(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: INT.dvi %%CreationDate: Sat Sep 11 11:21:26 1999 %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: C:\EMTEX\BIN\dvips32.exe -pj=tmp.mfj -K INT %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.09.11:1121 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 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newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (/WORK/CACHE/INT.dvi) @start /Fa 29 123 df<1378EA01FE1203EA07FFA25AA4EA07FEEA01EEEA000E131E13 1CA2133C13381378137013F0EA01E0EA03C0A2EA0780EA0F00121E5A5A5A1260101E798A 1F>44 D46 D65 D<94B912E0A31DC00500D90001EBC0004E5D6099C7FC17016062631703601A070507 5D60A2050F140F4E5CA2171F4E131F63173F95C7FC1A3F4D5D177EA205FE147F4D5D1601 5F040315FF4D5DA216074D5B040F93C8FC5F161F4D5B043F5D94C7FC5E04FE14074C5D15 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B89138F803FE14F0ED00F816005CA45CB3AA497E4813F8B612F8A428347DB32F>114 DI<14E0A61301A5 1303A21307A3130F131FA2133F137F13FF1203000F90B512E0B7FCA326003FE0C7FCB3A7 1638AC011F14786E1370A2010F14F06E13E0903807FC010103EB03C0903901FF07806DEB FF00EC3FFEEC07F8254B7EC92E>IIIIII<00 1FB712F0A301FCC7EA7FE001E014FF01804913C090C714804B1300001E5C4B5A485D4B5A 153F5E00384A5A15FF4A5B5E4A90C7FC5CC7485A5D4A5A143F5D4A5A4A5A5B5D4990C712 705B495A5C495A013F15E0495A5C495A5A4A13014890C7FC5A484814034914074848140F 003FED1FC04848147F49EB03FFB8FCA32C337DB235>II E /FI 10 118 df78 D80 D97 D102 D108 D111 D<023FECFF800003B5010713F0B6011F7F4C13FE9338FF 83FF923801FC07922603F00F1380DB07E014C0C64B5A011FEB0F806DEB1F00151E6D133E 153C037C6D138003786D130003F86D5A4B6D5A94C8FCA25DA35DA65DB3B3A2497F81017F 13FCB87EA63A4C7CCB43>114 D<91260FFF8013E049B5EAF8030107ECFE07013FECFF8F 90267FF80013DF2601FF80EB1FFF4848C71207D807F8140148488049157F4848153F4848 151FA2007F160F90C9FCA2481607A317037FA27F7FA213F86C6C92C7FC13FF14E06C13FE ECFFF06CECFF806C15F86C15FF6C16C06C16F06C826D81011F816D8101031680D9007F15 C0020715E0DA003F14F01501DB001F13F81603040013FC0078167F00F8163FEF1FFE170F 6C1607A21703A26C1601A37EA26D16FCA26D150318F87F17076D16F06DED0FE07F6DED1F C06DED7F80D99FC0903801FF00D90FF0EB07FE26FE07FEEB7FF8486CB65A48C615C04801 3F49C7FC48010313E0374F7ACC44>III E end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 1353 1034 a FI(Natural)59 b(Pro)5 b(ofs)372 1329 y FH(Alexander)40 b(A.)f(Razb)s(oro)m(v)1603 1286 y FG(\003)393 1479 y FH(Sc)m(ho)s(ol)h(of)f(Mathematics)222 1628 y(Institute)j(for)d (Adv)-7 b(anced)41 b(Study)453 1778 y(Princeton,)g(NJ)f(08540)891 1927 y(and)172 2077 y(Steklo)m(v)g(Mathematical)e(Institute)256 2226 y(V)-10 b(a)m(vilo)m(v)j(a)37 b(42,)j(117966,)e(GSP{1)525 2375 y(Mosco)m(w,)h(R)m(USSIA)2470 1329 y(Stev)m(en)j(Rudic)m(h)3226 1286 y FG(y)2043 1479 y FH(Computer)f(Science)g(Departmen)m(t)2133 1628 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(in)f(our)h(sense.)p 0 4580 1530 4 v 111 4641 a FA(\003)149 4671 y Fz(Supp)r(orted)g(b)n(y)f(the)h(gran)n(t)d(#)j(93-6-6)d(of)i (the)h(Alfred)f(P)-7 b(.)29 b(Sloan)g(F)-7 b(oundation,)29 b(b)n(y)g(the)h(gran)n(t)e(#)h(93-011-16015)24 b(of)0 4771 y(the)k(Russian)f(F)-7 b(oundation)27 b(for)g(F)-7 b(undamen)n(tal)28 b(Researc)n(h,)e(and)i(b)n(y)f(an)g(AMS-FSU)i(gran)n (t)115 4841 y FA(y)149 4871 y Fz(P)n(artially)d(supp)r(orted)i(b)n(y)f (NSF)h(gran)n(t)e(CCR-9119319)p eop %%Page: 2 2 2 1 bop 0 631 a Fy(1.)165 b(In)-5 b(tro)5 b(duction)0 850 y Fx(It)33 b(is)g(natural)g(to)g(ask)h(what)g(mak)m(es)g(lo)m(w)m (er)f(b)s(ound)h(questions)g(suc)m(h)h(as)f Fw(P)2831 797 y FB(?)2810 850 y Fx(=)29 b Fw(P)14 b(S)6 b(P)14 b(AC)7 b(E)f Fx(,)32 b Fw(P)3549 797 y FB(?)3528 850 y Fx(=)d Fw(N)10 b(P)k Fx(,)0 984 y(and)38 b Fw(P)330 932 y FB(?)309 984 y Fx(=)f Fw(N)10 b(C)46 b Fx(so)38 b(di\016cult)g(to)f(solv)m(e.)61 b(A)39 b(non-tec)m(hnical)e(reason)i (for)e(thinking)g(they)j(are)e(di\016cult)0 1105 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b(empt)m(y)f(in)m(tersection)g(with) 1864 5214 y(15)p eop %%Page: 16 16 16 15 bop 0 535 a Fj(n)64 631 y Fw(f)11 b Fx(\()p Fw(x)p Fx(\))271 531 y Fj(\014)271 581 y(\014)271 631 y(\014)21 b Fw(x)28 b Fu(2)h(f)p Fx(0)p Fw(;)17 b Fx(1)p Fu(g)740 595 y Fs(k)791 535 y Fj(o)846 631 y Fx(,)45 b(and)d(this)g(disjoin)m (tness)g(implies)e(that)i Fw(C)2480 646 y Fs(n)2568 631 y Fx(pro)m(vides)h(a)f(statistical)e(test)i(for)0 764 y Fw(f)11 b Fx(\()p Fr(x)p Fx(\),)32 b(with)966 884 y Fu(j)p Fp(P)p Fx([)p Fw(C)1168 899 y Fs(n)1215 884 y Fx(\()p Fr(f)1310 899 y Fs(n)1356 884 y Fx(\))c(=)f(1])22 b Fu(\000)h Fp(P)p Fx([)p Fw(C)1897 899 y Fs(n)1943 884 y Fx(\()p Fw(f)11 b Fx(\()p Fr(x)p Fx(\)\))27 b(=)h(1])o Fu(j)g(\025)g Fx(2)2636 843 y Fq(\000)p Fs(O)r FB(\()p Fs(n)p FB(\))2848 884 y Fw(:)825 b Fx(\(3\))0 1056 y(Note)33 b(that)f(this)g(test)i(is)e(computable)f(b)m(y)j(circuits)e(of)g(size)h (2)2274 1020 y Fs(O)r FB(\()p Fs(n)p FB(\))2431 1056 y Fx(.)146 1176 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Fw(C)1183 4746 y Fs(n)1230 4731 y Fx(\()p Fr(f)1325 4746 y Fs(n)1371 4731 y Fx(\))28 b(=)f(1])22 b Fu(\000)h Fp(P)p Fx([)p Fw(C)1912 4746 y Fs(n)1958 4731 y Fx(\()p Fw(G)2073 4746 y Fs(n;)p Fd(s)2177 4731 y Fx(\))28 b(=)f(1])p Fu(j)h Fw(<)f Fx(2)2630 4690 y Fq(\000)p Fs(!)r FB(\()p Fs(n)p FB(\))2833 4731 y Fw(:)840 b Fx(\(4\))0 4945 y Fv(Her)-5 b(e)35 b Fr(s)g Fv(is)f(a)h(r)-5 b(andom)34 b(se)-5 b(e)g(d)34 b(of)h(the)g(appr)-5 b(opriate)34 b(size.)1864 5214 y Fx(16)p eop %%Page: 17 17 17 16 bop 0 631 a Fp(Theorem)37 b(4.3.)49 b Fv(Ther)-5 b(e)34 b(is)h(no)f(lower)h(b)-5 b(ound)34 b(pr)-5 b(o)g(of)35 b(which)f(is)g Fw(AC)2538 595 y FB(0)2578 631 y Fv(-natur)-5 b(al)34 b(against)h Fw(AC)3434 595 y FB(0)3473 631 y Fx([2])p Fv(.)0 909 y Fp(Pro)s(of.)118 b Fx(Assume,)45 b(on)d(the)g(con)m(trary)-8 b(,)45 b(that)c(suc)m(h)i(a)f(pro)s(of)e (exists,)45 b(and)d(that)g Fw(C)3165 924 y Fs(n)3253 909 y Fx(has)g(the)g(same)0 1029 y(meaning)g(as)i(in)f(the)h(pro)s(of)e (of)h(Theorem)h(4.1.)76 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b(=q)t(pol)r(y)49 b Fx(as)d(the)h(class)f(of)g(non-uniform,)i(quasi-p)s (olynomial)42 b(size)0 2656 y(circuits,)32 b(i.e.,)g(size)h Fw(n)796 2620 y FB(log)14 b Fs(n)942 2594 y Fn(O)r Fm(\(1\))1077 2656 y Fx(.)0 2859 y Fp(Theorem)37 b(4.4.)49 b Fv(Ther)-5 b(e)43 b(is)f(no)h(lower)g(b)-5 b(ound)42 b(pr)-5 b(o)g(of)43 b(which)f(is)2476 2834 y Fx(~)2454 2859 y Fw(P)14 b(=q)t(pol)r(y)t Fv(-natur)-5 b(al)41 b(against)i Fw(P)8 b(=pol)r(y)0 2989 y Fv(unless)46 b Fw(H)8 b Fx(\()p Fw(G)507 3004 y Fs(k)549 2989 y Fx(\))50 b Fu(\024)g Fx(2)813 2953 y Fs(k)852 2930 y Fn(o)p Fm(\(1\))1015 2989 y Fv(for)d(every)f (pseudo-r)-5 b(andom)45 b(gener)-5 b(ator)47 b Fw(G)2663 3004 y Fs(k)2755 2989 y Fx(:)j Fu(f)p Fx(0)p Fw(;)17 b Fx(1)p Fu(g)3074 2953 y Fs(k)3165 2989 y Fu(\000)-16 b(!)49 b(f)p Fx(0)p Fw(;)17 b Fx(1)p Fu(g)3617 2953 y FB(2)p Fs(k)3740 2989 y Fv(in)0 3110 y Fw(P)8 b(=pol)r(y)t Fx(.)0 3399 y Ft(4.1.)137 b(Natural)36 b(pro)t(ofs)g(are)f(not)i (applicable)g(to)f(the)h(discrete)g(logarithm)351 3548 y(problem)0 3733 y Fx(It)c(is)f(p)s(ossible)g(\(though)h(w)m(e)g(are)g (una)m(w)m(are)h(of)e(an)m(y)h(suc)m(h)i(examples\))d(that)h(a)f(lo)m (w)m(er)h(b)s(ound)g(pro)s(of)e(for)0 3853 y(restricted)25 b(mo)s(dels)f(migh)m(t)f(b)s(e)i(natural,)g(but)h(cannot)e(b)s(e)h (applied)f(to)g(an)m(y)i(explicit)d(function.)41 b(In)25 b(other)0 3974 y(w)m(ords,)j(the)e(pro)s(of)e(migh)m(t)g(simply)g (argue)h(that)g(man)m(y)h(functions)f(are)g(complex)g(without)g(pro)m (viding)f(us)0 4094 y(with)34 b(an)m(y)g(explicit)f(examples)g(of)h (suc)m(h)h(functions.)48 b(Giv)m(en)33 b(our)h(hardness)i(assumption,)d (no)h(natural)0 4214 y(pro)s(of)27 b(can)g(pro)m(v)m(e)i(lo)m(w)m(er)f (b)s(ounds)g(against)f Fw(P)8 b(=pol)r(y)30 b Fx(whether)f(or)e(not)h (the)g(pro)s(of)e(mak)m(es)j(explicit)d(what)0 4335 y(the)j(hard)f (function)f(is.)42 b(Avi)28 b(Wigderson)g(has)g(p)s(oin)m(ted)g(out)g (that)g(if)f(w)m(e)i(restrict)f(ourselv)m(es)i(to)d(certain)0 4455 y(explicit)k(functions,)h(w)m(e)h(can)f(pro)m(v)m(e)h Fv(unc)-5 b(onditional)31 b Fx(results)i(in)e(the)i(st)m(yle)f(of)g (Theorem)g(4.1.)43 b(A)32 b(go)s(o)s(d)0 4575 y(example)38 b(of)h(suc)m(h)h(a)f(function)f(is)h(the)g(discrete)g(logarithm.)60 b(The)39 b(k)m(ey)i(p)s(oin)m(t)d(is)g(that)h(the)g(discrete)0 4696 y(logarithm)c(is)j(kno)m(wn)i(to)e(b)s(e)h(hard)f(on)h(a)m(v)m (erage)g(if)f(and)g(only)g(if)g(it)f(is)h(hard)h(in)e(the)i(w)m(orst)g (case.)63 b(In)0 4816 y(this)39 b(section,)h(w)m(e)g(sho)m(w)g(that)f (there)h(is)e(no)h(natural)f(pro)s(of)g(that)g(the)i(discrete)f (logarithm)d(requires)0 4937 y(exp)s(onen)m(tial-sized)c(circuits.)1864 5214 y(17)p eop %%Page: 18 18 18 17 bop 146 631 a Fx(Recall)29 b(from)f([9])i(that)g(for)f(a)h(prime) e Fw(p)i Fx(and)g(a)g(generator)f Fw(g)34 b Fx(for)29 b Fi(Z)2576 595 y Fq(\003)2576 656 y Fs(p)2613 631 y Fx(,)h(the)h(predicate)f Fw(B)3330 646 y Fs(p;g)3425 631 y Fx(\()p Fw(x)p Fx(\))g(on)g Fi(Z)3788 595 y Fq(\003)3788 656 y Fs(p)0 751 y Fx(is)24 b(de\014ned)j(to)d(b)s(e)h(1)g(if)e(log)936 775 y Fs(g)993 751 y Fw(x)28 b Fu(\024)g Fx(\()p Fw(p)6 b Fu(\000)g Fx(1\))p Fw(=)p Fx(2)26 b(and)f(0)f(otherwise.)42 b Fw(B)2367 766 y Fs(p;g)2462 751 y Fx(\()p Fw(x)p Fx(\))25 b(w)m(as)h(sho)m(wn)g(in)e([9])h(to)f(b)s(e)h(a)g(hard)0 872 y(bit)i(of)h(the)h(discrete)f(logarithm)d(problem.)41 b(W)-8 b(e)29 b(consider)f Fw(B)2240 887 y Fs(p;g)2335 872 y Fx(\()p Fw(x)p Fx(\))h(as)f(a)g(Bo)s(olean)f(function)h(in)f Fu(d)p Fx(log)17 b Fw(p)p Fu(e)0 992 y Fx(v)-5 b(ariables)35 b(\(extended)j(b)m(y)-8 b(,)38 b(sa)m(y)-8 b(,)38 b(zeros)f(on)f(those) h(inputs)g Fw(x)f Fx(whic)m(h)h(do)f(not)g(represen)m(t)i(an)f(in)m (teger)f(in)0 1112 y(the)f(range)f([1)p Fw(;)17 b(p)23 b Fu(\000)h Fx(1]\).)49 b(Our)34 b(principal)e(goal)h(in)h(this)g (section)g(is)g(to)g(sho)m(w)i(that)e Fv(no)i Fw(P)8 b(=pol)r(y)t Fv(-natur)-5 b(al)0 1233 y(pr)g(o)g(of)34 b(against)h(\\su\016ciently)f(lar)-5 b(ge")34 b(Bo)-5 b(ole)g(an)34 b(cir)-5 b(cuits)35 b(c)-5 b(an)34 b(b)-5 b(e)35 b(applie)-5 b(d)34 b(to)h Fw(B)2940 1248 y Fs(p;g)3035 1233 y Fx(\()p Fw(x)p Fx(\).)146 1375 y(T)-8 b(o)28 b(explain)e(the)i (meaning)e(of)h(\\su\016cien)m(tly)h(large",)f(w)m(e)h(need)g(a)f (couple)h(of)e(tec)m(hnical)h(de\014nitions.)0 1496 y(F)-8 b(or)34 b(an)i(in)m(teger-v)-5 b(alued)34 b(function)h Fw(t)p Fx(\()p Fw(n)p Fx(\),)h(let)f Fw(S)6 b(I)i(Z)f(E)f Fx(\()p Fw(t)p Fx(\()p Fw(n)p Fx(\)\))35 b(b)s(e)g(the)h(complexit)m(y) e(class)i(consisting)e(of)0 1616 y(all)c(functions)j Fu(f)p Fw(f)654 1631 y Fs(n)701 1616 y Fu(g)f Fx(whic)m(h)h(ha)m(v)m(e) h(circuit)e(size)h Fw(O)s Fx(\()p Fw(t)p Fx(\()p Fw(n)p Fx(\)\).)42 b(Let)1283 1815 y Fw(t)1318 1774 y Fq(\000)p FB(1)1413 1815 y Fx(\()p Fw(n)p Fx(\))28 b Fe(\012)f Fx(max)16 b Fu(f)9 b Fw(x)17 b Fu(j)k Fw(t)p Fx(\()p Fw(x)p Fx(\))29 b Fu(\024)f Fw(n)10 b Fu(g)16 b Fw(:)0 2015 y Fx(W)-8 b(e)33 b(sa)m(y)h(that)e Fw(t)p Fx(\()p Fw(n)p Fx(\))h(is)f Fv(half-exp)-5 b(onential)30 b Fx(if)i(it)f(is)h (non-decreasing)h(and)1467 2214 y Fw(t)1502 2173 y Fq(\000)p FB(1)1597 2214 y Fx(\()p Fw(n)1693 2173 y Fs(C)1752 2214 y Fx(\))28 b Fu(\024)g Fw(o)p Fx(\(log)16 b Fw(t)p Fx(\()p Fw(n)p Fx(\)\))1343 b(\(5\))0 2413 y(for)42 b(ev)m(ery)j Fw(C)52 b(>)45 b Fx(0.)75 b(The)43 b(meaning)f(of)g(this)h (de\014nition)f(is)g(that,)j(roughly)e(sp)s(eaking,)i(the)f(second)0 2534 y(iteration)37 b(of)h Fw(t)p Fx(\()p Fw(n)p Fx(\))h(should)g(gro)m (w)g(faster)g(than)f(the)i(exp)s(onen)m(t.)63 b(F)-8 b(or)38 b(example,)i Fw(t)p Fx(\()p Fw(n)p Fx(\))f(=)f(2)3413 2498 y Fs(n)3456 2474 y Fn(\017)3528 2534 y Fx(is)h(half-)0 2654 y(exp)s(onen)m(tial,)32 b(whereas)i Fw(t)p Fx(\()p Fw(n)p Fx(\))28 b(=)g(2)1269 2618 y FB(\(log)14 b Fs(n)p FB(\))1469 2595 y Fn(C)1557 2654 y Fx(is)32 b(not.)0 2859 y Fp(Theorem)37 b(4.5.)49 b Fv(L)-5 b(et)37 b Fw(t)p Fx(\()p Fw(n)p Fx(\))g Fv(b)-5 b(e)36 b(an)g(arbitr)-5 b(ary)36 b(half-exp)-5 b(onential)35 b(function.)49 b(Then)35 b(ther)-5 b(e)37 b(is)f(no)g(c)-5 b(om-)0 2979 y(binatorial)31 b(pr)-5 b(op)g(erty)31 b Fw(C)886 2994 y Fs(n)965 2979 y Fv(useful)g(against)g Fw(S)6 b(I)i(Z)f(E)f Fx(\()p Fw(t)p Fx(\()p Fw(n)p Fx(\)\))32 b Fv(and)f(satisfying)g Fw(P)8 b(=pol)r(y)t Fv(-c)-5 b(onstructivity)31 b(and)0 3100 y(lar)-5 b(geness)50 b(c)-5 b(onditions)51 b(such)g(that)1368 3033 y Fj(S)1438 3120 y Fs(n)p Fq(2)p Fs(!)1595 3100 y Fw(C)1665 3115 y Fs(n)1763 3100 y Fv(c)-5 b(ontains)51 b(in\014nitely)g(many)g(functions)g(of)h(the)g(form)0 3220 y Fw(B)74 3235 y Fs(p;g)169 3220 y Fx(\()p Fw(x)p Fx(\))p Fv(.)0 3469 y Fp(Pro)s(of.)69 b Fx(Assume)25 b(the)g(con)m(trary)-8 b(,)27 b(and)e(let)f Fu(f)p Fw(B)1727 3484 y Fs(p)1763 3492 y Fn(\027)1801 3484 y Fs(;g)1855 3492 y Fn(\027)1897 3469 y Fu(g)g Fx(b)s(e)h(an)g(in\014nite)e (sequence)28 b(con)m(tained)c(in)3482 3402 y Fj(S)3551 3489 y Fs(n)p Fq(2)p Fs(!)3708 3469 y Fw(C)3778 3484 y Fs(n)0 3589 y Fx(suc)m(h)38 b(that)f Fu(d)p Fx(log)17 b Fw(p)676 3604 y FB(1)715 3589 y Fu(e)36 b Fw(<)e Fu(d)p Fx(log)17 b Fw(p)1141 3604 y FB(2)1181 3589 y Fu(e)35 b Fw(<)g(:)17 b(:)g(:)36 b Fx(Let)h Fw(k)1752 3604 y Fs(\027)1830 3589 y Fe(\012)e Fu(d)p Fx(log)17 b Fw(p)2201 3604 y Fs(\027)2244 3589 y Fu(e)p Fx(.)57 b(Applying)36 b(the)h(usefulness)h(condition)0 3709 y(to)d(the)h(sequence)i Fw(f)748 3724 y Fs(n)831 3709 y Fx(obtained)d(from)f Fu(f)p Fw(B)1592 3724 y Fs(p)1628 3732 y Fn(\027)1666 3724 y Fs(;g)1720 3732 y Fn(\027)1762 3709 y Fu(g)h Fx(b)m(y)h(letting) e Fw(f)2345 3724 y Fs(n)2425 3709 y Fu(\021)f Fx(0)i(for)g(those)h Fw(n)g Fx(whic)m(h)g(are)f(not)g(of)0 3830 y(the)d(form)e Fu(d)p Fx(log)17 b Fw(p)632 3845 y Fs(\027)675 3830 y Fu(e)p Fx(,)32 b(w)m(e)g(will)d(\014nd)j(in)f Fu(f)p Fw(B)1534 3845 y Fs(p)1570 3853 y Fn(\027)1608 3845 y Fs(;g)1662 3853 y Fn(\027)1704 3830 y Fu(g)g Fx(an)g(in\014nite)f (subsequence)35 b(where)e(all)c(functions)j(ha)m(v)m(e)0 3950 y(the)f(circuit)d(size)j(at)f(least)f Fw(t)p Fx(\()p Fw(k)1115 3965 y Fs(\027)1158 3950 y Fx(\).)43 b(W.l.o.g.)f(w)m(e)31 b(ma)m(y)f(assume)g(that)g(this)g(is)f(the)i(case)g(for)e(our)h (original)0 4070 y(sequence.)146 4191 y(Let)45 b Fw(G)410 4206 y Fs(\027)502 4191 y Fx(:)k Fu(f)p Fx(0)p Fw(;)17 b Fx(1)p Fu(g)819 4142 y FB(2)p Fs(k)891 4150 y Fn(\027)983 4191 y Fu(\000)-16 b(!)48 b(f)p Fx(0)p Fw(;)17 b Fx(1)p Fu(g)1433 4142 y FB(4)p Fs(k)1505 4150 y Fn(\027)1593 4191 y Fx(b)s(e)45 b(the)g(standard)h(pseudo-random)e(generator)h(from) f([9])0 4311 y(based)33 b(up)s(on)g Fu(f)p Fw(B)642 4326 y Fs(p)678 4334 y Fn(\027)716 4326 y Fs(;g)770 4334 y Fn(\027)812 4311 y Fu(g)o Fx(.)44 b(It)32 b(is)g(easy)i(to)e(c)m(hec)m (k)i(that)e(the)h(pro)s(of)f(of)g([9,)g(Theorem)h(3])f(actually)f (extends)0 4432 y(to)h(sho)m(wing)h(that)f(the)h(circuit)f(size)h(of)f Fu(f)p Fw(B)1594 4447 y Fs(p)1630 4455 y Fn(\027)1668 4447 y Fs(;g)1722 4455 y Fn(\027)1764 4432 y Fu(g)g Fx(is)g(p)s (olynomial)d(in)j Fw(H)8 b Fx(\()p Fw(G)2766 4447 y Fs(\027)2808 4432 y Fx(\))22 b(+)g Fw(k)3017 4447 y Fs(\027)3060 4432 y Fx(.)44 b(Th)m(us,)34 b(w)m(e)g(ha)m(v)m(e)1360 4645 y Fw(t)p Fx(\()p Fw(k)1484 4660 y Fs(\027)1527 4645 y Fx(\))27 b Fu(\024)h Fx(\()p Fw(H)8 b Fx(\()p Fw(G)1939 4660 y Fs(\027)1982 4645 y Fx(\))22 b(+)g Fw(k)2191 4660 y Fs(\027)2234 4645 y Fx(\))2272 4597 y Fs(O)r FB(\(1\))2438 4645 y Fw(:)1235 b Fx(\(6\))146 4844 y(No)m(w)37 b(w)m(e)g(con)m(v)m (ert)h Fw(G)944 4859 y Fs(\027)1023 4844 y Fx(in)m(to)e(the)h (pseudo-random)f(function)f(generator)h Fw(f)2954 4859 y Fs(\027)3032 4844 y Fx(:)e Fu(f)o Fx(0)p Fw(;)17 b Fx(1)p Fu(g)3333 4796 y FB(2)p Fs(k)3405 4804 y Fn(\027)3482 4844 y Fu(\000)-16 b(!)33 b Fw(F)3739 4859 y Fs(n)3782 4867 y Fn(\027)0 4965 y Fx(as)g(in)f(the)h(pro)s(of)f(of)g(Theorem)h (4.1,)g(where)h Fw(n)1705 4980 y Fs(\027)1781 4965 y Fx(will)c(b)s(e)j(sp)s(eci\014ed)h(a)e(little)f(bit)h(later.)43 b(There)34 b(exists)f(a)1864 5214 y(18)p eop %%Page: 19 19 19 18 bop 0 631 a Fx(\014xed)35 b(constan)m(t)f Fw(C)i(>)29 b Fx(0)k(suc)m(h)i(that)e(for)g(almost)e(all)g Fw(\027)6 b Fx(,)35 b Fw(f)2122 646 y Fs(\027)2165 631 y Fx(\()p Fw(x)p Fx(\)\()p Fw(y)t Fx(\))d(is)h(computable)g(b)m(y)h(circuits)f (of)g(size)0 751 y(\()p Fw(k)89 766 y Fs(\027)154 751 y Fx(+)22 b Fw(n)310 766 y Fs(\027)353 751 y Fx(\))391 715 y Fs(C)450 751 y Fx(.)44 b(Let)33 b Fw(n)754 766 y Fs(\027)825 751 y Fe(\012)27 b Fw(t)987 715 y Fq(\000)p FB(1)1082 751 y Fx(\()p Fw(k)1174 715 y Fs(C)5 b FB(+1)1171 776 y Fs(\027)1323 751 y Fx(\))22 b(+)g(1.)146 872 y(\(5\))42 b(implies)d(that)i Fw(t)p Fx(\()p Fw(k)997 887 y Fs(\027)1041 872 y Fx(\))i Fw(>)g(k)1295 836 y Fs(C)5 b FB(+1)1292 896 y Fs(\027)1486 872 y Fx(for)41 b(almost)f(all)g Fw(\027)6 b Fx(,)44 b(since)f(otherwise)f(w)m(e)h(w)m(ould)e(ha)m(v)m(e)i Fw(k)3661 887 y Fs(\027)3748 872 y Fu(\024)0 1005 y Fw(t)35 968 y Fq(\000)p FB(1)146 908 y Fj(\020)196 1005 y Fw(k)250 968 y Fs(C)5 b FB(+1)247 1029 y Fs(\027)399 908 y Fj(\021)481 1005 y Fu(\024)32 b Fx(log)16 b Fw(t)p Fx(\()p Fw(k)856 1020 y Fs(\027)900 1005 y Fx(\))31 b Fu(\024)i Fx(\()p Fw(C)d 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Fw(n)125 2823 y Fs(\027)196 2808 y Fx(=)c Fw(t)335 2766 y Fq(\000)p FB(1)430 2808 y Fx(\()p Fw(k)522 2766 y Fs(C)5 b FB(+1)519 2832 y Fs(\027)671 2808 y Fx(\))22 b(+)g(1)27 b Fu(\024)i Fw(o)p Fx(\(log)16 b Fw(t)p Fx(\()p Fw(k)1362 2823 y Fs(\027)1405 2808 y Fx(\)\))28 b Fu(\024)g Fw(o)17 b Fx(\()o(log)g Fw(H)8 b Fx(\()p Fw(G)2062 2823 y Fs(\027)2104 2808 y Fx(\))23 b(+)f(log)16 b Fw(k)2456 2823 y Fs(\027)2499 2808 y Fx(\))28 b Fu(\024)g Fw(o)p Fx(\()p Fw(n)2813 2823 y Fs(\027)2856 2808 y Fx(\))22 b(+)g Fw(o)p Fx(\(log)17 b Fw(k)3293 2823 y Fs(\027)3336 2808 y Fx(\))27 b Fu(\024)h Fw(o)p Fx(\()p Fw(n)3649 2823 y Fs(\027)3693 2808 y Fx(\))p Fw(:)p 9 3020 43 43 v 0 3287 a Fp(Corollary)36 b(4.6.)49 b Fv(Ther)-5 b(e)47 b(is)g(no)g(c)-5 b(ombinatorial)46 b(pr)-5 b(op)g(erty)48 b Fw(C)2371 3302 y Fs(n)2465 3287 y Fv(useful)g(against)3106 3221 y Fj(T)3175 3308 y Fs(\017>)p FB(0)3315 3287 y Fw(S)6 b(I)i(Z)f(E)3600 3191 y Fj(\020)3649 3287 y Fx(2)3698 3251 y Fs(n)3741 3227 y Fn(\017)3775 3191 y Fj(\021)0 3420 y Fv(and)39 b(satisfying)g Fw(P)8 b(=pol)r(y)t Fv(-c)-5 b(onstructivity)40 b(and)f(lar)-5 b(geness)38 b(c)-5 b(onditions)39 b(such)g(that)3091 3353 y Fj(S)3160 3441 y Fs(n)p Fq(2)p Fs(!)3317 3420 y Fw(C)3387 3435 y Fs(n)3474 3420 y Fv(c)-5 b(ontains)0 3540 y(in\014nitely)34 b(many)h(functions)f (of)h(the)g(form)f Fw(B)1684 3555 y Fs(p;g)1780 3540 y Fx(\()p Fw(x)p Fx(\))p Fv(.)0 3848 y Fp(Pro)s(of.)384 3781 y Fj(T)453 3868 y Fs(\017>)p FB(0)592 3848 y Fw(S)6 b(I)i(Z)f(E)878 3751 y Fj(\020)927 3848 y Fx(2)976 3811 y Fs(n)1019 3788 y Fn(\017)1053 3751 y Fj(\021)1130 3848 y Fu(\023)28 b Fw(S)6 b(I)i(Z)f(E)1521 3726 y Fj(\022)1582 3848 y Fx(2)1631 3811 y FB(2)1666 3741 y Fq(p)p 1725 3741 129 3 v 46 x Fm(log)12 b Fn(n)1861 3726 y Fj(\023)1922 3848 y Fx(,)33 b(and)g Fw(t)p Fx(\()p Fw(n)p Fx(\))28 b(=)f(2)2521 3811 y FB(2)2556 3741 y Fq(p)p 2615 3741 V 46 x Fm(log)13 b Fn(n)2784 3848 y Fx(is)32 b(half-exp)s(onen)m(tial.) p 3603 3848 43 43 v 146 4053 a(It)27 b(is)e(easy)i(to)f(see)i(that)d (the)i(ab)s(o)m(v)m(e)g(pro)s(of)e(is)h(actually)f(v)-5 b(alid)24 b(for)i(an)g Fv(arbitr)-5 b(ary)26 b Fx(collection)e Fu(f)p Fw(f)3575 4068 y Fs(p;g)3671 4053 y Fu(g)h Fx(of)0 4173 y(functions)i(p)s(oly-time)d(non)m(uniformly)h(T)-8 b(uring)27 b(reducible)f(to)h(the)g(corresp)s(onding)g(discrete)g (logarithm)0 4294 y(problem)k(in)h(place)g(of)h Fu(f)p Fw(B)978 4309 y Fs(p;g)1073 4294 y Fu(g)p Fx(.)0 4625 y Fy(5.)165 b(One)54 b(prop)5 b(ert)-5 b(y)55 b(of)h(formal)g (complexit)-5 b(y)57 b(measures)0 4844 y Fx(A)33 b Fv(formal)h(c)-5 b(omplexity)34 b(me)-5 b(asur)g(e)32 b Fx(\(see)h(e.g.)44 b([38,)32 b(Section)h(8.8],)f([31]\))g(is)g(an)h(in)m(teger-v)-5 b(alued)31 b(function)0 4965 y Fw(\026)41 b Fx(on)h Fw(F)308 4980 y Fs(n)397 4965 y Fx(suc)m(h)h(that)f Fw(\026)p Fx(\()p Fw(f)11 b Fx(\))42 b Fu(\024)i Fx(1)e(for)f Fw(f)54 b Fu(2)44 b(f:)p Fw(x)1836 4980 y FB(1)1876 4965 y Fw(;)17 b(:)g(:)g(:)f(;)h Fu(:)p Fw(x)2216 4980 y Fs(n)2263 4965 y Fw(;)g(x)2362 4980 y FB(1)2402 4965 y Fw(;)g(:)g(:)g(:)f(;)h(x)2676 4980 y Fs(n)2723 4965 y Fu(g)41 b Fx(and)h Fw(\026)p Fx(\()p Fw(f)d Fu(\003)28 b Fw(g)t Fx(\))43 b Fu(\024)g Fw(\026)p Fx(\()p Fw(f)11 b Fx(\))28 b(+)1864 5214 y(19)p eop %%Page: 20 20 20 19 bop 0 631 a Fw(\026)p Fx(\()p Fw(g)t Fx(\))40 b(for)g(all)f Fw(f)5 b(;)17 b(g)46 b Fu(2)d Fw(F)889 646 y Fs(n)977 631 y Fx(and)e Fu(\003)h(2)h(f^)p Fw(;)17 b Fu(_g)p Fx(.)69 b(The)42 b(meaning)e(of)g(this)h(de\014nition)f(is)h(that)g(for)f(ev)m (ery)0 751 y(formal)33 b(complexit)m(y)i(measure)h Fw(\026)p Fx(,)f Fw(\026)p Fx(\()p Fw(f)11 b Fx(\))35 b(pro)m(vides)h(a)f(lo)m(w) m(er)h(b)s(ound)g(on)f(the)h(form)m(ula)d(size)j(of)f Fw(f)11 b Fx(,)36 b(and)0 872 y(actually)d(man)m(y)i(kno)m(wn)h(lo)m(w) m(er)f(b)s(ounds,)h(b)s(oth)f(for)f(monotone)g(and)h(non-monotone)f (form)m(ulae,)g(can)0 992 y(b)s(e)i(view)m(ed)h(from)d(this)i(p)s(ersp) s(ectiv)m(e.)54 b(See)37 b(the)f(ab)s(o)m(v)m(e-cited)g(sources)i(for)d (examples.)53 b(Also,)36 b(for)f(an)m(y)0 1112 y(appro)m(ximation)30 b(mo)s(del)h Fc(M)h Fx(\(see)i([39)o(,)f(32)o(])g(for)f(most)g(general) f(de\014nitions\),)h(w)m(e)i(ha)m(v)m(e)g Fw(\032)p Fx(\()p Fw(f)f Fu(\003)21 b Fw(g)t(;)c Fc(M)p Fx(\))27 b Fu(\024)0 1233 y Fw(\032)p Fx(\()p Fw(f)5 b(;)17 b Fc(M)p Fx(\))22 b(+)g Fw(\032)p Fx(\()p Fw(g)t(;)17 b Fc(M)p Fx(\))k(+)h(1,)32 b(hence)i Fw(\032)p Fx(\()p Fw(f)5 b(;)17 b Fc(M)p Fx(\))22 b(+)g(1)32 b(is)g(a)h(formal)d(complexit)m(y)i(measure.)146 1376 y(In)c(this)f(section)g(w)m(e)h(sho)m(w)g(that)f(an)m(y)h(formal)d (complexit)m(y)h(measure)i Fw(\026)f Fx(whic)m(h)g(tak)m(es)i(a)d (large)h(v)-5 b(alue)0 1497 y(at)29 b(a)g(single)f(function,)i(m)m(ust) f(tak)m(e)i(large)d(v)-5 b(alues)29 b(almost)f(ev)m(erywhere.)45 b(In)30 b(particular,)e(ev)m(ery)j(com)m(bi-)0 1617 y(natorial)h(prop)s (ert)m(y)k(based)g(on)e(suc)m(h)j(a)d(measure)h(automatically)c (satis\014es)36 b(the)f(largeness)g(condition)0 1737 y(in)d(the)h(de\014nition)e(of)h(natural)g(prop)s(ert)m(y)-8 b(.)146 1858 y(More)33 b(sp)s(eci\014cally)-8 b(,)32 b(w)m(e)i(ha)m(v)m(e)f(the)g(follo)m(wing:)0 2071 y Fp(Theorem)k(5.1.) 49 b Fv(L)-5 b(et)50 b Fw(\026)f Fv(b)-5 b(e)49 b(a)g(formal)f(c)-5 b(omplexity)49 b(me)-5 b(asur)g(e)49 b(on)g Fw(F)2648 2086 y Fs(n)2695 2071 y Fx(,)k Fv(and)48 b Fw(\026)p Fx(\()p Fw(f)11 b Fx(\))54 b(=)g Fw(t)c Fv(for)f(some)0 2192 y Fw(f)38 b Fu(2)28 b Fw(F)243 2207 y Fs(n)291 2192 y Fv(.)44 b(Then)p Fx(:)97 2384 y Fp(a\))49 b Fv(for)35 b(at)g(le)-5 b(ast)34 b Fx(1)p Fw(=)p Fx(4)g Fv(fr)-5 b(action)35 b(of)f(al)5 b(l)35 b(functions)f Fw(g)d Fu(2)d Fw(F)2194 2399 y Fs(n)2241 2384 y Fx(,)35 b Fw(\026)p Fx(\()p Fw(g)t Fx(\))27 b Fu(\025)h Fw(t=)p Fx(4;)89 2584 y Fp(b\))49 b Fv(for)35 b(any)f Fw(\017)28 b Fx(=)g Fw(\017)p Fx(\()p Fw(n)p Fx(\))35 b Fv(we)g(have)f(that)h(for)g(at)g (le)-5 b(ast)35 b Fx(\(1)21 b Fu(\000)i Fw(\017)p Fx(\))35 b Fv(fr)-5 b(action)35 b(of)f Fw(g)d Fu(2)d Fw(F)3060 2599 y Fs(n)3107 2584 y Fx(,)1389 2903 y Fw(\026)p Fx(\()p Fw(g)t Fx(\))e Fu(\025)i Fx(\012)1793 2707 y Fj(0)1793 2853 y(B)1793 2906 y(@)2116 2836 y Fw(t)p 1876 2880 515 4 v 1876 2910 a Fj(\020)1925 3006 y Fw(n)23 b Fx(+)f(log)2256 2967 y FB(1)p 2256 2983 36 4 v 2259 3040 a Fs(\017)2302 2910 y Fj(\021)2351 2933 y FB(2)2401 2707 y Fj(1)2401 2853 y(C)2401 2906 y(A)2495 2903 y Fu(\000)h Fw(n:)146 3259 y Fx(In)45 b(fact,)j(the)d(main)e(argumen)m(t)h(used)i(in)e(the)h (pro)s(of)f(of)g(this)g(theorem)h(is)f(v)-5 b(alid)43 b(for)h(arbitrary)0 3380 y(Bo)s(olean)21 b(algebras,)i(and)f(w)m(e)h (form)m(ulate)d(it)h(as)i(a)e(separate)i(result)f(since)g(this)g(migh)m (t)e(b)s(e)i(of)g(indep)s(enden)m(t)0 3500 y(in)m(terest.)0 3713 y Fp(Theorem)37 b(5.2.)49 b Fv(L)-5 b(et)35 b Fw(B)40 b Fv(b)-5 b(e)35 b(a)g(\014nite)f(Bo)-5 b(ole)g(an)34 b(algebr)-5 b(a)34 b(with)g Fw(N)46 b Fv(atoms)34 b(and)h Fw(S)e Fu(\022)28 b Fw(B)5 b Fv(.)97 3906 y Fp(a\))49 b Fv(if)35 b Fu(j)p Fw(S)6 b Fu(j)26 b Fw(>)601 3867 y FB(3)p 601 3883 V 601 3940 a(4)646 3906 y Fu(j)p Fw(B)5 b Fu(j)34 b Fv(then)h(every)g(element)f(of)h Fw(B)40 b Fv(c)-5 b(an)34 b(b)-5 b(e)35 b(r)-5 b(epr)g(esente)g(d)34 b(in)g(the)h(form)1184 4113 y Fx(\()p Fw(s)1268 4128 y FB(1)1330 4113 y Fu(^)22 b Fw(s)1464 4128 y FB(2)1504 4113 y Fx(\))g Fu(_)g Fx(\()p Fw(s)1736 4128 y FB(3)1798 4113 y Fu(^)h Fw(s)1933 4128 y FB(4)1972 4113 y Fx(\);)51 b Fw(s)2134 4128 y Fs(i)2190 4113 y Fu(2)28 b Fw(S)41 b Fx(\(1)27 b Fu(\024)i Fw(i)e Fu(\024)i Fx(4\);)815 b(\(9\))89 4360 y Fp(b\))49 b Fv(if)31 b Fw(S)38 b Fv(c)-5 b(ontains)31 b(al)5 b(l)31 b(atoms)g(and)g(c)-5 b(o)g(atoms)31 b(of)g Fw(B)37 b Fv(then)32 b(every)f(element)g(of)h Fw(B)k Fv(c)-5 b(an)31 b(b)-5 b(e)32 b(r)-5 b(epr)g(esente)g(d)244 4480 y(in)34 b(the)h(form)1884 4562 y Fs(`)1852 4587 y Fj(_)1841 4769 y Fs(i)p FB(=1)2019 4562 y Fs(`)1987 4587 y Fj(^)1972 4769 y Fs(j)t FB(=1)2111 4670 y Fw(s)2157 4685 y Fs(ij)2217 4670 y Fw(;)1408 b Fx(\(10\))244 4947 y Fv(wher)-5 b(e)34 b Fw(s)565 4962 y Fs(ij)653 4947 y Fu(2)28 b Fw(S)41 b Fv(and)34 b Fw(`)28 b Fu(\024)g Fw(O)1305 4851 y Fj(\020)1355 4947 y Fx(log)1507 4900 y Fs(N)7 b Fq(\001j)p Fs(B)s Fq(j)p 1507 4924 179 4 v 1553 4981 a(j)p Fs(S)t Fq(j)1695 4851 y Fj(\021)1745 4947 y Fv(.)1864 5214 y Fx(20)p eop %%Page: 21 21 21 20 bop 0 631 a Fp(Pro)s(of)34 b(of)g(Theorem)f(5.1)h(from)g(Theorem) f(5.2.)77 b Fx(Let)30 b Fw(S)j Fe(\012)27 b Fu(f)9 b Fw(g)20 b Fu(j)i Fw(\026)p Fx(\()p Fw(g)t Fx(\))k Fw(<)i(t=)p Fx(4)9 b Fu(g)30 b Fx(for)f(part)g(a\),)h(and)0 814 y Fw(S)36 b Fe(\012)226 667 y Fj(\()302 814 y Fw(g)369 664 y Fj(\014)369 714 y(\014)369 764 y(\014)369 814 y(\014)369 863 y(\014)418 814 y Fw(\026)p Fx(\()p Fw(g)t Fx(\))27 b Fu(\024)h Fw(\016)e Fu(\001)1033 774 y Fs(t)p 865 790 362 4 v 865 880 a Fx(\()903 872 y Fs(n)p FB(+log)1113 844 y Fm(1)p 1113 856 31 4 v 1115 898 a Fn(\017)1154 880 y Fx(\))1192 826 y Fm(2)1246 667 y Fj(\))1313 814 y Fx(,)35 b(where)g Fw(\016)j Fx(is)c(a)g(su\016cien)m(tly)h(small)c (constan)m(t,)36 b(for)e(part)f(b\).)49 b(Note)0 1002 y(that)28 b(in)f(part)h(b\))h(w)m(e)g(ma)m(y)f(assume)h(that)f Fw(\016)17 b Fu(\001)1808 963 y Fs(t)p 1640 979 362 4 v 1640 1068 a Fx(\()1678 1060 y Fs(n)p FB(+log)1889 1033 y Fm(1)p 1889 1045 31 4 v 1891 1086 a Fn(\017)1929 1068 y Fx(\))1967 1014 y Fm(2)2040 1002 y Fu(\025)28 b Fw(n)13 b Fx(+)g(1)28 b(since)h(otherwise)f(there)h(is)f(nothing)f(to)0 1176 y(pro)m(v)m(e.)50 b(Since)35 b Fw(\026)17 b Fx(\()676 1110 y Fj(V)745 1136 y Fs(n)745 1201 y(i)p FB(=1)880 1176 y Fw(p)929 1191 y Fs(i)957 1176 y Fx(\))31 b Fu(\024)g Fw(n)k Fx(and)f Fw(\026)17 b Fx(\()1532 1110 y Fj(W)1601 1136 y Fs(n)1601 1201 y(i)p FB(=1)1736 1176 y Fw(p)1785 1191 y Fs(i)1813 1176 y Fx(\))31 b Fu(\024)g Fw(n)p Fx(,)k(where)h Fw(p)2443 1191 y Fs(i)2505 1176 y Fx(is)e(either)g Fw(x)2938 1191 y Fs(i)3001 1176 y Fx(or)g Fu(:)p Fw(x)3243 1191 y Fs(i)3272 1176 y Fx(,)h(this)f(implies)0 1296 y(that)28 b Fw(S)34 b Fx(con)m(tains)28 b(all)e(atoms)i(and)g(coatoms)f(of)h Fw(F)1824 1311 y Fs(n)1871 1296 y Fx(,)h(the)g(latter)e(b)s(eing)g (view)m(ed)j(as)e(a)g(Bo)s(olean)f(algebra.)146 1417 y(No)m(w,)37 b(if)e Fu(j)p Fw(S)6 b Fu(j)32 b Fw(>)764 1378 y FB(3)p 764 1394 36 4 v 764 1451 a(4)809 1417 y Fu(j)p Fw(B)5 b Fu(j)35 b Fx(in)g(part)h(a\))f(or)g Fu(j)p Fw(S)6 b Fu(j)32 b(\025)h Fw(\017)p Fu(j)p Fw(B)5 b Fu(j)36 b Fx(in)e(part)i(b\),)g(then)g(w)m(e)h(w)m(ould)e(apply)h(Theorem)0 1537 y(5.2)h(and)h(represen)m(t)i Fw(f)49 b Fx(in)37 b(the)h(form)f(\(9\),)i(\(10\))e(resp)s(ectiv)m(ely)-8 b(.)60 b(This)38 b(represen)m(tation)h(in)e(b)s(oth)h(cases)0 1658 y(w)m(ould)33 b(imply)d(the)j(b)s(ound)g Fw(\026)p Fx(\()p Fw(f)11 b Fx(\))27 b Fw(<)g(t)p Fx(,)33 b(the)g(con)m (tradiction.)p 2212 1658 43 43 v 146 1827 a(No)m(w)g(w)m(e)h(pro)m(v)m (e)g(Theorem)f(5.2.)43 b(Denote)33 b(b)m(y)g Fr(b)f Fx(a)h(randomly)e (c)m(hosen)j(elemen)m(t)f(of)f Fw(B)5 b Fx(.)0 1996 y Fp(Pro)s(of)37 b(of)h(Theorem)f(5.2)g(a\).)81 b Fx(Fix)32 b Fw(b)1518 2011 y FB(0)1586 1996 y Fu(2)c Fw(B)37 b Fx(and)c(consider)g(the)g(represen)m(tation)1094 2214 y Fw(b)1135 2229 y FB(0)1203 2214 y Fx(=)27 b(\()p Fr(b)22 b Fu(^)h Fx(\()p Fu(:)p Fr(b)f Fu(\010)h Fw(b)1826 2229 y FB(0)1865 2214 y Fx(\)\))g Fu(_)f Fx(\()p Fu(:)p Fr(b)g Fu(^)h Fx(\()p Fr(b)f Fu(\010)h Fw(b)2572 2229 y FB(0)2612 2214 y Fx(\)\))16 b Fw(:)0 2431 y Fx(As)34 b(all)d(four)i(random)f(v)-5 b(ariables)32 b Fr(b)p Fw(;)17 b Fx(\()p Fu(:)p Fr(b)22 b Fu(\010)h Fw(b)1663 2446 y FB(0)1703 2431 y Fx(\))p Fw(;)17 b Fu(:)p Fr(b)p Fw(;)g Fx(\()p Fr(b)22 b Fu(\010)h Fw(b)2200 2446 y FB(0)2240 2431 y Fx(\))33 b(are)h(uniformly)d (distributed)h(on)i Fw(B)k Fx(and)0 2552 y Fu(j)p Fw(S)6 b Fu(j)27 b Fw(>)262 2512 y FB(3)p 262 2528 36 4 v 262 2586 a(4)308 2552 y Fu(j)p Fw(B)5 b Fu(j)p Fx(,)32 b(for)g(at)g(least)g (one)h(particular)e(c)m(hoice)i Fw(b)g Fx(of)f Fr(b)g Fx(w)m(e)i(ha)m(v)m(e)g Fw(b;)17 b Fx(\()p Fu(:)p Fw(b)23 b Fu(\010)f Fw(b)2948 2567 y FB(0)2988 2552 y Fx(\))p Fw(;)17 b Fu(:)p Fw(b;)g Fx(\()p Fw(b)23 b Fu(\010)g Fw(b)3464 2567 y FB(0)3504 2552 y Fx(\))k Fu(2)h Fw(S)6 b Fx(.)p 3765 2552 43 43 v 146 2721 a(F)-8 b(or)32 b(pro)m(ving)g(part) h(b\))f(of)g(Theorem)h(5.2)f(w)m(e)i(need)g(the)f(follo)m(wing)0 2946 y Fp(Lemma)k(5.3.)49 b Fv(L)-5 b(et)43 b Fw(B)48 b Fv(b)-5 b(e)42 b(a)g(\014nite)h(Bo)-5 b(ole)g(an)41 b(algebr)-5 b(a)42 b(with)g Fw(N)53 b Fv(atoms)42 b(and)g Fw(S)48 b Fu(\022)42 b Fw(B)5 b Fv(.)68 b(Then)42 b(ther)-5 b(e)0 3067 y(exists)36 b(a)g(subset)h Fw(S)705 3082 y FB(0)775 3067 y Fu(\022)31 b Fw(S)42 b Fv(of)36 b(c)-5 b(ar)g(dinality)36 b Fw(O)s Fx(\(log)16 b Fw(N)10 b Fx(\))37 b Fv(such)f(that)h Fu(^)p Fw(S)2553 3082 y FB(0)2629 3067 y Fv(c)-5 b(ontains)36 b(at)h(most)f Fw(O)3468 2970 y Fj(\020)3517 3067 y Fx(log)3670 3020 y Fq(j)p Fs(B)s Fq(j)p 3670 3044 96 4 v 3675 3101 a(j)p Fs(S)t Fq(j)3775 2970 y Fj(\021)0 3187 y Fv(atoms.)0 3462 y Fp(Pro)s(of)i(of)g(Lemma)g (5.3.)83 b Fx(Let)34 b(us)f(call)f(an)h(atom)f Fw(a)h Fv(go)-5 b(o)g(d)33 b Fx(if)f Fp(P)p Fx([)o Fw(a)c Fu(\024)g Fr(s)p Fx(])h Fu(\024)g Fx(2)p Fw(=)p Fx(3)j(and)i Fv(b)-5 b(ad)32 b Fx(otherwise.)0 3582 y(Here)h Fr(s)g Fx(is)f(pic)m(k)m(ed)i (at)e(random)g(from)f Fw(S)6 b Fx(.)146 3702 y(No)m(w,)34 b(the)f(standard)g(en)m(trop)m(y-coun)m(ting)g(argumen)m(t)f(giv)m(es)h (us)g(that)g(there)g(are)g(at)f(most)1651 3976 y Fw(O)1745 3830 y Fj( )1811 3976 y Fx(log)1963 3909 y Fu(j)p Fw(B)5 b Fu(j)p 1963 3953 135 4 v 1970 4045 a(j)p Fw(S)h Fu(j)2108 3830 y Fj(!)0 4250 y Fx(bad)28 b(atoms.)41 b(An)28 b(equally)g (standard)g(argumen)m(t)g(implies)d(that)j(if)f(w)m(e)i(tak)m(e)g(a)e (random)g(subset)j Fr(S)3587 4265 y FB(0)3654 4250 y Fu(\022)e Fw(S)0 4370 y Fx(of)34 b(cardinalit)m(y)f Fw(C)24 b Fx(log)17 b Fw(N)10 b Fx(,)35 b(the)h(constan)m(t)f Fw(C)42 b Fx(b)s(eing)34 b(su\016cien)m(tly)i(large,)e(then)i(for)e(an) m(y)h(go)s(o)s(d)f(atom)g Fw(a)p Fx(,)0 4491 y Fp(P)p Fx([)p Fw(a)28 b Fu(\024)g(^)p Fr(S)423 4506 y FB(0)463 4491 y Fx(])j Fw(<)h(N)717 4455 y Fq(\000)p FB(1)812 4491 y Fx(.)50 b(Hence,)37 b(for)e(at)g(least)f(one)h(particular)e(c)m (hoice)j Fw(S)2699 4506 y FB(0)2773 4491 y Fx(of)f Fr(S)2956 4506 y FB(0)2995 4491 y Fx(,)h Fu(^)p Fw(S)3184 4506 y FB(0)3258 4491 y Fx(con)m(tains)f(only)0 4611 y(bad)e(atoms,)f(and)g (the)h(lemma)e(follo)m(ws.)p 1502 4611 43 43 v 0 4830 a Fp(Pro)s(of)51 b(of)g(Theorem)g(5.2)g(b\).)130 b Fx(Denote)1775 4783 y Fq(j)p Fs(S)t Fq(j)p 1770 4806 96 4 v 1770 4864 a(j)p Fs(B)s Fq(j)1921 4830 y Fx(b)m(y)45 b Fw(\017)p Fx(.)79 b(Once)46 b(again,)g(\014x)f Fw(b)2968 4845 y FB(0)3055 4830 y Fu(2)k Fw(B)5 b Fx(.)79 b(Let)45 b(us)g(call)0 4964 y Fw(c)34 b Fu(\024)h Fw(b)229 4979 y FB(0)305 4964 y Fv(go)-5 b(o)g(d)36 b Fx(if)f Fp(P)p Fx([)11 b Fr(b)28 b Fu(2)g Fw(S)22 b Fu(j)12 b Fr(b)22 b Fu(^)g Fw(b)1233 4979 y FB(0)1301 4964 y Fx(=)28 b Fw(c)10 b Fx(])34 b Fu(\025)1643 4924 y Fs(\017)p 1639 4940 36 4 v 1639 4998 a FB(2)1721 4964 y Fx(and)i Fv(b)-5 b(ad)36 b Fx(otherwise.)56 b(Note)36 b(that)g Fr(b)25 b Fu(^)g Fw(b)3237 4979 y FB(0)3313 4964 y Fx(is)36 b Fv(uniformly)1864 5214 y Fx(21)p eop %%Page: 22 22 22 21 bop 0 631 a Fx(distributed)32 b(on)h(the)g(Bo)s(olean)e(algebra)g Fw(B)1593 646 y FB(0)1661 631 y Fe(\012)c Fu(f)8 b Fw(c)17 b Fu(j)22 b Fw(c)27 b Fu(\024)h Fw(b)2170 646 y FB(0)2220 631 y Fu(g)p Fx(.)44 b(Hence)1539 874 y Fp(P)p Fx([)p Fr(c)32 b Fx(is)g(go)s(o)s(d)o(])c Fu(\025)2205 807 y Fw(\017)p 2200 851 49 4 v 2200 942 a Fx(2)2259 874 y Fw(;)1366 b Fx(\(11\))0 1119 y(where)34 b Fr(c)e Fx(is)h(c)m(hosen)h (from)d Fw(B)1080 1134 y FB(0)1152 1119 y Fx(at)h(random.)146 1239 y(No)m(w,)44 b(\014x)d(a)f(go)s(o)s(d)g Fw(c)h Fu(2)h Fw(B)1153 1254 y FB(0)1193 1239 y Fx(.)68 b(The)41 b(set)h Fw(B)5 b Fx(\()p Fw(c)p Fx(\))41 b Fe(\012)g Fu(f)9 b Fw(b)28 b Fu(2)g Fw(B)22 b Fu(j)f Fw(b)i Fu(^)g Fw(b)2597 1254 y FB(0)2664 1239 y Fx(=)28 b Fw(c)10 b Fu(g)40 b Fx(is)g(a)h(Bo)s(olean)e(algebra.)0 1360 y(Applying)e(Lemma)f(5.3)h(to) h(this)f(algebra)g(and)g(to)h Fw(S)k Fx(:=)36 b Fw(S)c Fu(\\)26 b Fw(B)5 b Fx(\()p Fw(c)p Fx(\),)39 b(w)m(e)g(come)e(up)h (with)f Fw(S)3453 1375 y FB(0)3529 1360 y Fu(\022)g Fw(S)43 b Fx(of)0 1493 y(cardinalit)m(y)c Fw(O)s Fx(\(log)16 b Fw(N)10 b Fx(\))41 b(suc)m(h)h(that)f Fw(c)h Fu(\024)g(^)p Fw(S)1698 1508 y FB(0)1778 1493 y Fx(and)f(\()p Fu(^)p Fw(S)2140 1508 y FB(0)2180 1493 y Fu(n)p Fw(c)p Fx(\))g(has)g(at)f (most)h Fw(O)3001 1396 y Fj(\020)3051 1493 y Fx(log)3204 1453 y FB(1)p 3204 1469 36 4 v 3207 1527 a Fs(\017)3249 1396 y Fj(\021)3339 1493 y Fx(atoms.)68 b(W)-8 b(e)0 1613 y(extend)37 b Fw(S)380 1628 y FB(0)456 1613 y Fx(b)m(y)f (including)e(to)i(it)f(the)h(corresp)s(onding)g(coatoms)f(and)h(\014nd) g(that)g(ev)m(ery)i(go)s(o)s(d)c Fw(c)g Fu(2)f Fw(B)3785 1628 y FB(0)0 1733 y Fx(can)g(b)s(e)g(represen)m(ted)i(in)d(the)h(form) 1342 1667 y Fj(V)1411 1693 y Fs(`)1411 1758 y(j)t FB(=1)1554 1733 y Fw(s)1600 1748 y Fs(j)1637 1733 y Fw(;)49 b(s)1759 1748 y Fs(j)1823 1733 y Fu(2)28 b Fw(S;)50 b(`)27 b Fu(\024)i Fw(O)2321 1637 y Fj(\020)2371 1733 y Fx(log)2524 1694 y Fs(N)p 2524 1710 64 4 v 2541 1768 a(\017)2597 1637 y Fj(\021)2646 1733 y Fx(.)146 1854 y(Next)43 b(w)m(e)h(apply)d(the)i (dual)e(v)m(ersion)i(of)f(Lemma)e(5.3)i(to)g(the)h(Bo)s(olean)d (algebra)h Fw(B)3331 1869 y FB(0)3413 1854 y Fx(and)h Fw(S)50 b Fx(:=)0 1974 y Fu(f)8 b Fw(c)28 b Fu(2)g Fw(B)296 1989 y FB(0)352 1974 y Fu(j)22 b Fw(c)33 b Fx(is)f(go)s(o)s(d)9 b Fu(g)o Fx(.)43 b(In)31 b(view)g(of)g(\(11\),)f(the)h(same)g(argumen)m (t)f(as)h(ab)s(o)m(v)m(e)h(yields)e(that)g Fw(b)3352 1989 y FB(0)3420 1974 y Fx(=)3524 1908 y Fj(W)3593 1934 y Fs(`)3593 1999 y(i)p FB(=1)3728 1974 y Fw(c)3770 1989 y Fs(i)3798 1974 y Fx(,)0 2095 y(where)k Fw(c)324 2110 y Fs(i)384 2095 y Fx(are)f(either)f(go)s(o)s(d)g(or)g(atoms.)43 b(The)33 b(statemen)m(t)g(follo)m(ws.)p 2485 2095 43 43 v 0 2477 a Fy(6.)165 b(Conclusion)0 2696 y Fx(W)-8 b(e)26 b(do)g(not)f(conclude)i(that)e(researc)m(hers)j(should)e(giv)m (e)g(up)g(on)f(pro)m(ving)h(serious)g(lo)m(w)m(er)g(b)s(ounds.)42 b(Quite)0 2817 y(the)33 b(con)m(trary)-8 b(,)33 b(b)m(y)g(classifying)d 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y(26)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF