)f Fw(b)1950 2106 y Fy(1)1989 2092 y FA(0)p Fw(b)2073 2106 y Fy(2)2113 2092 y FA(0)15 b Fw(:)g(:)g(:)i(b)2334 2106 y Fx(n)2381 2092 y FA(0,)36 b(outputs)f Fw(b)2865 2106 y Fy(1)2920 2092 y Fw(:)15 b(:)g(:)h(b)3080 2106 y Fx(n)3127 2092 y FA(.)36 b(Then)505 2200 y Fw(K)582 2214 y Fx(M)662 2200 y FA(\()p Fw(A)795 2188 y Fz(\026)864 2200 y Fw(n)p FA(\))30 b(=)g Fw(K)7 b FA(\()p Fw(B)1308 2188 y Fz(\026)1376 2200 y FA(2)p Fw(n)p FA(\),)34 b(so)f(for)h(eac)m (h)g Fw(c)g FA(there)f(are)h(in\014nitely)c(man)m(y)k Fw(n)f FA(suc)m(h)505 2308 y(that)e Fw(K)779 2322 y Fx(M)859 2308 y FA(\()p Fw(A)987 2296 y Fz(\026)1051 2308 y Fw(n)p FA(\))25 b Fw(<)g(n)19 b Fr(\000)h Fw(c)p FA(,)31 b(and)f(hence)h Fw(A)f FA(is)f(not)i(1-random.)588 2416 y(W)-8 b(e)30 b(w)m(ould)d(lik)m(e)h(to)h(pro)m(v)m(e)h(the)e(analogs)h(of)g(our)f (basic)g(results)f(that)i(martingale)505 2524 y(randomness,)h(test)i (set)f(randomness,)f(and)g(incompressibilit)m(y)d(all)i(coincide.)i (Un-)505 2632 y(fortunately)-8 b(,)32 b(the)f(pro)s(of)f(breaks)h(do)m (wn)g(for)g(the)g(martingale)g(case.)h(Consider)d(the)505 2740 y(pro)s(of)f(that)h(if)f(a)h(set)g(is)f(Martin-L\177)-45 b(of)28 b(random)g(then)g(no)g(c.e.)i(martingale)e(succeeds)505 2847 y(on)36 b(it.)g(W)-8 b(e)37 b(are)f(giv)m(en)g(a)g(c.e.)h (martingale)e Fw(d)p FA(,)i(and)e(when)g(w)m(e)h(see)h Fw(d)p FA(\()p Fw(\033)s FA(\))e Fw(>)f FA(2)3212 2814 y Fx(k)3291 2847 y FA(w)m(e)505 2955 y(put)d Fw(\033)k FA(in)m(to)d Fw(U)1008 2970 y Fx(k)1050 2955 y FA(.)g(No)m(w)g(imagine) f(w)m(e)h(are)g(follo)m(wing)e(the)i(same)g(pro)s(of)f(metho)s(d)g(for) 505 3063 y(the)42 b Fw(s)g(<)h FA(1)f(case.)g(The)f(problem)e(is)h (that)i Fw(d)f FA(is)g(only)f(c.e.)i(W)-8 b(e)42 b(migh)m(t)f(see)h (that)505 3171 y Fw(d)552 3185 y Fx(t)582 3171 y FA(\()p Fw(\033)s FA(0\))28 b Fw(>)e FA(2)922 3138 y Fx(k)997 3171 y FA(and)k(put)h Fw(\033)s FA(0)g(in)m(to)g Fw(V)1711 3186 y Fx(k)1754 3171 y FA(.)g(A)m(t)h(some)g(later)f(stage)i Fw(u)26 b(>)g(t)p FA(,)31 b(w)m(e)h(migh)m(t)f(see)505 3281 y(that)d Fw(d)746 3295 y Fx(u)792 3281 y FA(\()p Fw(\033)s FA(\))e Fw(>)f FA(2)1084 3248 y Fx(k)1127 3281 y FA(.)j(W)-8 b(e)28 b(w)m(ould)f(lik)m(e)f(to)j(put)d Fw(\033)31 b FA(in)m(to)d Fw(V)2349 3296 y Fx(k)2391 3281 y FA(,)g(but)f(need)g(to)h(k)m(eep)g(the)g(set)505 3389 y(pre\014x-free.)34 b(In)f(the)h Fw(s)d FA(=)g(1)j(case)h(w)m(e)g (can)f(do)g(this)f(b)m(y)g(putting)g Fw(\033)s FA(1)i(in)m(to)f Fw(V)3143 3404 y Fx(k)3185 3389 y FA(.)g(But)505 3498 y(in)29 b(the)i Fw(s)25 b(<)g FA(1)31 b(case,)g(2\(2)1350 3465 y Fq(\000)p Fx(s)p Fy(\()p Fq(j)p Fx(\033)r Fq(j)p Fy(+1\))1670 3498 y FA(\))g(migh)m(t)f(b)s(e)g(m)m(uc)m(h)g(bigger)g (than)g(2)2892 3465 y Fq(\000)p Fx(s)p Fq(j)p Fx(\033)r Fq(j)3066 3498 y FA(.)588 3606 y(Another)i(approac)m(h,)g(tak)m(en)g(b) m(y)f(Lutz)h([83)q(,)f(84)q(],)h(is)e(to)i(de\014ne)f(partial)f (random-)505 3714 y(ness)41 b(using)e(martingales)g(and)h(orders,)h(as) f(in)g(the)g(dev)m(elopmen)m(t)h(of)g(Hausdor\013)505 3822 y(dimension.)32 b(Recall)i(that)h(Sc)m(hnorr)e(pro)m(v)m(ed)i (that)g(a)g(set)f(is)g(1-random)g(iff)f(no)h(c.e.)505 3930 y(martingale)j(succeeds)g(on)g Fw(A)p FA(.)g(No)m(w)h(w)m(e)f(w)m (an)m(t)h(to)g(sa)m(y)f(that)h(no)f(c.e.)h(martingale)505 4038 y Fs(quickly)g FA(succeeds)31 b(on)f Fw(A)p FA(.)588 4209 y FB(Definition)35 b FA(15.11)h(\(Calude,)30 b(Staiger,)h(and)e(T) -8 b(erwijn)29 b([19)q(,)i(133)q(]\))p FB(.)563 4342 y FA(\(i\))42 b(A)32 b Fs(str)-5 b(ong)36 b(Martin-L\177)-46 b(of)34 b Fw(s)p Fs(-test)41 b FA(is)31 b(a)h(computable)g(collection)g (of)g(c.e.)i(sets)e(of)701 4450 y(strings)e Fr(f)p Fw(V)1094 4465 y Fx(k)1138 4450 y Fr(g)1183 4465 y Fx(k)r Fq(2)p Fx(!)1351 4450 y FA(\()p Fs(not)35 b(ne)-5 b(c)g(essarily)35 b(pr)-5 b(e\014x-fr)g(e)g(e)7 b FA(\))34 b(suc)m(h)d(that)i(for)e(all)g (pre\014x-)701 4569 y(free)f(subsets)1201 4546 y Fp(b)1190 4569 y Fw(V)1243 4584 y Fx(k)1311 4569 y Fr(\022)25 b Fw(V)1460 4584 y Fx(k)1502 4569 y FA(,)1623 4658 y Fp(X)1604 4875 y Fx(\033)r Fq(2)1702 4858 y Ff(b)1693 4875 y Fx(V)1734 4887 y Fl(k)1788 4745 y FA(2)1833 4707 y Fq(\000)p Fx(s)p Fq(j)p Fx(\033)r Fq(j)2032 4745 y Fz(6)g FA(2)2173 4707 y Fq(\000)p Fx(k)2271 4745 y Fw(:)538 5026 y FA(\(ii\))41 b(W)-8 b(e)33 b(sa)m(y)f(that)h Fw(A)f FA(is)f Fs(str)-5 b(ongly)36 b(Martin-L\177)-46 b(of)34 b Fw(s)p Fs(-r)-5 b(andom)41 b FA(if)31 b Fw(A)38 b(=)-55 b Fr(2)2906 4958 y Fp(T)2982 5053 y Fx(k)3040 4958 y Fp(S)3116 5053 y Fx(\033)r Fq(2)p Fx(V)3246 5065 y Fl(k)3289 5026 y FA([)p Fw(\033)s FA(])701 5141 y(for)30 b(ev)m(ery)h(strong)f(Martin-L\177)-45 b(of)31 b Fw(s)p FA(-test)g Fr(f)p Fw(V)2173 5156 y Fx(k)2216 5141 y Fr(g)2261 5156 y Fx(k)r Fq(2)p Fx(!)2397 5141 y FA(.)p eop %%Page: 78 78 78 77 bop 505 363 a FD(78)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)513 541 y FA(\(iii\))40 b(W)-8 b(e)28 b(sa)m(y)h(that)f Fw(A)g FA(is)e Fs(Schnorr)32 b Fw(s)p Fs(-r)-5 b(andom)36 b FA(if)27 b(for)g(an)m(y)h(c.e.)h(\(sup)s(er\)martingale)701 649 y Fw(d)p FA(,)1510 831 y(lim)15 b(sup)1557 903 y Fx(n)p Fq(!1)1813 769 y Fw(d)p FA(\()p Fw(A)1990 757 y Fz(\026)2053 769 y Fw(n)p FA(\))p 1813 810 330 4 v 1843 897 a(2)1888 871 y Fy(\(1)p Fq(\000)p Fx(s)p Fy(\))p Fx(n)2178 831 y Fw(<)25 b Fr(1)p Fw(:)515 1027 y FA(\(iv\))42 b(W)-8 b(e)33 b(sa)m(y)f(that)h Fw(A)f FA(is)f Fs(Lutz)j Fw(s)p Fs(-r)-5 b(andom)41 b FA(if)31 b(for)h(an)m(y)g Fw(s)p FA(-\(sup)s(er\)gale)f Fw(d)p FA(,)i(w)m(e)f(ha)m(v)m(e)701 1135 y Fw(A)j(=)-55 b Fr(2)25 b Fw(S)5 b FA([)p Fw(d)p FA(].)588 1308 y FB(Theorem)34 b FA(15.12)i(\(Calude,)30 b(Staiger,)h(and)e(T)-8 b(erwijn)29 b([19)q(,)i(133)q(]\))p FB(.)46 b Fs(Fix)k(a)g(c)-5 b(om-)505 1416 y(putable)35 b Fw(s)f Fs(with)h FA(0)28 b Fw(<)g(s)f Fz(6)h FA(1)p Fs(.)34 b(Then)h(a)f(set)g Fw(A)h Fs(is)f(str)-5 b(ongly)36 b(Martin-L\177)-46 b(of)34 b Fw(s)p Fs(-r)-5 b(andom)505 1524 y(iff)33 b Fw(A)f Fs(is)h(Schnorr)h Fw(s)p Fs(-r)-5 b(andom)34 b(iff)e Fw(A)h Fs(is)g(Lutz)f Fw(s)p Fs(-r)-5 b(andom.)588 1697 y FA(It)32 b(is)e(also)h(p)s(ossible)e(to)j(giv)m(e)g (a)f(mac)m(hine)g(c)m(haracterization)i(of)e(strong)h(Martin-)505 1805 y(L\177)-45 b(of)40 b Fw(s)p FA(-randomness.)f(W)-8 b(e)40 b(sa)m(y)h(that)f(a)g(mac)m(hine)f Fw(M)50 b FA(is)38 b Fw(s)p Fs(-me)-5 b(asur)g(able)48 b FA(if)38 b(for)i(all)505 1913 y(pre\014x-free)30 b Fw(V)46 b Fr(\022)25 b FA(dom)o(\()p Fw(M)10 b FA(\),)32 b(w)m(e)f(ha)m(v)m(e)g Fw(\026)1924 1927 y Fx(s)1961 1913 y FA(\()1996 1844 y Fp(S)2072 1939 y Fx(\033)r Fq(2)p Fx(V)2223 1913 y FA([)p Fw(\033)s FA(]\))26 b Fz(6)f FA(1.)588 2086 y FB(Theorem)34 b FA(15.13)i(\(Do)m (wney)-8 b(,)32 b(Reid,)e(and)g(T)-8 b(erwijn)54 b(\(see)31 b([112)r(]\)\))p FB(.)46 b Fs(A)41 b(set)g Fw(A)g Fs(is)505 2193 y(str)-5 b(ongly)51 b(Martin-L\177)-46 b(of)50 b Fw(s)p Fs(-r)-5 b(andom)50 b(iff)f Fw(K)1983 2207 y Fx(M)2062 2193 y FA(\()p Fw(A)2221 2181 y Fz(\026)2314 2193 y Fw(n)p FA(\))55 b Fz(>)g Fw(n)31 b Fr(\000)h Fw(O)s FA(\(1\))50 b Fs(for)g(al)5 b(l)49 b Fw(s)p Fs(-)505 2301 y(me)-5 b(asur)g(able)35 b(machines)e Fw(M)10 b Fs(.)588 2474 y FA(Reimann)45 b(and)g(Stephan)f(\(unpublished;)e(see)k([114)q(]\))h (ha)m(v)m(e)f(recen)m(tly)g(sho)m(wn)505 2582 y(that)d(w)m(eak)g (Martin-L\177)-45 b(of)42 b Fw(s)p FA(-randomness)f(do)s(es)g(not)i (imply)c(strong)k(Martin-L\177)-45 b(of)505 2690 y Fw(s)p FA(-randomness.)1708 2964 y Fy(REFERENCES)605 3132 y Fv([1])27 b(K.)j(Am)n(b)r(os-Spies)f(and)h(A.)h(Ku)n(\024)-36 b(cera,)30 b Fe(R)l(andomness)k(in)d(c)l(omputability)i(the)l(ory)p Fv(,)g(in)d Fe(Com-)505 3223 y(putability)g(The)l(ory)g(and)f(its)g (Applic)l(ations)h(\(Boulder,)g(CO,)e(1999\))i Fv(\(P)-6 b(.)28 b(A.)f(Cholak,)h(S.)f(Lempp,)505 3314 y(M.)e(Lerman,)f(and)g(R.) g(A.)g(Shore,)g(eds.\),)h(Con)n(temp)r(orary)f(Mathematics)h(257,)g (American)f(Math-)505 3406 y(ematical)j(So)r(ciet)n(y)-6 b(,)25 b(2000,)j(1{14.)605 3497 y([2])f(K.)d(Am)n(b)r(os-Spies,)f(W.)h (Merkle,)g(J.)h(Reimann,)e(and)h(F.)g(Stephan,)f Fe(Hausdor\013)28 b(Dimension)505 3588 y(in)36 b(Exp)l(onential)h(Time)p Fv(,)d(in)g Fe(Computational)j(Complexity)f(2001)p Fv(,)g(210{217,)h (IEEE)e(Computer)505 3680 y(So)r(ciet)n(y)-6 b(,)26 b(2001.)605 3771 y([3])h(L.)20 b(An)n(tunes)e(and)h(L.)h(F)-6 b(ortno)n(w,)21 b Fe(Sophistic)l(ation)i(r)l(evisite)l(d)p Fv(,)e(in)f Fe(A)n(utomata,)j(L)l(anguages)h(and)505 3862 y(Pr)l(o)l(gr)l(amming,) 34 b(30th)f(International)h(Col)t(lo)l(quium,)e(ICALP)g(2003,)h (Eindhoven,)g(The)h(Nether-)505 3954 y(lands,)g(June)h(30{July)g(4,)f (2003)43 b Fv(\(J.)34 b(C.)f(M.)h(Baeten,)f(J.)h(K.)f(Lenstra,)g(J.)h (P)n(arro)n(w,)g(and)e(G.)i(J.)505 4045 y(W)-6 b(o)r(eginger,)28 b(eds.\),)e(Lect.)g(Notes)g(in)f(Comput.)g(Sci.)h(2719,)i(Springer-V)-6 b(erlag,)26 b(2003,)h(267{277.)605 4136 y([4])g(K.)d(B.)g(A)n(threy)n (a,)f(J.)i(M.)f(Hitc)n(hco)r(c)n(k,)g(J.)h(H.)f(Lutz,)f(and)h(E.)g(Ma)n (y)n(ordomo,)h Fe(E\013e)l(ctive)i(str)l(ong)505 4228 y(dimension)21 b(in)e(algorithmic)i(information)f(and)h(c)l (omputational)g(c)l(omplexity)p Fv(,)f(to)d(app)r(ear)i(in)e(SIAM)505 4319 y(J.)43 b(Comput.)f(\(extended)f(abstract)i(in)f Fe(Pr)l(o)l(c)l(e)l(e)l(dings)j(of)e(the)g(Twenty-First)i(Symp)l(osium) e(on)505 4410 y(The)l(or)l(etic)l(al)37 b(Asp)l(e)l(cts)g(of)d (Computer)i(Scienc)l(e)g(\(Montp)l(el)t(lier,)f(F)-6 b(r)l(anc)l(e,)36 b(Mar)l(ch)g(25{27,)f(2004\))p Fv(,)505 4502 y(Springer-V)-6 b(erlag,)26 b(2004,)i(632{643\).)605 4593 y([5])f(G.)g(Barmpalias,)h Fe(Computably)h(enumer)l(able)h(sets)f (in)f(the)i(Solovay)f(and)f(the)i(str)l(ong)g(we)l(ak)505 4684 y(truth)25 b(table)e(de)l(gr)l(e)l(es)p Fv(,)g(in)d Fe(New)j(Computational)h(Par)l(adigms:)f(First)g(Confer)l(enc)l(e)h(on) f(Computabil-)505 4776 y(ity)g(in)g(Eur)l(op)l(e,)g(CiE)f(2005,)h(A)n (mster)l(dam,)h(The)f(Netherlands,)h(June)g(8-12,)f(2005)31 b Fv(\(S.)21 b(B.)g(Co)r(op)r(er,)505 4867 y(B.)e(L\177)-38 b(ow)n(e,)19 b(and)e(L.)h(T)-6 b(oren)n(vliet,)18 b(eds.\),)h(Lect.)f (Notes)g(in)f(Comp.)h(Sci.)g(3526,)i(Springer-V)-6 b(erlag,)18 b(2005,)505 4958 y(8{17.)605 5050 y([6])27 b(G.)k(Barmpalias)h(and)f (A.)f(E.)i(M.)f(Lewis,)h Fe(A)h(c.e.)e(r)l(e)l(al)i(that)h(c)l(annot)g (b)l(e)e(sw-c)l(ompute)l(d)j(by)505 5141 y(any)28 b Fv(\012)g Fe(numb)l(er)p Fv(,)f(to)f(app)r(ear.)p eop %%Page: 79 79 79 78 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(79)605 541 y Fv([7])27 b(J.)k(M.)f(Barzdins,)i Fe(Complexity)g(of) f(pr)l(o)l(gr)l(ams)j(to)e(determine)h(whether)g(natur)l(al)g(numb)l (ers)505 633 y(not)27 b(gr)l(e)l(ater)h(than)f Fm(n)f Fe(b)l(elong)h(to)g(a)f(r)l(e)l(cursively)i(enumer)l(able)f(set)p Fv(,)e(So)n(viet)f(Math.)h(Dokl.)f(9)g(\(1968\),)505 724 y(1251{1254.)605 815 y([8])j(V.)g(Bec)n(her)h(and)g(G.)g(Chaitin,)h Fe(A)n(nother)i(example)f(of)f(higher)h(or)l(der)h(r)l(andomness)p Fv(,)f(F)-6 b(und.)505 907 y(Inform.)26 b(51)g(\(2002\),)h(325{338.)605 998 y([9])g(V.)36 b(Bec)n(her,)i(S.)f(Daicz,)h(and)e(G.)i(Chaitin,)g Fe(A)g(highly)f(r)l(andom)i(numb)l(er)p Fv(,)f(in)f Fe(Combina-)505 1089 y(torics,)30 b(Computability)f(and)g(L)l(o)l(gic)g(\(Constant)-30 b(\030)m(a,)28 b(2001\))j Fv(\(C.)c(S.)g(Calude,)h(M.)f(J.)h(Dinneen,)e (and)505 1181 y(S.)i(Sburlan,)g(eds.\),)g(Springer)g(Ser.)g(Discrete)h (Math.)f(Theor.)h(Comput.)f(Sci.,)g(Springer-V)-6 b(erlag,)505 1272 y(London,)26 b(2001,)i(55{68.)605 1363 y([10])f(V.)33 b(Bec)n(her)g(and)f(S.)g(Grigorie\013,)j Fe(R)l(andom)g(r)l(e)l(als)f (and)h(p)l(ossibly)f(in\014nite)h(c)l(omputations)505 1455 y(p)l(art)29 b(I:)e(r)l(andomness)i(in)e Fc(;)1298 1423 y Fj(0)1321 1455 y Fv(,)f(J.)g(Sym)n(b)r(olic)f(Logic)i(70)f (\(2005\),)i(891{913.)605 1546 y([11])f(B.)22 b(R.)g(C.)g(Bedregal)i (and)d(A.)g(Nies,)i Fe(L)l(owness)i(pr)l(op)l(erties)h(of)e(r)l(e)l (als)h(and)f(hyp)l(er-immunity)p Fv(,)505 1637 y(in)c Fe(WOLLIC)h(2003)p Fv(,)g(Electronic)g(Notes)f(in)g(Theoretical)i (Computer)d(Science)h(84,)g(Elsevier,)i(2003,)505 1729 y Fb(http://www1.elsevier.com/ge)q(j-ng/)q(31/2)q(9/23)q(/142/)q(23/s)q (how/P)q(rodu)q(cts/)q(notes)q(/)505 1820 y(index.htt#001)p Fv(.)605 1911 y([12])27 b(R.)45 b(Beigel,)i(H.)e(Buhrman,)f(P)-6 b(.)45 b(F)-6 b(ejer,)46 b(L.)f(F)-6 b(ortno)n(w,)46 b(P)-6 b(.)45 b(Grab)r(o)n(wski,)i(L.)e(Longpr)n(\023)-36 b(e,)505 2003 y(A.)56 b(Muc)n(hnik,)g(F.)g(Stephan,)f(and)h(L.)g(T)-6 b(oren)n(vliet,)56 b Fe(Enumer)l(ations)h(of)e(the)h(Kolmo)l(gor)l(ov) 505 2094 y(function)p Fv(,)c(Electronic)g(Collo)r(quium)f(on)g (Computational)g(Complexit)n(y)-6 b(,)50 b(TR04-015,)j(2004,)505 2185 y Fb(http://eccc.uni-trier.de/ec)q(cc-re)q(port)q(s/20)q(04/TR)q (04-0)q(15/)p Fv(.)605 2277 y([13])27 b(J.-Y.)34 b(Cai)h(and)e(J.)i (Hartmanis,)e Fe(On)i(Hausdor\013)h(and)g(top)l(olo)l(gic)l(al)g (dimensions)f(of)g(the)505 2368 y(Kolmo)l(gor)l(ov)29 b(c)l(omplexity)g(of)e(the)h(r)l(e)l(al)g(line)p Fv(,)e(J.)g(Comput.)f (System)f(Sci.)i(49)h(\(1994\),)g(605{619.)605 2459 y([14])g(C.)22 b(S.)e(Calude,)i Fe(Information)h(and)g(R)l(andomness,)h(A)n(n)f(A)n (lgorithmic)g(Persp)l(e)l(ctive)p Fv(,)h(Sprin-)505 2551 y(ger-V)-6 b(erlag,)27 b(1994.)h(Second)d(edition)h(2002.)605 2642 y([15])h(C.)i(S.)g(Calude)g(and)f(R.)g(J.)h(Coles,)h Fe(Pr)l(o)l(gr)l(am-size)i(c)l(omplexity)f(of)f(initial)f(se)l(gments)j (and)505 2733 y(domination)23 b(r)l(e)l(ducibility)p Fv(,)f(in)e Fe(Jewels)k(ar)l(e)f(F)-6 b(or)l(ever)31 b Fv(\(J.)21 b(Karh)n(um\177)-38 b(aki,)19 b(H.)i(Maurer,)g(G.)g(P\025) -38 b(aun,)21 b(and)505 2825 y(G.)27 b(Rozen)n(b)r(erg,)f(eds.\),)g (Springer-V)-6 b(erlag,)26 b(1999,)h(225{237.)605 2916 y([16])g(C.)33 b(S.)f(Calude,)h(R.)f(Coles,)i(P)-6 b(.)32 b(H.)g(Hertling,)g(and)g(B.)h(Khoussaino)n(v,)f Fe(De)l(gr)l(e)l(e-the) l(or)l(etic)505 3007 y(asp)l(e)l(cts)25 b(of)d(c)l(omputably)i(enumer)l (able)g(r)l(e)l(als)p Fv(,)d(in)f Fe(Mo)l(dels)j(and)g(Computability)28 b Fv(\(S.)20 b(B.)g(Co)r(op)r(er)i(and)505 3099 y(J.)33 b(K.)f(T)-6 b(russ,)33 b(eds.\),)f(London)g(Math.)h(So)r(c.)f(Lecture)g (Note)g(Ser.)g(259,)i(Cam)n(bridge)e(Univ)n(ersit)n(y)505 3190 y(Press,)27 b(1999,)h(23{39.)605 3281 y([17])f(C.)19 b(S.)f(Calude,)g(P)-6 b(.)19 b(H.)e(Hertling,)i(B.)f(Khoussaino)n(v,)h (and)e(Y.)h(W)-6 b(ang,)18 b Fe(R)l(e)l(cursively)k(enumer-)505 3372 y(able)g(r)l(e)l(als)g(and)f(Chaitin)g Fv(\012)h Fe(numb)l(ers)p Fv(,)e(Theoret.)g(Comput.)e(Sci.)h(255)h(\(2001\),)g (125{149)i(\(extended)505 3464 y(abstract)g(in)g(ST)-6 b(A)n(CS)20 b(98,)i(Lect.)g(Notes)g(in)f(Comput.)g(Sci.)g(1373,)j (Springer-V)-6 b(erlag,)21 b(Berlin,)i(1998,)505 3555 y(596{606\).)605 3646 y([18])k(C.)d(S.)g(Calude)g(and)f(A.)g(Nies,)h Fe(Chaitin)h Fv(\012)h Fe(numb)l(ers)h(and)f(str)l(ong)h(r)l(e)l (ducibilities)p Fv(,)d(J.)g(Univ.)505 3738 y(Comp.)i(Sci.)g(3)g (\(1998\),)h(1162{1166.)605 3829 y([19])g(C.)e(S.)g(Calude,)h(L.)f (Staiger,)g(and)g(S.)f(A.)h(T)-6 b(erwijn,)26 b Fe(On)g(p)l(artial)i(r) l(andomness)p Fv(,)f(Ann.)c(Pure)505 3920 y(Appl.)j(Logic,)h(in)e (press.)605 4012 y([20])i(G.)22 b(J.)f(Chaitin,)h Fe(A)h(the)l(ory)i (of)e(pr)l(o)l(gr)l(am)i(size)f(formal)t(ly)e(identic)l(al)h(to)h (information)f(the)l(ory)p Fv(,)505 4103 y(J.)k(Asso)r(c.)g(Comput.)e (Mac)n(h.)h(22)g(\(1975\),)i(329{340.)605 4194 y([21])f(G.)33 b(J.)g(Chaitin,)h Fe(A)n(lgorithmic)g(information)g(the)l(ory)p Fv(,)g(IBM)f(Journal)h(of)f(Researc)n(h)g(and)505 4286 y(Dev)n(elopmen)n(t)24 b(21)j(\(1977\),)g(350{359,)i(496.)605 4377 y([22])e(G.)g(J.)h(Chaitin,)g Fe(Inc)l(ompleteness)i(the)l(or)l (ems)h(for)d(r)l(andom)i(r)l(e)l(als)p Fv(,)e(Adv.)e(in)h(Appl.)f (Math.)505 4468 y(8)g(\(1987\),)i(119{146.)605 4560 y([23])f(G.)35 b(J.)f(Chaitin,)h Fe(A)n(lgorithmic)h(Information)f(The)l(ory)p Fv(,)g(Cam)n(bridge)f(Univ)n(ersit)n(y)f(Press,)505 4651 y(1987.)605 4742 y([24])27 b(A.)35 b(V.)f(Cherno)n(v,)h(An.)f(A.)h(Muc) n(hnik,)f(A.)h(E.)g(Romashc)n(henk)n(o,)f(A.)g(Shen,)h(and)f(N.)h(K.) 505 4834 y(V)-6 b(ereshc)n(hagin,)25 b Fe(Upp)l(er)j(semi-lattic)l(e)g (of)e(binary)h(strings)h(with)f(the)g(r)l(elation)h(\\x)f(is)f(simple)h (c)l(ondi-)505 4925 y(tional)h(to)g(y")p Fv(,)f(Theor.)f(Comput.)f (Sci.)h(271)h(\(2002\),)g(69{95.)605 5016 y([25])g(P)-6 b(.)19 b(Cholak,)i(R.)e(Coles,)i(R.)e(Do)n(wney)-6 b(,)18 b(and)h(E.)h(Herrmann,)e Fe(A)n(utomorphisms)23 b(of)e(the)i(lattic)l (e)505 5108 y(of)37 b Fv(\005)663 5076 y Fn(0)663 5121 y(1)734 5108 y Fe(classes:)h(p)l(erfe)l(ct)h(thin)e(classes)h(and)g (a.n.c.)e(de)l(gr)l(e)l(es)p Fv(,)i(T)-6 b(rans.)37 b(Amer.)e(Math.)i (So)r(c.)f(353)p eop %%Page: 80 80 80 79 bop 505 363 a FD(80)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y Fv(\(2001\),)28 b(4899{4924.)605 633 y([26])f(B.)j(F.)g(Csima,)h Fe(Applic)l(ations)h(of)e(Computability)i(The)l(ory)g(to)g(Prime)f(Mo)l (dels)h(and)f(Dif-)505 724 y(fer)l(ential)d(Ge)l(ometry)p Fv(,)g(Ph.D.)e(Dissertation,)h(The)f(Univ)n(ersit)n(y)f(of)h(Chicago,)i (2003.)605 815 y([27])f(B.)36 b(F.)g(Csima)g(and)f(A.)g(Mon)n(talb\023) -38 b(an,)36 b Fe(A)h(Minimal)e(Pair)h(of)g(K-de)l(gr)l(e)l(es)p Fv(,)j(Pro)r(c.)d(Amer.)505 907 y(Math.)27 b(So)r(c.,)f(in)g(press.)605 998 y([28])h(G.)35 b(Da)n(vie,)g Fe(Char)l(acterising)i(the)g (Martin-L\177)-39 b(of)36 b(r)l(andom)g(se)l(quenc)l(es)i(using)e(c)l (omputably)505 1089 y(enumer)l(able)29 b(sets)g(of)e(me)l(asur)l(e)i (one)p Fv(,)e(Inform.)e(Pro)r(cess.)j(Lett.)e(92)g(\(2004\),)h (157{160.)605 1181 y([29])g(R.)k(G.)h(Do)n(wney)-6 b(,)31 b Fe(Some)i(c)l(omputability-the)l(or)l(etic)k(asp)l(e)l(cts)e(of)d(r)l (e)l(als)i(and)f(r)l(andomness)p Fv(,)505 1272 y(in)e Fe(The)h(Notr)l(e)i(Dame)e(L)l(e)l(ctur)l(es)39 b Fv(\(P)-6 b(.)31 b(Cholak,)g(ed.\),)g(to)g(app)r(ear)g(in)g(the)f(Lect.)h(Notes)g (Log.)g(18,)505 1363 y(Asso)r(c.)c(for)g(Sym)n(b)r(ol.)d(Logic,)j (Urbana,)f(IL,)g(2005,)h(97{148.)605 1455 y([30])g(R.)d(Do)n(wney)g (and)f(E.)i(Gri\016ths,)f Fe(Schnorr)k(r)l(andomness)p Fv(,)e(J.)f(Sym)n(b)r(olic)e(Logic)i(69)g(\(2004\),)505 1546 y(533{554.)605 1637 y([31])i(R.)e(Do)n(wney)-6 b(,)25 b(E.)h(Gri\016ths,)g(and)f(G.)h(LaF)-6 b(orte,)26 b Fe(On)h(Schnorr)i (and)f(c)l(omputable)g(r)l(andom-)505 1729 y(ness,)h(martingales,)f (and)g(machines)p Fv(,)e(Math.)h(Logic)f(Quart.)g(50)g(\(2004\),)i (613{627.)605 1820 y([32])f(R.)c(Do)n(wney)-6 b(,)23 b(E.)h(Gri\016ths,)g(and)f(S.)h(Reid,)f Fe(On)i(Kurtz)i(r)l(andomness)p Fv(,)f(Theoret.)e(Comput.)505 1911 y(Sci.)i(321)h(\(2004\),)h(249{270.) 605 2003 y([33])f(R.)34 b(Do)n(wney)f(and)h(D.)f(R.)h(Hirsc)n(hfeldt,)g Fe(A)n(lgorithmic)i(R)l(andomness)g(and)g(Complexity)p Fv(,)505 2094 y(Springer-V)-6 b(erlag,)26 b(to)g(app)r(ear.)605 2185 y([34])h(R.)d(Do)n(wney)-6 b(,)23 b(D.)h(R.)g(Hirsc)n(hfeldt,)g (and)g(G.)g(LaF)-6 b(orte,)25 b Fe(R)l(andomness)j(and)e(r)l(e)l (ducibility)p Fv(,)f(J.)505 2277 y(Comput.)30 b(System)e(Sci.)j(68)f (\(2004\),)i(96{114)g(\(extended)d(abstract)i(in)f Fe(Mathematic)l(al)i (F)-6 b(ounda-)505 2368 y(tions)28 b(of)e(Computer)i(Scienc)l(e)g(2001) 36 b Fv(\(J.)25 b(Sgall,)h(A.)f(Pultr,)g(and)g(P)-6 b(.)25 b(Kolman,)g(eds.\),)g(Lect.)g(Notes)505 2459 y(in)h(Comput.)f(Sci.)h (2136,)i(Springer-V)-6 b(erlag,)26 b(2001,)h(316{327\).)605 2551 y([35])g(R.)i(Do)n(wney)-6 b(,)28 b(D.)g(R.)h(Hirsc)n(hfeldt,)g (and)f(G.)h(LaF)-6 b(orte,)30 b Fe(Unde)l(cidability)h(of)e(the)j (structur)l(e)505 2642 y(of)c(the)g(Solovay)g(de)l(gr)l(e)l(es)i(of)d (c.e.)g(r)l(e)l(als)p Fv(,)g(in)f(preparation.)605 2733 y([36])h(R.)h(Do)n(wney)-6 b(,)29 b(D.)f(R.)h(Hirsc)n(hfeldt,)g(J.)g (S.)g(Miller,)h(and)e(A.)h(Nies,)g Fe(R)l(elativizing)i(Chaitin)-8 b('s)505 2825 y(halting)28 b(pr)l(ob)l(ability)p Fv(,)f(to)f(app)r (ear.)605 2916 y([37])h(R.)34 b(Do)n(wney)-6 b(,)34 b(D.)g(R.)g(Hirsc)n (hfeldt,)h(and)f(A.)g(Nies,)h Fe(R)l(andomness,)h(c)l(omputability,)g (and)505 3007 y(density)p Fv(,)e(SIAM)d(J.)i(Comput.)e(31)i(\(2002\),)g (1169{1183)i(\(extended)c(abstract)i(in)e Fe(ST)-6 b(A)n(CS)34 b(2001)505 3099 y(Pr)l(o)l(c)l(e)l(e)l(dings)48 b Fv(\(A.)40 b(F)-6 b(erreira)40 b(and)f(H.)h(Reic)n(hel,)g(eds.\),)g(Lect.)g(Notes) f(in)h(Comput.)f(Sci.)h(2010,)505 3190 y(Springer-V)-6 b(erlag,)26 b(2001,)i(195{201\).)605 3281 y([38])f(R.)k(Do)n(wney)-6 b(,)30 b(D.)h(R.)g(Hirsc)n(hfeldt,)g(A.)g(Nies,)g(and)g(F.)g(Stephan,)f Fe(T)-6 b(rivial)32 b(r)l(e)l(als)p Fv(,)g(in)e Fe(Pr)l(o-)505 3372 y(c)l(e)l(e)l(dings)k(of)e(the)h(7th)g(and)g(8th)g(Asian)f(L)l(o)l (gic)h(Confer)l(enc)l(es)38 b Fv(\(R.)30 b(Do)n(wney)-6 b(,)31 b(D.)f(Dec)n(heng,)h(T.)h(S.)505 3464 y(Ping,)22 b(Q.)f(Y.)f(Hui,)h(and)f(M.)i(Y)-6 b(asugi,)21 b(eds.\),)g(Singap)r (ore)h(Univ)n(ersit)n(y)e(Press)h(and)g(W)-6 b(orld)20 b(Scien)n(ti\014c,)505 3555 y(2003,)25 b(103{131)g(\(extended)c (abstract)i(in)g(Electronic)g(Notes)g(in)f(Theoretical)j(Computer)c (Science)505 3646 y(66)27 b(\(2002\),)g(no.)f(1\).)605 3738 y([39])h(R.)19 b(Do)n(wney)-6 b(,)19 b(C.)h(Jo)r(c)n(kusc)n(h,)f (and)g(M.)h(Stob,)f Fe(A)n(rr)l(ay)k(nonr)l(e)l(cursive)h(sets)f(and)f (multiple)f(p)l(er-)505 3829 y(mitting)31 b(ar)l(guments)p Fv(,)g(in)e Fe(R)l(e)l(cursion)j(The)l(ory)g(We)l(ek)f(\(Ob)l (erwolfach,)g(1989\))i Fv(\(K.)c(Am)n(b)r(os-Spies,)505 3920 y(G.)23 b(H.)f(M)r(\177)-41 b(uller,)24 b(and)e(G.)g(E.)h(Sac)n (ks,)f(eds.\),)g(Lect.)h(Notes)f(in)g(Math.)g(1432,)i(Springer-V)-6 b(erlag,)23 b(1990,)505 4012 y(141{174.)605 4103 y([40])k(R.)i(Do)n (wney)-6 b(,)28 b(C.)h(Jo)r(c)n(kusc)n(h,)g(and)g(M.)g(Stob,)f Fe(A)n(rr)l(ay)k(nonr)l(e)l(cursive)h(de)l(gr)l(e)l(es)f(and)f(gener-) 505 4194 y(icity)p Fv(,)24 b(in)f Fe(Computability,)j(Enumer)l (ability,)f(Unsolvability)31 b Fv(\(S.)23 b(B.)h(Co)r(op)r(er,)h(T.)f (A.)f(Slaman,)g(and)505 4286 y(S.)28 b(S.)g(W)-6 b(ainer,)28 b(eds.\),)g(London)f(Math.)h(So)r(c.)h(Lect.)f(Notes)g(Series)g(224,)h (Cam)n(bridge)f(Univ)n(ersit)n(y)505 4377 y(Press,)f(1996,)h(93{104.) 605 4468 y([41])f(R.)e(Do)n(wney)f(and)h(J.)h(S.)e(Miller,)j Fe(A)g(b)l(asis)g(the)l(or)l(em)i(for)e Fv(\005)2449 4437 y Fn(0)2449 4482 y(1)2510 4468 y Fe(classes)h(of)e(p)l(ositive)i (me)l(asur)l(e)505 4560 y(and)g(jump)f(inversion)i(for)e(r)l(andom)i(r) l(e)l(als)p Fv(,)d(to)g(app)r(ear)g(in)g(Pro)r(c.)h(Amer.)e(Math.)h(So) r(c.)605 4651 y([42])h(R.)f(Do)n(wney)-6 b(,)25 b(J.)h(S.)g(Miller,)h (and)f(J.)g(Reimann,)f Fe(Finite)i(r)l(andomness)p Fv(,)h(in)e (preparation.)605 4742 y([43])h(R.)21 b(Do)n(wney)g(and)f(Y.)h(Y)-6 b(ang,)21 b Fe(R)l(elative)j(c)l(omputability)g(vs)g(r)l(elative)g(c)l (omplexity)p Fv(,)e(in)f(prepa-)505 4834 y(ration.)605 4925 y([44])27 b(K.)38 b(F)-6 b(alconer,)39 b Fe(F)-6 b(r)l(actal)40 b(Ge)l(ometry.)g(Mathematic)l(al)g(F)-6 b(oundations)40 b(and)f(Applic)l(ations)p Fv(,)505 5016 y(Wiley)26 b(&)g(Sons,)g(1990.)p eop %%Page: 81 81 81 80 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(81)605 541 y Fv([45])27 b(P)-6 b(.)31 b(G\023)-38 b(acs,)33 b Fe(On)f(the)h(symmetry)h(of)e(algorithmic)h(information)p Fv(,)e(So)n(viet)g(Math.)h(Dokl.)f(15)505 633 y(\(1974\),)d(1477{1480.) 605 724 y([46])f(P)-6 b(.)28 b(G\023)-38 b(acs,)30 b Fe(Every)h(se)l(quenc)l(e)g(is)f(r)l(e)l(ducible)h(to)f(a)g(r)l(andom)g (one)p Fv(,)f(Inform.)f(and)g(Con)n(trol)h(70)505 815 y(\(1986\),)f(186{192.)605 907 y([47])f(H.)j(Gaifman)h(and)f(M.)h (Snir,)f Fe(Pr)l(ob)l(abilities)i(over)h(rich)f(languages)p Fv(,)g(J.)f(Sym)n(b)r(olic)e(Logic)505 998 y(47)e(\(1982\),)g(495{548.) 605 1089 y([48])g(F.)e(Hausdor\013,)f Fe(Dimension)i(und)g(\177)-39 b(au\031er)l(es)28 b(Ma\031)p Fv(,)c(Mathematisc)n(he)h(Annalen)e(79)i (\(1919\),)505 1181 y(157{179.)605 1272 y([49])i(G.)19 b(Hjorth)f(and)g(A.)g(Nies,)h Fe(R)l(andomness)j(in)e(e\013e)l(ctive)i (descriptive)g(set)g(the)l(ory)p Fv(,)e(to)f(app)r(ear.)605 1363 y([50])27 b(D.)k(R.)h(Hirsc)n(hfeldt,)g(A.)f(Nies,)h(and)f(F.)h (Stephan,)e Fe(Using)j(r)l(andom)h(sets)g(as)f(or)l(acles)p Fv(,)g(to)505 1455 y(app)r(ear.)605 1546 y([51])27 b(J.)h(M.)h(Hitc)n (hco)r(c)n(k,)e(J.)i(H.)e(Lutz)g(and)h(S.)f(A.)h(T)-6 b(erwijn,)29 b Fe(The)g(arithmetic)l(al)i(c)l(omplexity)f(of)505 1637 y(dimension)e(and)g(r)l(andomness)p Fv(,)g(A)n(CM)e(T)-6 b(rans.)26 b(on)g(Comput.)f(Logic,)i(in)f(press.)605 1729 y([52])h(S.)e(Ishm)n(ukhameto)n(v,)d Fe(We)l(ak)27 b(r)l(e)l(cursive)i(de)l(gr)l(e)l(es)h(and)d(a)g(pr)l(oblem)h(of)e(Sp)l (e)l(ctor)p Fv(,)i(in)d Fe(R)l(e)l(cur-)505 1820 y(sion)34 b(The)l(ory)g(and)g(Complexity)39 b Fv(\(M.)33 b(Arslano)n(v)f(and)g (S.)f(Lempp,)g(eds.\),)i(de)f(Gruyter,)f(Berlin,)505 1911 y(1999,)d(81{88.)605 2003 y([53])f(C.)c(G.)g(Jo)r(c)n(kusc)n(h,)g (Jr.,)h Fe(The)h(de)l(gr)l(e)l(es)i(of)d(bi-immune)g(sets)p Fv(,)g(Z.)f(Math.)g(Logik)g(Grundlagen)505 2094 y(Math.)k(15)f (\(1969\))h(135{140.)605 2185 y([54])g(C.)35 b(G.)g(Jo)r(c)n(kusc)n(h,) f(Jr.,)i Fe(Thr)l(e)l(e)g(e)l(asy)h(c)l(onstructions)h(of)d(r)l(e)l (cursively)i(enumer)l(able)g(sets)p Fv(,)505 2277 y(in)f Fe(L)l(o)l(gic)i(Y)-6 b(e)l(ar)37 b(1979{80)i(\(Pr)l(o)l(c.)e(Seminars) h(and)f(Conf.)f(Math.)h(L)l(o)l(gic,)g(Univ.)f(Conne)l(cticut,)505 2368 y(Storrs,)25 b(Conn.,)e(1979/80\))j Fv(\(M.)c(Lerman,)g(J.)g(H.)f (Sc)n(hmerl,)g(and)g(R.)g(I.)h(Soare,)g(eds.\),)g(Lect.)g(Notes)505 2459 y(in)k(Math.)g(859,)h(Springer-V)-6 b(erlag,)26 b(1981,)i(83{91.)605 2551 y([55])f(C.)h(G.)h(Jo)r(c)n(kusc)n(h,)f(Jr.,) h(M.)f(Lerman,)f(R.)g(I.)h(Soare,)h(and)e(R.)h(Solo)n(v)l(a)n(y)-6 b(,)27 b Fe(R)l(e)l(cursively)k(enu-)505 2642 y(mer)l(able)22 b(sets)g(mo)l(dulo)f(iter)l(ate)l(d)i(jumps)e(and)g(extensions)i(of)e (A)n(rslanov's)h(c)l(ompleteness)h(criterion)p Fv(,)505 2733 y(J.)k(Sym)n(b)r(olic)d(Logic)j(54)g(\(1989\),)g(1288{1323.)605 2825 y([56])g(C.)33 b(G.)f(Jo)r(c)n(kusc)n(h,)g(Jr.)h(and)e(R.)h(A.)g (Shore,)g Fe(Pseudo-jump)i(op)l(er)l(ators)i(I:)c(the)i(r.e.)f(c)l(ase) p Fv(,)505 2916 y(T)-6 b(rans.)27 b(Amer.)e(Math.)h(So)r(c.)g(275)h (\(1983\),)g(599{609.)605 3007 y([57])g(C.)g(G.)g(Jo)r(c)n(kusc)n(h,)f (Jr.,)h(and)f(R.)g(I.)g(Soare,)h(\005)2064 2975 y Fn(0)2064 3020 y(1)2126 3007 y Fe(classes)j(and)e(de)l(gr)l(e)l(es)i(of)e(the)l (ories)p Fv(,)g(T)-6 b(rans.)505 3099 y(Amer.)25 b(Math.)h(So)r(c.)h (173)f(\(1972\),)i(33{56.)605 3190 y([58])f(T.)20 b(Kamae,)f Fe(Subse)l(quenc)l(es)24 b(of)d(normal)h(se)l(quenc)l(es)p Fv(,)g(Israel)d(J.)h(Math.)g(16)f(\(1973\))h(121{149.)605 3281 y([59])27 b(H.)22 b(P)-6 b(.)23 b(Katse\013,)g Fe(Complexity)i (dips)g(in)f(r)l(andom)i(in\014nite)e(binary)h(se)l(quenc)l(es)p Fv(,)h(Inform.)c(and)505 3372 y(Con)n(trol)27 b(38)f(\(1978\),)i (258{263.)605 3464 y([60])f(S.)33 b(M.)h(Kautz,)f Fe(De)l(gr)l(e)l(es)j (of)e(R)l(andom)h(Sets)p Fv(,)g(PhD)d(Dissertation,)j(Cornell)g(Univ)n (ersit)n(y)-6 b(,)505 3555 y(1991.)605 3646 y([61])27 b(A.)j(S.)f(Kec)n(hris)h(and)f(Y.)h(Mosc)n(ho)n(v)l(akis,)h(eds.,)f Fe(Cab)l(al)h(Seminar)h(76{77)p Fv(,)f(Lect.)f(Notes)g(in)505 3738 y(Math.)d(689,)g(Springer-V)-6 b(erlag,)26 b(1978.)605 3829 y([62])h(B.)34 b(Kjos-Hanssen,)g(A.)g(Nies,)g(and)f(F.)g(Stephan,) g Fe(L)l(owness)j(for)f(the)g(class)h(of)e(Schnorr)505 3920 y(r)l(andom)29 b(r)l(e)l(als)p Fv(,)e(to)f(app)r(ear.)605 4012 y([63])h(K.-I)f(Ko,)i Fe(On)h(the)g(notion)h(of)e(in\014nite)h (pseudor)l(andom)i(se)l(quenc)l(es)p Fv(,)f(Theoret.)e(Comput.)505 4103 y(Sci.)e(48)h(\(1986\))g(9{33.)605 4194 y([64])g(A.)g(N.)g (Kolmogoro)n(v,)h Fe(Thr)l(e)l(e)i(appr)l(o)l(aches)h(to)f(the)g (quantitative)h(de\014nition)e(of)f(informa-)505 4286 y(tion)p Fv(,)21 b(Problems)f(of)h(Information)e(T)-6 b(ransmission)21 b(\(Problem)n(y)e(P)n(eredac)n(hi)h(Informatsii\))h(1) f(\(1965\),)505 4377 y(1{7.)605 4468 y([65])27 b(L.)f(G.)g(Kraft,)g Fe(A)h(Devic)l(e)h(for)f(Quantizing,)g(Gr)l(ouping,)h(and)g(Co)l(ding)f (A)n(mplitude)h(Mo)l(du-)505 4560 y(late)l(d)h(Pulses)p Fv(,)d(M.Sc.)h(Thesis,)g(MIT,)g(1949.)605 4651 y([66])g(A.)32 b(Ku)n(\024)-36 b(cera,)33 b Fe(Me)l(asur)l(e,)h Fv(\005)1544 4619 y Fn(0)1544 4664 y(1)1579 4651 y Fe(-classes)h(and)f(c)l(omplete)h (extensions)g(of)e(P)-6 b(A)p Fv(,)33 b(in)f Fe(R)l(e)l(cursion)505 4742 y(The)l(ory)d(We)l(ek)k Fv(\(H.-D.)24 b(Ebbinghaus,)i(G.)f(H.)g(M) r(\177)-41 b(uller,)27 b(and)e(G.)h(E.)f(Sac)n(ks,)g(eds.\),)h(Lect.)f (Notes)h(in)505 4834 y(Math.)h(1141,)g(Springer-V)-6 b(erlag,)26 b(1985,)i(245{259.)605 4925 y([67])f(A.)f(Ku)n(\024)-36 b(cera,)26 b Fe(On)h(r)l(elative)i(r)l(andomness)p Fv(,)f(Ann.)c(Pure)i (Appl.)f(Logic)i(63)g(\(1993\))f(61{67.)605 5016 y([68])h(A.)g(Ku)n (\024)-36 b(cera)27 b(and)g(T.)h(A.)f(Slaman,)g Fe(R)l(andomness)k(and) e(r)l(e)l(cursive)i(enumer)l(ability)p Fv(,)d(SIAM)505 5108 y(J.)f(Comput.)e(31)h(\(2001\))h(199{211.)p eop %%Page: 82 82 82 81 bop 505 363 a FD(82)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)605 541 y Fv([69])27 b(A.)c(Ku)n(\024)-36 b(cera)22 b(and)h(S.)f(A.)h(T)-6 b(erwijn,)24 b Fe(L)l(owness)i(for)f(the)h(class)g(of)e(r)l(andom)i (sets)p Fv(,)e(J.)g(Sym)n(b)r(olic)505 633 y(Logic)j(64)g(\(1999\))g (1396{1402.)605 724 y([70])g(M.)20 b(Kummer,)d Fe(Kolmo)l(gor)l(ov)23 b(c)l(omplexity)g(and)f(instanc)l(e)i(c)l(omplexity)e(of)g(r)l(e)l (cursively)i(enu-)505 815 y(mer)l(able)29 b(sets)p Fv(,)e(SIAM)e(J.)h (Comput.)f(25)i(\(1996\),)g(1123{1143.)605 907 y([71])g(S.)g(A.)g (Kurtz,)g Fe(R)l(andomness)k(and)e(Genericity)h(in)f(the)h(De)l(gr)l(e) l(es)g(of)f(Unsolvability)p Fv(,)f(PhD)505 998 y(Dissertation,)g(Univ)n (ersit)n(y)c(of)j(Illinois,)g(1981.)605 1089 y([72])g(M.)j(v)l(an)e (Lam)n(balgen,)h Fe(R)l(andom)h(Se)l(quenc)l(es)p Fv(,)i(PhD)c (Dissertation,)j(Univ)n(ersit)n(y)c(of)j(Ams-)505 1181 y(terdam,)25 b(1987.)605 1272 y([73])i(M.)k(v)l(an)f(Lam)n(balgen,)g Fe(The)i(axiomatization)h(of)f(r)l(andomness)p Fv(,)g(J.)g(Sym)n(b)r (olic)d(Logic)i(55,)505 1363 y(1990,)d(1143{1167.)605 1455 y([74])f(K.)j(de)g(Leeu)n(w,)h(E.)g(F.)g(Mo)r(ore,)h(C.)f(F.)f (Shannon,)g(and)g(N.)g(Shapiro,)h Fe(Computability)h(by)505 1546 y(pr)l(ob)l(abilistic)23 b(machines)p Fv(,)e(in)f Fe(A)n(utomata)j(Studies)p Fv(,)e(Annals)f(of)g(Mathematics)g(Studies)g (34,)g(Prince-)505 1637 y(ton)26 b(Univ)n(ersit)n(y)f(Press,)i(1956,)g (183{212.)605 1729 y([75])g(L.)21 b(A.)g(Levin,)f Fe(Some)k(The)l(or)l (ems)g(on)f(the)h(A)n(lgorithmic)f(Appr)l(o)l(ach)i(to)f(Pr)l(ob)l (ability)f(The)l(ory)505 1820 y(and)28 b(Information)g(The)l(ory)p Fv(,)f(Dissertation)g(in)f(Mathematics,)g(Mosco)n(w,)i(1971.)605 1911 y([76])f(L.)d(A.)f(Levin,)g Fe(On)i(the)h(notion)g(of)e(a)i(r)l (andom)g(se)l(quenc)l(e)p Fv(,)f(So)n(viet)e(Math.)h(Dokl.)f(14)h (\(1973\))505 2003 y(1413{1416.)605 2094 y([77])j(L.)f(A.)g(Levin,)g Fe(L)l(aws)i(of)f(information)h(c)l(onservation)i(\(non-gr)l(owth\))h (and)d(asp)l(e)l(cts)i(of)e(the)505 2185 y(foundation)g(of)f(pr)l(ob)l (ability)h(the)l(ory)p Fv(,)f(Problems)f(Informat.)f(T)-6 b(ransmission)26 b(10)g(\(1974\),)g(206{210.)605 2277 y([78])h(A.)k(E.)g(M.)g(Lewis)h(and)f(G.)g(Barmpalias,)h Fe(R)l(andomness)i(and)e(the)h(Lipschitz)g(de)l(gr)l(e)l(es)i(of)505 2368 y(c)l(omputability)p Fv(,)27 b(to)f(app)r(ear.)605 2459 y([79])h(M.)j(Li)f(and)g(P)-6 b(.)29 b(Vit\023)-38 b(an)n(yi,)29 b Fe(A)n(n)h(Intr)l(o)l(duction)j(to)e(Kolmo)l(gor)l(ov)h (Complexity)f(and)g(its)g(Ap-)505 2551 y(plic)l(ations)p Fv(,)c(2nd)e(edition,)i(Springer-V)-6 b(erlag,)26 b(1997.)605 2642 y([80])h(D.)j(Lo)n(v)n(eland,)g Fe(A)i(variant)h(of)e(the)h(Kolmo) l(gor)l(ov)h(c)l(onc)l(ept)h(of)d(c)l(omplexity)p Fv(,)g(Inform.)f(and) 505 2733 y(Con)n(trol)d(15)f(\(1969\),)i(510{526.)605 2825 y([81])f(J.)j(H.)e(Lutz,)h Fe(Cate)l(gory)j(and)f(me)l(asur)l(e)h (in)e(c)l(omplexity)h(classes)p Fv(,)f(SIAM)e(J.)i(Comput.)e(19)505 2916 y(\(1990\),)g(1100{1131.)605 3007 y([82])f(J.)d(H.)e(Lutz,)h Fe(A)n(lmost)i(everywher)l(e)j(high)d(nonuniform)f(c)l(omplexity)p Fv(,)g(J.)g(Comput.)e(System)505 3099 y(Sci.)k(44)h(\(1992\),)g (220{258.)605 3190 y([83])g(J.)21 b(H.)g(Lutz,)f Fe(Dimension)i(in)h(c) l(omplexity)g(classes)p Fv(,)g(SIAM)c(J.)j(Comput.)d(32)j(\(2003\),)g (1236{)505 3281 y(1259)j(\(extended)c(abstract)j(in)e Fe(15th)k(A)n(nnual)f(IEEE)g(Confer)l(enc)l(e)h(on)f(Computational)h (Complex-)505 3372 y(ity)d(\(Flor)l(enc)l(e,)g(2000\))h Fv(\(F.)c(Titsw)n(orth,)h(ed.\),)f(IEEE)g(Computer)f(So)r(c.,)i(Los)f (Alamitos,)h(CA,)f(2000,)505 3464 y(158{169\).)605 3555 y([84])27 b(J.)35 b(H.)f(Lutz,)h Fe(The)h(dimensions)g(of)f(individual) g(strings)i(and)f(se)l(quenc)l(es)p Fv(,)h(Inform.)d(and)505 3646 y(Comput.)22 b(187)i(\(2003\),)g(49{79)h(\(preliminary)d(v)n (ersion:)h Fe(Gales)i(and)g(the)h(c)l(onstructive)h(dimension)505 3738 y(of)38 b(individual)g(se)l(quenc)l(es)p Fv(,)i(in)d Fe(Pr)l(o)l(c.)h(27th)h(International)h(Col)t(lo)l(quium)d(on)i(A)n (utomata,)g(L)l(an-)505 3829 y(guages,)d(and)e(Pr)l(o)l(gr)l(amming)41 b Fv(\(U.)33 b(Mon)n(tanari,)h(J.)g(D.)e(P)-6 b(.)33 b(Rolim,)g(E.)h(W)-6 b(elzl,)33 b(eds.\),)h(Springer-)505 3920 y(V)-6 b(erlag,)27 b(2000,)g(902{913\).)605 4012 y([85])g(J.)f(H.)g(Lutz,)g Fe(E\013e)l(ctive)i(fr)l(actal)g(dimensions) p Fv(,)f(Math.)f(Logic)h(Quart.)f(51)g(\(2005\),)h(62{72.)605 4103 y([86])g(P)-6 b(.)43 b(Martin-L\177)-38 b(of,)44 b Fe(The)f(de\014nition)h(of)f(r)l(andom)h(se)l(quenc)l(es)p Fv(,)h(Inform.)d(and)h(Con)n(trol)g(9)505 4194 y(\(1966\),)28 b(602{619.)605 4286 y([87])f(E.)19 b(Ma)n(y)n(ordomo,)f Fe(A)j(Kolmo)l(gor)l(ov)h(c)l(omplexity)g(char)l(acterization)h(of)e(c) l(onstructive)i(Haus-)505 4377 y(dor\013)29 b(dimension)p Fv(,)d(Inform.)f(Pro)r(cess.)j(Lett.)d(84)i(\(2002\))g(1{3.)605 4468 y([88])g(W.)34 b(Merkle,)h Fe(The)g(c)l(omplexity)h(of)f(sto)l (chastic)j(se)l(quenc)l(es)p Fv(,)e(to)e(app)r(ear)h(in)f(J.)g(Comput.) 505 4560 y(Syst.)20 b(Sci.)h(\(preliminary)e(v)n(ersion)i(in)f Fe(Confer)l(enc)l(e)k(on)e(Computational)i(Complexity)f(2003)p Fv(,)e(IEEE)505 4651 y(Computer)k(So)r(ciet)n(y)h(Press,)h(2003,)g (230{235\).)605 4742 y([89])g(W.)h(Merkle)g(and)f(N.)h(Mihailo)n(vi)n (\023)-36 b(c,)29 b Fe(On)g(the)i(c)l(onstruction)h(of)d(e\013e)l (ctive)h(r)l(andom)g(sets)p Fv(,)g(J.)505 4834 y(Sym)n(b)r(olic)f (Logic)h(69)g(\(2004\),)g(862-878)h(\(preliminary)e(v)n(ersion)g(in)g Fe(Mathematic)l(al)j(F)-6 b(oundations)505 4925 y(of)26 b(Computer)h(Scienc)l(e)g(2002)34 b Fv(\(K.)24 b(Diks)g(and)f(W.)h (Rytter,)f(eds.\),)i(Lect.)f(Notes)g(in)g(Comput.)f(Sci.)505 5016 y(2420,)28 b(Springer-V)-6 b(erlag,)26 b(2002,)h(568{580\).)p eop %%Page: 83 83 83 82 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(83)605 541 y Fv([90])27 b(W.)19 b(Merkle,)h(N.)f(Mihailo)n(vi)n (\023)-36 b(c,)21 b(and)d(T.)i(Slaman,)e Fe(Some)k(r)l(esults)h(on)e (e\013e)l(ctive)i(r)l(andomness)p Fv(,)505 633 y(to)30 b(app)r(ear)g(in)g(Theory)g(Comput.)f(Syst.)g(\(preliminary)g(v)n (ersion)h(in)g Fe(International)i(Col)t(lo)l(quium)505 724 y(on)i(A)n(utomata,)g(L)l(anguages)h(and)f(Pr)l(o)l(gr)l(amming)g (2004)p Fv(,)f(Lect.)f(Notes)g(in)f(Comput.)h(Sci.)g(3142,)505 815 y(Springer-V)-6 b(erlag,)26 b(2004,)i(983-995\).)605 907 y([91])f(W.)35 b(Merkle,)g(J.)g(S.)f(Miller,)j(A.)d(Nies,)h(J.)g (Reimann,)e(and)h(F.)h(Stephan,)f Fe(Kolmo)l(gor)l(ov-)505 998 y(L)l(oveland)e(r)l(andomness)g(and)f(sto)l(chasticity)p Fv(,)h(Ann.)c(Pure)h(Appl.)f(Logic,)i(in)f(press)g(\(preliminary)505 1089 y(v)n(ersion)37 b(in)g(ST)-6 b(A)n(CS)35 b(2005,)k(Lecture)d (Notes)h(in)f(Computer)g(Science)h(3404,)h(Springer-V)-6 b(erlag,)505 1181 y(2005,)28 b(422{433\).)605 1272 y([92])f(J.)g(S.)g (Miller,)h Fe(Every)i(2-r)l(andom)f(r)l(e)l(al)g(is)f(Kolmo)l(gor)l(ov) i(r)l(andom)p Fv(,)d(J.)h(Sym)n(b)r(olic)d(Logic)j(69)505 1363 y(\(2004\),)g(907{913.)605 1455 y([93])f(J.)j(S.)f(Miller,)i Fe(The)g Fm(K)5 b Fe(-de)l(gr)l(e)l(es,)34 b(low)c(for)h Fm(K)36 b Fe(de)l(gr)l(e)l(es,)d(and)e(we)l(akly)g(low)g(for)g Fm(K)36 b Fe(or)l(acles)p Fv(,)505 1546 y(in)26 b(preparation.)605 1637 y([94])h(J.)37 b(S.)f(Miller)i(and)d(A.)h(Nies,)h Fe(R)l(andomness)i(and)f(c)l(omputability:)f(op)l(en)h(questions)p Fv(,)g(to)505 1729 y(app)r(ear.)605 1820 y([95])27 b(J.)20 b(S.)g(Miller)h(and)e(L.)g(Y)-6 b(u,)19 b Fe(On)j(initial)e(se)l(gment) k(c)l(omplexity)e(and)h(de)l(gr)l(e)l(es)h(of)d(r)l(andomness)p Fv(,)505 1911 y(to)26 b(app)r(ear)g(in)g(T)-6 b(rans.)26 b(Amer.)f(Math.)h(So)r(c.)605 2003 y([96])h(J.)g(S.)g(Miller)h(and)e (L.)h(Y)-6 b(u,)26 b Fe(Oscil)t(lation)i(in)g(the)h(initial)e(se)l (gment)j(c)l(omplexity)g(of)e(r)l(andom)505 2094 y(r)l(e)l(als)p Fv(,)f(to)f(app)r(ear.)605 2185 y([97])h(W.)e(Miller)i(and)e(D.)g(A.)g (Martin,)h Fe(The)i(de)l(gr)l(e)l(es)h(of)e(hyp)l(erimmune)h(sets)p Fv(,)f(Z.)e(Math.)h(Logik)505 2277 y(Grundlag.)h(Math.)f(14)g (\(1968\),)i(159{166.)605 2368 y([98])f(R.)c(v)n(on)g(Mises,)j Fe(Grund)t(lagen)h(der)f(Wahrscheinlichkeitsr)l(e)l(chnung)p Fv(,)h(Math.)d(Z.)g(5)g(\(1919\),)505 2459 y(52{99.)605 2551 y([99])j(A.)22 b(A.)g(Muc)n(hnik,)f(A.)h(L.)g(Semeno)n(v,)f(and)g (V.)h(A.)g(Usp)r(ensky)-6 b(,)21 b Fe(Mathematic)l(al)k(metaphysics)505 2642 y(of)j(r)l(andomness)p Fv(,)g(Theoret.)e(Comput.)f(Sci.)h(207)h (\(1998\),)g(263{317.)605 2733 y([100])g(A.)h(Nabuto)n(vsky)e(and)h(S.) g(W)-6 b(ein)n(b)r(erger,)28 b Fe(The)i(fr)l(actal)g(natur)l(e)h(of)e (R)n(iem/Di\013)f(I)p Fv(,)f(Geom.)505 2825 y(Dedicata)g(101)f (\(2003\),)i(1{54.)605 2916 y([101])f(A.)19 b(Nies,)g Fe(E\013e)l(ctively)j(dense)g(Bo)l(ole)l(an)g(algebr)l(as)g(and)g (their)f(applic)l(ations)p Fv(,)f(T)-6 b(rans.)19 b(Amer.)505 3007 y(Math.)27 b(So)r(c.)f(352)h(\(2000\),)g(4989{5012.)605 3099 y([102])g(A.)f(Nies,)g Fe(R)l(e)l(als)i(which)g(c)l(ompute)h (little)p Fv(,)d(to)g(app)r(ear.)605 3190 y([103])h(A.)f(Nies,)g Fe(L)l(owness)j(pr)l(op)l(erties)h(and)e(r)l(andomness)p Fv(,)g(Adv.)d(Math.,)h(in)g(press.)605 3281 y([104])h(A.)f(Nies,)g Fe(L)l(ow)i(for)g(r)l(andom)g(sets:)g(the)h(story)p Fv(,)e(preprin)n (t,)f(a)n(v)l(ailable)g(at)505 3372 y Fb(http://www.cs.auckland.ac.n)q (z/~ni)q(es)p Fv(.)605 3464 y([105])h(A.)f(Nies,)g Fe(Computability)i (and)g(R)l(andomness)p Fv(,)g(to)e(app)r(ear.)605 3555 y([106])h(A.)k(Nies,)h Fe(Eliminating)f(c)l(onc)l(epts:)j Fm(K)5 b Fe(-trivial)33 b(e)l(quals)h(low)e(for)g(r)l(andom)p Fv(,)g(presen)n(tation)505 3646 y(material)27 b(of)f(talks)g(at)g(IMS)f (Singap)r(ore,)i(2005,)h(a)n(v)l(ailable)e(at)505 3738 y Fb(http://www.cs.auckland.ac.n)q(z/\~ni)q(es)p Fv(.)605 3829 y([107])h(A.)19 b(Nies,)g(F.)g(Stephan,)f(and)g(S.)h(A.)f(T)-6 b(erwijn,)20 b Fe(R)l(andomness,)i(r)l(elativization,)g(and)f(T)-6 b(uring)505 3920 y(de)l(gr)l(e)l(es)p Fv(,)29 b(J.)d(Sym)n(b)r(olic)f (Logic)i(70)f(\(2005\),)h(515{535.)605 4012 y([108])g(P)-6 b(.)27 b(G.)h(Odifreddi,)f Fe(Classic)l(al)i(R)l(e)l(cursion)h(The)l (ory,)f(V)-6 b(ol.)28 b(1)p Fv(,)f(Studies)g(in)f(Logic)i(and)f(the)505 4103 y(F)-6 b(oundations)26 b(of)h(Mathematics)f(125,)h(North-Holland,) e(1989.)605 4194 y([109])i(P)-6 b(.)27 b(G.)h(Odifreddi,)f Fe(Classic)l(al)i(R)l(e)l(cursion)h(The)l(ory,)f(V)-6 b(ol.)28 b(2)p Fv(,)f(Studies)g(in)f(Logic)i(and)f(the)505 4286 y(F)-6 b(oundations)26 b(of)h(Mathematics)f(143,)h(North-Holland,) e(1999.)605 4377 y([110])i(J.)g(C.)f(Oxtob)n(y)-6 b(,)24 b Fe(Me)l(asur)l(e)29 b(and)f(Cate)l(gory)p Fv(,)g(2nd)e(edition,)g (Springer-V)-6 b(erlag,)26 b(1980.)605 4468 y([111])h(J.)36 b(Raisonnier,)f Fe(A)h(mathematic)l(al)g(pr)l(o)l(of)h(of)e(S.)g (Shelah's)h(the)l(or)l(em)i(on)d(the)i(me)l(asur)l(e)505 4560 y(pr)l(oblem)29 b(and)f(r)l(elate)l(d)h(r)l(esults)p Fv(,)e(Israel)g(J.)f(Math.)g(48)h(\(1984\))g(48{56.)605 4651 y([112])g(S.)18 b(Reid,)f Fe(The)j(Classes)g(of)g(A)n(lgorithmic)l (al)t(ly)f(R)l(andom)h(R)l(e)l(als)p Fv(,)e(Masters)h(Thesis,)g (Victoria)505 4742 y(Univ)n(ersit)n(y)25 b(of)i(W)-6 b(ellington,)26 b(2003.)605 4834 y([113])h(J.)e(Reimann,)d Fe(Computability)k(and)g(F)-6 b(r)l(actal)27 b(Dimension)p Fv(,)d(PhD)f(Dissertation,)i(Univ)n(er-)505 4925 y(sit)n(y)h(of)g (Heidelb)r(erg,)h(2004.)605 5016 y([114])g(J.)42 b(Reimann)d(and)h(F.)h (Stephan,)f Fe(On)h(Hier)l(ar)l(chies)i(of)e(R)l(andomness)h(T)-6 b(ests)p Fv(,)43 b(trans-)505 5108 y(parencies)h(for)h(a)e(talk)g(b)n (y)g(Stephan)f(at)h(the)g(9th)g(Asian)h(Logic)g(Conference,)h(a)n(v)l (ailable)f(at)p eop %%Page: 84 84 84 83 bop 505 363 a FD(84)277 b FC(R.)29 b(DO)n(WNEY,)g(D.)f(R.)i (HIRSCHFELDT,)f(A.)f(NIES,)h(AND)g(S.)f(A.)h(TER)-7 b(WIJN)505 541 y Fb(http://www.math.uni-heidelb)q(erg.d)q(e/lo)q(gic/)q(reima)q (nn/l)q(ectur)q(es.h)q(tml)p Fv(.)605 633 y([115])27 b(G.)42 b(E.)g(Sac)n(ks,)f Fe(De)l(gr)l(e)l(es)i(of)f(Unsolvability)p Fv(,)g(Annals)f(of)g(Mathematics)h(Studies)e(55,)505 724 y(Princeton)27 b(Univ)n(ersit)n(y)d(Press,)j(1963.)605 815 y([116])g(B.)20 b(Y.)e(Ry)n(abk)n(o,)g Fe(Co)l(ding)j(of)g(c)l (ombinatorial)h(sour)l(c)l(es)h(and)f(Hausdor\013)g(dimension)p Fv(,)d(Dokl.)505 907 y(Ak)l(ad.)25 b(Nauk)g(SSSR)f(277)j(\(1984\),)g (1066{1070.)605 998 y([117])g(B.)34 b(Y.)e(Ry)n(abk)n(o,)g Fe(Noise-fr)l(e)l(e)j(c)l(o)l(ding)g(of)f(c)l(ombinatorial)g(sour)l(c)l (es,)i(Hausdor\013)f(dimen-)505 1089 y(sion)k(and)h(Kolmo)l(gor)l(ov)f (c)l(omplexity)p Fv(,)h(Problems)d(of)i(Information)f(T)-6 b(ransmission)38 b(\(Problem)n(y)505 1181 y(P)n(eredac)n(hi)27 b(Informatsii\))f(22)g(\(1986\),)h(16{26.)605 1272 y([118])g(C.-P)-6 b(.)19 b(Sc)n(hnorr,)f Fe(A)i(uni\014e)l(d)h(appr)l(o)l(ach)h(to)f(the) g(de\014nition)g(of)f(a)h(r)l(andom)g(se)l(quenc)l(e)p Fv(,)g(Math-)505 1363 y(ematical)27 b(Systems)d(Theory)i(5)g(\(1971\),) h(246{258.)605 1455 y([119])g(C.-P)-6 b(.)28 b(Sc)n(hnorr,)f Fe(Zuf\177)-39 b(al)t(ligkeit)28 b(und)h(Wahrscheinlichkeit)p Fv(,)g(Lect.)e(Notes)g(in)g(Math.)h(218,)505 1546 y(Springer-V)-6 b(erlag,)26 b(1971.)605 1637 y([120])h(C.)c(E.)f(Shannon,)f Fe(The)k(mathematic)l(al)f(the)l(ory)i(of)e(c)l(ommunic)l(ation)p Fv(,)e(Bell)h(System)d(T)-6 b(ec)n(h.)505 1729 y(J.)27 b(27)f(\(1948\))h(379{423,)i(623{656.)605 1820 y([121])e(J.)21 b(H.)f(Silv)n(er,)g Fe(Counting)j(the)g(numb)l(er)g(of)f(e)l(quivalenc) l(e)i(classes)f(of)f(Bor)l(el)h(and)f(c)l(o)l(analytic)505 1911 y(e)l(quivalenc)l(e)29 b(r)l(elations)p Fv(,)f(Ann.)c(Math.)j (Logic)f(18)h(\(1980\),)g(1{28.)605 2003 y([122])g(R.)f(I.)g(Soare,)g Fe(R)l(e)l(cursively)j(Enumer)l(able)g(Sets)f(and)g(De)l(gr)l(e)l(es)p Fv(,)g(Springer-V)-6 b(erlag,)26 b(1987.)605 2094 y([123])h(R.)37 b(I.)g(Soare,)h Fe(Computability)g(the)l(ory)i(and)e(di\013er)l(ential) h(ge)l(ometry)p Fv(,)g(Bull.)f(Sym)n(b)r(olic)505 2185 y(Logic)27 b(10)g(\(2004\),)g(457{486.)605 2277 y([124])g(R.)f(J.)h (Solomono\013,)f Fe(A)i(pr)l(eliminary)g(r)l(ep)l(ort)i(on)e(a)g(gener) l(al)h(the)l(ory)g(of)f(inductive)g(infer-)505 2368 y(enc)l(e)p Fv(,)f(T)-6 b(ec)n(h.)26 b(Rep)r(ort)g(ZTB-138,)h(Zator)f(Compan)n(y)-6 b(,)25 b(Cam)n(bridge,)h(Mass.,)h(No)n(v)n(em)n(b)r(er)d(1960.)605 2459 y([125])j(R.)f(Solo)n(v)l(a)n(y)-6 b(,)26 b Fe(A)i(mo)l(del)g(of)f (set)i(the)l(ory)h(in)d(which)h(every)h(set)g(of)e(r)l(e)l(als)i(is)f (L)l(eb)l(esgue)i(me)l(a-)505 2551 y(sur)l(able)p Fv(,)e(Ann.)d(of)h (Math.)g(92)h(\(1970\),)g(1{56.)605 2642 y([126])g(R.)18 b(Solo)n(v)l(a)n(y)-6 b(,)17 b Fe(Dr)l(aft)j(of)f(a)h(p)l(ap)l(er)i (\(or)e(series)h(of)f(p)l(ap)l(ers\))i(on)e(Chaitin)-8 b('s)19 b(work)p Fv(,)g(unpublished)505 2733 y(man)n(uscript,)25 b(IBM)h(Thomas)g(J.)h(W)-6 b(atson)25 b(Researc)n(h)h(Cen)n(ter,)g(New) g(Y)-6 b(ork,)25 b(Ma)n(y)h(1975,)h(215)g(pp.)605 2825 y([127])g(L.)19 b(Staiger,)g Fe(Kolmo)l(gor)l(ov)j(c)l(omplexity)g(and) f(Hausdor\013)h(dimension)p Fv(,)d(Inform.)e(and)h(Com-)505 2916 y(put.)26 b(103)g(\(1993\),)i(159{194.)605 3007 y([128])f(L.)i(Staiger,)g Fe(A)h(tight)g(upp)l(er)h(b)l(ound)g(on)f (Kolmo)l(gor)l(ov)h(c)l(omplexity)f(and)h(uniformly)e(op-)505 3099 y(timal)e(pr)l(e)l(diction)p Fv(,)h(Theory)e(of)g(Computing)f (Systems)f(31)j(\(1998\),)g(215{229.)605 3190 y([129])g(L.)20 b(Staiger,)g Fe(Constructive)k(dimension)d(e)l(quals)i(Kolmo)l(gor)l (ov)f(c)l(omplexity)p Fv(,)f(Inform.)d(Pro-)505 3281 y(cess.)34 b(Lett.)e(93)h(\(2005\),)h(149{153)h(\(preliminary)d(v)n (ersion:)h(Researc)n(h)f(Rep)r(ort)g(CDMTCS-210,)505 3372 y(Univ)n(ersit)n(y)25 b(of)i(Auc)n(kland,)d(Jan)n(uary)i(2003\).) 605 3464 y([130])h(F.)48 b(Stephan,)f Fe(Martin-L\177)-39 b(of)48 b(r)l(andom)g(and)g(P)-6 b(A-c)l(omplete)49 b(sets)p Fv(,)g(F)-6 b(orsc)n(h)n(ungsb)r(eric)n(h)n(t)505 3555 y(Mathematisc)n(he)26 b(Logik)g(58)h(/)f(2002,)h(Univ)n(ersit\177)-38 b(at)26 b(Heidelb)r(erg,)g(2002.)605 3646 y([131])h(K.)e(T)-6 b(adaki,)26 b Fe(A)g(gener)l(alization)i(of)f(Chaitin)-8 b('s)26 b(halting)h(pr)l(ob)l(ability)h Fv(\012)f Fe(and)g(halting)g (self-)505 3738 y(similar)g(sets)p Fv(,)h(Hokk)l(aido)d(Math.)h(J.)g (31)h(\(2002\),)g(219{253.)605 3829 y([132])g(S.)35 b(A.)g(T)-6 b(erwijn,)36 b Fe(Computability)g(and)h(Me)l(asur)l(e)p Fv(,)f(PhD)f(Dissertation,)h(Univ)n(ersit)n(y)e(of)505 3920 y(Amsterdam/ILLC,)25 b(1998.)605 4012 y([133])i(S.)22 b(A.)g(T)-6 b(erwijn,)23 b Fe(Complexity)h(and)g(R)l(andomness)p Fv(,)g(Rendicon)n(ti)d(del)h(Seminario)f(Matem-)505 4103 y(atico)31 b(di)e(T)-6 b(orino)31 b(62)f(\(2004\))h(1{38)g (\(preliminary)d(v)n(ersion:)j(Researc)n(h)e(Rep)r(ort)g(CDMTCS-212,) 505 4194 y(Univ)n(ersit)n(y)c(of)i(Auc)n(kland,)d(Marc)n(h)i(2003\).) 605 4286 y([134])h(S.)34 b(A.)f(T)-6 b(erwijn)34 b(and)f(D.)g(Zam)n(b)r (ella,)h Fe(Computational)h(r)l(andomness)h(and)f(lowness)p Fv(,)g(J.)505 4377 y(Sym)n(b)r(olic)25 b(Logic)i(66)f(\(2001\))h (1199{1205.)605 4475 y([135])g(J.)g(Ville,)1089 4456 y Fe(\023)1078 4475 y(Etude)h(Critique)g(de)g(la)f(Notion)h(de)g(Col)t (le)l(ctif)p Fv(,)e(Gauthier-Villars,)h(1939.)605 4566 y([136])g(Y.)i(W)-6 b(ang,)28 b Fe(R)l(andomness)k(and)e(Complexity)p Fv(,)f(PhD)f(Dissertation,)i(Univ)n(ersit)n(y)d(of)i(Hei-)505 4657 y(delb)r(erg,)e(1996.)605 4749 y([137])g(Y.)33 b(W)-6 b(ang,)32 b Fe(A)i(sep)l(ar)l(ation)h(of)f(two)g(r)l(andomness)h(c)l (onc)l(epts)p Fv(,)g(Information)d(Pro)r(cessing)505 4840 y(Letters)26 b(69)h(\(1999\),)g(115{118.)605 4931 y([138])g(L.)f(Y)-6 b(u,)25 b Fe(When)j(van)g(L)l(amb)l(algen)-8 b('s)28 b(The)l(or)l(em)h(fails)p Fv(,)c(to)h(app)r(ear)g(in)f(Pro)r (c.)i(Amer.)e(Math.)505 5023 y(So)r(c.)p eop %%Page: 85 85 85 84 bop 1448 363 a FC(CALIBRA)-5 b(TING)31 b(RANDOMNESS)888 b FD(85)605 541 y Fv([139])27 b(L.)32 b(Y)-6 b(u)31 b(and)g(D.)g(Ding,) h Fe(Ther)l(e)i(is)f(no)g(sw-c)l(omplete)i(c.e.)e(r)l(e)l(al)p Fv(,)f(J.)g(Sym)n(b)r(olic)f(Logic)i(69)505 633 y(\(2004\),)28 b(1163{1170.)605 726 y([140])f(L.)f(Y)-6 b(u)24 b(and)h(D.)g(Ding,)h Fe(Ther)l(e)i(ar)l(e)g Fv(2)1852 694 y Fj(@)1890 704 y Fa(0)1954 726 y Fe(many)g Fm(H)6 b Fe(-de)l(gr)l(e)l(es)29 b(in)e(the)h(r)l(andom)g(r)l(e)l(als)p Fv(,)e(Pro)r(c.)505 817 y(Amer.)f(Math.)h(So)r(c.)h(132)f(\(2004\),)i(2461{2464.)605 909 y([141])f(L.)g(Y)-6 b(u,)24 b(D.)i(Ding,)g(and)g(R.)f(Do)n(wney)-6 b(,)25 b Fe(The)j(Kolmo)l(gor)l(ov)h(c)l(omplexity)g(of)e(r)l(andom)h (r)l(e)l(als)p Fv(,)505 1000 y(Ann.)d(Pure)h(Appl.)f(Logic)i(129)g (\(2004\),)g(163{180.)605 1091 y([142])g(D.)22 b(Zam)n(b)r(ella,)h Fe(On)h(se)l(quenc)l(es)j(with)d(simple)g(initial)f(se)l(gments)p Fv(,)i(ILLC)d(tec)n(hnical)g(rep)r(ort)505 1183 y(ML-1990-05,)28 b(Univ)n(ersit)n(y)d(of)h(Amsterdam,)e(1990.)605 1274 y([143])j(A.)k(K.)g(Zv)n(onkin)e(and)i(L.)f(A.)h(Levin,)g 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Fy(INSTITUTE)f(OF)e (DISCRETE)h(MA)-6 b(THEMA)g(TICS)23 b(AND)g(GEOMETR)-6 b(Y)683 3282 y(TECHNICAL)23 b(UNIVERSITY)h(OF)f(VIENNA)753 3374 y(WIEDNER)h(HA)n(UPTSTRASSE)g(8-10)f(/)h(E104)824 3465 y(A)f(-)g(1040)i(VIENNA)894 3556 y(A)n(USTRIA)588 3648 y Fe(E-mail)8 b Fv(:)26 b(terwijn@logic.at)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF </y)r>