(original) (raw)

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4695 y Ft(j)p Fu(\))906 b Fy(\(1\))0 4870 y(It)23 b(remains)h(to)e(compute)j(the)e(sizes)h(of)f(the)g(tw)o (o)f(sets.)29 b(Since)24 b(the)f(coordinates)j(of)d Fv(x)f Fy(and)h Fv(y)i Fy(outside)g Fv(S)i Fy(do)c(not)g(determine)0 4983 y(whether)i(the)e(pair)h Fv(x;)15 b(y)26 b Fy(belongs)g(to)d(one)h (of)g(these)g(sets,)g(we)f(ha)n(v)o(e)720 5246 y Ft(j)p Fv(Z)807 5260 y Fp(1)847 5246 y Ft(j)i Fu(=)g(2)1038 5209 y Fs(n)p FH(\000)p Fs(s)1193 5246 y Ft(\001)c Fu(2)1284 5209 y Fs(n)p FH(\000)p Fs(s)1439 5246 y Ft(\001)1484 5078 y Fi(0)1484 5227 y(@)1703 5141 y Fs(s)1660 5166 y Fi(X)1557 5348 y Fs(i)p Fp(=0)p Fs(;i)h(ev)r(en)1898 5103 y Fi( )1963 5185 y Fv(s)1969 5309 y(i)2006 5103 y Fi(!)2252 5141 y Fs(i)2204 5166 y Fi(X)2087 5350 y Fs(k)r Fp(=0)p Fs(;k)h(ev)r(en)2457 5103 y Fi( )2532 5185 y Fv(i)2522 5309 y(k)2572 5103 y Fi(!)2659 5141 y Fs(s)p FH(\000)p Fs(i)2655 5166 y Fi(X)2653 5348 y Fs(j)t Fp(=0)2791 5103 y Fi( )2857 5185 y Fv(s)d Ft(\000)f Fv(i)2928 5309 y(j)3042 5103 y Fi(!)3108 5078 y(1)3108 5227 y(A)3794 5246 y Fy(\(2\))1927 5589 y(6)p eop %%Page: 7 7 7 6 bop 0 91 a Fy(and)742 273 y Ft(j)p Fv(Z)829 287 y Fp(2)869 273 y Ft(j)25 b Fu(=)g(2)1060 235 y Fs(n)p FH(\000)p Fs(s)1215 273 y Ft(\001)c Fu(2)1306 235 y Fs(n)p FH(\000)p Fs(s)1461 273 y Ft(\001)1506 104 y Fi(0)1506 253 y(@)1706 167 y Fs(s)1663 192 y Fi(X)1579 376 y Fs(i)p Fp(=0)p Fs(;i)i(odd)1882 129 y Fi( )1947 211 y Fv(s)1953 335 y(i)1990 129 y Fi(!)2123 167 y Fs(i)2075 192 y Fi(X)2071 376 y Fs(k)r Fp(=0)2215 129 y Fi( )2290 211 y Fv(i)2280 335 y(k)2331 129 y Fi(!)2527 167 y Fs(s)p FH(\000)p Fs(i)2523 192 y Fi(X)2411 374 y Fs(j)t Fp(=0)p Fs(;j)j(ev)r(en)2769 129 y Fi( )2835 211 y Fv(s)19 b Ft(\000)h Fv(i)2906 335 y(j)3020 129 y Fi(!)3085 104 y(1)3085 253 y(A)3794 273 y Fy(\(3\))0 519 y(The)j(\002rst)g(e)o(xpression)j(equals)661 713 y Fu(2)706 675 y Fp(2)p Fs(n)p FH(\000)p Fp(2)p Fs(s)932 713 y Ft(\001)21 b Fu(\(2)1058 675 y Fp(2)p Fs(s)p FH(\000)p Fp(2)1241 713 y Fu(+)f(2)1377 675 y Fs(s)p FH(\000)p 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b(suf)n(\002ciently)h(close)g(to)f(a)f(parity)i (function\).)52 b(What)31 b(we)f(w)o(ould)h(lik)o(e)g(to)g(do)g(is)f (check)i(whether)g Fv(g)h Fy(is)e(a)f(singleton,)0 1849 y(by)36 b(selecting)i(a)d(sample)h(of)f(pairs)i Fv(x;)15 b(y)38 b Fy(and)e(checking)i(whether)e(it)g(contains)h(a)e(violating)j (pair)f(with)e(respect)i(to)f Fv(g)s Fy(.)0 1962 y(Observ)o(e)e(that,)i (since)f(the)f(distance)h(between)g(functions)h(is)d(measured)i(with)e (respect)i(to)f(the)f(uniform)i(distrib)n(ution,)0 2075 y(for)28 b(a)g(uniformly)i(selected)g(pair)f Fv(x;)15 b(y)s Fy(,)28 b(with)g(probability)j(at)d(least)h Fu(\(1)24 b Ft(\000)f Fv(\016)s Fu(\))2450 2042 y Fp(2)2491 2075 y Fy(,)28 b(both)h Fv(f)10 b Fu(\()p Fv(x)p Fu(\))34 b(=)f Fv(g)s Fu(\()p Fv(x)p Fu(\))c Fy(and)f Fv(f)10 b Fu(\()p Fv(y)s Fu(\))34 b(=)f Fv(g)s Fu(\()p Fv(y)s Fu(\))p Fy(.)0 2188 y(Ho)n(we)n(v)o(er)l(,)38 b(we)d(cannot)j(mak)o(e)e (a)f(similar)h(claim)g(about)h Fv(f)10 b Fu(\()p Fv(x)29 b Ft(^)g Fv(y)s Fu(\))35 b Fy(and)h Fv(g)s 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Fv(f)33 b Fy(is)25 b Fv(\017)p Fy(-close)h(to)0 204 y Fv(g)s Fy(.)41 b(By)27 b(Claim)g(2,)i(the)f(probability)j(that)d(a)f(uniformly)j(selected)g (pair)e Fv(x;)15 b(y)30 b Fy(is)e(a)f(violating)j(pair)f(with)e (respect)j(to)d Fv(g)k Fy(is)c(at)0 317 y(least)22 b Fu(1)p Fv(=)p Fu(8)p Fy(.)29 b(Gi)n(v)o(en)21 b(such)h(a)f(pair)l(,)i (the)e(probability)k(that)d(the)g(self-corrector)j(returns)e(the)f(v)n (alue)g(of)f Fv(g)j Fy(on)e(all)f(the)h(three)g(calls)0 430 y(\(that)k(is,)f Fv(b)338 444 y Fs(x)410 430 y Fu(=)j Fv(g)s Fu(\()p Fv(x)p Fu(\))p Fy(,)e Fv(b)765 444 y Fs(y)834 430 y Fu(=)i Fv(g)s Fu(\()p Fv(y)s Fu(\))p Fy(,)e(and)f Fv(b)1340 444 y Fs(x)p FH(^)p Fs(y)1497 430 y Fu(=)i Fv(g)s Fu(\()p Fv(x)c Ft(^)e Fv(y)s Fu(\))p Fy(\),)k(is)g(at)f(least)i Fu(\(1)d Ft(\000)e Fu(1)p Fv(=)p Fu(10\))2769 397 y Fp(3)2838 430 y Fv(>)28 b Fu(7)p Fv(=)p Fu(10)p Fy(.)35 b(The)24 b(probability)29 b(that)0 543 y(the)f(algorithm)i(obtains)f(a)f (violating)i(pair)e(with)g(respect)h(to)f Fv(g)j Fw(and)g Fy(all)d(calls)g(to)g(the)g(self)g(corrector)i(return)g(the)e(correct)0 656 y(v)n(alue,)d(is)f(greater)i(than)g Fu(1)p Fv(=)p Fu(16)p Fy(.)32 b(Therefore,)26 b(a)e(sample)h(of)f(32)h(pairs)g(will)f (ensure)i(that)f(a)f(violation)j Fv(b)3217 670 y Fs(x)p FH(^)p Fs(y)3372 656 y Ft(6)p Fu(=)f Fv(b)3508 670 y Fs(x)3573 656 y Ft(^)21 b Fv(b)3694 670 y Fs(y)3759 656 y Fy(will)0 769 y(be)j(found)h(with)f(probability)j(at)d(least)h Fu(5)p Fv(=)p Fu(6)p Fy(.)30 b(The)24 b(total)g(probability)j(that)e Fv(f)32 b Fy(is)24 b(accepted,)i(despite)g(being)f Fv(\017)p Fy(-f)o(ar)f(from)g(an)o(y)0 882 y(singleton,)i(is)d(hence)i(at)e(most) h Fu(1)p Fv(=)p Fu(10)e(+)e(1)p Fv(=)p Fu(6)26 b Fv(<)f Fu(1)p Fv(=)p Fu(3)p Fy(.)141 1014 y(The)f(query)i(comple)o(xity)h(of)e (the)g(algorithm)h(is)f(dominated)i(by)e(the)g(query)h(comple)o(xity)g (of)f(the)g(parity)h(tester)g(which)0 1127 y(is)d Fv(O)s Fu(\(1)p Fv(=\017)p Fu(\))p Fy(.)30 b(The)23 b(second)i(stage)f(tak)o (es)h(constant)h(time.)p 1876 1127 67 67 v 0 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Fu(2)1915 5299 y Fs(n)2020 5263 y Ft(\000)20 b Fu(2)2156 5226 y FH(\000)p Fs(k)2254 5141 y Fi(\014)2254 5191 y(\014)2254 5240 y(\014)2254 5290 y(\014)2302 5263 y Fu(+)g Fv(k)s(\021)2501 5198 y Fn(1)p 2501 5210 31 4 v 2501 5252 a(2)2546 5263 y Fv(:)1927 5589 y Fy(9)p eop %%Page: 10 10 10 9 bop 0 91 a Fj(Lemma)23 b(8)46 b Fw(Let)28 b Fv(f)45 b Fu(:)36 b Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)929 49 y Fs(n)1012 91 y Ft(!)35 b(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)31 b Fw(be)e(a)g(function)i(for)f(whic)o(h)f Ft(j)p Fu(Pr)o([)p Fv(f)46 b Fu(=)35 b(1])25 b Ft(\000)f Fu(2)2861 58 y FH(\000)p Fs(k)2959 91 y Ft(j)36 b Fv(<)f Fu(2)3171 58 y FH(\000)p Fs(k)r FH(\000)p Fp(3)3359 91 y Fw(.)45 b(Suppose)32 b(that)0 204 y(ther)m(e)24 b(e)n(xists)h(a)e(function)j Fv(g)j Fu(:)c Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1163 162 y Fs(n)1235 204 y Ft(!)25 b(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)25 b Fw(suc)o(h)f(that:)114 411 y(1.)45 b Fu(dist)o(\()p Fv(f)5 b(;)15 b(g)s Fu(\))26 b Ft(\024)f Fu(2)747 378 y FH(\000)p Fs(k)r 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b Fu(\()p Fv(x)p Fu(\)])35 b Fy(the)g(e)o(xpectation)j(of)d Fv(f)44 b Fy(o)o(v)o(er)35 b(the)0 5173 y(whole)24 b(space)h Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)690 5140 y Fs(n)739 5173 y Fy(.)1905 5589 y(10)p eop %%Page: 11 11 11 10 bop 141 91 a Fy(F)o(or)31 b Fu(1)41 b Ft(\024)g Fv(i)g Ft(\024)g Fv(n)p Fy(,)32 b(let)g Fv(r)952 105 y Fs(i)1021 91 y Fu(:)41 b Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1307 58 y Fs(n)1398 91 y Ft(!)40 b(f\000)p Fu(1)p Fv(;)15 b Fu(1)p Ft(g)33 b Fy(be)f(a)g(function)i(de\002ned)f(by)f Fv(r)2838 105 y Fs(i)2866 91 y Fu(\()p Fv(x)2953 105 y Fp(1)2993 91 y Fv(;)15 b(:::;)g(x)3200 105 y Fs(n)3248 91 y Fu(\))41 b(=)g(1)32 b Fy(if)f Fv(x)3651 105 y Fs(i)3720 91 y Fu(=)41 b(0)p Fy(,)0 204 y(and)36 b Fv(r)207 218 y Fs(i)235 204 y Fu(\()p Fv(x)322 218 y Fp(1)362 204 y Fv(;)15 b(:::;)g(x)569 218 y Fs(n)617 204 y Fu(\))48 b(=)e Ft(\000)p Fu(1)35 b Fy(otherwise.)66 b(F)o(or)34 b(an)o(y)i Fv(S)52 b Ft(\022)47 b Fu([)p Fv(n)p Fu(])p Fy(,)37 b(de\002ne)f(a)f(function)i Fv(w)2857 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b(will)g(need)h(the)g(follo)n(wing)h(important)g (property)h(of)d(con)l(v)n(olution:)1453 1913 y Fi(d)1384 1945 y Fu(\()p Fv(f)30 b Ft(\003)21 b Fv(g)s Fu(\))q(\()p Fv(S)5 b Fu(\))26 b(=)1914 1921 y(^)1895 1945 y Fv(f)9 b Fu(\()p Fv(S)c Fu(\))21 b Ft(\001)i Fu(^)-48 b Fv(g)t Fu(\()p Fv(S)5 b Fu(\))p Fv(:)0 2149 y Fy(And)23 b(we)g(will)g(also)h (need)h(se)n(v)o(eral)f(simple)g(f)o(acts)h(about)g(F)o(ourier)f(coef)n (\002cients.)30 b(F)o(or)23 b(an)o(y)h(tw)o(o)f(functions)j Fv(f)5 b(;)15 b(g)s Fy(:)1179 2354 y Ff(E)1248 2368 y Fs(x)1292 2354 y Fu([)p Fv(f)10 b Fu(\()p Fv(x)p Fu(\))20 b Ft(\001)h Fv(g)s Fu(\()p Fv(x)p Fu(\)])27 b(=)1907 2273 y Fi(X)1875 2461 y Fs(S)t FH(\022)p Fp([)p Fs(n)p Fp(])2094 2330 y Fu(^)2074 2354 y Fv(f)9 b Fu(\()p Fv(S)c Fu(\))21 b Ft(\001)j Fu(^)-48 b Fv(g)s Fu(\()p Fv(S)5 b Fu(\))26 b Fv(:)0 2648 y Fy(In)d(particular)l(,)k(P)o(arse)n(v)n(al') -5 b(s)24 b(equality)i(holds:)1392 2852 y Ff(E)1461 2866 y Fs(x)1505 2852 y Fu([)p Fv(f)1585 2814 y Fp(2)1624 2852 y 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Ft(2)d Fv(F)1059 4990 y Fp(1)1122 4976 y Fy(and)f Fv(z)29 b Ft(2)c(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1653 4943 y Fs(n)1702 4976 y Fy(,)23 b(and)h(checks)h(whether)g Fv(h)p Fu(\()p Fv(x)20 b Ft(\010)g Fv(y)j Ft(\010)d Fv(z)t Fu(\))h Ft(\001)g Fv(h)p Fu(\()p Fv(z)t Fu(\))26 b(=)f(1)p Fy(.)p 0 5038 1560 4 v 100 5092 a Fn(4)134 5123 y Fq(Note)20 b(that)g(W)-6 b(alsh)19 b(functions)i(are)e(essentially)h(the)g(parity) g(functions)g(on)h Fk(f)p Fe(0)p Fl(;)13 b Fe(1)p Fk(g)2261 5092 y Fh(n)2305 5123 y Fq(,)19 b(b)o(ut)h(written)f(in)g(a)h Fm(multiplicative)f(notation)p Fq(.)27 b(If)19 b(we)g(de\002ne)16 5215 y Fe(~)-54 b Fl(w)55 5223 y Fh(S)100 5215 y Fe(\()p Fl(x)p Fe(\))17 b Fq(to)i(be)g Fe(0)h Fq(if)e Fl(w)564 5223 y Fh(S)608 5215 y Fe(\()p Fl(x)p Fe(\))j(=)g(1)p Fq(,)e(and)g Fe(1)g Fq(if)g Fl(w)1193 5223 y Fh(S)1237 5215 y Fe(\()p Fl(x)p Fe(\))i(=)g Fk(\000)p Fe(1)p Fq(,)d(then)i(the)f (functions)35 b Fe(~)-53 b Fl(w)2188 5223 y Fh(S)2251 5215 y Fq(are)19 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1032 y Fu([\()p Fv(h)e Ft(\003)g Fv(h)p Fu(\)\()p Fv(x)g Ft(\010)f Fv(y)s Fu(\)])26 b(=)1641 971 y(2)1686 938 y Fp(2)p Fs(n)p 1611 1011 188 4 v 1611 1094 a Ft(j)p Fv(F)1694 1108 y Fp(1)1734 1094 y Ft(j)1759 1068 y Fp(2)1829 1032 y Ft(\001)20 b Ff(E)1943 1046 y Fs(x;y)2044 1032 y Fu([)p Fv(f)10 b Fu(\()p Fv(x)p Fu(\))20 b Ft(\001)h Fv(f)10 b Fu(\()p Fv(y)s Fu(\))20 b Ft(\001)g Fu(\()p Fv(h)h Ft(\003)g Fv(h)p Fu(\)\()p Fv(x)g Ft(\010)f Fv(y)s Fu(\)])p Fv(:)0 1299 y Fy(Recall)k(that)g Fv(f)35 b Fu(=)600 1263 y Fp(1)p FH(\000)p Fs(h)p 600 1278 131 4 v 647 1330 a Fp(2)763 1299 y Fy(and)24 b(hence)135 1505 y Fu(2)180 1472 y Fp(2)p Fs(n)p 105 1546 188 4 v 105 1629 a Ft(j)p Fv(F)188 1643 y Fp(1)228 1629 y Ft(j)253 1603 y Fp(2)323 1567 y Ft(\001)c Ff(E)437 1581 y Fs(x;y)537 1567 y Fu([)p Fv(f)10 b Fu(\()p Fv(x)p Fu(\))21 b Ft(\001)f Fv(f)10 b Fu(\()p Fv(y)s Fu(\))20 b Ft(\001)h Fu(\()p Fv(h)g Ft(\003)f Fv(h)p Fu(\)\()p Fv(x)i Ft(\010)e Fv(y)s Fu(\)])178 1815 y(=)361 1753 y(1)p 341 1794 86 4 v 341 1877 a Fv(p)387 1851 y Fp(2)457 1815 y Ft(\001)g Ff(E)571 1829 y Fs(x;y)687 1696 y Fi(\024)740 1753 y Fu(1)h Ft(\000)f Fv(h)p Fu(\()p Fv(x)p Fu(\))p 740 1794 332 4 v 884 1877 a(2)1102 1815 y Ft(\001)1158 1753 y Fu(1)g Ft(\000)g Fv(h)p Fu(\()p Fv(y)s Fu(\))p 1158 1794 328 4 v 1299 1877 a(2)1515 1815 y Ft(\001)h Fu(\()p Fv(h)g Ft(\003)f Fv(h)p Fu(\)\()p Fv(x)i Ft(\010)e Fv(y)s Fu(\))2104 1696 y Fi(\025)178 2048 y Fu(=)384 1987 y(1)p 341 2027 131 4 v 341 2111 a(4)p Fv(p)432 2084 y Fp(2)482 2048 y Ff(E)551 2062 y Fs(x;y)667 2048 y Fu([\()p Fv(h)h Ft(\003)f Fv(h)p Fu(\)\()p Fv(x)i Ft(\010)d Fv(y)s Fu(\))46 b Ft(\000)f Fu(\()p Fv(h)p Fu(\()p Fv(x)p Fu(\))22 b(+)e Fv(h)p Fu(\()p Fv(y)s Fu(\)\))h Ft(\001)g Fu(\()p Fv(h)g Ft(\003)f Fv(h)p Fu(\)\()p Fv(x)i Ft(\010)d Fv(y)s Fu(\))46 b(+)f Fv(h)p Fu(\()p Fv(x)p Fu(\))22 b Ft(\001)e Fv(h)p Fu(\()p Fv(y)s Fu(\))h Ft(\001)g Fu(\()p Fv(h)f Ft(\003)h Fv(h)p Fu(\)\()p Fv(x)g Ft(\010)f Fv(y)s Fu(\)])41 b Fv(:)0 2290 y Fy(W)-7 b(e)23 b(open)h(the)g(brack)o(ets)i(and)e(compute)g(the)g(e)o (xpectations)j(one)d(by)g(one.)88 2494 y Ff(E)157 2508 y Fs(x;y)258 2494 y Fu([\()p Fv(h)d Ft(\003)g Fv(h)p Fu(\)\()p Fv(x)g Ft(\010)f Fv(y)s Fu(\))25 b(=)g Ff(E)1016 2508 y Fs(y)1057 2494 y Fu([)p Ff(E)1151 2508 y Fs(x)1195 2494 y Fu([\()p Fv(h)c Ft(\003)g Fv(h)p Fu(\)\()p Fv(x)g Ft(\010)f Fv(y)s Fu(\)]])26 b(=)f Ff(E)2004 2508 y Fs(y)2045 2494 y Fu([)p Ff(E)2139 2508 y Fs(z)2179 2494 y Fu([\()p Fv(h)c Ft(\003)g Fv(h)p Fu(\)\()p Fv(z)t Fu(\)]])27 b(=)e Ff(E)2823 2508 y Fs(z)2862 2494 y Fu([\()p Fv(h)d Ft(\003)e Fv(h)p Fu(\)\()p Fv(z)t Fu(\)])28 b(=)3413 2470 y(^)3412 2494 y Fv(h)3464 2457 y Fp(2)3504 2494 y Fu(\()p Ft(;)p Fu(\))p Fv(;)0 2698 y Fy(and)161 2903 y Ff(E)230 2917 y Fs(x;y)331 2903 y Fu([)p Fv(h)p Fu(\()p Fv(x)p Fu(\))21 b Ft(\001)g Fu(\()p Fv(h)f Ft(\003)h Fv(h)p Fu(\)\()p Fv(x)g Ft(\010)f Fv(y)s Fu(\)])26 b(=)f Ff(E)1355 2917 y Fs(y)1396 2903 y Fu([)p Ff(E)1490 2917 y Fs(x)1534 2903 y Fu([)p Fv(h)p Fu(\()p Fv(x)p Fu(\))c Ft(\001)g Fu(\()p Fv(h)g Ft(\003)f Fv(h)p Fu(\)\()p Fv(x)i Ft(\010)e Fv(y)s Fu(\)]])25 b(=)g Ff(E)2583 2917 y Fs(y)2625 2903 y Fu([\()p Fv(h)c Ft(\003)f Fv(h)h Ft(\003)g Fv(h)p Fu(\)\()p Fv(y)s Fu(\)])26 b(=)3315 2879 y(^)3314 2903 y Fv(h)3366 2865 y Fp(3)3406 2903 y Fu(\()p Ft(;)p Fu(\))h Fv(:)0 3107 y Fy(Similarly)-6 b(,)1177 3311 y Ff(E)1246 3325 y Fs(x;y)1347 3311 y Fu([)p Fv(h)p Fu(\()p Fv(y)s Fu(\))21 b Ft(\001)g Fu(\()p Fv(h)g Ft(\003)f Fv(h)p Fu(\)\()p Fv(x)i Ft(\010)d Fv(y)s Fu(\)]])26 b(=)2324 3287 y(^)2323 3311 y Fv(h)2375 3274 y Fp(3)2415 3311 y Fu(\()p Ft(;)p Fu(\))p Fv(:)0 3515 y Fy(Finally)-6 b(,)527 3720 y Ff(E)596 3734 y Fs(x;y)696 3720 y Fu([)p Fv(h)p Fu(\()p Fv(x)p Fu(\))22 b Ft(\001)e Fv(h)p Fu(\()p Fv(y)s Fu(\))h Ft(\001)g Fu(\()p Fv(h)g Ft(\003)g Fv(h)p Fu(\)\()p Fv(x)g Ft(\010)f Fv(y)s Fu(\)]])83 b(=)g Ff(E)2098 3734 y Fs(y)2139 3720 y Fu([)p Fv(h)p Fu(\()p Fv(y)s Fu(\))22 b Ft(\001)e Ff(E)2470 3734 y Fs(x)2514 3720 y Fu([)p Fv(h)p Fu(\()p Fv(x)p Fu(\))h Ft(\001)g Fu(\()p Fv(h)g Ft(\003)f Fv(h)p Fu(\)\()p Fv(x)h Ft(\010)f Fv(y)s Fu(\)]])1875 3857 y(=)83 b Ff(E)2098 3871 y Fs(y)2139 3857 y Fu([)p Fv(h)p Fu(\()p Fv(y)s Fu(\))22 b Ft(\001)e Fu(\()p Fv(h)h Ft(\003)g Fv(h)f Ft(\003)h Fv(h)p Fu(\)\()p Fv(y)s Fu(\)])1875 4006 y(=)83 b(\()p Fv(h)21 b Ft(\003)g Fv(h)f Ft(\003)h Fv(h)g Ft(\003)f Fv(h)p Fu(\)\(0\))27 b(=)2804 3925 y Fi(X)2841 4109 y Fs(S)2940 3982 y Fu(^)2939 4006 y Fv(h)2991 3969 y Fp(4)3031 4006 y Fu(\()p Fv(S)5 b Fu(\))26 b Fv(:)141 4331 y Fy(Observ)o(e)d (that)622 4307 y Fu(^)621 4331 y Fv(h)q Fu(\()p Ft(;)p Fu(\))j(=)f Ff(E)980 4345 y Fs(x)1024 4331 y Fu([)p Fv(h)p Fu(\()p Fv(x)p Fu(\)])h(=)f(1)14 b Ft(\000)1525 4286 y Fp(2)p FH(j)p Fs(F)1625 4295 y Fn(1)1659 4286 y FH(j)p 1525 4310 154 4 v 1563 4362 a Fp(2)1598 4343 y Fh(n)1714 4331 y Fu(=)25 b(1)14 b Ft(\000)g Fu(2)p Fv(p)p Fy(,)21 b(and)i(that)f Fv(h)k Fu(=)f(1)14 b Ft(\000)g Fu(2)p Fv(f)c Fy(.)27 b(Therefore,)3271 4307 y Fu(^)3270 4331 y Fv(h)p Fu(\()p Fv(S)5 b Fu(\))26 b(=)f Ft(\000)p Fu(2)3711 4307 y(^)3691 4331 y Fv(f)10 b Fu(\()p Fv(S)5 b Fu(\))p Fy(,)0 4458 y(for)24 b(all)f Fv(S)31 b Ft(6)p Fu(=)25 b Ft(;)p Fy(.)j(Note)23 b(also)i(that)1068 4434 y Fu(^)1049 4458 y Fv(f)9 b Fu(\()p Ft(;)p Fu(\))26 b(=)f Fv(p)p Fy(.)j(T)-7 b(aking)24 b(all)g(this)g(into)g(consideration)j(we)c(ha)n (v)o(e:)422 4727 y Fu(1)e Ft(\000)e Fu(2)p Fv(\021)87 b Fu(=)962 4666 y(1)p 919 4706 131 4 v 919 4790 a(4)p Fv(p)1010 4763 y Fp(2)1075 4583 y Fi( )1140 4727 y Fu(\(1)22 b Ft(\000)d Fu(2)p Fv(p)p Fu(\))1458 4690 y Fp(2)1519 4727 y Ft(\000)h Fu(2\(1)h Ft(\000)f Fu(2)p Fv(p)p Fu(\))1973 4690 y Fp(3)2033 4727 y Fu(+)g(\(1)h Ft(\000)f Fu(2)p Fv(p)p Fu(\))2442 4690 y Fp(4)2502 4727 y Fu(+)g(16)h Ft(\001)2750 4646 y Fi(X)2786 4830 y Fs(S)2904 4703 y Fu(^)2885 4727 y Fv(f)2940 4690 y Fp(4)2978 4727 y Fu(\()p Fv(S)5 b Fu(\))21 b Ft(\000)f Fu(16)p Fv(p)3357 4690 y Fp(4)3397 4583 y Fi(!)755 5020 y Fu(=)83 b(\(1)21 b Ft(\000)f Fu(2)p Fv(p)p Fu(\))1227 4983 y Fp(2)1287 5020 y Ft(\000)g Fu(4)p Fv(p)1469 4983 y Fp(2)1529 5020 y Fu(+)g(4)1675 4894 y Fi(P)1763 4982 y Fs(S)1848 4935 y Fu(^)1829 4959 y Fv(f)1884 4926 y Fp(4)1923 4959 y Fu(\()p Fv(S)5 b Fu(\))p 1675 4999 379 4 v 1822 5083 a Fv(p)1868 5056 y Fp(2)755 5282 y Fu(=)83 b(1)21 b Ft(\000)f Fu(4)p Fv(p)g Fu(+)g(4)1323 5156 y Fi(P)1411 5244 y Fs(S)1496 5197 y Fu(^)1477 5221 y Fv(f)1532 5188 y Fp(4)1571 5221 y Fu(\()p Fv(S)5 b Fu(\))p 1323 5261 V 1470 5344 a Fv(p)1516 5318 y Fp(2)1738 5282 y Fv(:)1905 5589 y Fy(12)p eop %%Page: 13 13 13 12 bop 0 91 a Fy(Claim)23 b(9)g(follo)n(ws.)p 710 91 67 67 v 141 262 a(Assume)j(that)g Fv(\021)i Fy(is)d(small,)h(in)f (particular)j Fv(\021)k Ft(\024)c Fv(p)1713 229 y Fp(2)1752 262 y Fy(.)34 b(In)25 b(this)h(case,)g(Claim)f(9)g(implies)i(that)3040 198 y Fi(P)3127 285 y Fs(S)3213 238 y Fu(^)3193 262 y Fv(f)3248 229 y Fp(4)3287 262 y Fu(\()p Fv(S)5 b Fu(\))25 b Fy(is)g(v)o(ery)h(close)0 375 y(to)d Fv(p)139 342 y Fp(3)179 375 y Fy(.)28 b(Note)23 b(that,)h(since)g Fv(f)32 b Fy(is)24 b(nonne)o(gati)n(v)o(e,)h(for)f(an)o(y)f Fv(S)31 b Ft(\022)25 b Fu([)p Fv(n)p Fu(])p Fy(:)1456 522 y Fu(^)1436 546 y Fv(f)10 b Fu(\()p Fv(S)5 b Fu(\))25 b Ft(\024)g(j)1788 522 y Fu(^)1768 546 y Fv(f)10 b Fu(\()p Fv(S)5 b Fu(\))p Ft(j)26 b(\024)2121 522 y Fu(^)2101 546 y Fv(f)9 b Fu(\()p Ft(;)p Fu(\))27 b(=)e Fv(p:)0 716 y Fy(W)-7 b(e)20 b(no)n(w)h(sho)n(w)g (that)g(for)g(man)o(y)g(subsets)i Fv(S)5 b Fy(,)21 b Ft(j)1459 692 y Fu(^)1440 716 y Fv(f)9 b Fu(\()p Fv(S)c Fu(\))p Ft(j)21 b Fy(is)g(actually)i(v)o(ery)f(close)g(to)f Fv(p)p Fy(.)27 b(W)-7 b(e)20 b(later)h(use)h(these)g(subsets)h(to)e (de\002ne)0 829 y(the)j(af)n(\002ne)f(subspace)j Fv(G)d Fy(that)h(is)g(close)g(to)g Fv(F)1410 843 y Fp(1)1450 829 y Fy(.)k(T)-7 b(o)22 b(this)i(end)g(we)f(de\002ne)h(the)g(follo)n (wing)h(collections)h(of)e(subsets:)639 1019 y(small)861 969 y Fp(def)875 1019 y Fu(=)38 b Ft(f)p Fv(S)31 b Fu(:)25 b Ft(j)1211 995 y Fu(^)1191 1019 y Fv(f)10 b Fu(\()p Fv(S)5 b Fu(\))p Ft(j)26 b Fv(<)f(p)20 b Ft(\000)g Fv(\021)1739 955 y Fn(1)p 1739 967 31 4 v 1739 1008 a(2)1784 1019 y Ft(g)p Fv(;)106 b Fy(big)2102 969 y Fp(def)2115 1019 y Fu(=)38 b Ft(f)p Fv(S)31 b Fu(:)26 b Ft(j)2451 995 y Fu(^)2432 1019 y Fv(f)9 b Fu(\()p Fv(S)c Fu(\))p Ft(j)26 b(\025)f Fv(p)20 b Ft(\000)g Fv(\021)2979 955 y Fn(1)p 2980 967 V 2980 1008 a(2)3024 1019 y Ft(g)p Fv(:)0 1218 y Fj(Claim)j(10)46 b Fw(If)24 b Ft(j)p Fv(p)c Ft(\000)g Fu(2)702 1185 y FH(\000)p Fs(k)800 1218 y Ft(j)25 b(\024)g Fu(2)991 1185 y FH(\000)p Fp(\()p Fs(k)r Fp(+3\))1257 1218 y Fw(and)f Fv(\021)29 b Ft(\024)c Fv(p)1632 1185 y Fp(2)1671 1218 y Fw(,)d(then)j Ft(j)p Fu(big)p Ft(j)h(\025)f Fu(2)2235 1185 y Fs(k)r FH(\000)p Fp(1)2388 1218 y Fu(+)20 b(1)p Fw(.)0 1416 y Fj(Pr)n(oof:)47 b Fy(Using)24 b(Claim)f(9,)g(we)g (ha)n(v)o(e:)792 1639 y Fv(p)838 1601 y Fp(3)897 1639 y Ft(\000)998 1577 y Fv(p)1044 1544 y Fp(2)1083 1577 y Fv(\021)p 998 1618 134 4 v 1042 1701 a Fu(2)1225 1639 y(=)1378 1558 y Fi(X)1415 1742 y Fs(S)1533 1615 y Fu(^)1513 1639 y Fv(f)1568 1601 y Fp(4)1607 1639 y Fu(\()p Fv(S)5 b Fu(\))1225 1872 y(=)1464 1792 y Fi(X)1378 1992 y Fs(S)t FH(2)p Fy(small)1704 1848 y Fu(^)1684 1872 y Fv(f)1739 1835 y Fp(4)1778 1872 y Fu(\()p Fv(S)g Fu(\))21 b(+)2066 1792 y Fi(X)2021 1992 y Fs(S)t FH(2)p Fy(big)2265 1848 y Fu(^)2246 1872 y Fv(f)2301 1835 y Fp(4)2340 1872 y Fu(\()p Fv(S)5 b Fu(\))1225 2150 y Ft(\024)82 b Fu(\()p Fv(p)21 b Ft(\000)f Fv(\021)1629 2085 y Fn(1)p 1629 2097 31 4 v 1629 2138 a(2)1674 2150 y Fu(\))1709 2112 y Fp(2)1769 2150 y Ft(\001)1899 2069 y Fi(X)1814 2269 y Fs(S)t FH(2)p Fy(small)2139 2126 y Fu(^)2120 2150 y Fv(f)2175 2112 y Fp(2)2214 2150 y Fu(\()p Fv(S)5 b Fu(\))20 b(+)g Fv(p)2502 2112 y Fp(2)2562 2150 y Ft(\001)2652 2069 y Fi(X)2607 2269 y Fs(S)t FH(2)p Fy(big)2852 2126 y Fu(^)2832 2150 y Fv(f)2887 2112 y Fp(2)2926 2150 y Fu(\()p Fv(S)5 b Fu(\))26 b Fv(:)0 2451 y Fy(Let)d Fv(r)28 b Fu(=)309 2386 y Fi(P)396 2481 y Fs(S)t FH(2)p Fy(small)726 2427 y Fu(^)706 2451 y Fv(f)761 2418 y Fp(2)800 2451 y Fu(\()p Fv(S)5 b Fu(\))p Fy(.)29 b(Using)24 b(Equation)h(\(4\),)1725 2386 y Fi(P)1813 2473 y Fs(S)1898 2427 y Fu(^)1879 2451 y Fv(f)1934 2418 y Fp(2)1972 2451 y Fu(\()p Fv(S)5 b Fu(\))26 b(=)f Ff(E)2294 2465 y Fs(x)2338 2451 y Fu([)p Fv(f)2418 2418 y Fp(2)2457 2451 y Fu(\()p Fv(x)p Fu(\)])h(=)f Fv(p)p Fy(.)i(Thus,)1017 2652 y Fv(p)1063 2614 y Fp(3)1123 2652 y Ft(\000)1224 2590 y Fv(\021)p 1224 2631 49 4 v 1225 2714 a Fu(2)1282 2652 y Fv(p)1328 2614 y Fp(2)1450 2652 y Ft(\024)83 b Fu(\()p Fv(p)20 b Ft(\000)g Fv(\021)1854 2587 y Fn(1)p 1854 2599 31 4 v 1854 2640 a(2)1899 2652 y Fu(\))1934 2614 y Fp(2)1974 2652 y Fv(r)i Fu(+)e Fv(p)2174 2614 y Fp(2)2214 2652 y Fu(\()p Fv(p)g Ft(\000)g Fv(r)s Fu(\))1450 2844 y(=)83 b Fv(p)1650 2807 y Fp(3)1709 2844 y Ft(\000)20 b Fv(r)j Ft(\001)d Fu(\()p Fv(p)1990 2807 y Fp(2)2050 2844 y Ft(\000)g Fu(\()p Fv(p)g Ft(\000)g Fv(\021)2391 2780 y Fn(1)p 2392 2792 V 2392 2833 a(2)2436 2844 y Fu(\))2471 2807 y Fp(2)2511 2844 y Fu(\))1450 2997 y(=)83 b Fv(p)1650 2959 y Fp(3)1709 2997 y Ft(\000)20 b Fv(r)s(\021)1902 2932 y Fn(1)p 1902 2944 V 1902 2986 a(2)1947 2997 y Fu(\(2)p Fv(p)g Ft(\000)g Fv(\021)2242 2932 y Fn(1)p 2243 2944 V 2243 2986 a(2)2287 2997 y Fu(\))26 b Ft(\024)f Fv(p)2490 2959 y Fp(3)2549 2997 y Ft(\000)20 b Fv(r)s(p\021)2788 2932 y Fn(1)p 2788 2944 V 2788 2986 a(2)2857 2997 y Fv(:)0 3210 y Fy(The)j(last)h(inequality)i(is)e(based)h (on)e(our)h(assumption)i(that)e Fv(\021)29 b Ft(\024)c Fv(p)2082 3177 y Fp(2)2121 3210 y Fy(.)j(Therefore,)d Fv(r)i Ft(\024)2750 3169 y Fs(\021)2798 3118 y Fn(1)p 2798 3127 V 2798 3162 a(2)p 2750 3189 93 4 v 2779 3241 a Fp(2)2853 3210 y Fv(p)p Fy(.)g(It)c(follo)n(ws)i(that:)1245 3464 y Fv(p)20 b Ft(\000)g Fv(r)28 b Fu(=)1612 3383 y Fi(X)1567 3583 y Fs(S)t FH(2)p Fy(big)1811 3440 y Fu(^)1792 3464 y Fv(f)1847 3426 y Fp(2)1885 3464 y Fu(\()p Fv(S)5 b Fu(\))26 b Ft(\025)f Fv(p)20 b Ft(\000)2305 3402 y Fv(\021)2363 3339 y Fn(1)p 2363 3351 31 4 v 2363 3392 a(2)p 2305 3442 103 4 v 2334 3526 a Fu(2)2418 3464 y Fv(p:)0 3764 y Fy(But)174 3740 y Fu(^)154 3764 y Fv(f)9 b Fu(\()p Fv(S)c Fu(\))26 b Ft(\024)f Fv(p)p Fy(,)d(and)i(therefore:) 1005 4042 y Ft(j)p Fy(big)q Ft(j)76 b(\025)1404 3977 y Fv(p)20 b Ft(\000)1571 3936 y Fs(\021)1618 3885 y Fn(1)p 1618 3894 31 4 v 1618 3929 a(2)p 1571 3956 93 4 v 1599 4008 a Fp(2)1673 3977 y Fv(p)p 1404 4021 315 4 v 1519 4105 a(p)1565 4079 y Fp(2)1804 4042 y Fu(=)1961 3977 y(1)g Ft(\000)2127 3936 y Fs(\021)2175 3885 y Fn(1)p 2175 3894 31 4 v 2175 3929 a(2)p 2127 3956 93 4 v 2156 4008 a Fp(2)p 1961 4021 269 4 v 2072 4105 a Fv(p)2315 4042 y Ft(\025)2472 3981 y Fu(1)p 2472 4021 46 4 v 2472 4105 a Fv(p)2548 4042 y Ft(\000)2648 3981 y Fu(1)p 2648 4021 V 2648 4105 a(2)2704 4042 y Fv(:)0 4271 y Fy(Since)k Ft(j)p Fv(p)c Ft(\000)g Fu(2)452 4238 y FH(\000)p Fs(k)550 4271 y Ft(j)25 b Fv(<)g Fu(2)741 4238 y FH(\000)p Fp(\()p Fs(k)r Fp(+3\))984 4271 y Fy(,)e(Claim)g(10)g(follo)n(ws)p 1757 4271 67 67 v 0 4423 a(Observ)o(e)j(that)g(in)f(a)g(set)h(of)f (size)h Ft(\025)i Fu(2)1183 4390 y Fs(k)r FH(\000)p Fp(1)1338 4423 y Fu(+)21 b(1)k Fy(there)i(are)e(at)g(least)h Fv(k)i Fy(linearly)f(independent)i(v)o(ectors)e(\(otherwise)g(the)f(set)0 4536 y(lies)c(in)g(a)f(span)h(of)g(at)f(most)h Fv(k)16 b Ft(\000)d Fu(1)21 b Fy(v)o(ectors,)i(which)f(is,)f(ob)o(viously)-6 b(,)25 b(of)c(size)h Fu(2)2432 4503 y Fs(k)r FH(\000)p Fp(1)2566 4536 y Fy(\).)27 b(W)-7 b(e)21 b(use)h(this)g(in)f(the)h (follo)n(wing)h(claim.)0 4715 y Fj(Claim)g(11)46 b Fw(Suppose)28 b(that)e Ft(j)p Fv(p)c Ft(\000)g Fu(2)1121 4682 y FH(\000)p Fs(k)1219 4715 y Ft(j)29 b(\024)g Fu(2)1418 4682 y FH(\000)p Fp(\()p Fs(k)r Fp(+3\))1686 4715 y Fw(and)d Fv(\021)32 b Ft(\024)d Fv(p)2070 4682 y Fp(2)2109 4715 y Fw(.)34 b(Let)25 b Fv(S)2363 4729 y Fp(1)2402 4715 y Fv(;)15 b(:)g(:)g(:)i(;)e(S)2660 4730 y Fs(k)2732 4715 y Ft(2)29 b Fy(big)c Fw(be)h Fv(k)i Fw(linearly)g(independent)0 4828 y(vector)o(s.)j(Let)22 b Fv(\014)502 4842 y Fs(i)556 4828 y Fu(=)j(0)e Fw(if)813 4804 y Fu(^)794 4828 y Fv(f)9 b Fu(\()p Fv(S)939 4842 y Fs(i)967 4828 y Fu(\))26 b Fv(>)f Fu(0)p Fw(,)e(and)h Fv(\014)1425 4842 y Fs(i)1479 4828 y Fu(=)h(1)e Fw(if)1736 4804 y Fu(^)1716 4828 y Fv(f)9 b Fu(\()p Fv(S)1861 4842 y Fs(i)1890 4828 y Fu(\))25 b Fv(<)g Fu(0)p Fw(.)k(De\002ne)1287 5021 y Fv(G)1384 4971 y Fp(def)1397 5021 y Fu(=)39 b Ft(f)p Fv(y)51 b Fu(:)63 b Ft(h)q Fv(y)s(;)15 b(S)1916 5035 y Fs(i)1944 5021 y Ft(i)26 b Fu(=)e Fv(\014)2151 5035 y Fs(i)2180 5021 y Fv(;)38 b(i)26 b Fu(=)f(1)p Fv(:::k)s Ft(g)0 5191 y Fw(wher)m(e)36 b Ft(h)p Fv(y)s(;)15 b(T)e Ft(i)49 b Fu(=)646 5127 y Fi(L)738 5154 y Fs(n)738 5214 y(j)t Fp(=1)880 5191 y Fv(y)925 5205 y Fs(j)990 5191 y Ft(\001)30 b Fv(T)1098 5205 y Fs(j)1135 5191 y Fw(.)65 b(Then)36 b Fv(G)f Fw(is)h(an)g(af)n (\002ne)i(subspace)g(of)e(dimension)i Fv(n)29 b Ft(\000)g Fv(k)s Fw(,)38 b(and)f Fu(dist)o(\()p Fv(G;)15 b(F)3705 5205 y Fp(1)3746 5191 y Fu(\))48 b Ft(\024)0 5340 y(j)p Fv(p)20 b Ft(\000)g Fu(2)227 5307 y FH(\000)p Fs(k)325 5340 y Ft(j)h Fu(+)e Fv(k)s(\021)569 5277 y Fn(1)p 570 5289 31 4 v 570 5330 a(2)615 5340 y Fw(,)j(wher)m(e)i Fu(dist)o(\()p Fv(G;)15 b(F)1253 5354 y Fp(1)1294 5340 y Fu(\))1354 5290 y Fp(def)1368 5340 y Fu(=)38 b(2)1522 5307 y FH(\000)p Fs(n)1644 5340 y Ft(\001)21 b Fu(\()p Ft(j)p Fv(G)g Ft(n)g Fv(F)1966 5354 y Fp(1)2006 5340 y Ft(j)f Fu(+)g Ft(j)p Fv(F)2225 5354 y Fp(1)2285 5340 y Ft(n)h Fv(G)p Ft(j)p Fu(\))p Fw(.)1905 5589 y Fy(13)p eop %%Page: 14 14 14 13 bop 0 91 a Fj(Pr)n(oof:)53 b Fy(Let)25 b Fv(M)36 b Fy(be)26 b(a)g Fv(k)g Ft(\002)c Fv(n)j Fy(matrix)h(with)h(ro)n(ws)e Fv(S)1695 105 y Fp(1)1735 91 y Fv(;)15 b(:::;)g(S)1946 106 y Fs(k)1990 91 y Fy(.)36 b(Consider)28 b(the)e(linear)i (transformation)h(from)e Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)3755 58 y Fs(n)3829 91 y Fy(to)0 204 y Ft(f)p Fu(0)p Fv(;)g Fu(1)p Ft(g)220 171 y Fs(k)265 204 y Fy(,)27 b(taking)h Fv(x)f Fy(to)g Fv(M)10 b(x)p Fy(.)38 b(Since)27 b(the)h(ro)n(ws)e(of)h Fv(M)37 b Fy(are)27 b(linearly)i(independent,)i(the)d(rank)f(of)g Fv(M)37 b Fy(is)27 b Fv(k)i Fy(and)f(the)f(linear)0 317 y(transformation)35 b(is)c Fw(onto)p Fy(.)53 b(The)31 b(set)g Fv(G)g Fy(is)g(a)g(pre-image)i(of)e(the)h(v)o(ector)g Fu(\()p Fv(\014)2449 331 y Fp(1)2490 317 y Fv(;)15 b(:::;)g(\014)2696 332 y Fs(k)2740 317 y Fu(\))31 b Fy(in)g Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)3127 284 y Fs(k)3203 317 y Fy(and)31 b(therefore)j(is)d(an)0 430 y(af)n(\002ne)23 b(subspace)j(of)e (dimension)h Fv(n)20 b Ft(\000)g Fv(k)s Fy(.)141 563 y(W)-7 b(e)23 b(turn)h(to)f(the)h(second)h(part)f(of)g(the)g(claim.)29 b(W)-7 b(e)22 b(\002rst)h(sho)n(w)h(that)g(for)g(an)o(y)f(\002x)o(ed)g Fu(1)j Ft(\024)f Fv(i)h Ft(\024)f Fv(k)s Fy(:)1227 756 y Ft(j)p Fv(F)1310 770 y Fp(1)1371 756 y Ft(\\)19 b(f)q Fv(y)51 b Fu(:)63 b Ft(h)p Fv(y)s(;)15 b(S)1860 770 y Fs(i)1889 756 y Ft(i)25 b(6)p Fu(=)g Fv(\014)2096 770 y Fs(i)2125 756 y Ft(g)15 b(j)26 b(\024)f Fu(2)2377 718 y Fs(n)p FH(\000)p Fp(1)2514 756 y Fv(\021)2562 718 y Fp(1)p Fs(=)p Fp(2)0 949 y Fy(Since)f Fv(f)32 b Fy(is)23 b(the)h(characteristic)j(function)f(of)d Fv(F)1508 963 y Fp(1)1548 949 y Fy(,)619 1152 y Fu(^)600 1176 y Fv(f)9 b Fu(\()p Fv(S)745 1190 y Fs(i)773 1176 y Fu(\))26 b(=)963 1114 y(1)p 940 1155 93 4 v 940 1238 a(2)985 1212 y Fs(n)1080 1095 y Fi(X)1057 1279 y Fs(x)p FH(2)p Fs(F)1189 1288 y Fn(1)1223 1176 y Fu(\()p Ft(\000)p Fu(1\))1409 1138 y FH(h)r Fs(x;S)1541 1148 y Fh(i)1566 1138 y FH(i)1681 1176 y Fu(=)1868 1114 y(1)p 1845 1155 V 1845 1238 a(2)1890 1212 y Fs(n)1962 1082 y Fi(\020)2012 1176 y Ft(j)p Fv(F)2095 1190 y Fp(1)2135 1176 y Ft(j)20 b(\000)g Fu(2)2316 1078 y Fi(\014)2316 1128 y(\014)2316 1178 y(\014)2360 1176 y Ft(f)p Fv(y)28 b Ft(2)d Fv(F)2622 1190 y Fp(1)2687 1176 y Fu(:)h Ft(h)p Fv(y)s(;)15 b(S)2917 1190 y Fs(i)2945 1176 y Ft(i)26 b Fu(=)f(1)p Ft(g)3208 1078 y Fi(\014)3208 1128 y(\014)3208 1178 y(\014)3235 1082 y(\021)1681 1449 y Fu(=)83 b Fv(p)20 b Ft(\000)2070 1388 y Fu(1)p 2002 1428 183 4 v 2002 1512 a(2)2047 1486 y Fs(n)p FH(\000)p Fp(1)2194 1352 y Fi(\014)2194 1402 y(\014)2194 1452 y(\014)2237 1449 y Ft(f)p Fv(y)28 b Ft(2)d Fv(F)2499 1463 y Fp(1)2564 1449 y Fu(:)h Ft(h)p Fv(y)s(;)15 b(S)2794 1463 y Fs(i)2822 1449 y Ft(i)26 b Fu(=)f(1)p Ft(g)3085 1352 y Fi(\014)3085 1402 y(\014)3085 1452 y(\014)3113 1449 y Fv(:)0 1688 y Fy(Assume)f(that)g Fv(\014)526 1702 y Fs(i)580 1688 y Fu(=)h(0)p Fy(,)d(and)i(therefore,)1314 1664 y Fu(^)1294 1688 y Fv(f)9 b Fu(\()p Fv(S)1439 1702 y Fs(i)1468 1688 y Fu(\))25 b Ft(\025)g Fv(p)20 b Ft(\000)g Fv(\021)1829 1655 y Fp(1)p Fs(=)p Fp(2)1939 1688 y Fy(.)28 b(It)c(follo)n(ws)g (that,)513 1799 y Fi(\014)513 1849 y(\014)513 1899 y(\014)556 1897 y Ft(f)q Fv(y)k Ft(2)c Fv(F)818 1911 y Fp(1)909 1897 y Fu(:)51 b Ft(h)p Fv(y)s(;)15 b(S)1164 1911 y Fs(i)1192 1897 y Ft(i)26 b Fu(=)f(1)p Ft(g)1455 1799 y Fi(\014)1455 1849 y(\014)1455 1899 y(\014)1558 1897 y Fu(=)1705 1799 y Fi(\014)1705 1849 y(\014)1705 1899 y(\014)1732 1897 y Fv(F)1790 1911 y Fp(1)1850 1897 y Ft(\\)20 b(f)q Fv(y)28 b Fu(:)d Ft(h)q Fv(y)s(;)15 b(S)2280 1911 y Fs(i)2308 1897 y Ft(i)25 b(6)p Fu(=)g Fv(\014)2515 1911 y Fs(i)2544 1897 y Ft(g)2605 1799 y Fi(\014)2605 1849 y(\014)2605 1899 y(\014)2708 1897 y Ft(\024)75 b Fu(2)2899 1859 y Fs(n)p FH(\000)p Fp(1)3037 1897 y Fv(\021)3085 1859 y Fp(1)p Fs(=)p Fp(2)3195 1897 y Fv(:)0 2132 y Fy(The)23 b(case)h(that)g Fv(\014)553 2146 y Fs(i)607 2132 y Fu(=)h(1)e Fy(and)945 2108 y Fu(^)925 2132 y Fv(f)10 b Fu(\()p Fv(S)1071 2146 y Fs(i)1099 2132 y Fu(\))26 b Ft(\024)e(\000)p Fu(\()p Fv(p)c Ft(\000)g Fv(\021)1576 2069 y Fn(1)p 1577 2081 31 4 v 1577 2123 a(2)1621 2132 y Fu(\))j Fy(is)h(similar)-5 b(.)29 b(Therefore,)c(we)e(ha)n(v)o(e:)980 2404 y Ft(j)p Fv(F)1063 2418 y Fp(1)1123 2404 y Ft(\\)d Fv(G)p Ft(j)84 b Fu(=)1537 2307 y Fi(\014)1537 2356 y(\014)1537 2406 y(\014)1565 2404 y Fv(F)1623 2418 y Fp(1)1683 2404 y Ft(\\)1802 2298 y Fs(k)1775 2323 y Fi(\\)1764 2505 y Fs(i)p Fp(=1)1893 2404 y Ft(f)q Fv(y)51 b Fu(:)63 b Ft(h)p Fv(y)s(;)15 b(S)2302 2418 y Fs(i)2331 2404 y Ft(i)25 b Fu(=)g Fv(\014)2538 2418 y Fs(i)2567 2404 y Ft(g)2627 2307 y Fi(\014)2627 2356 y(\014)2627 2406 y(\014)1384 2707 y Ft(\025)82 b(j)p Fv(F)1620 2721 y Fp(1)1661 2707 y Ft(j)20 b(\000)1838 2601 y Fs(k)1797 2626 y Fi(X)1800 2809 y Fs(i)p Fp(=1)1932 2610 y Fi(\014)1932 2660 y(\014)1932 2709 y(\014)1960 2707 y Fv(F)2018 2721 y Fp(1)2078 2707 y Ft(\\)g(f)p Fv(y)51 b Fu(:)63 b Ft(h)q Fv(y)s(;)15 b(S)2568 2721 y Fs(i)2596 2707 y Ft(i)25 b(6)p Fu(=)g Fv(\014)2803 2721 y Fs(i)2832 2707 y Ft(g)2893 2610 y Fi(\014)2893 2660 y(\014)2893 2709 y(\014)1384 2947 y Ft(\025)82 b(j)p Fv(F)1620 2961 y Fp(1)1661 2947 y Ft(j)20 b(\000)g Fu(2)1842 2909 y Fs(n)p FH(\000)p Fp(1)1980 2947 y Fv(k)s(\021)2088 2882 y Fn(1)p 2088 2894 V 2088 2935 a(2)2133 2947 y Fv(:)0 3140 y Fy(So)1082 3350 y Fu(dist)o(\()p Fv(G;)15 b(F)1433 3364 y Fp(1)1474 3350 y Fu(\))83 b(=)1779 3289 y(1)p 1756 3329 93 4 v 1756 3413 a(2)1801 3386 y Fs(n)1873 3350 y Fu(\()q Ft(j)p Fv(G)p Ft(j)21 b Fu(+)f Ft(j)p Fv(F)2225 3364 y Fp(1)2265 3350 y Ft(j)g(\000)g Fu(2)p Ft(j)p Fv(G)i Ft(\\)e Fv(F)2703 3364 y Fp(1)2742 3350 y Ft(j)p Fu(\))1592 3566 y Ft(\024)1779 3504 y Fu(1)p 1756 3545 V 1756 3628 a(2)1801 3602 y Fs(n)1873 3472 y Fi(\020)1923 3566 y Fu(\()p Ft(j)p Fv(G)p Ft(j)i(\000)d(j)p Fv(F)2274 3580 y Fp(1)2315 3566 y Ft(j)p Fu(\))h(+)g(2)2531 3528 y Fs(n)2579 3566 y Fv(k)s(\021)2687 3501 y Fn(1)p 2687 3513 31 4 v 2687 3554 a(2)2732 3472 y Fi(\021)1592 3791 y Fu(=)1777 3730 y(1)p 1756 3770 89 4 v 1756 3854 a(2)1801 3827 y Fs(k)1874 3791 y Ft(\000)1975 3730 y(j)p Fv(F)2058 3744 y Fp(1)2098 3730 y Ft(j)p 1975 3770 149 4 v 2003 3854 a Fu(2)2048 3827 y Fs(n)2154 3791 y Fu(+)g Fv(k)s(\021)2353 3727 y Fn(1)p 2353 3739 31 4 v 2353 3780 a(2)1592 3984 y Ft(\024)1746 3887 y Fi(\014)1746 3937 y(\014)1746 3986 y(\014)1774 3984 y Fv(p)g Ft(\000)f Fu(2)1975 3947 y FH(\000)p Fs(k)2073 3887 y Fi(\014)2073 3937 y(\014)2073 3986 y(\014)2121 3984 y Fu(+)h Fv(k)s(\021)2320 3920 y Fn(1)p 2321 3932 V 2321 3973 a(2)0 4191 y Fy(and)k(we)f(are)g (done.)p 720 4191 67 67 v 0 4370 a Fj(Pr)n(oof)h(of)g(Lemma)e(7:)75 b Fy(The)23 b(proof)h(follo)n(ws)h(immediately)g(from)e(Claims)h(10)f (and)h(11.)p 2926 4370 V 0 4637 a Fo(4.2)99 b(Analysis)24 b(of)h(the)h(Closur)n(e-Under)l(-Intersection)i(T)-9 b(est)0 4831 y Fy(W)i(e)23 b(\002rst)g(recall)h(se)n(v)o(eral)h(simple) f(properties)i(of)e(af)n(\002ne)f(spaces.)0 5029 y Fj(Claim)g(12)46 b Fw(Let)29 b Fv(H)35 b Fw(be)29 b(an)g(af)n(\002ne)i(subspace)h(suc)o (h)d(that)h Fv(H)42 b Fu(=)36 b Fv(V)44 b Ft(\010)24 b Fv(x)p Fw(,)29 b(wher)m(e)h Fv(x)35 b Ft(2)g(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)3013 4987 y Fs(n)3089 5029 y Fw(and)29 b Fv(V)56 b Ft(\022)35 b(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)3690 4987 y Fs(n)3766 5029 y Fw(is)29 b(a)0 5142 y(linear)c(subspace)o(.)31 b(Then,)114 5340 y(1.)45 b Fv(x)25 b Ft(2)g Fv(H)7 b Fw(.)1905 5589 y Fy(14)p eop %%Page: 15 15 15 14 bop 114 91 a Fw(2.)45 b(F)-10 b(or)29 b(e)o(very)g Fv(z)38 b Ft(2)c Fv(H)h Fw(we)28 b(have)h Fv(H)41 b Fu(=)34 b Fv(V)44 b Ft(\010)24 b Fv(z)t Fw(.)43 b(By)28 b(the)g(de\002nition)j (of)e(the)g Ft(\010)e Fw(oper)o(ator)-10 b(,)31 b(we)d(thus)h(also)g (have)h(that)227 204 y Fv(V)46 b Fu(=)25 b Fv(H)i Ft(\010)20 b Fv(z)t Fw(,)j(for)g(e)o(very)i Fv(z)k Ft(2)c Fv(H)7 b Fw(.)114 392 y(3.)45 b Ft(j)p Fv(H)7 b Ft(j)26 b Fu(=)f Ft(j)p Fv(V)20 b Ft(j)26 b Fu(=)f(2)772 359 y Fs(dimV)956 392 y Fw(.)0 624 y Fj(Claim)e(13)46 b Fw(Let)23 b Fv(H)7 b Fw(,)23 b Fv(H)747 591 y FH(0)793 624 y Fw(be)g(two)g(af)n(\002ne)i (subspaces)i(of)c Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1983 591 y Fs(n)2032 624 y Fw(,)23 b(suc)o(h)h(that)g Fv(H)32 b Ft(6\022)25 b Fv(H)2717 591 y FH(0)2741 624 y Fw(.)i(Then:)1674 816 y Ft(j)p Fv(H)g Ft(\\)20 b Fv(H)1966 783 y FH(0)1989 816 y Ft(j)p 1674 857 341 4 v 1777 940 a(j)p Fv(H)7 b Ft(j)2050 878 y(\024)2156 816 y Fu(1)p 2156 857 46 4 v 2156 940 a(2)2211 878 y Fv(:)0 1160 y Fj(Pr)n(oof:)57 b Fy(The)28 b(claim)g(follo)n(ws)h(from)f(the)h(corresponding)j (property)f(of)d(linear)i(subspaces,)i(namely)d Fv(V)54 b Ft(6\022)34 b Fv(V)3556 1127 y FH(0)3607 1160 y Fy(implies:)10 1239 y FH(j)p Fs(V)15 b FH(\\)p Fs(V)190 1216 y Fr(0)213 1239 y FH(j)p 10 1263 223 4 v 73 1316 a(j)p Fs(V)g FH(j)267 1284 y Ft(\024)373 1248 y Fp(1)p 373 1263 36 4 v 373 1315 a(2)419 1284 y Fv(:)p 542 1284 67 67 v 0 1466 a Fy(The)23 b(follo)n(wing)i(corollary)h(is)d(immediate:)0 1698 y Fj(Cor)n(ollary)j(14)46 b Fw(Let)28 b Fv(H)7 b Fw(,)28 b Fv(H)897 1665 y FH(0)948 1698 y Fw(be)g(two)f(af)n(\002ne)j (subspaces)h(of)d Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2161 1665 y Fs(n)2238 1698 y Fw(suc)o(h)28 b(that)h Fv(H)2682 1665 y FH(0)2739 1698 y Ft(\022)34 b Fv(H)7 b Fw(.)42 b(Then)28 b(either)h Fv(H)3523 1665 y FH(0)3580 1698 y Fu(=)34 b Fv(H)7 b Fw(,)28 b(or)0 1811 y Ft(j)p Fv(H)108 1778 y FH(0)131 1811 y Ft(j)e(\024)f(j)p Fv(H)7 b Ft(j)p Fv(=)p Fu(2)p Fw(.)0 2043 y Fj(Claim)23 b(15)46 b Fw(Let)26 b Fv(H)7 b Fw(,)27 b Fv(H)754 2010 y FH(0)803 2043 y Fw(be)f(two)g(af)n(\002ne)i(subspaces)i(of)c Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2008 2010 y Fs(n)2083 2043 y Fw(suc)o(h)27 b(that)g Fv(H)2524 2010 y FH(0)2578 2043 y Ft(\022)k Fv(H)i Fw(and)27 b(let)f Fv(y)34 b Ft(2)c Fv(H)3320 2010 y FH(0)3344 2043 y Fw(.)37 b(Denote)27 b(by)f Fv(V)3877 2010 y FH(0)0 2156 y Fw(the)e(linear)h(subspace)h(suc)o(h)e(that)g Fv(H)1165 2123 y FH(0)1213 2156 y Fu(=)h Fv(V)1383 2123 y FH(0)1426 2156 y Ft(\010)20 b Fv(y)s Fw(,)i(and)i(by)g Fv(V)43 b Fw(the)24 b(linear)h(subspace)h(suc)o(h)e(that)g Fv(H)32 b Fu(=)25 b Fv(V)40 b Ft(\010)20 b Fv(y)s Fw(.)28 b(Then:)114 2363 y(1.)45 b Fv(V)301 2330 y FH(0)349 2363 y Ft(\022)25 b Fv(V)20 b Fw(.)114 2551 y(2.)45 b(F)-10 b(or)24 b(any)g Fv(x)h Ft(2)g Fv(V)43 b Fw(we)23 b(have)h Fu(\()p Fv(H)1226 2518 y FH(0)1270 2551 y Ft(\010)c Fv(x)p Fu(\))25 b Ft(\022)g Fv(H)7 b Fw(,)23 b(and)h(for)f(any)h Fv(x)i Ft(62)f Fv(V)43 b Fw(we)22 b(have)j Fu(\()p Fv(H)2835 2518 y FH(0)2878 2551 y Ft(\010)20 b Fv(x)p Fu(\))h Ft(\\)f Fv(H)32 b Fu(=)25 b Ft(;)p Fw(.)0 2783 y Fj(Pr)n(oof:)47 b Fy(By)23 b(de\002nition,)i Fv(V)887 2750 y FH(0)936 2783 y Fu(=)g Fv(H)1115 2750 y FH(0)1158 2783 y Ft(\010)20 b Fv(y)28 b Ft(\022)d Fv(H)i Ft(\010)20 b Fv(y)28 b Fu(=)d Fv(V)20 b Fy(.)28 b(This)c(pro)o(v)o(es)g(the)g(\002rst)f(part)h(of)g (the)f(lemma.)141 2915 y(No)n(w)f(let)g Fv(x)k Ft(2)f Fv(V)20 b Fy(.)27 b(Since)c Fv(V)43 b Fy(and)23 b Fv(V)1283 2882 y FH(0)1328 2915 y Fy(are)g(linear)h(subspaces)i(and)d Fv(V)2302 2882 y FH(0)2351 2915 y Ft(\022)i Fv(V)20 b Fy(,)i(then)i Fu(\()p Fv(V)2852 2882 y FH(0)2893 2915 y Ft(\010)17 b Fv(x)p Fu(\))26 b Ft(\022)f Fv(V)20 b Fy(.)28 b(Thus,)22 b Fv(H)3623 2882 y FH(0)3664 2915 y Ft(\010)17 b Fv(x)25 b Fu(=)0 3028 y(\()p Fv(V)109 2995 y FH(0)147 3028 y Ft(\010)15 b Fv(y)s Fu(\))g Ft(\010)g Fv(x)24 b Fu(=)h(\()p Fv(V)698 2995 y FH(0)736 3028 y Ft(\010)15 b Fv(x)p Fu(\))g Ft(\010)g Fv(y)27 b Ft(\022)e Fv(V)35 b Ft(\010)15 b Fv(y)28 b Fu(=)d Fv(H)7 b Fy(.)27 b(On)22 b(the)g(other)h(hand,)g(let)f Fv(x)j Ft(62)g Fv(V)c Fy(.)27 b(Observ)o(e)c(that)f Fu(\()p Fv(V)3336 2995 y FH(0)3374 3028 y Ft(\010)15 b Fv(x)p Fu(\))g Ft(\\)g Fv(V)45 b Fu(=)25 b Ft(;)p Fy(.)0 3141 y(Since)f Fv(H)308 3108 y FH(0)351 3141 y Ft(\010)19 b Fv(x)25 b Fu(=)g(\()p Fv(V)723 3108 y FH(0)766 3141 y Ft(\010)19 b Fv(y)s Fu(\))h Ft(\010)g Fv(x)25 b Fu(=)g(\()p Fv(V)1332 3108 y FH(0)1375 3141 y Ft(\010)19 b Fv(x)p Fu(\))h Ft(\010)g Fv(y)s Fy(,)i(we)h(get)g (that)h Fu(\()p Fv(H)2295 3108 y FH(0)2339 3141 y Ft(\010)19 b Fv(x)p Fu(\))h Ft(\\)g Fv(H)32 b Fu(=)25 b(\()p Fv(V)2929 3108 y FH(0)2972 3141 y Ft(\010)20 b Fv(x)p Fu(\))g Ft(\010)g Fv(y)i Ft(\\)e Fu(\()p Fv(V)40 b Ft(\010)19 b Fv(y)s Fu(\))26 b(=)f Ft(;)p Fy(.)0 3254 y(This)e(concludes)k(the)c(proof)i (of)f(the)f(claim.)p 1485 3254 V 0 3486 a(T)-7 b(o)18 b(pro)o(v)o(e)h(Lemma)f(8)h(we)f(will)h(need)h(se)n(v)o(eral)g (auxiliary)h(claims.)28 b(The)19 b(\002rst)f(claim)i(relates)g(af)n (\002ne)f(spaces)i(that)e(correspond)0 3599 y(to)k Fv(k)s Fy(-monomials)j(and)e(monotonicity)-6 b(.)0 3831 y Fj(Claim)23 b(16)46 b Fw(Let)22 b Fv(H)28 b Fw(be)23 b(an)f(af)n(\002ne)i(subspace) g(of)f Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1747 3798 y Fs(n)1817 3831 y Fw(of)22 b(size)h Fu(2)2112 3798 y Fs(n)p FH(\000)p Fs(k)2253 3831 y Fw(.)k(Assume)22 b(also)h(that)g Fv(H)29 b Fw(is)22 b(monotone)o(.)29 b(Namely)-5 b(,)22 b(if)0 3944 y Fv(x)j Ft(2)g Fv(H)30 b Fw(and)24 b Fv(y)k Ft(\027)d Fv(x)p Fw(,)d(then)j Fv(y)j Ft(2)d Fv(H)7 b Fw(.)27 b(Then)d Fv(H)32 b Fu(=)25 b Ft(f)p Fv(x)h Fu(:)f Fv(x)1800 3958 y Fs(i)1824 3967 y Fn(1)1888 3944 y Fu(=)g(1)p Fv(:::x)2156 3958 y Fs(i)2180 3970 y Fh(k)2249 3944 y Fu(=)g(1)p Ft(g)p Fw(,)e(for)h(some)g(subset)h Fv(i)3101 3958 y Fp(1)3140 3944 y Fv(:::i)3246 3959 y Fs(k)3313 3944 y Fw(of)e(coor)m(dinates.)0 4176 y Fj(Pr)n(oof:)53 b Fy(Let)25 b Fv(V)46 b Fy(be)26 b(an)h Fv(n)22 b Ft(\000)g Fv(k)28 b Fy(dimensional)h(linear)f(subspace)g(and)f(let)g Fv(y)33 b Ft(2)d(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2729 4143 y Fs(n)2804 4176 y Fy(be)26 b(such)h(that)g Fv(H)37 b Fu(=)30 b Fv(V)43 b Ft(\010)22 b Fv(y)s Fy(.)36 b(Let)0 4289 y Fv(v)44 4303 y Fp(1)84 4289 y Fv(:::v)203 4304 y Fs(n)p FH(\000)p Fs(k)374 4289 y Fy(be)31 b(a)f(basis)i(of)f Fv(V)20 b Fy(.)50 b(Consider)33 b(an)e Fu(\()p Fv(n)25 b Ft(\000)h Fv(k)s Fu(\))g Ft(\002)f Fv(n)30 b Fy(matrix)i(with)e(ro)n (ws)h Fv(v)2711 4303 y Fp(1)2750 4289 y Fv(:::v)2869 4304 y Fs(n)p FH(\000)p Fs(k)3011 4289 y Fy(.)50 b(Its)31 b(rank)g(is)g Fv(n)25 b Ft(\000)h Fv(k)s Fy(,)32 b(and)0 4402 y(therefore)g(it)e(has)g Fv(n)24 b Ft(\000)h Fv(k)32 b Fy(linearly)g(independent)h(columns.)49 b(W)l(ithout)31 b(loss)g(of)f(generality)-6 b(,)33 b(these)e(are)f(the)g(\002rst)g Fv(n)24 b Ft(\000)h Fv(k)0 4515 y Fy(columns.)30 b(Therefore)23 b(the)g(restriction)i(of)d(the)h(ro)n(ws)f(to)g(the)h(\002rst)f Fv(n)15 b Ft(\000)h Fv(k)24 b Fy(coordinates)i(is)c(a)g(basis)h(of)g Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)3409 4482 y Fs(n)p FH(\000)p Fs(k)3551 4515 y Fy(,)22 b(and)g(thus)0 4628 y(it)h(spans)i(all)f(the)g(v)o(ectors)h(in)e Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1146 4586 y Fs(n)p FH(\000)p Fs(k)1309 4628 y Fy(and)24 b(in)f(particular)k(the)c(\002rst)h Fv(n)c Ft(\000)g Fv(k)26 b Fy(coordinates)g(of)e Fv(y)s Fy(.)k(It)23 b(follo)n(ws)i(that)f(there)g(is)0 4741 y(a)i(v)o(ector)i Fv(v)35 b Ft(2)c Fv(V)20 b Fy(,)26 b(namely)i(a)e(linear)i(combination)i(of)d(the)g(ro)n(ws,)g(that)g(is)g (equal)h(to)e Fv(y)j Fy(on)e(the)g(\002rst)g Fv(n)22 b Ft(\000)g Fv(k)29 b Fy(coordinates.)0 4854 y(Therefore,)c Fv(z)k Fu(=)c(\()p Fv(v)f Ft(\010)c Fv(y)s Fu(\))26 b Ft(2)e Fv(H)30 b Fy(is)23 b Fu(0)h Fy(on)f(the)h(\002rst)f Fv(n)d Ft(\000)g Fv(k)26 b Fy(coordinates.)141 4986 y(Since)37 b Fv(H)42 b Fy(is)36 b(monotone,)41 b(if)36 b Ft(j)p Fv(z)t Ft(j)50 b Fv(<)e(k)s Fy(,)39 b(or)d(there)h(e)o(xists)h(a)e Fv(z)2178 4953 y FH(0)2250 4986 y Ft(6\027)49 b Fv(z)39 b Fy(such)e(that)g Fv(z)2871 4953 y FH(0)2944 4986 y Ft(2)48 b Fv(H)7 b Fy(,)39 b(then)e Ft(j)p Fv(H)7 b Ft(j)49 b Fv(>)f Fu(2)3736 4953 y Fs(n)p FH(\000)p Fs(k)3877 4986 y Fy(,)0 5099 y(contradicting)36 b(our)d(assumption)i(on)e Fv(H)7 b Fy(.)55 b(Hence)33 b Fv(H)49 b Fu(=)41 b Ft(f)p Fv(x)i Fu(:)f Fv(x)2154 5113 y Fs(i)2178 5122 y Fn(1)2258 5099 y Fu(=)g(1)p Fv(;)15 b Ft(\001)g(\001)g(\001)i Fv(;)e(x)2670 5113 y Fs(i)2694 5125 y Fh(k)2779 5099 y Fu(=)41 b(1)p Ft(g)33 b Fy(where)g Fv(i)3299 5113 y Fp(1)3338 5099 y Fv(;)15 b(:)g(:)g(:)i(;)e(i)3571 5114 y Fs(k)3646 5099 y Fy(are)33 b(the)0 5212 y(coordinates)27 b(on)c(which)h Fv(z)j Fy(is)d(1.)p 1125 5212 V 1905 5589 a(15)p eop %%Page: 16 16 16 15 bop 141 91 a Fy(Recall)33 b(that)f(by)g(the)h(premise)g(of)f (Lemma)f(8,)i(there)g(e)o(xists)g(a)f(function)i Fv(g)h Fy(such)e(that)g Fu(dist)o(\()p Fv(f)5 b(;)15 b(g)s Fu(\))42 b Ft(\024)e Fu(2)3524 58 y FH(\000)p Fs(k)r FH(\000)p Fp(3)3713 91 y Fy(,)33 b(and)0 227 y Fv(G)71 241 y Fp(1)155 177 y(def)168 227 y Fu(=)57 b Ft(f)p Fv(x)44 b Fu(:)h Fv(g)s Fu(\()p Fv(x)p Fu(\))g(=)e(1)p Ft(g)34 b Fy(is)g(an)f(af)n (\002ne)h(subspace)i(of)d(dimension)j Fv(n)27 b Ft(\000)h Fv(k)s Fy(.)58 b(Claim)33 b(16)h(implies)g(that)g(if)g Fv(g)i Fy(is)e(not)g(a)0 340 y Fv(k)s Fy(-monomial,)25 b(then)f(the)g(af)n(\002ne)g(subspace)i Fv(G)1458 354 y Fp(1)1520 340 y Fy(cannot)g(be)d(monotone.)31 b(W)-7 b(e)23 b(shall)h(use)g(this,)g(together)i(with)e(the)f(f)o(act)i(that)0 453 y Fv(f)31 b Fy(and)22 b Fv(g)k Fy(are)c(close,)h(to)f(pro)o(v)o(e)h (that)g(there)g(are)g(man)o(y)f(pairs)h Fv(x)i Ft(2)g Fv(F)2113 467 y Fp(1)2153 453 y Fy(,)c Fv(y)28 b Ft(2)d(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2578 411 y Fs(n)2647 453 y Fy(such)23 b(that)g Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)15 b Ft(^)g Fv(y)s Fu(\))p Fy(.)27 b(T)-7 b(o)22 b(this)0 565 y(end)i(we)f(de\002ne)h(the) f(follo)n(wing)i(subsets.)0 804 y Fj(De\002nition)e(9)46 b Fw(Let)22 b Fv(x)k Ft(2)f(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1020 771 y Fs(n)1091 804 y Fw(and)25 b Fv(z)k Ft(2)c Fu(2)1453 771 y Fs(x)1497 804 y Fw(.)j(De\002ne)c Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))2114 754 y Fp(def)2128 804 y Fu(=)38 b Ft(f)p Fv(y)s Ft(j)23 b Fv(x)d Ft(^)g Fv(y)28 b Fu(=)d Fv(z)t Ft(g)p Fw(.)141 1024 y Fy(W)-7 b(e)21 b(shall)i(sho)n(w)e(that)i(for)e(man)o(y)h(pairs)h Fu(\()p Fv(x;)15 b(z)t Fu(\))p Fy(,)22 b(with)g Fv(x)j Ft(2)g Fv(G)2058 1038 y Fp(1)2119 1024 y Fy(and)d Fv(z)29 b Ft(2)c Fu(2)2473 991 y Fs(x)2517 1024 y Fy(,)c(the)h(function)i Fv(g)h Fy(is)d(f)o(ar)f(from)h(constant)i(on)0 1136 y Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))p Fy(.)30 b(Since)24 b(the)g(functions)i Fv(f)32 b Fy(and)24 b Fv(g)j Fy(are)c(close)i(to)f (each)g(other)l(,)h(this)f(will)f(imply)h(the)g(e)o(xistence)i(of)d (man)o(y)h(violating)0 1249 y(pairs,)g(as)g(desired.)30 b(First,)23 b(we)g(pro)o(v)o(e)h(some)g(properties)i(of)d(the)h (subsets)i Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))p Fy(.)0 1469 y Fj(Claim)23 b(17)46 b Fw(F)-10 b(or)36 b(e)o(very)g Fv(x)48 b Ft(2)g(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1217 1436 y Fs(n)1301 1469 y Fw(and)36 b Fv(z)52 b Ft(2)47 b Fu(2)1719 1436 y Fs(x)1764 1469 y Fw(,)37 b Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))37 b Fw(is)e(an)h(af)n(\002ne)h(subspace)i (of)c Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)3290 1436 y Fs(n)3374 1469 y Fw(of)36 b(size)g Fu(2)3696 1436 y Fs(n)p FH(\000j)p Fs(x)p FH(j)3877 1469 y Fw(.)0 1582 y(Furthermor)m(e)o(,)25 b(for)f(e)o(very)g Fv(x)h Ft(2)g(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1241 1549 y Fs(n)1290 1582 y Fw(,)23 b(the)g(af)n(\002ne)i (subspaces)i Ft(f)p Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))p Ft(g)2453 1596 y Fs(z)s FH(2)p Fp(2)2571 1576 y Fh(x)2639 1582 y Fw(partition)26 b Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)3200 1549 y Fs(n)3249 1582 y Fw(.)0 1801 y Fj(Pr)n(oof:)68 b Fy(These)34 b(f)o(acts)h(about)g Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))35 b Fy(follo)n(w)f(easily)h(from)f(the)g(follo)n(wing)i (observ)n(ation:)53 b(for)34 b(a)f(\002x)o(ed)h Fv(x)p Fy(,)h(the)f(map)0 1914 y Fv(m)80 1928 y Fs(x)149 1914 y Fu(:)25 b Fv(y)k Ft(!)c Fv(x)20 b Ft(^)g Fv(y)25 b Fy(is)f(a)f(linear)h(map)g(from)f Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1587 1881 y Fs(n)1659 1914 y Fy(to)23 b Fu(2)1797 1881 y Fs(x)1842 1914 y Fy(,)f(and)i Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))27 b(=)e Fv(m)2523 1881 y FH(\000)p Fp(1)2523 1936 y Fs(x)2617 1914 y Fu(\()p Fv(z)t Fu(\))p Fy(.)p 2861 1914 67 67 v 0 2133 a Fj(Claim)e(18)46 b Fw(Let)23 b Fv(x)j Ft(2)e Fv(G)769 2147 y Fp(1)832 2133 y Fw(be)g(suc)o(h)g(that)g(ther)m(e)g(e)n(xists)h Fv(z)k Ft(2)c Fu(2)1924 2100 y Fs(x)1991 2133 y Fw(for)f(whic)o(h)g Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))27 b Ft(\022)e Fv(G)2832 2147 y Fp(1)2872 2133 y Fw(.)j(Then,)23 b Fv(G)p Fu(\()p Fv(x;)15 b(x)p Fu(\))27 b Ft(\022)e Fv(G)3629 2147 y Fp(1)3668 2133 y Fw(.)0 2352 y Fj(Pr)n(oof:)47 b Fy(W)-7 b(e)22 b(\002rst)h(sho)n(w)h(that)g Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))21 b Ft(\010)f Fv(x)f Ft(\010)h Fv(z)29 b Ft(\022)c Fv(G)1759 2366 y Fp(1)1799 2352 y Fy(.)j(Since)c Fv(G)2146 2366 y Fp(1)2208 2352 y Fy(is)f(an)h(af)n(\002ne)f(subspace,) j(by)d(F)o(act)g(5,)g(is)g(it)g(enough)i(to)0 2465 y(sho)n(w)g(that)g Fv(x)g Fy(and)g Fv(z)k Fy(lie)c(in)g Fv(G)958 2479 y Fp(1)997 2465 y Fy(,)g(and)g(that)h Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))26 b Fy(is)f(a)f(subset)j(of)e Fv(G)2238 2479 y Fp(1)2278 2465 y Fy(.)32 b(T)-7 b(aking)25 b(into)h(account)h (the)e(assumptions)j(of)d(the)0 2578 y(Claim,)e(we)f(only)j(need)f(to)g (sho)n(w)f Fv(z)30 b Ft(2)24 b Fv(G)1309 2592 y Fp(1)1349 2578 y Fy(.)k(Since)c Fv(z)29 b Ft(\026)c Fv(x)p Fy(,)e(we)g(ha)n(v)o (e)h Fv(z)g Ft(^)c Fv(x)25 b Fu(=)g Fv(z)t Fy(.)j(Hence,)c Fv(z)30 b Ft(2)24 b Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))27 b Ft(\022)e Fv(G)3535 2592 y Fp(1)3575 2578 y Fy(.)141 2711 y(Ne)o(xt,)d(we)g(sho)n(w)g(that)h Fv(G)p Fu(\()p Fv(x;)15 b(x)p Fu(\))26 b Ft(\022)f Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))j Ft(\010)d Fv(x)h Ft(\010)g Fv(z)t Fy(.)27 b(T)-7 b(ak)o(e)22 b Fv(y)29 b Ft(2)24 b Fv(G)p Fu(\()p Fv(x;)15 b(x)p Fu(\))p Fy(.)29 b(No)n(w)-6 b(,)21 b(de\002ne)i Fv(y)3095 2678 y FH(0)3140 2711 y Fy(as)f(follo)n(ws.)29 b(If)22 b Fv(z)3682 2725 y Fs(i)3736 2711 y Fu(=)j(1)p Fy(,)0 2824 y(then)k Fv(y)232 2791 y FH(0)229 2848 y Fs(i)292 2824 y Fu(=)34 b(1)28 b Fy(\(in)h(this)g(case)h(al)o(w)o(ays)f Fv(x)1273 2838 y Fs(i)1336 2824 y Fu(=)34 b(1)p Fy(\).)44 b(If)29 b Fv(z)1714 2838 y Fs(i)1777 2824 y Fu(=)34 b(0)28 b Fy(and)h Fv(x)2166 2838 y Fs(i)2229 2824 y Fu(=)35 b(1)p Fy(,)29 b(then)g Fv(y)2664 2791 y FH(0)2661 2848 y Fs(i)2724 2824 y Fu(=)34 b(0)p Fy(,)29 b(and)g(if)g Fv(z)3211 2838 y Fs(i)3274 2824 y Fu(=)34 b(0)29 b Fy(and)g Fv(x)3664 2838 y Fs(i)3726 2824 y Fu(=)35 b(0)p Fy(,)0 2936 y(then)28 b Fv(y)231 2903 y FH(0)228 2961 y Fs(i)289 2936 y Fu(=)k Fv(y)437 2950 y Fs(i)465 2936 y Fy(.)40 b(Thus,)29 b Fv(y)809 2903 y FH(0)855 2936 y Ft(^)23 b Fv(x)33 b Fu(=)f Fv(z)f Fy(and)d(so)g Fv(y)1514 2903 y FH(0)1570 2936 y Ft(2)k Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))p Fy(.)42 b(It)28 b(is)f(also)h(easy)h(to)e(v)o(erify)i(that)f Fv(y)3090 2903 y FH(0)3136 2936 y Ft(\010)23 b Fv(x)g Ft(\010)g Fv(z)37 b Fu(=)c Fv(y)s Fy(.)40 b(\(Note)0 3049 y(that)26 b Fv(y)31 b Ft(\027)d Fv(x)p Fy(,)d(and)h(therefore)h Fv(x)996 3063 y Fs(i)1053 3049 y Fu(=)h(1)d Fy(implies)h Fv(y)1559 3063 y Fs(i)1615 3049 y Fu(=)j(1)p Fy(\).)k(Hence,)26 b Fv(y)32 b Ft(2)c Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))23 b Ft(\010)e Fv(x)h Ft(\010)f Fv(z)t Fy(.)33 b(Since)26 b(we)e(ha)n(v)o(e)i(sho)n(wn)g(that)0 3162 y Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))22 b Ft(\010)e Fv(x)g Ft(\010)g Fv(z)30 b Ft(\022)24 b Fv(G)793 3176 y Fp(1)833 3162 y Fy(,)f(the)h(claim)f(follo)n(ws.)p 1635 3162 V 0 3340 a(W)-7 b(e)23 b(shall)h(be)g(interested)i(in)d(the)h(follo)n(wing)h (set:)1346 3551 y Ft(X)1449 3501 y Fp(def)1463 3551 y Fu(=)38 b Ft(f)p Fv(x)26 b Ft(2)f Fv(G)1852 3565 y Fp(1)1917 3551 y Fu(:)48 b Fv(G)p Fu(\()p Fv(x;)15 b(x)p Fu(\))27 b Ft(\022)e Fv(G)2469 3565 y Fp(1)2509 3551 y Ft(g)1240 b Fy(\(5\))0 3743 y(Thus)24 b Ft(X)36 b Fy(consists)26 b(of)e(those)h Fv(x)h Ft(2)g Fv(G)1167 3757 y Fp(1)1230 3743 y Fy(for)e(which)g(e)n(v)o(ery)h Fv(y)k Ft(\027)c Fv(x)e Fy(is)h(in)g Fv(G)2320 3757 y Fp(1)2360 3743 y Fy(.)29 b(Since,)24 b(by)g(Claim)g(16,)f Fv(G)3227 3757 y Fp(1)3291 3743 y Fy(is)g(not)i(monotone,)0 3856 y(necessarily)31 b Ft(X)47 b(6)p Fu(=)33 b Fv(G)718 3870 y Fp(1)758 3856 y Fy(.)42 b(As)27 b(we)g(sho)n(w)h(momentarily)-6 b(,)31 b Ft(X)40 b Fy(is)28 b(actually)i(signi\002cantly)h(smaller)e(than)g Fv(G)3346 3870 y Fp(1)3385 3856 y Fy(,)g(and)f(we)f(shall)0 3968 y(e)o(xploit)e(this)f(in)f(our)h(proof.)0 4188 y Fj(Claim)f(19)46 b Fw(The)24 b(set)f Ft(X)36 b Fw(is)24 b(an)f(af)n(\002ne)i(subspace)h(of)e Fv(G)1721 4202 y Fp(1)1761 4188 y Fw(.)k(Furthermor)m(e)o(,)c(if)g Fv(g)i Fw(is)d(not)h(a)f Fv(k)s Fw(-monomial)i(then)f Ft(jX)13 b(j)26 b(\024)3661 4152 y Fp(1)p 3661 4167 36 4 v 3661 4219 a(2)3707 4188 y Ft(j)p Fv(G)3803 4202 y Fp(1)3843 4188 y Ft(j)p Fw(.)0 4407 y Fj(Pr)n(oof:)46 b Fy(By)23 b(F)o(act)f(5,)h(in)g(order)h(to)f(pro)o(v)o(e)h(the)f(\002rst)g(part)h (of)f(the)g(lemma)g(it)g(suf)n(\002ces)h(to)f(sho)n(w)g(that)g(for)h(e) n(v)o(ery)f Fv(x)3511 4374 y Fp(1)3551 4407 y Fv(;)15 b(x)3643 4374 y Fp(2)3682 4407 y Fv(;)g(x)3774 4374 y Fp(3)3839 4407 y Ft(2)0 4520 y(X)e Fy(,)24 b(we)g(ha)n(v)o(e)h Fv(x)499 4487 y Fp(1)560 4520 y Ft(\010)c Fv(x)704 4487 y Fp(2)764 4520 y Ft(\010)g Fv(x)908 4487 y Fp(3)975 4520 y Ft(2)27 b(X)13 b Fy(.)31 b(Let)25 b(us)f(\002x)g Fv(x)1617 4487 y Fp(1)1657 4520 y Fv(;)15 b(x)1749 4487 y Fp(2)1788 4520 y Fv(;)g(x)1880 4487 y Fp(3)1948 4520 y Ft(2)27 b(X)13 b Fy(,)24 b(and)h(let)g Fv(x)i Fu(=)g Fv(x)2660 4487 y Fp(1)2721 4520 y Ft(\010)21 b Fv(x)2865 4487 y Fp(2)2925 4520 y Ft(\010)g Fv(x)3069 4487 y Fp(3)3108 4520 y Fy(.)32 b(T)-7 b(o)24 b(sho)n(w)g(that)h Fv(x)j Ft(2)f(X)0 4633 y Fy(we)h(ha)n(v)o(e)i(to)e(sho)n(w)h(that)h Fv(G)p Fu(\()p Fv(x;)15 b(x)p Fu(\))36 b Ft(\022)f Fv(G)1310 4647 y Fp(1)1350 4633 y Fy(.)44 b(Namely)-6 b(,)30 b(that)f(for)h(e)n (v)o(ery)f Fv(y)38 b Ft(\027)d Fv(x)p Fy(,)29 b(we)f(ha)n(v)o(e)h Fv(y)38 b Ft(2)d Fv(G)3149 4647 y Fp(1)3189 4633 y Fy(.)44 b(Let)28 b Fv(y)38 b Ft(\027)d Fv(x)p Fy(.)44 b(Then)0 4746 y(there)29 b(e)o(xist)g Fv(y)455 4713 y Fp(1)494 4746 y Fv(;)15 b(y)582 4713 y Fp(2)622 4746 y Fv(;)g(y)710 4713 y Fp(3)777 4746 y Fy(such)29 b(that)f Fv(y)37 b Fu(=)c Fv(y)1368 4713 y Fp(1)1431 4746 y Ft(\010)24 b Fv(y)1574 4713 y Fp(2)1637 4746 y Ft(\010)f Fv(y)1779 4713 y Fp(3)1818 4746 y Fy(,)28 b(where)h Fv(y)2167 4713 y Fs(j)2237 4746 y Ft(\027)k Fv(x)2393 4760 y Fs(j)2457 4746 y Fy(for)c Fv(j)39 b Fu(=)33 b(1)p Fv(:::)p Fu(3)p Fy(.)44 b(\(T)-7 b(o)27 b(v)o(erify)i(this,)h(choose)g(a)0 4869 y(coordinate)e Fv(i)p Fy(:)56 b(\(1\))25 b(If)f Fv(y)779 4883 y Fs(i)835 4869 y Fu(=)j Fv(x)985 4883 y Fs(i)1014 4869 y Fy(:)k(set)25 b Fv(y)1243 4825 y Fs(j)1240 4895 y(i)1307 4869 y Fu(=)j Fv(x)1458 4825 y Fs(j)1458 4895 y(i)1518 4869 y Fy(for)d(all)g Fv(j)5 b Fy(.)57 b(\(2\))25 b(If)g Fv(y)2145 4883 y Fs(i)2201 4869 y Fu(=)i(1)e Fy(and)g Fv(x)2576 4883 y Fs(i)2632 4869 y Fu(=)i(0)p Fy(:)32 b(Set)24 b Fv(y)3020 4825 y Fs(j)3017 4895 y(i)3085 4869 y Fu(=)j(1)e Fy(for)g(all)g Fv(j)5 b Fy(.\))33 b(That)25 b(is,)0 4982 y Fv(y)48 4949 y Fs(j)110 4982 y Ft(2)f Fv(G)p Fu(\()p Fv(x)353 4949 y Fs(j)391 4982 y Fv(;)15 b(x)483 4949 y Fs(j)520 4982 y Fu(\))25 b Ft(\022)g Fv(G)747 4996 y Fp(1)787 4982 y Fy(.)j(Therefore)d Fv(y)1267 4949 y Fs(j)1329 4982 y Ft(2)g Fv(G)1486 4996 y Fp(1)1548 4982 y Fy(for)f(all)g Fv(j)5 b Fy(,)23 b(and)h(so)f Fv(y)28 b Fu(=)d Fv(y)2353 4949 y Fp(1)2413 4982 y Ft(\010)20 b Fv(y)2552 4949 y Fp(2)2611 4982 y Ft(\010)g Fv(y)2750 4949 y Fp(3)2815 4982 y Ft(2)k Fv(G)2971 4996 y Fp(1)3011 4982 y Fy(.)141 5114 y(By)29 b(Corollary)h(14,)h(since)f Ft(X)42 b Fy(is)29 b(an)g(af)n(\002ne)g(subspace)j(of)d Fv(G)2086 5128 y Fp(1)2126 5114 y Fy(,)g(either)i Ft(X)49 b Fu(=)35 b Fv(G)2705 5128 y Fp(1)2745 5114 y Fy(,)29 b(or)h Ft(jX)13 b(j)36 b(\024)3182 5078 y Fp(1)p 3182 5093 V 3182 5146 a(2)3228 5114 y Ft(j)p Fv(G)3324 5128 y Fp(1)3364 5114 y Ft(j)p Fy(.)45 b(If)29 b Ft(X)49 b Fu(=)35 b Fv(G)3837 5128 y Fp(1)3877 5114 y Fy(,)0 5227 y(then)26 b(for)f(an)o(y)g Fv(x)j Ft(2)f Fv(G)704 5241 y Fp(1)768 5227 y Fy(we)d(ha)n(v)o(e)i Fv(G)p Fu(\()p Fv(x;)15 b(x)p Fu(\))29 b(=)f Ft(f)p Fv(y)j Fu(:)52 b Fv(y)31 b Ft(\027)d Fv(x)p Ft(g)g(\022)g Fv(G)2172 5241 y Fp(1)2212 5227 y Fy(,)c(namely)i Fv(G)2622 5241 y Fp(1)2686 5227 y Fy(is)e(monotone.)35 b(By)25 b(Claim)f(16,)h Fv(g)j Fy(is)d(a)0 5340 y Fv(k)s Fy(-monomial,)g(which)f(contradicts)i(our)e (assumptions.)32 b(Therefore,)24 b Ft(jX)13 b(j)26 b(\024)2469 5304 y Fp(1)p 2469 5319 V 2469 5371 a(2)2514 5340 y Ft(j)p Fv(G)2610 5354 y Fp(1)2651 5340 y Ft(j)p Fy(.)p 2803 5340 67 67 v 1905 5589 a(16)p eop %%Page: 17 17 17 16 bop 141 91 a Fy(In)28 b(the)h(ne)o(xt)g(claim)f(we)g(sho)n(w)g (that)h(for)g(e)n(v)o(ery)f Fv(x)35 b Ft(2)e Fv(G)1921 105 y Fp(1)1985 91 y Ft(n)24 b(X)13 b Fy(,)29 b(the)g(function)h Fv(g)h Fy(is)e(f)o(ar)f(from)g(constant)j(on)d Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))p Fy(,)0 204 y(for)27 b(man)o(y)g Fv(z)36 b Ft(2)c Fu(2)575 171 y Fs(x)619 204 y Fy(.)38 b(Observ)o(e)28 b(that)g(this)f(is)g(tri)n(vially)i(true)f(if)f Fv(g)i Fy(is)e(a)g(monotone)i(monomial,)f(since)g(in)f(this)h(case)g (the)f(set)0 317 y Fv(G)71 331 y Fp(1)131 317 y Ft(n)21 b(X)36 b Fy(is)23 b(empty)-6 b(.)0 543 y Fj(Claim)23 b(20)46 b Fw(F)-10 b(or)24 b(e)o(very)g Fv(x)h Ft(2)g Fv(G)998 557 y Fp(1)1058 543 y Ft(n)c(X)13 b Fw(,)23 b(and)h(for)g(any)g(\002xed)g(function)i Fv(h)f Fu(:)h Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2555 501 y Fs(n)2627 543 y Ft(!)25 b(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)s Fw(,)1203 718 y Fu(1)p 1161 758 129 4 v 1161 844 a(2)1206 818 y FH(j)p Fs(x)p FH(j)1320 779 y Ft(\001)1384 698 y Fi(X)1365 879 y Fs(z)s FH(2)p Fp(2)1483 861 y Fh(x)1537 779 y Fu(Pr)1635 798 y Fs(y)r FH(2)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))1928 779 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))27 b Ft(6)p Fu(=)e Fv(h)p Fu(\()p Fv(z)t Fu(\)])h Ft(\025)f Fu(2)2600 742 y FH(\000)p Fs(k)2724 779 y Fv(:)0 1081 y Fj(Pr)n(oof:)58 b Fy(Let)29 b(us)g(\002x)f Fv(x)36 b Ft(2)f Fv(G)940 1095 y Fp(1)1004 1081 y Ft(n)25 b(X)41 b Fy(and)30 b(a)e(function)k Fv(h)p Fy(.)45 b(F)o(or)28 b(e)n(v)o(ery)h Fv(z)40 b Ft(2)35 b Fu(2)2462 1048 y Fs(x)2507 1081 y Fy(,)29 b(if)g Fv(h)p Fu(\()p Fv(z)t Fu(\))37 b(=)e(0)29 b Fy(then)h Fu(Pr)3310 1100 y Fs(y)r FH(2)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))3603 1081 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))37 b Ft(6)p Fu(=)0 1219 y Fv(h)p Fu(\()p Fv(z)t Fu(\)])27 b(=)326 1175 y FH(j)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))p FH(\\)p Fs(G)653 1184 y Fn(1)687 1175 y FH(j)p 326 1198 381 4 v 394 1251 a(j)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))p FH(j)739 1219 y Fy(and)d(if)f Fv(h)p Fu(\()p Fv(z)t Fu(\))k(=)e(1)e Fy(then)h Fu(Pr)1607 1237 y Fs(y)r FH(2)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))1900 1219 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))j Ft(6)p Fu(=)e Fv(h)p Fu(\()p Fv(z)t Fu(\)])h(=)2537 1175 y FH(j)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))p FH(n)p Fs(G)2852 1184 y Fn(1)2887 1175 y FH(j)p 2537 1198 369 4 v 2600 1251 a(j)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))p FH(j)2941 1219 y Fu(=)f(1)c Ft(\000)3204 1175 y FH(j)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))p FH(\\)p Fs(G)3531 1184 y Fn(1)3565 1175 y FH(j)p 3204 1198 381 4 v 3272 1251 a(j)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))p FH(j)3594 1219 y Fy(.)28 b(Hence,)634 1491 y Fu(Pr)731 1509 y Fs(y)r FH(2)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))1024 1491 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))f Ft(6)p Fu(=)e Fv(h)p Fu(\()p Fv(z)t Fu(\)])i Ft(\025)e Fu(min)1819 1372 y Fi(\032)1891 1429 y Ft(j)p Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))22 b Ft(\\)e Fv(G)2369 1443 y Fp(1)2409 1429 y Ft(j)p 1891 1470 544 4 v 1997 1553 a(j)p Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))p Ft(j)2444 1491 y Fv(;)g Fu(1)21 b Ft(\000)2651 1429 y(j)p Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))22 b Ft(\\)e Fv(G)3129 1443 y Fp(1)3169 1429 y Ft(j)p 2651 1470 V 2757 1553 a(j)p Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))p Ft(j)3204 1372 y Fi(\033)3794 1491 y Fy(\(6\))0 1737 y(But,)25 b(for)g(all)g Fv(z)32 b Ft(2)27 b Fu(2)631 1704 y Fs(x)676 1737 y Fy(,)d Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))30 b Ft(6\022)d Fv(G)1201 1751 y Fp(1)1265 1737 y Fy(\(otherwise,)g(by)e(Claim)g(18,)g(we)f(w)o (ould)h(ha)n(v)o(e)h Fv(G)p Fu(\()p Fv(x;)15 b(x)p Fu(\))29 b Ft(\022)f Fv(G)3253 1751 y Fp(1)3293 1737 y Fy(,)c(and)i(so)f Fv(x)i Ft(2)h(X)13 b Fy(\).)0 1862 y(Thus,)23 b(by)h(Claim)f(13,)732 1818 y FH(j)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))p FH(\\)p Fs(G)1059 1827 y Fn(1)1093 1818 y FH(j)p 732 1841 381 4 v 801 1894 a(j)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))p FH(j)1148 1862 y Ft(\024)1254 1827 y Fp(1)p 1254 1842 36 4 v 1254 1894 a(2)1299 1862 y Fy(.)28 b(Combining)d(this)f(with)g(Equation)h (\(6\),)347 2073 y Fu(1)p 305 2113 129 4 v 305 2199 a(2)350 2173 y FH(j)p Fs(x)p FH(j)464 2134 y Ft(\001)528 2053 y Fi(X)510 2234 y Fs(z)s FH(2)p Fp(2)628 2215 y Fh(x)682 2134 y Fu(Pr)779 2153 y Fs(y)r FH(2)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))1072 2134 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))i Ft(6)p Fu(=)e Fv(h)p Fu(\()p Fv(z)t Fu(\)])85 b Ft(\025)1867 2073 y Fu(1)p 1825 2113 V 1825 2199 a(2)1870 2173 y FH(j)p Fs(x)p FH(j)1984 2134 y Ft(\001)2048 2053 y Fi(X)2030 2234 y Fs(z)s FH(2)p Fp(2)2148 2215 y Fh(x)2212 2073 y Ft(j)p Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))22 b Ft(\\)e Fv(G)2690 2087 y Fp(1)2730 2073 y Ft(j)p 2212 2113 544 4 v 2317 2196 a(j)p Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))p Ft(j)1662 2369 y Fu(=)82 b(2)1860 2332 y FH(\000)p Fs(n)1983 2369 y Ft(\001)2047 2288 y Fi(X)2028 2470 y Fs(z)s FH(2)p Fp(2)2146 2451 y Fh(x)2200 2369 y Ft(j)p Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))23 b Ft(\\)c Fv(G)2678 2383 y Fp(1)2718 2369 y Ft(j)51 b Fu(=)f(2)2960 2332 y FH(\000)p Fs(n)3083 2369 y Ft(\001)20 b(j)p Fv(G)3224 2383 y Fp(1)3265 2369 y Ft(j)50 b Fu(=)g(2)3506 2332 y FH(\000)p Fs(k)0 2658 y Fy(In)22 b(the)g(last)g(sequence)i(of)e (steps)h(we)e(ha)n(v)o(e)h(used)h(the)f(follo)n(wing:)51 b(\(1\))22 b Ft(j)p Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))p Ft(j)28 b Fu(=)d(2)2716 2625 y Fs(n)p FH(\000j)p Fs(x)p FH(j)2918 2658 y Fy(for)d(e)n(v)o(ery)g Fv(z)j Fy(\(Claim)d(17\);)44 b(\(2\))0 2771 y(F)o(or)23 b(e)n(v)o(ery)h Fv(x)p Fy(,)e(the)i(subsets) h Fv(G)p Fu(\()p Fv(x;)15 b(z)t Fu(\))25 b Fy(form)e(a)g(partition)j (of)e Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2103 2729 y Fs(n)2172 2771 y Fy(\(Claim)24 b(17\);)g(\(3\))g Fv(G)2817 2785 y Fp(1)2879 2771 y Fy(is)g(of)f(size)h Fu(2)3270 2738 y Fs(n)p FH(\000)p Fs(k)3411 2771 y Fy(.)p 3538 2771 67 67 v 141 2951 a(If)31 b(the)h(closure-under)n(-inte)q(rse)q (ctio)q(n)37 b(test)32 b(in)f(Step)g(3)g(of)h(Algorithm)g(2)f(w)o(as)g (performed)j(on)d Fv(g)j Fy(and)e(not)g(on)f Fv(f)10 b Fy(,)32 b(we)0 3064 y(w)o(ould)25 b(be)f(done.)31 b(Indeed,)25 b(Claims)f(19)g(and)h(20)f(imply)g(that)h(if)f Fv(g)j Fy(is)c(not)i(a)e Fv(k)s Fy(-monomial)j(then)f Fu(Pr)3125 3084 y Fs(x)p FH(2)p Fs(G)3267 3093 y Fn(1)3301 3084 y Fs(;y)r FH(2f)p Fp(0)p Fs(;)p Fp(1)p FH(g)3566 3054 y Fh(n)3613 3064 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))i Ft(6)p Fu(=)0 3191 y Fv(g)s Fu(\()p Fv(x)12 b Ft(^)g Fv(y)s Fu(\)])27 b Ft(\025)e Fu(2)494 3158 y FH(\000)p Fp(\()p Fs(k)r Fp(+1\))737 3191 y Fy(.)i(Therefore,)d(uniformly)f (picking)g Fv(x)i Ft(2)g Fv(G)2101 3205 y Fp(1)2141 3191 y Fv(;)15 b(y)29 b Ft(2)c(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2563 3149 y Fs(n)2630 3191 y Fy(and)22 b(checking)i(that)e Fv(g)s Fu(\()p Fv(y)s Fu(\))27 b(=)e Fv(g)s Fu(\()p Fv(x)12 b Ft(^)g Fv(y)s Fu(\))p Fy(,)0 3304 y(we)23 b(w)o(ould)h(detect)h(a)e (violation)j(with)d(probability)k(at)c(least)i Fu(2)1956 3271 y FH(\000)p Fp(\()p Fs(k)r Fp(+1\))2199 3304 y Fy(.)141 3437 y(Ho)n(we)n(v)o(er)l(,)33 b(the)f(test)g(is)f(performed)i(on)f Fv(f)10 b Fy(,)32 b(and)f Fv(g)k Fy(and)d Fv(f)40 b Fy(might)31 b(dif)n(fer)i(\(though)g(the)f(distance)h(between)g(them)e(is)0 3550 y(bounded\).)f(Consequently)-6 b(,)24 b(we)c(need)h(to)f(relate)h (between)h(tw)o(o)e(dif)n(ferent)i(probabilities.)31 b(This)21 b(is)f(done)h(in)f(the)h(follo)n(wing)0 3663 y(claim.)0 3888 y Fj(Claim)i(21)46 b Fu(Pr)494 3902 y Fs(x)p FH(2)p Fs(F)626 3911 y Fn(1)660 3902 y Fs(;y)736 3888 y Fu(\()p Fv(f)10 b Fu(\()p Fv(y)s Fu(\))26 b Ft(6)p Fu(=)f Fv(f)10 b Fu(\()p Fv(x)19 b Ft(^)h Fv(y)s Fu(\)])26 b Ft(\025)1548 3853 y Fp(1)p 1548 3868 36 4 v 1548 3920 a(2)1608 3794 y Fi(\020)1658 3888 y Fu(Pr)1755 3902 y Fs(x)p FH(2)p Fs(F)1887 3911 y Fn(1)1921 3902 y FH(\\)p Fs(G)2023 3911 y Fn(1)2058 3902 y Fs(;y)2134 3888 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))h Ft(6)p Fu(=)e Fv(f)10 b Fu(\()p Fv(x)19 b Ft(^)h Fv(y)s Fu(\)])h Ft(\000)f Fu(2)2953 3855 y FH(\000)p Fs(k)r FH(\000)p Fp(3)3141 3794 y Fi(\021)3206 3888 y Fv(:)0 4141 y Fj(Pr)n(oof:)47 b Fy(Recall)24 b(that)g Fu(dist)o(\()p Fv(f)5 b(;)15 b(g)s Fu(\))26 b Ft(\024)f Fu(2)1225 4108 y FH(\000)p Fs(k)r FH(\000)p Fp(3)1414 4141 y Fy(,)d(and)i(that)g Ft(j)p Fv(F)1855 4155 y Fp(1)1895 4141 y Ft(j)i(\025)f Fu(2)2087 4108 y Fs(n)p FH(\000)p Fs(k)2248 4141 y Ft(\000)20 b Fu(2)2384 4108 y Fs(n)p FH(\000)p Fs(k)r FH(\000)p Fp(3)2615 4141 y Fy(.)28 b(Thus:)530 4339 y Fu(Pr)628 4353 y Fs(x)p FH(2)p Fs(F)760 4362 y Fn(1)794 4353 y Fs(;y)870 4339 y Fu([)p Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)g Fv(y)s Fu(\)])83 b Ft(\025)g Fu(Pr)1875 4353 y Fs(x)p FH(2)p Fs(F)2007 4362 y Fn(1)2041 4353 y Fs(;y)2117 4339 y Fu([)p Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)g Fv(y)s Fu(\))p Fv(;)38 b(x)25 b Ft(2)g Fv(G)3059 4353 y Fp(1)3099 4339 y Fu(])1623 4529 y(=)83 b(Pr)1875 4543 y Fs(x)p FH(2)p Fs(F)2007 4552 y Fn(1)2041 4543 y FH(\\)p Fs(G)2143 4552 y Fn(1)2177 4543 y Fs(;y)2253 4529 y Fu([)q Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)f Fv(y)s Fu(\)])i Ft(\001)2999 4467 y(j)p Fv(F)3082 4481 y Fp(1)3143 4467 y Ft(\\)f Fv(G)3295 4481 y Fp(1)3334 4467 y Ft(j)p 2999 4508 361 4 v 3105 4591 a(j)p Fv(F)3188 4605 y Fp(1)3228 4591 y Ft(j)1623 4767 y Fv(>)1787 4705 y Fu(1)p 1787 4746 46 4 v 1787 4829 a(2)1863 4767 y Ft(\001)g Fu(Pr)2006 4781 y Fs(x)p FH(2)p Fs(F)2138 4790 y Fn(1)2172 4781 y FH(\\)p Fs(G)2274 4790 y Fn(1)2308 4781 y Fs(;y)2384 4767 y Fu([)q Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)f Fv(y)s Fu(\)])d Fv(:)141 5004 y Fy(Also,)407 5202 y Fu(Pr)505 5216 y Fs(x)p FH(2)p Fs(F)637 5225 y Fn(1)671 5216 y FH(\\)p Fs(G)773 5225 y Fn(1)808 5216 y Fs(;y)884 5202 y Fu([)p Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)g Fv(y)s Fu(\)])83 b Ft(\025)g Fu(Pr)1888 5216 y Fs(x)p FH(2)p Fs(F)2020 5225 y Fn(1)2054 5216 y FH(\\)p Fs(G)2156 5225 y Fn(1)2191 5216 y Fs(;y)2267 5202 y Fu([)p Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)g Fv(y)s Fu(\))p Fv(;)38 b(f)10 b Fu(\()p Fv(y)s Fu(\))25 b(=)g Fv(g)s Fu(\()p Fv(y)s Fu(\)])1637 5340 y(=)83 b(Pr)1888 5354 y Fs(x)p FH(2)p Fs(F)2020 5363 y Fn(1)2054 5354 y FH(\\)p Fs(G)2156 5363 y Fn(1)2191 5354 y Fs(;y)2267 5340 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))27 b Ft(6)p Fu(=)e Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)f Fv(y)s Fu(\))p Fv(;)39 b(f)10 b Fu(\()p Fv(y)s Fu(\))25 b(=)g Fv(g)s Fu(\()p Fv(y)s Fu(\)])16 b Fv(:)1905 5589 y Fy(17)p eop %%Page: 18 18 18 17 bop 0 91 a Fy(On)23 b(the)h(other)g(hand,)606 296 y Fu(Pr)703 310 y Fs(x)p FH(2)p Fs(F)835 319 y Fn(1)869 310 y FH(\\)p Fs(G)971 319 y Fn(1)1006 310 y Fs(;y)1082 296 y Fu([)q Fv(g)s Fu(\()p Fv(y)s Fu(\))i Ft(6)p Fu(=)f Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)f Fv(y)s Fu(\)])689 433 y Ft(\024)83 b Fu(Pr)940 447 y Fs(x)p FH(2)p Fs(F)1072 456 y Fn(1)1106 447 y FH(\\)p Fs(G)1208 456 y Fn(1)1243 447 y Fs(;y)1319 433 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))27 b Ft(6)p Fu(=)e Fv(f)10 b Fu(\()p Fv(x)19 b Ft(^)h Fv(y)s Fu(\))p Fv(;)39 b(f)10 b Fu(\()p Fv(y)s Fu(\))25 b(=)g Fv(g)s Fu(\()p Fv(y)s Fu(\)])c(+)f(Pr)2713 447 y Fs(y)2769 433 y Fu([)p Fv(f)10 b Fu(\()p Fv(y)s Fu(\))26 b Ft(6)p Fu(=)e Fv(g)s Fu(\()p Fv(y)s Fu(\)])689 571 y Ft(\024)83 b Fu(Pr)940 585 y Fs(x)p FH(2)p Fs(F)1072 594 y Fn(1)1106 585 y FH(\\)p Fs(G)1208 594 y Fn(1)1243 585 y Fs(;y)1319 571 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))27 b Ft(6)p Fu(=)e Fv(f)10 b Fu(\()p Fv(x)19 b Ft(^)h Fv(y)s Fu(\))p Fv(;)39 b(f)10 b Fu(\()p Fv(y)s Fu(\))25 b(=)g Fv(g)s Fu(\()p Fv(y)s Fu(\)])c(+)f(2)2660 534 y FH(\000)p Fs(k)r FH(\000)p Fp(3)2848 571 y Fv(:)0 775 y Fy(Hence,)475 1008 y Fu(Pr)572 1022 y Fs(x)p FH(2)p Fs(F)704 1031 y Fn(1)738 1022 y Fs(;y)815 1008 y Fu([)p Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)g Fv(y)s Fu(\)])83 b Fv(>)1732 946 y Fu(1)p 1732 987 46 4 v 1732 1070 a(2)1787 1008 y(Pr)1885 1022 y Fs(x)p FH(2)p Fs(F)2017 1031 y Fn(1)2051 1022 y FH(\\)p Fs(G)2153 1031 y Fn(1)2187 1022 y Fs(;y)2263 1008 y Fu([)q Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)19 b Ft(^)h Fv(y)s Fu(\)])1568 1224 y Ft(\025)1732 1162 y Fu(1)p 1732 1203 V 1732 1286 a(2)1802 1129 y Fi(\020)1852 1224 y Fu(Pr)1949 1238 y Fs(x)p FH(2)p Fs(F)2081 1247 y Fn(1)2115 1238 y FH(\\)p Fs(G)2217 1247 y Fn(1)2252 1238 y Fs(;y)2328 1224 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))27 b Ft(6)p Fu(=)e Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)f Fv(y)s Fu(\)])i Ft(\000)f Fu(2)3147 1186 y FH(\000)p Fs(k)r FH(\000)p Fp(3)3335 1129 y Fi(\021)3400 1224 y Fv(:)p 76 1452 67 67 v 0 1584 a Fy(W)-7 b(e)23 b(are)g(no)n(w)g(ready)i (to)e(pro)o(v)o(e)h(Lemma)f(8.)0 1717 y Fj(Pr)n(oof)h(of)g(Lemma)e(8:) 75 b Fy(By)22 b(Claim)h(21,)h(it)f(suf)n(\002ces)h(to)g(bound)h(the)f (follo)n(wing)h(probability:)260 1956 y Fu(Pr)358 1970 y Fs(x)p FH(2)p Fs(F)490 1979 y Fn(1)524 1970 y FH(\\)p Fs(G)626 1979 y Fn(1)661 1970 y Fs(;y)737 1956 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))h Ft(6)p Fu(=)f Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)g Fv(y)s Fu(\)])83 b(=)1804 1895 y(1)p 1646 1935 361 4 v 1646 2019 a Ft(j)p Fv(F)1729 2033 y Fp(1)1790 2019 y Ft(\\)19 b Fv(G)1941 2033 y Fp(1)1981 2019 y Ft(j)2123 1876 y Fi(X)2032 2059 y Fs(x)p FH(2)p Fs(F)2164 2068 y Fn(1)2198 2059 y FH(\\)p Fs(G)2300 2068 y Fn(1)2401 1895 y Fu(1)p 2359 1935 129 4 v 2359 2021 a(2)2404 1995 y FH(j)p Fs(x)p FH(j)2531 1876 y Fi(X)2513 2057 y Fs(z)s FH(2)p Fp(2)2631 2038 y Fh(x)2685 1956 y Fu(Pr)2783 1975 y Fs(y)r FH(2)p Fs(G)p Fp(\()p Fs(x;z)s Fp(\))3091 1956 y Fu([)p Fv(g)s Fu(\()p Fv(y)s Fu(\))27 b Ft(6)p Fu(=)e Fv(f)10 b Fu(\()p Fv(z)t Fu(\)])15 b Fv(:)155 b Fy(\(7\))0 2239 y(Let)27 b Ft(X)41 b Fy(be)28 b(the)g(set)g(de\002ned)h(in)f(Equation)h(\(5\).)43 b(If)27 b(we)g(replace)j(the)e(summation)i(o)o(v)o(er)e Fv(x)33 b Ft(2)g Fv(F)3055 2253 y Fp(1)3118 2239 y Ft(\\)24 b Fv(G)3274 2253 y Fp(1)3341 2239 y Fy(in)k(Equation)h(\(7\))0 2351 y(with)23 b(a)g(summation)i(o)o(v)o(er)f Fv(x)h Ft(2)g Fu(\()p Fv(F)1112 2365 y Fp(1)1172 2351 y Ft(\\)20 b Fv(G)1324 2365 y Fp(1)1364 2351 y Fu(\))h Ft(n)f(X)13 b Fy(,)23 b(then,)h(by)g(Claim)f(20,)g(we)g(obtain)759 2594 y Fu(Pr)856 2608 y Fs(x)p FH(2)p Fs(F)988 2617 y Fn(1)1022 2608 y FH(\\)p Fs(G)1124 2617 y Fn(1)1159 2608 y Fs(;y)1235 2594 y Fu([)q Fv(g)s Fu(\()p Fv(y)s Fu(\))j Ft(6)p Fu(=)f Fv(f)10 b Fu(\()p Fv(x)20 b Ft(^)g Fv(y)s Fu(\)])83 b Ft(\025)2302 2532 y Fu(1)p 2145 2573 361 4 v 2145 2656 a Ft(j)p Fv(F)2228 2670 y Fp(1)2288 2656 y Ft(\\)20 b Fv(G)2440 2670 y Fp(1)2480 2656 y Ft(j)2697 2513 y Fi(X)2530 2701 y Fs(x)p FH(2)p Fp(\()p Fs(F)2689 2710 y Fn(1)2724 2701 y FH(\\)p Fs(G)2826 2710 y Fn(1)2860 2701 y Fp(\))p FH(nX)2998 2594 y Fu(2)3043 2556 y FH(\000)p Fs(k)1981 2889 y Fu(=)2145 2828 y Ft(j)p Fu(\()p Fv(F)2263 2842 y Fp(1)2323 2828 y Ft(\\)g Fv(G)2475 2842 y Fp(1)2515 2828 y Fu(\))h Ft(n)f(X)13 b(j)p 2145 2868 596 4 v 2262 2951 a(j)p Fv(F)2345 2965 y Fp(1)2405 2951 y Ft(\\)20 b Fv(G)2557 2965 y Fp(1)2597 2951 y Ft(j)2770 2889 y(\001)g Fu(2)2860 2852 y FH(\000)p Fs(k)2958 2889 y Fv(:)811 b Fy(\(8\))0 3168 y(Ho)n(we)n(v)o(er)l(,)71 b(by)62 b(Claim)f(19,)72 b Ft(jX)13 b(j)97 b(\024)1438 3124 y FH(j)p Fs(G)1513 3133 y Fn(1)1548 3124 y FH(j)p 1438 3147 129 4 v 1485 3199 a Fp(2)1673 3168 y Fu(=)f(2)1885 3135 y Fs(n)p FH(\000)p Fs(k)r FH(\000)p Fp(1)2116 3168 y Fy(.)143 b(Also)62 b Ft(X)110 b(\022)96 b Fv(G)2928 3182 y Fp(1)3029 3168 y Fy(and)62 b Fu(Pr)o([)p Fv(f)106 b Ft(6)p Fu(=)96 b Fv(g)s Fu(])h(=)31 3245 y Fp(1)p 10 3260 78 4 v 10 3312 a(2)45 3293 y Fh(n)113 3281 y Fu(\()p Ft(j)p Fv(F)231 3295 y Fp(1)292 3281 y Ft(n)20 b Fv(G)428 3295 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Fu(\))41 b(=)f Ft(f)p Fv(i)p Ft(g)p Fy(.)53 b(Then,)34 b(for)e(e)n(v)o(ery)0 204 y(neighbor)24 b Fv(y)k Ft(2)d Fv(N)10 b Fu(\()p Fv(x)p Fu(\))p Fy(,)22 b(we)f(must)h(ha)n(v)o(e)g Fv(i)k Ft(2)f Fv(S)5 b Fu(\()p Fv(y)s Fu(\))21 b Fy(\(or)h(else)h Fv(S)5 b Fu(\()p Fv(x)14 b Ft(^)g Fv(y)s Fu(\))21 b Fy(w)o(ould)h(be)g(empty) -6 b(,)22 b(implying)i(that)e Fv(f)10 b Fu(\()p Fv(x)k Ft(^)g Fv(y)s Fu(\))25 b(=)g(0)p Fy(\).)0 317 y(Notice)35 b(that)g(the)g(con)l(v)o(erse)i(statement)g(holds)f(as)e(well,)j(that)e (is,)i Fv(i)47 b Ft(2)e Fv(S)5 b Fu(\()p Fv(y)s Fu(\))34 b Fy(implies)i(that)f Fv(x)f Fy(and)h Fv(y)i Fy(are)e(neighbors.)0 430 y(Therefore,)30 b(the)e(set)g(of)g(neighbors)i(of)e Fv(x)f Fy(is)h(e)o(xactly)h(the)f(set)g(of)g(all)g(strings)h (satisfying)i(the)d(term)f Fv(T)3242 444 y Fs(i)3271 430 y Fy(.)40 b(The)28 b(goal)g(of)g(the)0 543 y(algorithm)e(will)e(be) h(to)f(\002nd)g(at)h(most)f Fv(`)g Fy(such)h(single-term)i (representati)n(v)o(es)h Fv(x)f Ft(2)g(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2845 501 y Fs(n)2892 543 y Fy(,)24 b(and)g(for)h(each)g (such)h Fv(x)d Fy(to)i(test)0 656 y(that)j(its)g(set)g(of)g(neighbors)i Fv(N)10 b Fu(\()p Fv(x)p Fu(\))28 b Fy(satis\002es)h(some)e(common)h (term.)42 b(W)-7 b(e)26 b(shall)j(sho)n(w)f(that)g(if)f Fv(f)37 b Fy(is)27 b(in)h(f)o(act)g(a)f(monotone)0 769 y Fv(`)p Fy(-term)d(DNF)-7 b(,)21 b(then)k(all)f(these)h(tests)g(pass)f (with)g(high)h(probability)-6 b(.)33 b(On)23 b(the)i(other)g(hand,)f (if)g(all)g(the)g(tests)h(pass)g(with)e(high)0 882 y(probability)-6 b(,)27 b(then)d Fv(f)32 b Fy(is)23 b(close)i(to)e(some)h(monotone)h Fv(`)p Fy(-term)f(DNF)-7 b(.)141 1014 y(W)g(e)26 b(start)i(with)e(a)h (high-le)n(v)o(el)i(description)h(of)d(the)g(algorithm,)i(and)e(then)h (sho)n(w)f(ho)n(w)f(to)h(implement)h(its)f(main)g(step)0 1127 y(of)c(\002nding)i(single-term)h(representati)n(v)o(es.)0 1349 y Fj(Algorithm)e(3)46 b Fy(T)-6 b(est)23 b(for)g(Monotone)j Fv(`)p Fy(-term)d(DNF)114 1548 y Fw(1.)45 b Fv(R)26 b Ft( )f(;)p Fw(.)k FE(R)21 b FF(is)g(designated)e(to)i(be)f(a)g(set)h (of)f(single-term)f(representati)n(v)o(es)f(for)i FE(f)9 b FF(.)114 1732 y Fw(2.)45 b(F)-10 b(or)24 b Fv(i)h Fu(=)g(1)f Fw(to)f Fv(`)d Fu(+)g(1)j Fw(\(T)-5 b(ry)23 b(to)h(add)g Fv(`)f Fw(single-term)i(r)m(epr)m(esentatives)j(to)23 b Fv(R)q Fw(\):)276 1928 y(\(a\))46 b(T)-8 b(ak)o(e)24 b(a)h(uniform)g(sample)h Fv(U)1353 1895 y Fs(i)1404 1928 y Fw(of)f(size)g Fv(m)1739 1942 y Fp(1)1805 1928 y Fu(=)i(\002)1989 1834 y Fi(\020)2049 1887 y Fs(`)11 b Fp(log)i Fs(`)p 2049 1907 173 4 v 2121 1959 a(\017)2231 1834 y Fi(\021)2304 1928 y Fw(strings.)34 b(Let)24 b Fv(W)2845 1895 y Fs(i)2899 1928 y Fu(=)j(\()p Fv(U)3104 1895 y Fs(i)3154 1928 y Ft(\\)20 b Fv(F)3293 1942 y Fp(1)3333 1928 y Fu(\))i Ft(n)f Fv(N)10 b Fu(\()p Fv(R)q Fu(\))p Fw(.)31 b(That)427 2065 y(is,)f Fv(W)639 2032 y Fs(i)694 2065 y Fw(consists)h(of)e (strings)h Fv(x)e Fw(in)h(the)g(sample)g(suc)o(h)g(that)g Fv(f)10 b Fu(\()p Fv(x)p Fu(\))35 b(=)g(1)p Fw(,)29 b(and)g Fv(x)f Fw(is)h(not)g(a)f(neighbor)j(of)e(any)427 2178 y(string)c(alr)m(eady)g(in)f Fv(R)q Fw(.)427 2306 y FF(Observ)o(e)e (that)h(if)g(the)g(strings)g(in)g FE(R)i FF(are)d(in)h(f)o(act)h (single-term)d(representati)n(v)o(es,)h(then)g(e)n(v)o(ery)g FE(x)28 b FC(2)h FE(W)3415 2276 y FB(i)3466 2306 y FF(satis\002es)24 b(only)427 2419 y(terms)d(not)e(satis\002ed)i(by)f(the)g(representati)n (v)o(es)f(in)h FE(R)q FF(.)276 2562 y Fw(\(b\))46 b(If)24 b Fv(i)h Fu(=)g Fv(`)20 b Fu(+)g(1)j Fw(and)h Fv(W)1133 2529 y Fs(i)1186 2562 y Ft(6)p Fu(=)h Ft(;)p Fw(,)e(then)h Fx(reject)p Fw(.)427 2690 y FF(If)c(there)g(are)g(more)g(than)f FE(`)i FF(single)f(term)g(representati)n(v)o(es)e(for)i FE(f)29 b FF(then)20 b(necessarily)f FE(f)30 b FF(is)21 b(not)f(an)g FE(`)p FF(-term)f(DNF)-7 b(.)281 2846 y Fw(\(c\))46 b(Else)o(,)23 b(if)712 2802 y FH(j)p Fs(W)809 2779 y Fh(i)834 2802 y FH(j)p 712 2825 142 4 v 734 2878 a Fs(m)796 2887 y Fn(1)889 2846 y Fv(<)998 2811 y Fs(\017)p 995 2826 36 4 v 995 2878 a Fp(4)1063 2846 y Fw(then)h Fx(go)g(to)h(Step)g(3)p Fw(.)427 2974 y FF(The)20 b(current)f(set)i(of) f(representati)n(v)o(es)e(already)h(\223co)o(v)o(ers\224)g(almost)h (all)h(of)f FE(F)2614 2986 y Fg(1)2652 2974 y FF(.)276 3131 y Fw(\(d\))46 b(Else)605 3037 y Fi(\020)665 3087 y FH(j)p Fs(W)762 3063 y Fh(i)787 3087 y FH(j)p 665 3110 142 4 v 687 3162 a Fs(m)749 3171 y Fn(1)842 3131 y Ft(\025)951 3095 y Fs(\017)p 948 3110 36 4 v 948 3162 a Fp(4)1016 3131 y Fv(and)22 b(i)k Ft(\024)f Fv(`)1379 3037 y Fi(\021)1428 3131 y Fw(,)d(use)g Fv(W)1714 3098 y Fs(i)1763 3131 y Fw(in)g(or)m(der)h(to)f(\002nd)h(a)f(string)h Fv(x)2677 3098 y Fs(i)2727 3131 y Fw(that)f(is)g(designated)k(to)c(be)g(a)g (single-)427 3254 y(term)h(r)m(epr)m(esentative)j(of)d(a)f(term)h(not)g (yet)g(r)m(epr)m(esented)j(in)c Fv(R)q Fw(.)28 b(This)22 b(step)i(will)e(be)h(described)j(subsequently)-5 b(.)114 3438 y(3.)45 b(F)-10 b(or)23 b(eac)o(h)g(string)h Fv(x)854 3405 y Fs(i)907 3438 y Ft(2)h Fv(R)q Fw(,)d(let)h(the)g(function)h Fv(f)1728 3405 y Fs(i)1781 3438 y Fu(:)i Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2054 3396 y Fs(n)2126 3438 y Ft(7!)25 b(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)24 b Fw(be)f(de\002ned)h(as)f (follows:)30 b Fv(f)3357 3405 y Fs(i)3384 3438 y Fu(\()p Fv(y)s Fu(\))c(=)f(1)d Fw(if)h(and)227 3551 y(only)i(if)e Fv(y)28 b Ft(2)d Fv(N)10 b Fu(\()p Fv(x)809 3518 y Fs(i)837 3551 y Fu(\))p Fw(.)227 3699 y FF(As)21 b(observ)o(ed)d(pre)n(viously) -5 b(,)18 b(if)j FE(x)1163 3669 y FB(i)1211 3699 y FF(is)g(in)g(f)o (act)f(a)h(single-term)e(representati)n(v)o(e,)f(then)h FE(f)2691 3669 y FB(i)2739 3699 y FF(is)i(a)g(monomial.)114 3884 y Fw(4.)45 b(F)-10 b(or)22 b(eac)o(h)g Fv(f)621 3851 y Fs(i)649 3884 y Fw(,)f(test)h(that)h(it)e(is)h(monomial,)h (using)g(distance)h(par)o(ameter)f Fv(\017)2541 3851 y FH(0)2590 3884 y Fu(=)2713 3848 y Fs(\017)p 2696 3863 65 4 v 2696 3915 a Fp(2)p Fs(`)2791 3884 y Fw(and)f(con\002dence)j Fu(1)14 b Ft(\000)3526 3848 y Fp(1)p 3512 3863 V 3512 3915 a(6)p Fs(`)3607 3884 y Fw(\(instead)227 3997 y(of)331 3961 y Fp(2)p 331 3976 36 4 v 331 4028 a(3)399 3997 y Fw(\227)22 b(this)i(can)g(simply)g(be)g(done)g(by)g Fv(O)s Fu(\(log)17 b Fv(`)p Fu(\))23 b Fw(r)m(epeated)i(applications)i(of)d (eac)o(h)g(test\).)227 4145 y FF(Note)g(that)g(we)h(do)f(not)f(specify) h(the)g(size)h(of)e(the)h(monomial,)g(and)f(so)h(we)h(need)e(to)i (apply)e(the)h(appropriate)d(v)n(ariant)j(of)f(our)227 4258 y(test,)e(as)g(described)e(in)h(Subsection)f(4.3.)114 4442 y Fw(5.)45 b(If)24 b(any)g(of)f(the)h(tests)g(fail)g(then)g Fx(reject)p Fw(,)f(otherwise)i Fx(accept)p Fw(.)141 4664 y Fy(The)c(heart)h(of)f(the)h(algorithm)h(lies)e(in)g(\002nding)h(a)f (ne)n(w)g(representati)n(v)o(e)j(in)d(each)h(iteration)i(of)d(Step)g (2.)28 b(This)21 b(procedure)0 4777 y(will)i(be)h(described)i(and)e (analyzed)i(shortly)-6 b(.)30 b(In)24 b(particular)l(,)i(we)d(shall)h (pro)o(v)o(e)g(the)g(follo)n(wing)h(lemma.)0 4976 y Fj(Lemma)e(22)46 b Fw(Suppose)33 b(that)e Fv(f)40 b Fw(is)30 b(an)h Fv(`)p Fw(-term)g(monotone)h(DNF)-12 b(,)29 b(and)i(let)g Fv(R)39 b Ft(\032)g(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2860 4934 y Fs(n)2937 4976 y Fw(be)31 b(a)f(subset)i(of)f(single-term)0 5089 y(r)m(epr)m(esentatives)d(for)d Fv(f)33 b Fw(suc)o(h)25 b(that)g Fu(Pr)14 b([)q Fv(x)25 b Ft(2)g Fv(F)1493 5103 y Fp(1)1553 5089 y Ft(n)20 b Fv(N)10 b Fu(\()p Fv(R)q Fu(\)])28 b Ft(\025)e Fv(\017=)p Fu(8)p Fw(.)32 b(Let)23 b Fv(U)2384 5056 y Fs(i)2436 5089 y Fw(be)h(a)g(uniformly)i(selected)h (sample)e(of)f Fv(m)3763 5103 y Fp(1)3829 5089 y Fu(=)0 5217 y(\002)86 5123 y Fi(\020)145 5177 y Fs(`)12 b Fp(log)h Fs(`)p 145 5196 173 4 v 217 5249 a(\017)328 5123 y Fi(\021)397 5217 y Fw(strings,)23 b(and)e(let)g Fv(W)1054 5184 y Fs(i)1107 5217 y Fu(=)k(\()p Fv(U)1310 5184 y Fs(i)1348 5217 y Ft(\\)10 b Fv(F)1477 5231 y Fp(1)1517 5217 y Fu(\))g Ft(n)g Fv(N)g Fu(\()p Fv(R)q Fu(\))p Fw(.)28 b(Then)20 b(ther)m(e)i(e)n(xists)g(a)e(pr)l(ocedur)m(e)j(that)f(r)m(eceives)g Fv(W)3542 5184 y Fs(i)3590 5217 y Fw(as)e(input,)0 5340 y(for)k(whic)o(h)f(the)h(following)h(holds:)1905 5589 y Fy(20)p eop %%Page: 21 21 21 20 bop 114 91 a Fw(1.)45 b(W)-5 b(ith)20 b(pr)l(obability)j(at)c (least)i Fu(1)6 b Ft(\000)1265 56 y Fp(1)p 1250 71 65 4 v 1250 123 a(6)p Fs(`)1324 91 y Fw(,)20 b(tak)o(en)h(o)o(ver)f(the)g (c)o(hoice)h(of)e Fv(U)2307 58 y Fs(i)2354 91 y Fw(and)i(the)f (internal)i(coin)e(\003ips)g(of)g(the)g(pr)l(ocedur)m(e)o(,)227 204 y(the)h(pr)l(ocedur)m(e)h(r)m(eturns)g(a)d(string)j Fv(x)1370 171 y Fs(i)1417 204 y Fw(that)f(is)f(a)f(single)j(term)e(r)m (epr)m(esentative)j(for)e Fv(f)28 b Fw(of)20 b(a)g(term)g(not)g(yet)h (r)m(epr)m(esented)227 317 y(in)j Fv(R)q Fw(.)j(That)d(is,)f Ft(j)p Fv(S)5 b Fu(\()p Fv(x)910 284 y Fs(i)939 317 y Fu(\))p Ft(j)25 b Fu(=)g(1)f Fw(and)g Fv(S)5 b Fu(\()p Fv(x)1496 284 y Fs(i)1524 317 y Fu(\))21 b Ft(\\)f Fv(S)5 b Fu(\()p Fv(R)q Fu(\))25 b(=)g Ft(;)p Fw(.)114 505 y(2.)45 b(The)23 b(query)i(comple)n(xity)h(of)d(the)h(pr)l(ocedur)m(e)i(is)d Fv(O)s Fu(\()p Fv(`)15 b Fu(log)2008 468 y Fp(2)2063 505 y Fv(`=\017)p Fu(\))p Fw(.)141 712 y Fy(Conditioned)26 b(on)e(the)g(abo)o(v)o(e)g(lemma)f(we)g(can)h(pro)o(v)o(e)g(the)g (follo)n(wing)h(theorem.)0 919 y Fj(Theor)n(em)f(3)45 b Fw(Algorithm)30 b(3)e(is)g(a)h(testing)h(algorithm)g(for)f Fv(`)p Fw(-term)f(DNF)-12 b(.)54 b(The)28 b(query)i(comple)n(xity)h(of) d(the)h(algorithm)h(is)21 1009 y Fu(~)0 1032 y Fv(O)s Fu(\()p Fv(`)145 999 y Fp(4)185 1032 y Fv(=\017)267 999 y Fp(3)306 1032 y Fu(\))p Fw(.)0 1262 y Fj(Pr)n(oof:)47 b Fy(W)-7 b(e)23 b(shall)h(use)g(the)g(follo)n(wing)h(notation:)31 b(for)24 b(an)o(y)g(set)g Fv(R)i Ft(\032)e(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2449 1220 y Fs(n)2496 1262 y Fy(,)22 b(let)31 b Fu(\026)-52 b Fv(p)o Fu(\()p Fv(R)q Fu(\))2866 1212 y Fp(def)2880 1262 y Fu(=)38 b(Pr)o([)p Fv(x)26 b Ft(2)f Fv(F)3333 1276 y Fp(1)3393 1262 y Ft(n)20 b Fv(N)10 b Fu(\()p Fv(R)q Fu(\)])p Fy(.)141 1394 y(Suppose)36 b Fv(f)43 b Fy(is)35 b(a)f(monotone)j Fv(`)p Fy(-term)e(DNF)-7 b(,)32 b(and)j(consider)i(each)e(iteration)i(of)e(Step)g(2.)62 b(By)34 b(Lemma)f(22,)38 b(if)c(all)0 1507 y(strings)e(in)f Fv(R)g Fy(are)g(single-term)i(representati)n(v)o(es)h(for)d Fv(f)39 b Fy(and)g Fu(\026)-52 b Fv(p)o Fu(\()p Fv(R)q Fu(\))39 b Ft(\025)g Fv(\017=)p Fu(8)p Fy(,)32 b(then)f(with)g (probability)j(at)c(least)i Fu(1)26 b Ft(\000)3818 1471 y Fp(1)p 3803 1486 V 3803 1539 a(6)p Fs(`)3877 1507 y Fy(,)0 1620 y(the)f(procedure)j(for)d(\002nding)h(a)f(single)h(term)f (representati)n(v)o(e)k(in)c(f)o(act)g(returns)i(a)e(ne)n(w)f (representati)n(v)o(e)29 b(\(of)c(a)g(term)f(not)i(yet)0 1733 y(represented)f(in)c Fv(R)q Fy(\).)27 b(Hence,)22 b(the)g(probability)j(that,)d(for)g(some)g(iteration)h Fv(i)p Fy(,)f(the)f(string)i Fv(x)2839 1700 y Fs(i)2888 1733 y Fy(returned)h(by)d(the)h(procedure)i(is)0 1846 y(not)i(a)e(single-term)k(representati)n(v)o(e,)h(is)c(at)g(most)g Fu(1)p Fv(=)p Fu(6)p Fy(.)34 b(Conditioned)28 b(on)e(such)g(an)f(e)n(v) o(ent)g(not)h(occurring,)i(the)d(algorithm)0 1959 y(completes)g(Step)f (2)f(with)g(a)g(set)h Fv(R)g Fy(that)g(contains)h(at)f(most)f Fv(`)g Fy(single-term)j(representati)n(v)o(es)h(for)d Fv(f)10 b Fy(.)141 2091 y(In)22 b(such)g(a)f(case,)h(by)g(the)f (de\002nition)j(of)d(single-term)j(representati)n(v)o(es,)h(each)d Fv(f)2657 2058 y Fs(i)2705 2091 y Fy(de\002ned)h(in)e(Step)g(3)h(is)f (a)g(\(monotone\))0 2204 y(monomial.)50 b(F)o(or)29 b(each)i(\002x)o (ed)f Fv(f)1055 2171 y Fs(i)1082 2204 y Fy(,)h(the)g(probability)i (that)e(it)f(f)o(ails)h(the)f(monomial)h(test)g(is)f(at)g(most)3246 2168 y Fp(1)p 3231 2183 V 3231 2236 a(6)p Fs(`)3306 2204 y Fy(.)48 b(By)29 b(applying)k(a)0 2317 y(union)24 b(bound,)g(the)f (probability)j(that)e(an)o(y)f(one)g(of)g(the)g Fv(f)1805 2284 y Fs(i)1832 2317 y Fy(')-5 b(s)23 b(f)o(ail,)g(is)g(at)f(most)2459 2281 y Fp(1)p 2459 2296 36 4 v 2459 2348 a(6)2504 2317 y Fy(.)28 b(Adding)c(up)f(the)g(error)h(probabilities,)i(we)0 2430 y(obtain)f(that)f Fv(f)32 b Fy(is)23 b(accepted)j(with)e (probability)i(at)e(least)g Fu(2)p Fv(=)p Fu(3)p Fy(.)141 2587 y(W)-7 b(e)25 b(no)n(w)h(turn)h(to)f(the)g(case)h(in)f(which)h Fv(f)34 b Fy(is)26 b Fv(\017)p Fy(-f)o(ar)h(from)f(being)h(a)f (monotone)i Fv(`)d Fy(term)h(DNF)-7 b(.)23 b(Consider)28 b(the)e(v)n(alue)h(of)8 2700 y Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))26 b Fy(at)h(the)g(start)h(of)f(each)g(iteration)j Fv(i)c Fy(of)h(Step)g(2.)38 b(Observ)o(e)28 b(that)35 b Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))27 b Fy(does)g(not)h (increase)h(with)d Fv(i)p Fy(.)39 b(If)34 b Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))32 b Fv(>)f(\017=)p Fu(2)p Fy(,)0 2832 y(then,)f(by)e(the)h(multiplicati)n(v)o(e)i(Chernof)n(f)e (bound,)i(the)e(probability)i(that)2365 2788 y FH(j)p Fs(W)2462 2764 y Fh(i)2488 2788 y FH(j)p 2365 2811 142 4 v 2388 2863 a Fs(m)2450 2872 y Fn(1)2551 2832 y Ft(\024)j Fv(\017=)p Fu(4)29 b Fy(\(causing)h(the)f(algorithm)h(to)e(e)o(xit)0 2960 y(Step)c(2\))g(is)g(smaller)h(than)863 2924 y Fp(1)p 848 2939 65 4 v 848 2991 a(6)p Fs(`)923 2960 y Fy(.)k(Hence,)24 b(the)h(probability)i(that)e(the)f(algorithm)i(completes)g(Step)e(2)f (without)i(rejecting)i(and)0 3072 y(with)c(a)g(set)h Fv(R)g Fy(for)f(which)32 b Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))26 b Fv(>)f(\017=)p Fu(2)p Fy(,)e(is)g(at)h(most)f Fu(1)p Fv(=)p Fu(6)p Fy(.)141 3205 y(Conditioned)28 b(on)d(such)h(an)f (e)n(v)o(ent)g(not)h(occurring,)h(consider)g(the)f(functions)h Fv(f)2669 3172 y Fs(i)2721 3205 y Fy(de\002ned)f(in)f(Step)g(3.)32 b(W)-7 b(e)24 b(claim)i(that)0 3318 y(at)e(least)h(one)g(of)g(these)g (functions)i(is)1218 3282 y Fs(\017)p 1200 3297 V 1200 3349 a Fp(2)p Fs(`)1274 3318 y Fy(-f)o(ar)e(from)f(being)i(a)e (monomial.)32 b(T)-7 b(o)24 b(v)o(erify)h(this,)g(assume)g(in)f (contradiction)29 b(that)0 3431 y(all)i(these)i Ft(j)p Fv(R)q Ft(j)40 b(\024)f Fv(`)30 b Fy(functions)k(are)1215 3395 y Fs(\017)p 1197 3410 V 1197 3462 a Fp(2)p Fs(`)1272 3431 y Fy(-close)e(to)f(being)i(monomials.)53 b(F)o(or)30 b(each)i(such)h(function,)i(let)c Fv(g)3406 3398 y Fs(i)3465 3431 y Fy(be)h(a)f(closest)0 3553 y(monomial)22 b(and)h(let)e Fv(g)29 b Fu(=)c Fv(g)867 3520 y Fp(1)920 3553 y Ft(_)13 b Fv(g)1040 3520 y Fp(2)1093 3553 y Ft(_)g Fv(:)i(:)g(:)e Ft(_)g Fv(g)1405 3520 y FH(j)p Fs(R)p FH(j)1502 3553 y Fy(.)27 b(Then)22 b Fu(dist)o(\()p Fv(f)5 b(;)15 b(g)s Fu(\))26 b Ft(\024)f(j)p Fv(R)q Ft(j)13 b(\001)2434 3517 y Fs(\017)p 2416 3532 V 2416 3585 a Fp(2)p Fs(`)2503 3553 y Fu(+)21 b(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))25 b Ft(\024)g Fv(\017)p Fy(,)c(contradicting)26 b(the)c(f)o(act)g(that)0 3666 y Fv(f)35 b Fy(is)27 b Fv(\017)p Fy(-f)o(ar)h(from)f(an)o(y)g Fv(`)p Fy(-term)g(DNF)-7 b(.)24 b(Thus,)k(let)f Fv(f)1608 3633 y Fs(t)1663 3666 y Fy(be)g(one)g(of)g(the)h Fv(f)2227 3633 y Fs(i)2254 3666 y Fy(')-5 b(s)27 b(that)h(is)2618 3630 y Fs(\017)p 2600 3645 V 2600 3698 a Fp(2)p Fs(`)2674 3666 y Fy(-f)o(ar)g(from)f(being)h(a)f(monomial.)40 b(The)0 3779 y(probability)29 b(that)d(the)h(the)f(monomial)g(test)h(does)f (not)h(reject)f Fv(f)2017 3746 y Fs(t)2071 3779 y Fy(is)g(at)g(most) 2474 3743 y Fp(1)p 2460 3758 V 2460 3810 a(6)p Fs(`)2534 3779 y Fy(.)34 b(Adding)27 b(up)f(the)g(error)h(probabilities,)j Fv(f)0 3892 y Fy(is)23 b(rejected)j(with)d(probability)k(at)c(least)i Fu(2)p Fv(=)p Fu(3)p Fy(.)141 4049 y(Finally)-6 b(,)29 b(we)d(bound)i(the)g(query)g(comple)o(xity)h(of)e(the)g(algorithm.)41 b(There)27 b(are)g(at)g(most)g Fv(`)22 b Fu(+)h(1)k Fy(iterations)i(in) e(Step)g(2.)0 4162 y(In)c(each)g(iteration,)i Fv(m)714 4176 y Fp(1)779 4162 y Fu(=)g Fv(O)s Fu(\()p Fv(`)15 b Fu(log)i Fv(`=\017)p Fu(\))22 b Fy(strings)j(are)e(queried)i(in)d (Step)h(2a.)29 b(By)22 b(Lemma)g(22,)h Fv(O)s Fu(\()p Fv(`)15 b Fu(log)3292 4125 y Fp(2)3347 4162 y Fv(`=\017)p Fu(\))22 b Fy(strings)j(are)0 4275 y(queried)k(by)f(the)g(procedure)i (for)e(\002nding)h(a)e(ne)n(w)g(representati)n(v)o(e)k(that)d(is)g (called)g(in)g(Step)f(2d.)42 b(By)27 b(Theorem)h(2,)g(testing)0 4388 y(each)g(of)e(the)h(at)g(most)g Fv(`)f Fy(functions)j Fv(f)1209 4355 y Fs(i)1263 4388 y Fy(requires)1602 4365 y Fu(~)1582 4388 y Fv(O)r Fu(\(1)p Fv(=)p Fu(\()p Fv(\017)1850 4355 y FH(0)1875 4388 y Fu(\))1910 4355 y Fp(3)1950 4388 y Fu(\))23 b Ft(\001)g Fv(O)s Fu(\(log)17 b Fv(`)p Fu(\))31 b(=)2523 4365 y(~)2502 4388 y Fv(O)s Fu(\()p Fv(`)2647 4355 y Fp(3)2687 4388 y Fv(=\017)2769 4355 y Fp(3)2809 4388 y Fu(\))26 b Fy(queries.)40 b(Therefore,)29 b(the)e(total)0 4501 y(number)d(of)g(queries)h(is)788 4478 y Fu(~)768 4501 y Fv(O)r Fu(\()p Fv(`)912 4468 y Fp(4)952 4501 y Fv(=\017)1034 4468 y Fp(3)1074 4501 y Fu(\))p Fy(.)p 1236 4501 67 67 v 0 4770 a Fo(5.1)99 b(Finding)25 b(New)g(Repr)n (esentati)o(v)o(es)0 4964 y Fy(Suppose)j(that)f Fv(f)35 b Fy(is)27 b(a)f(monotone)j Fv(`)p Fy(-term)d(DNF)-7 b(,)24 b(and)j(consider)i(an)e(arbitrary)i(iteration)g Fv(i)d Fy(in)g(Step)h(2)f(of)h(the)g(algorithm.)0 5077 y(Assume)f(that)g Fv(R)31 b Ft(\032)e(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)901 5035 y Fs(n)973 5077 y Fy(is)26 b(a)f(subset)i(of)f (single-term)i(representati)n(v)o(es)i(for)c Fv(f)10 b Fy(,)25 b(such)h(that)h Fu(Pr)o([)p Fv(x)j Ft(2)f Fv(F)3422 5091 y Fp(1)3484 5077 y Ft(n)22 b Fv(N)10 b Fu(\()p Fv(R)q Fu(\)])30 b Ft(\025)0 5212 y Fv(\017=)p Fu(8)p Fy(.)47 b(Let)p 348 5140 83 4 v 30 w Fv(N)10 b Fu(\()p Fv(R)q Fu(\))608 5162 y Fp(def)621 5212 y Fu(=)50 b Fv(F)800 5226 y Fp(1)865 5212 y Ft(n)25 b Fv(N)10 b Fu(\()p Fv(R)q Fu(\))29 b Fy(be)h(the)g(set)g(of)g(all)g(the)g(strings)h(that)g(are)f (not)g(neighbors)i(of)e(an)o(y)g(string)h(in)f Fv(R)q Fy(,)g(and)1905 5589 y(21)p eop %%Page: 22 22 22 21 bop 0 104 a Fy(let)p 125 32 61 4 v 35 w Fv(S)5 b Fu(\()p Fv(R)q Fu(\))373 54 y Fp(def)386 104 y Fu(=)60 b Ft(f)p Fu(1)p Fv(;)15 b(:)g(:)g(:)i(;)e(`)p Ft(g)30 b(n)f Fv(S)5 b Fu(\()p Fv(R)q Fu(\))35 b Fy(be)g(the)g(set)g(of)g (indices)i(of)e(terms)g(not)g(yet)g(represented)k(in)34 b Fv(R)q Fy(.)62 b(By)35 b(de\002nition,)0 217 y Fv(W)99 184 y Fs(i)152 217 y Ft(\022)p 248 145 83 4 v 25 w Fv(N)10 b Fu(\()p Fv(R)q Fu(\))p Fy(,)23 b(and)h(for)f(e)n(v)o(ery)h Fv(x)i Ft(2)f Fv(W)1283 184 y Fs(i)1333 217 y Fy(we)e(ha)n(v)o(e)h Fv(S)5 b Fu(\()p Fv(x)p Fu(\))26 b Ft(\022)p 1958 145 61 4 v 25 w Fv(S)5 b Fu(\()p Fv(R)q Fu(\))p Fy(.)141 350 y(Gi)n(v)o(en)22 b(a)f(string)j Fv(x)723 364 y Fp(0)787 350 y Ft(2)h Fv(W)972 317 y Fs(i)1000 350 y Fy(,)c(we)g(shall)i(try)f (to)g(\223remo)o(v)o(e\224)h(terms)f(from)g Fv(S)5 b Fu(\()p Fv(x)2519 364 y Fp(0)2559 350 y Fu(\))p Fy(,)21 b(until)i(we)e(are)i(left)f(with)g(a)f(single)j(term.)0 463 y(More)29 b(precisely)-6 b(,)32 b(we)27 b(produce)k(a)d(sequence)j (of)e(strings)h Fv(x)1908 477 y Fp(0)1947 463 y Fv(;)15 b(:)g(:)g(:)i(;)e(x)2201 477 y Fs(r)2239 463 y Fy(,)29 b(where)g Fv(x)2593 477 y Fp(0)2667 463 y Ft(2)34 b Fv(W)2861 430 y Fs(i)2889 463 y Fy(,)28 b(such)i(that)f Ft(;)35 b(6)p Fu(=)f Fv(S)5 b Fu(\()p Fv(x)3632 477 y Fs(j)t Fp(+1)3759 463 y Fu(\))35 b Ft(\022)0 576 y Fv(S)5 b Fu(\()p Fv(x)148 590 y Fs(j)185 576 y Fu(\))p Fy(,)23 b(and)h(in)g(particular)j Ft(j)p Fv(S)5 b Fu(\()p Fv(x)1059 590 y Fs(r)1097 576 y Fu(\))p Ft(j)27 b Fu(=)f(1)p Fy(.)j(The)23 b(aim)h(is)g(to)g(decrease)i(the)e(size)g(of)g Fv(S)5 b Fu(\()p Fv(x)2761 590 y Fs(j)2798 576 y Fu(\))23 b Fy(by)h(a)g(constant)i(f)o(actor)f(for)f(most)0 689 y Fv(j)5 b Fy(')-5 b(s.)28 b(This)22 b(will)f(ensure)i(that)f(for)g Fv(r)27 b Fu(=)e(\002\(log)17 b Fv(`)p Fu(\))p Fy(,)k(the)h(\002nal)f (string)i Fv(x)2154 703 y Fs(r)2212 689 y Fy(is)f(a)f(single-term)i (representati)n(v)o(e)i(as)d(desired.)30 b(Ho)n(w)0 801 y(is)c(such)h(a)f(sequence)i(obtained?)40 b(Gi)n(v)o(en)26 b(a)g(string)h Fv(y)1679 815 y Fs(j)1746 801 y Ft(2)i Fv(N)10 b Fu(\()p Fv(x)2006 815 y Fs(j)2043 801 y Fu(\))p Fy(,)26 b(de\002ne)h Fv(x)2427 815 y Fs(j)t Fp(+1)2584 801 y Fu(=)j Fv(x)2737 815 y Fs(j)2795 801 y Ft(^)22 b Fv(y)2923 815 y Fs(j)2959 801 y Fy(.)36 b(Then)26 b Fv(f)10 b Fu(\()p Fv(x)3372 815 y Fs(j)t Fp(+1)3499 801 y Fu(\))30 b(=)g(1)c Fy(\(i.e.,)0 914 y Fv(S)5 b Fu(\()p Fv(x)148 928 y Fs(j)t Fp(+1)275 914 y Fu(\))33 b Ft(6)p Fu(=)f Ft(;)p Fy(\),)c(and)g Fv(S)5 b Fu(\()p Fv(x)878 928 y Fs(j)t Fp(+1)1005 914 y Fu(\))32 b(=)g Fv(S)5 b Fu(\()p Fv(x)1323 928 y Fs(j)1360 914 y Fu(\))24 b Ft(\\)e Fv(S)5 b Fu(\()p Fv(y)1643 928 y Fs(j)1680 914 y Fu(\))32 b Ft(\022)h Fv(S)5 b Fu(\()p Fv(x)1999 928 y Fs(j)2035 914 y Fu(\))p Fy(.)40 b(The)27 b(string)i Fv(y)2580 928 y Fs(j)2642 914 y Fy(is)f(acquired)h(by)f(uniformly)h(selecting)0 1027 y(a)24 b(suf)n(\002ciently)j(lar)n(ge)f(sample)g(from)f Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1404 985 y Fs(n)1451 1027 y Fy(,)24 b(and)h(picking)i(the)e(\002rst)g(string)h(in)f(the)g (sample)g(that)h(belongs)g(to)f Fv(N)10 b Fu(\()p Fv(x)3725 1041 y Fs(j)3762 1027 y Fu(\))p Fy(,)24 b(if)0 1140 y(such)g(e)o (xists.)30 b(The)23 b(e)o(xact)h(procedure)i(follo)n(ws.)0 1322 y Fj(Pr)n(ocedur)n(e)e(f)n(or)g(\002nding)e(a)h(new)f(r)n(epr)n (esentati)o(v)o(e,)27 b(gi)o(v)o(en)d Fv(W)2026 1289 y Fs(i)2079 1322 y Ft(\022)p 2175 1250 83 4 v 25 w Fv(N)10 b Fu(\()p Fv(R)q Fu(\))114 1530 y Fw(1.)45 b(Let)23 b(the)h(strings)h (in)f Fv(W)963 1497 y Fs(i)1013 1530 y Fw(be)g(denoted)h Fv(w)1497 1544 y Fp(1)1537 1530 y Fv(;)15 b(:)g(:)g(:)h(;)f(w)1803 1550 y FH(j)p Fs(W)1900 1531 y Fh(i)1926 1550 y FH(j)1950 1530 y Fw(.)114 1717 y(2.)45 b(Uniformly)31 b(and)g(independently)j (select)d Fv(r)40 b Fu(=)d(\002\(log)16 b Fv(`)p Fu(\))30 b Fw(samples,)i Fv(Y)2500 1731 y Fp(0)2539 1717 y Fv(;)15 b Ft(\001)g(\001)g(\001)i Fv(;)e(Y)2794 1731 y Fs(r)r FH(\000)p Fp(1)2922 1717 y Fw(,)31 b(eac)o(h)f(consisting)j(of)d Fv(m)3753 1731 y Fp(2)3829 1717 y Fu(=)227 1830 y Fv(O)s Fu(\()p Fv(`)15 b Fu(log)i Fv(`=\017)p Fu(\))23 b Fw(strings)j(fr)l(om) d Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1381 1788 y Fs(n)1428 1830 y Fw(.)114 2018 y(3.)45 b Fv(f)10 b(ound)25 b Ft( )g Fv(F)13 b(ALS)5 b(E)g Fw(;)23 b Fv(t)j Ft( )f Fu(0)p Fw(;)114 2205 y(4.)45 b(While)24 b Fv(f)10 b(ound)25 b Ft(6)p Fu(=)g Fv(T)13 b(R)q(U)d(E)27 b Fw(and)d Fv(t)h(<)g Ft(j)p Fv(W)1575 2172 y Fs(i)1603 2205 y Ft(j)e Fw(do:)276 2393 y(\(a\))46 b Fv(t)25 b Ft( )h Fv(t)20 b Fu(+)f(1)p Fw(;)24 b Fv(x)896 2407 y Fp(0)961 2393 y Ft( )h Fv(w)1142 2407 y Fs(t)1171 2393 y Fw(.)276 2539 y(\(b\))46 b(F)-10 b(or)24 b Fv(j)31 b Fu(=)25 b(1)e Fw(to)g Fv(r)504 2685 y Fw(i.)45 b(If)24 b Fv(Y)729 2699 y Fs(j)t FH(\000)p Fp(1)875 2685 y Ft(\\)c Fv(N)10 b Fu(\()p Fv(x)1126 2699 y Fs(j)t FH(\000)p Fp(1)1253 2685 y Fu(\))26 b(=)f Ft(;)e Fw(then)h(e)n(xit)g(the)g(\223for\224)f(loop)h(and)g(go)g(to)f(\(4a\).) 479 2815 y(ii.)45 b(Otherwise)o(,)25 b(pic)n(k)f(the)g(\002r)o(st)g (string)h Fv(y)1765 2829 y Fs(j)t FH(\000)p Fp(1)1916 2815 y Ft(2)g Fv(Y)2055 2829 y Fs(j)t FH(\000)p Fp(1)2202 2815 y Ft(\\)20 b Fv(N)10 b Fu(\()p Fv(x)2453 2829 y Fs(j)t FH(\000)p Fp(1)2580 2815 y Fu(\))p Fw(,)23 b(and)h(let)f Fv(x)2985 2829 y Fs(j)3047 2815 y Fu(=)i Fv(x)3195 2829 y Fs(j)t FH(\000)p Fp(1)3342 2815 y Ft(^)20 b Fv(y)3468 2829 y Fs(j)t FH(\000)p Fp(1)3594 2815 y Fw(.)281 2961 y(\(c\))46 b(If)24 b Fv(j)30 b Fu(=)25 b Fv(r)h Fw(then)e Fv(f)10 b(ound)24 b Ft( )i Fv(T)13 b(R)q(U)d(E)5 b Fw(.)114 3149 y(5.)45 b(if)24 b Fv(f)10 b(ound)24 b Fu(=)h Fv(T)13 b(R)q(U)d(E)5 b Fw(,)22 b(r)m(eturn)j Fv(x)1298 3163 y Fs(r)1336 3149 y Fw(,)d(else)i(r)m(eturn)h(an)f(arbitr)o(ary)h (string)o(.)141 3455 y Fy(W)-7 b(e)20 b(\002rst)h(pro)o(v)o(e)g(that)g (if)g Fv(Y)951 3469 y Fs(j)1008 3455 y Fy(intersects)i Fv(N)10 b Fu(\()p Fv(x)1541 3469 y Fs(j)1578 3455 y Fu(\))p Fy(,)20 b(the)i(probability)i(that)d(the)g(size)h(of)f Fv(S)5 b Fu(\()p Fv(x)2900 3469 y Fs(j)t Fp(+1)3027 3455 y Fu(\))20 b Fy(is)h(signi\002cantly)i(smaller)0 3568 y(than)30 b(that)h(of)e Fv(S)5 b Fu(\()p Fv(x)603 3582 y Fs(j)640 3568 y Fu(\))29 b Fy(is)h(bounded)i(a)o(w)o(ay)d(from)h Fu(0)p Fy(.)47 b(Observ)o(e)30 b(that)h(since)g(the)f(sample)g Fv(Y)2866 3582 y Fs(j)2931 3568 y Fy(is)g(uniformly)h(distrib)n(uted)i (in)0 3681 y Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)222 3639 y Fs(n)269 3681 y Fy(,)23 b Fv(Y)368 3695 y Fs(j)424 3681 y Ft(\\)d Fv(N)10 b Fu(\()p Fv(x)675 3695 y Fs(j)712 3681 y Fu(\))23 b Fy(is)g(uniformly)j(distrib)n(uted)g(in)e Fv(N)10 b Fu(\()p Fv(x)1913 3695 y Fs(j)1950 3681 y Fu(\))p Fy(.)0 3913 y Fj(Claim)23 b(23)46 b Fw(Let)32 b Fv(x)596 3927 y Fs(j)664 3913 y Fw(be)h(a)f(\002xed)h(string)o(.)56 b(W)-5 b(ith)32 b(pr)l(obability)k(at)c(least)2295 3878 y Fp(1)p 2295 3893 36 4 v 2295 3945 a(3)2372 3913 y Fw(o)o(ver)h(the)g (uniform)g(c)o(hoice)h(of)e(a)g(string)i Fv(y)3762 3927 y Fs(j)3839 3913 y Ft(2)0 4026 y Fv(N)10 b Fu(\()p Fv(x)170 4040 y Fs(j)207 4026 y Fu(\))p Fw(,)23 b(if)g Fv(x)413 4040 y Fs(j)t Fp(+1)565 4026 y Fu(=)i Fv(x)713 4040 y Fs(j)769 4026 y Ft(^)20 b Fv(y)895 4040 y Fs(j)954 4026 y Fw(then)k Ft(j)p Fv(S)5 b Fu(\()p Fv(x)1306 4040 y Fs(j)t Fp(+1)1434 4026 y Fu(\))p Ft(j)25 b(\024)g Fu(1)c(+)1782 3990 y Fp(3)p 1782 4005 V 1782 4058 a(4)1847 4026 y Ft(\001)g Fu(\()p Ft(j)p Fv(S)5 b Fu(\()p Fv(x)2101 4040 y Fs(j)2138 4026 y Fu(\))p Ft(j)21 b(\000)f Fu(1\))p Fw(.)0 4258 y Fj(Pr)n(oof:)60 b Fy(W)l(ithout)31 b(loss)g(of)f(generality)-6 b(,)34 b(let)c Fv(S)5 b Fu(\()p Fv(x)1582 4272 y Fs(j)1618 4258 y Fu(\))38 b(=)e Ft(f)p Fu(1)p Fv(;)15 b(:)g(:)g(:)j(;)d(t)p Ft(g)p Fy(.)48 b(W)-7 b(e)29 b(partition)j(the)e(set)g(of)g(neighbors)i Fv(N)10 b Fu(\()p Fv(x)3657 4272 y Fs(j)3694 4258 y Fu(\))30 b Fy(into)0 4371 y(disjoint)37 b(subsets)h Fv(N)678 4385 y Fs(i)706 4371 y Fu(\()p Fv(x)793 4385 y Fs(j)830 4371 y Fu(\))p Fy(,)f(for)f Fu(1)47 b Ft(\024)g Fv(i)g Ft(\024)g Fv(t)p Fy(,)37 b(where)f Fv(N)1895 4385 y Fs(i)1923 4371 y Fu(\()p Fv(x)2010 4385 y Fs(j)2047 4371 y Fu(\))47 b(=)g Ft(f)p Fv(y)k Fu(:)94 b Fv(i)48 b Ft(2)e Fv(S)5 b Fu(\()p Fv(y)s Fu(\))24 b Fy(and)g(for)f(e)n(v)o(ery)h Fv(i)3430 4338 y FH(0)3501 4371 y Fv(<)47 b(i;)63 b(i)3769 4338 y FH(0)3849 4371 y Fv(=)-55 b Ft(2)0 4484 y Fv(S)5 b Fu(\()p Fv(y)s Fu(\))p Ft(g)p Fy(.)29 b(Since)24 b Fv(y)546 4498 y Fs(j)605 4484 y Fy(is)g(uniformly)h(distrib)n(uted)i (in)d Fv(N)10 b Fu(\()p Fv(x)1749 4498 y Fs(j)1786 4484 y Fu(\))p Fy(,)23 b(we)g(can)h(vie)n(w)f(it)h(as)f(being)i(selected)h (by)e(\002rst)f(choosing)j Fv(i)d Fy(with)0 4616 y(probability)432 4569 y FH(j)p Fs(N)508 4579 y Fh(i)534 4569 y Fp(\()p Fs(x)601 4579 y Fh(j)633 4569 y Fp(\))p FH(j)p 432 4595 249 4 v 461 4648 a Fs(N)7 b Fp(\()p Fs(x)591 4658 y Fh(j)624 4648 y Fp(\))690 4616 y Fy(,)23 b(and)h(then)g(selecting)i Fv(y)g Fy(uniformly)f(in)e Fv(N)2038 4630 y Fs(i)2067 4616 y Fu(\()p Fv(x)2154 4630 y Fs(j)2190 4616 y Fu(\))p Fy(.)141 4764 y(Consider)33 b(the)e(case)h Fv(y)873 4778 y Fs(j)949 4764 y Ft(2)39 b Fv(N)1122 4778 y Fp(1)1162 4764 y Fu(\()p Fv(x)1249 4778 y Fs(j)1285 4764 y Fu(\))p Fy(.)52 b(In)31 b(order)h(to)f(select)i(a)e(string)h(uniformly)h(in)e Fv(N)2938 4778 y Fp(1)2978 4764 y Fu(\()p Fv(x)3065 4778 y Fs(j)3102 4764 y Fu(\))p Fy(,)h(we)f(\002rst)g(set)g(to)g(1)g(all)0 4877 y(bits)25 b(corresponding)j(to)c(the)h(v)n(ariables)h(in)e Fv(T)1423 4891 y Fp(1)1463 4877 y Fy(,)f(and)i(then)f(let)h(the)f (remaining)i(bits)f(to)f(be)g(0)g(or)g(1)g(with)g(equal)h(probability) -6 b(.)0 4990 y(Since)24 b(for)f(e)n(v)o(ery)h Fv(i)i Ft(6)p Fu(=)f(1)e Fy(there)i(is)e(at)g(least)i(one)f(v)n(ariable)h (that)f(appears)h(in)f Fv(T)2434 5004 y Fs(i)2485 4990 y Fy(and)g(not)g(in)f Fv(T)2924 5004 y Fp(1)2964 4990 y Fy(,)f(we)h(ha)n(v)o(e)h(that)1295 5229 y Fu(Pr)15 b([)p Fv(T)1486 5243 y Fs(i)1515 5229 y Fu(\()p Fv(y)1595 5243 y Fs(j)1631 5229 y Fu(\))26 b(=)f(0)e Ft(j)g Fv(y)1949 5243 y Fs(j)2010 5229 y Ft(2)i Fv(N)2169 5243 y Fp(1)2208 5229 y Fu(\()p Fv(x)2295 5243 y Fs(j)2332 5229 y Fu(\)])h Ft(\025)2524 5168 y Fu(1)p 2524 5208 46 4 v 2524 5292 a(2)2579 5229 y Fv(:)1905 5589 y Fy(22)p eop %%Page: 23 23 23 22 bop 141 91 a Fy(It)27 b(follo)n(ws)g(that)h(the)f(e)o(xpected)i (number)f(of)f(indices)h Fv(i)k Ft(2)f Fv(S)5 b Fu(\()p Fv(x)2159 105 y Fs(j)2196 91 y Fu(\))p Fy(,)27 b Fv(i)32 b Ft(6)p Fu(=)f(1)p Fy(,)c(for)g(which)h Fv(T)2975 105 y Fs(i)3003 91 y Fu(\()p Fv(y)3083 105 y Fs(j)3120 91 y Fu(\))j(=)h(1)26 b Fy(is)h(at)g(most)3752 56 y Fs(t)p FH(\000)p Fp(1)p 3752 71 116 4 v 3792 123 a(2)3877 91 y Fy(.)0 204 y(By)22 b(Mark)o(o)o(v')-5 b(s)25 b(inequality)-6 b(,)25 b(the)e(probability)k(that)c(there)h(are)f(more)g(than)h Fu(\(1)18 b Ft(\000)g Fv(\013)p Fu(\)\()p Fv(t)h Ft(\000)f Fu(1\))23 b Fy(terms)g Fv(T)3173 218 y Fs(i)3201 204 y Fy(,)f Fv(i)k Ft(6)p Fu(=)f(1)p Fy(,)d(satis\002ed)i(by)0 317 y(a)f(uniformly)j(selected)g Fv(y)811 331 y Fs(j)873 317 y Ft(2)g Fv(N)1033 331 y Fp(1)1072 317 y Fu(\()p Fv(x)1159 331 y Fs(j)1196 317 y Fu(\))p Fy(,)e(is)f(at)h(most)1755 281 y Fp(1)p 1660 296 226 4 v 1660 349 a(2\(1)p FH(\000)p Fs(\013)p Fp(\))1896 317 y Fy(.)29 b(Setting)24 b Fv(\013)j Fu(=)f(1)p Fv(=)p Fu(4)p Fy(,)e(we)f(get)h(that,)h(with)e(probability) 28 b(at)23 b(least)10 417 y Fp(1)p 10 432 36 4 v 10 484 a(3)82 453 y Fy(o)o(v)o(er)k(the)h(choice)h(of)e(a)g(uniformly)j (selected)f Fv(y)1593 467 y Fs(j)1662 453 y Ft(2)j Fv(N)1828 467 y Fp(1)1868 453 y Fu(\()p Fv(x)1955 467 y Fs(j)1991 453 y Fu(\))p Fy(,)c(we)f(ha)n(v)o(e)h Ft(j)p Fv(S)5 b Fu(\()p Fv(x)2578 467 y Fs(j)t Fp(+1)2705 453 y Fu(\))p Ft(j)33 b(\024)g Fu(1)23 b(+)3074 417 y Fp(3)p 3074 432 V 3074 484 a(4)3142 453 y Ft(\001)h Fu(\()p Ft(j)p Fv(S)5 b Fu(\()p Fv(x)3399 467 y Fs(j)3436 453 y Fu(\))p Ft(j)24 b(\000)f Fu(1\))p Fy(.)40 b(It)27 b(is)0 566 y(easy)f(to)g(see)g(that)g (for)g(an)o(y)g Fv(N)943 580 y Fs(i)971 566 y Fu(\()p Fv(x)1058 580 y Fs(j)1095 566 y Fu(\))p Fy(,)f Fv(i)30 b(>)e Fu(1)p Fy(,)e(this)g(probability)j(is)d(at)f(least)h(as)g(lar)n (ge.)36 b(In)26 b(particular)l(,)j(note)d(that)g(for)g Fv(i)k Fu(=)e Fv(t)p Fy(,)0 679 y(for)c(an)o(y)f Fv(y)326 693 y Fs(j)388 679 y Ft(2)i Fv(N)547 693 y Fs(t)576 679 y Fu(\()p Fv(x)663 693 y Fs(j)700 679 y Fu(\))p Fy(,)e Ft(j)p Fv(S)5 b Fu(\()p Fv(x)954 693 y Fs(j)t Fp(+1)1081 679 y Fu(\))p Ft(j)26 b Fu(=)f(1)p Fy(.)p 1435 679 67 67 v 141 811 a(The)e(ne)o(xt)h(corollary)i(follo)n(ws)e(directly)i (from)d(Claim)g(23)h(and)g(the)g(f)o(act)g(that)g Ft(j)p Fv(S)5 b Fu(\()p Fv(x)2764 825 y Fp(0)2804 811 y Fu(\))p Ft(j)26 b(\024)f Fv(`)p Fy(.)0 1004 y Fj(Cor)n(ollary)h(24)46 b Fw(Let)26 b Fv(r)34 b Fu(=)c Fv(c)23 b Ft(\001)g Fu(log)16 b Fv(`)p Fw(,)27 b(wher)m(e)f Fv(c)g Fw(is)h(a)f(suf)n(\002ciently)k (lar)m(g)o(e)d(constant,)j(and)d(let)g Fv(x)2971 1018 y Fp(0)3036 1004 y Fw(be)f(a)h(\002xed)g(string)h(in)f Fv(W)3850 971 y Fs(i)3877 1004 y Fw(.)0 1117 y(Consider)f(the)e (following)i(pr)l(ocess,)g(consisting)h(of)e Fv(r)h Fw(steps,)f(wher)m (e)f(in)g(the)h Fv(j)5 b Fw(')l(s)25 b(step)g(we)e(uniformly)j(and)f (independently)0 1230 y(select)j(a)f(string)i Fv(y)589 1244 y Fs(j)t FH(\000)p Fp(1)747 1230 y Ft(2)i Fv(N)10 b Fu(\()p Fv(x)1009 1244 y Fs(j)t FH(\000)p Fp(1)1136 1230 y Fu(\))27 b Fw(and)h(set)f Fv(x)1540 1244 y Fs(j)1609 1230 y Fu(=)k Fv(x)1763 1244 y Fs(j)t FH(\000)p Fp(1)1913 1230 y Ft(^)22 b Fv(y)2041 1244 y Fs(j)t FH(\000)p Fp(1)2168 1230 y Fw(.)38 b(Then,)28 b(with)f(pr)l(obability)j(at)e(least)g Fu(1)23 b Ft(\000)g Fu(p)s(oly\(1)p Fv(=`)p Fu(\))0 1343 y Fw(o)o(ver)h(the)g(c)o(hoice)h(of)e Fv(y)714 1357 y Fp(0)753 1343 y Fv(;)15 b(:)g(:)g(:)i(;)e(y)1000 1357 y Fs(r)r FH(\000)p Fp(1)1128 1343 y Fw(,)22 b(we)h(obtain)i Ft(j)p Fv(S)5 b Fu(\()p Fv(x)1725 1357 y Fs(r)1764 1343 y Fu(\))p Ft(j)26 b Fu(=)f(1)p Fw(.)0 1580 y Fy(Finally)-6 b(,)23 b(we)f(bound)i(the)f(size)g(of)g(a)f(sample)h Fv(Y)1461 1594 y Fs(j)1519 1580 y Fy(suf)n(\002cient)h(for)f(acquiring) i(a)d(string)i Fv(y)2706 1594 y Fs(j)2768 1580 y Ft(2)g Fv(N)10 b Fu(\()p Fv(x)3023 1594 y Fs(j)3060 1580 y Fu(\))22 b Fy(with)h(high)g(probability)-6 b(.)0 1693 y(W)f(e)23 b(\002rst)g(de\002ne)h(a)f(\223good)i(initial)g(string\224)g Fv(x)1416 1707 y Fp(0)1455 1693 y Fy(.)j(This)23 b(is)h(a)f(string)i (that)f(satis\002es)g(only)h(\223lar)n(ge\224)g(monomials.)0 1908 y Fj(De\002nition)e(12)46 b Fw(A)23 b(string)i Fv(x)909 1922 y Fp(0)971 1908 y Fw(will)f(be)f(called)j(a)d Fx(good)h(initial)h (str)q(ing)f Fw(if)g(for)g(e)o(very)g Fv(i)i Ft(2)g Fv(S)5 b Fu(\()p Fv(x)2973 1922 y Fp(0)3012 1908 y Fu(\))p Fw(,)24 b Fu(Pr)o([)p Fv(T)3269 1922 y Fs(i)3323 1908 y Fu(=)i(1])g Ft(\025)3658 1872 y Fs(\017)p 3623 1887 100 4 v 3623 1940 a Fp(16)p Fs(`)3732 1908 y Fw(.)j(Let)0 2052 y Fu(Go)s(o)s(d)243 2002 y Fp(def)256 2052 y Fu(=)366 1958 y Fi(n)421 2052 y Fv(x)c Ft(2)p 584 1980 83 4 v 25 w Fv(N)10 b Fu(\()p Fv(R)q Fu(\))23 b Fw(and)h Fv(x)f Fw(is)g(a)h(good)g(initial)h(string) 1872 1958 y Fi(o)1928 2052 y Fw(.)0 2294 y Fj(Claim)e(25)46 b Fw(Suppose)30 b Fu(Pr)o([)p Fv(x)j Ft(2)p 1026 2222 V 32 w Fv(N)10 b Fu(\()p Fv(R)q Fu(\)])33 b Ft(\025)1423 2258 y Fs(\017)p 1420 2273 36 4 v 1420 2325 a Fp(8)1465 2294 y Fw(.)40 b(Then)27 b(the)h(pr)l(obability)-5 b(,)31 b(tak)o(en)e(o)o(ver)f(the)g(c)o(hoices)g(of)g Fv(U)3334 2261 y Fs(i)3362 2294 y Fw(,)f(that)h Fv(W)3679 2261 y Fs(i)3733 2294 y Fw(does)0 2407 y(not)c(contain)h(any)f(good)h (initial)g(strings,)g(is)e(at)h(most)1739 2371 y Fp(1)p 1707 2386 100 4 v 1707 2438 a(18)p Fs(`)1816 2407 y Fw(.)0 2653 y Fj(Pr)n(oof:)47 b Fy(Recall)24 b(that)32 b Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))917 2603 y Fp(def)930 2653 y Fu(=)38 b(Pr[)p Fv(x)25 b Ft(2)p 1325 2581 83 4 v 25 w Fv(N)10 b Fu(\()p Fv(R)q Fu(\)])p Fy(.)29 b(F)o(or)22 b(an)o(y)i Fv(i)i Ft(2)p 2067 2581 61 4 v 24 w Fv(S)5 b Fu(\()p Fv(R)q Fu(\))p Fy(,)23 b(consider)j(the)e(e)n(v)o(ent)1286 2863 y Fv(E)1353 2877 y Fs(i)1406 2813 y Fp(def)1420 2863 y Fu(=)1529 2769 y Fi(n)1584 2863 y Fv(x)i Ft(2)p 1748 2791 83 4 v 25 w Fv(N)9 b Fu(\()p Fv(R)q Fu(\))24 b Fy(and)g Fv(T)2201 2877 y Fs(i)2229 2863 y Fu(\()p Fv(x)p Fu(\))i(=)f(1)2518 2769 y Fi(o)2589 2863 y Fv(:)0 3095 y Fy(By)e(de\002nition,)33 b Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))25 b(=)g(Pr)943 3001 y Fi(h)982 3031 y(S)1051 3125 y Fs(i)p FH(2)p 1122 3070 47 4 v Fs(S)t Fp(\()p Fs(R)p Fp(\))1297 3095 y Fv(E)1364 3109 y Fs(i)1392 3001 y Fi(i)1431 3095 y Fy(.)j(Let)p 1078 3287 61 4 v 1078 3359 a Fv(S)1139 3374 y Fp(small)1304 3359 y Fu(\()p Fv(R)q Fu(\))e(=)1566 3240 y Fi(\032)1628 3359 y Fv(i)g Ft(2)p 1771 3287 V 25 w Fv(S)5 b Fu(\()p Fv(R)q Fu(\))23 b Fy(and)h Fu(Pr)o([)p Fv(E)2338 3373 y Fs(i)2367 3359 y Fu(])h Ft(\024)2531 3298 y Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))p 2523 3338 187 4 v 2575 3422 a(2)p Fv(`)2719 3240 y Fi(\033)2797 3359 y Fv(:)0 3616 y Fy(Clearly\241)31 b(for)f(an)o(y)f(term)h Fv(i)p Fy(,)g Fu(Pr[)p Fv(T)1076 3630 y Fs(i)1141 3616 y Fu(=)36 b(1])i Ft(\025)e Fu(Pr[)p Fv(E)1653 3630 y Fs(i)1681 3616 y Fu(])p Fy(.)47 b(Therefore,)32 b(if)e Fv(x)36 b Ft(2)2458 3522 y Fi(\020)2507 3552 y(S)2576 3647 y Fs(i)p FH(2)p 2647 3592 47 4 v Fs(S)t Fp(\()p Fs(R)p Fp(\))2822 3616 y Fv(E)2889 3630 y Fs(i)2917 3522 y Fi(\021)2992 3616 y Ft(n)3062 3522 y Fi(\020)3111 3552 y(S)3180 3647 y Fs(i)p FH(2)p 3251 3592 V Fs(S)3298 3659 y Fn(small)3442 3647 y Fp(\()p Fs(R)p Fp(\))3570 3616 y Fv(E)3637 3630 y Fs(i)3665 3522 y Fi(\021)3743 3616 y Fy(then)0 3771 y Fv(S)5 b Fu(\()p Fv(x)p Fu(\))29 b Ft(\022)p 310 3699 61 4 v 27 w Fv(S)5 b Fu(\()p Fv(R)q Fu(\))22 b Ft(n)p 600 3699 V 22 w Fv(S)661 3786 y Fp(small)826 3771 y Fu(\()p Fv(R)q Fu(\))p Fy(,)j(and)h(therefore)h(for)e(all)g Fv(i)k Ft(2)e Fv(S)5 b Fu(\()p Fv(x)p Fu(\))25 b Fy(we)f(ha)n(v)o(e)i Fu(Pr[)p Fv(T)2622 3785 y Fs(i)2650 3771 y Fu(\()p Fv(x)p Fu(\))j(=)f(1])g Ft(\025)g Fu(Pr[)p Fv(E)3287 3785 y Fs(i)3315 3771 y Fu(])g Ft(\025)3483 3726 y Fp(\026)-41 b Fs(p)p Fp(\()p Fs(R)p Fp(\))p 3477 3750 144 4 v 3517 3802 a(2)p Fs(`)3659 3771 y Ft(\025)3803 3735 y Fs(\017)p 3768 3750 100 4 v 3768 3802 a Fp(16)p Fs(`)3877 3771 y Fy(.)0 3884 y(Thus,)23 b Fv(x)i Ft(2)g Fu(Go)s(o)s(d)p Fy(.)j(Therefore,)933 4176 y Fu(Pr)o([Go)s(o)s(d])84 b Ft(\025)e Fu(Pr)1648 3982 y Fi(2)1648 4129 y(6)1648 4182 y(4)1703 3982 y(0)1703 4129 y(B)1703 4182 y(@)1843 4095 y([)1776 4296 y Fs(i)p FH(2)p 1847 4240 47 4 v Fs(S)t Fp(\()p Fs(R)p Fp(\))2017 4176 y Fv(E)2084 4190 y Fs(i)2113 3982 y Fi(1)2113 4129 y(C)2113 4182 y(A)2205 4176 y Ft(n)2271 3982 y Fi(0)2271 4129 y(B)2271 4182 y(@)2483 4095 y([)2344 4296 y Fs(i)p FH(2)p 2415 4240 V Fs(S)2461 4308 y Fn(small)2605 4296 y Fp(\()p Fs(R)p Fp(\))2729 4176 y Fv(E)2796 4190 y Fs(i)2824 3982 y Fi(1)2824 4129 y(C)2824 4182 y(A)2897 3982 y(3)2897 4129 y(7)2897 4182 y(5)1382 4558 y Ft(\025)g Fu(Pr)1648 4364 y Fi(2)1648 4510 y(6)1648 4564 y(4)1770 4477 y([)1703 4678 y Fs(i)p FH(2)p 1774 4622 V Fs(S)t Fp(\()p Fs(R)p Fp(\))1945 4558 y Fv(E)2012 4572 y Fs(i)2040 4364 y Fi(3)2040 4510 y(7)2040 4564 y(5)2115 4558 y Ft(\000)20 b Fu(Pr)2319 4364 y Fi(2)2319 4510 y(6)2319 4564 y(4)2513 4477 y([)2374 4678 y Fs(i)p FH(2)p 2445 4622 V Fs(S)2492 4690 y Fn(small)2636 4678 y Fp(\()p Fs(R)p Fp(\))2759 4558 y Fv(E)2826 4572 y Fs(i)2855 4364 y Fi(3)2855 4510 y(7)2855 4564 y(5)1382 4873 y Ft(\025)90 b Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))21 b Ft(\000)f Fv(`)g Ft(\001)1954 4811 y Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))p 1946 4852 187 4 v 1998 4935 a(2)p Fv(`)2243 4873 y Fu(=)2433 4811 y(\026)g Fv(p)p Fu(\()p Fv(R)q Fu(\))p 2425 4852 V 2495 4935 a(2)2621 4873 y Fv(:)0 5095 y Fy(Since)31 b Fu(\026)-53 b Fv(p)p Fu(\()p Fv(R)q Fu(\))26 b Ft(\025)545 5059 y Fs(\017)p 541 5074 36 4 v 541 5126 a Fp(8)587 5095 y Fy(,)c(and)h(the)g(size)h(of)f(the)g(sample)h Fv(U)1665 5062 y Fs(i)1715 5095 y Fy(is)f Fu(\002\()p Fv(`)15 b Fu(log)i Fv(`=\017)p Fu(\))p Fy(,)22 b(the)h(probability)j (that)e Fv(W)3102 5062 y Fs(i)3152 5095 y Fy(does)g(not)f(contain)i Fw(any)0 5208 y Fy(initial)g(good)f(strings)i(is,)d(for)g(a)h(suf)n (\002ciently)h(lar)n(ge)g(constant)h(in)d(the)h Fu(\002\()p Ft(\001)p Fu(\))f Fy(notation,)j(smaller)e(than)3232 5172 y Fp(1)p 3200 5187 100 4 v 3200 5239 a(18)p Fs(`)3310 5208 y Fy(.)p 3436 5208 67 67 v 141 5340 a(The)f(ne)o(xt)h(claim)g (follo)n(ws)g(from)f(the)h(de\002nition)i(of)d(a)g(good)i(initial)g (string.)1905 5589 y(23)p eop %%Page: 24 24 24 23 bop 0 91 a Fj(Claim)23 b(26)46 b Fw(Let)27 b Fv(m)619 105 y Fp(2)690 91 y Fu(=)k Fv(c)23 b Ft(\001)g Fv(`)15 b Fu(log)i Fv(`=\017)p Fw(,)27 b(wher)m(e)g Fv(c)f Fw(is)h(a)g(suf)n (\002ciently)j(lar)m(g)o(e)e(constant,)i(and)e(suppose)h Fv(x)3251 105 y Fp(0)3316 91 y Fw(is)e(a)g(good)h(initial)0 204 y(string)o(.)i(Then,)23 b(for)h(eac)o(h)g Fu(1)h Ft(\024)g Fv(j)31 b Ft(\024)25 b Fv(r)s Fw(,)d(the)i(pr)l(obability)j (that)d(a)f(sample)h Fv(Y)2352 218 y Fs(j)2411 204 y Fw(of)f Fv(m)2584 218 y Fp(2)2646 204 y Fw(strings)j(inter)o(sects)g Fv(N)10 b Fu(\()p Fv(x)3457 218 y Fs(j)3494 204 y Fu(\))23 b Fw(is)g(at)g(least)0 317 y Fu(1)e Ft(\000)216 281 y Fp(1)p 167 296 134 4 v 167 350 a(18)p Fs(`)266 331 y Fn(2)310 317 y Fw(.)0 524 y Fj(Pr)n(oof)32 b(of)f(Lemma)f(22:)106 b Fy(By)30 b(the)h(premise)i(of)e(the)g(lemma,)i Fu(Pr)o([)p Fv(x)40 b Ft(2)p 2366 452 83 4 v 39 w Fv(N)10 b Fu(\()p Fv(R)q Fu(\)])40 b Ft(\025)f Fv(\017=)p Fu(8)p Fy(.)52 b(By)30 b(Claim)h(25,)i(the)e(set)g Fv(W)3872 491 y Fs(i)0 637 y Fy(contains)f(a)d(good)h(initial)h(string)g(with)e(probability)k (at)d(least)g Fu(1)c Ft(\000)2182 601 y Fp(1)p 2150 616 100 4 v 2150 669 a(18)p Fs(`)2260 637 y Fy(.)40 b(Conditioned)30 b(on)e(this)g(e)n(v)o(ent,)g(let)g(us)g(\002x)e(such)j(a)0 750 y(string)h Fv(x)287 764 y Fp(0)326 750 y Fy(,)e(and)h(consider)h (the)f(e)o(x)o(ecution)h(of)e(Step)g(4b)h(in)f(the)g(procedure.)46 b(By)28 b(Claim)f(26,)i(the)g(probability)i(that)e(there)0 863 y(e)o(xists)g Fv(j)41 b Ft(\024)34 b Fv(r)c Fy(for)f(which)g(the)g (sample)g Fv(Y)1348 877 y Fs(j)1412 863 y Fy(does)h(not)f(contain)h(a)e (string)i(in)e Fv(N)10 b Fu(\()p Fv(x)2617 877 y Fs(j)2654 863 y Fu(\))28 b Fy(is)h(at)f(most)3146 827 y Fp(1)p 3114 842 V 3114 894 a(18)p Fs(`)3223 863 y Fy(.)43 b(Since)29 b(the)g(strings)0 976 y(in)h Fv(Y)153 990 y Fs(j)219 976 y Fy(are)g(uniformly)i(selected)g(from)e Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1502 934 y Fs(n)1549 976 y Fy(,)30 b(the)h(strings)g(in)f Fv(Y)2167 990 y Fs(j)2229 976 y Ft(\\)24 b Fv(N)10 b Fu(\()p Fv(x)2484 990 y Fs(j)2521 976 y Fu(\))30 b Fy(are)g(uniformly)i(distrib)n(uted)h(in)d Fv(N)10 b Fu(\()p Fv(x)3805 990 y Fs(j)3842 976 y Fu(\))p Fy(.)0 1089 y(Hence,)22 b(conditioned)k(on)c(each)h Fv(Y)1080 1103 y Fs(j)1137 1089 y Fy(containing)i(a)d(string)h(from)f Fv(N)10 b Fu(\()p Fv(x)2200 1103 y Fs(j)2237 1089 y Fu(\))p Fy(,)21 b(we)h(can)g(apply)h(Corollary)h(24)e(and)h(get)f(that)g(with)0 1202 y(probability)35 b(at)d(least)g Fu(1)27 b Ft(\000)935 1166 y Fp(1)p 903 1181 V 903 1233 a(18)p Fs(`)1012 1202 y Fy(,)33 b Ft(j)p Fv(S)5 b Fu(\()p Fv(x)1241 1216 y Fs(r)1280 1202 y Fu(\))p Ft(j)41 b Fu(=)f(1)p Fy(.)53 b(Since)32 b Fv(x)1898 1216 y Fp(0)1978 1202 y Ft(2)p 2079 1130 83 4 v 40 w Fv(N)10 b Fu(\()p Fv(R)q Fu(\))p Fy(,)34 b(necessarily)g Fv(x)2845 1216 y Fs(r)2924 1202 y Ft(2)p 3025 1130 V 40 w Fv(N)10 b Fu(\()p Fv(R)q Fu(\))p Fy(.)53 b(Therefore,)35 b(with)0 1315 y(probability)d(at)c(least)i Fu(1)24 b Ft(\000)g Fu(3)g Ft(\001)1039 1279 y Fp(1)p 1007 1294 100 4 v 1007 1346 a(18)p Fs(`)1151 1315 y Fu(=)35 b(1)24 b Ft(\000)1445 1279 y Fp(1)p 1431 1294 65 4 v 1431 1346 a(6)p Fs(`)1505 1315 y Fy(,)29 b(tak)o(en)g(o)o(v)o(er)g(the) g(choices)h(of)f Fv(U)2582 1282 y Fs(i)2638 1315 y Fy(and)g(the)g (samples)h Fv(Y)3310 1329 y Fs(j)3346 1315 y Fy(,)f(the)g(procedure)0 1428 y(returns)c(a)e(string)i Fv(x)620 1442 y Fs(r)681 1428 y Fy(that)f(is)f(a)g(single-term)j(representati)n(v)o(e)h(for)d Fv(f)32 b Fy(of)23 b(a)g(term)h(not)g(yet)f(represented)k(in)d Fv(f)10 b Fy(.)0 1560 y(The)23 b(number)i(of)e(queries)i(performed)h (is)d Fv(r)g Ft(\001)d Fv(m)1522 1574 y Fp(2)1587 1560 y Fu(=)25 b Fv(O)s Fu(\()p Fv(`)15 b Fu(log)1960 1523 y Fp(2)2015 1560 y Fv(`=\017)p Fu(\))p Fy(.)p 2297 1560 67 67 v 0 1872 a Fz(6)119 b(T)-11 b(esting)30 b(Singletons)h(W)n (ithout)e(T)-11 b(esting)30 b(Linearity)0 2099 y Fy(Recall)19 b(that)h(by)f(Claim)f(1)g(an)h(alternati)n(v)o(e)i(characterization)j (of)19 b(singletons)i(is)e(that)g Fv(P)13 b(r)s Fu([)p Fv(f)35 b Fu(=)25 b(1])h(=)e(1)p Fv(=)p Fu(2)p Fy(,)d(and)e (furthermore)0 2212 y(that)g(there)g(are)f(no)h(violating)i(pairs)e Fv(x;)c(y)28 b Ft(2)d(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)1596 2179 y Fs(n)1645 2212 y Fy(.)26 b(That)19 b(is,)f(there)i(are)e(no)g Fv(x;)d(y)21 b Fy(such)e(that)g Fv(f)10 b Fu(\()p Fv(x)q Ft(^)q Fv(y)s Fu(\))24 b Ft(6)p Fu(=)h Fv(f)10 b Fu(\()p Fv(x)p Fu(\))q Ft(^)q Fv(f)g Fu(\()p Fv(y)s Fu(\))p Fy(.)25 b(W)-7 b(e)0 2325 y(sho)n(w)26 b(that)i(the)e(follo)n(wing)i(simple)g (algorithm)g(that)f(checks)h(these)g(properties,)h(is)e(a)f(testing)i (algorithm)h(for)d(singletons)0 2438 y(if)i Fv(f)38 b Fy(is)28 b(not)h(too)h(f)o(ar)f(from)f(a)h(singleton)i(function.)46 b(Let)29 b Ft(F)1889 2452 y Fc(sing)2055 2438 y Fy(denote)h(the)f (class)h(of)e(singletons.)48 b(The)28 b(algorithm)j(will)0 2551 y(recei)n(v)o(e)c(a)e(v)n(alue)h Fv(\015)615 2565 y Fp(0)680 2551 y Fy(such)g(that)h Fu(min)1184 2565 y Fs(g)r FH(2F)1317 2573 y Fb(sing)1441 2551 y Fu(dist)o(\()p Fv(f)5 b(;)15 b(g)s Fu(\))30 b Ft(\024)1934 2515 y Fp(1)p 1934 2530 36 4 v 1934 2582 a(2)2001 2551 y Ft(\000)22 b Fv(\015)2141 2565 y Fp(0)2180 2551 y Fy(.)35 b(That)25 b(is,)h Fv(\015)2585 2565 y Fp(0)2650 2551 y Fy(is)f(a)g(lo)n(wer)h (bound)h(on)f(the)g(dif)n(ference)0 2663 y(between)f Fu(1)p Fv(=)p Fu(2)f Fy(and)g(the)g(distance)h(of)f Fv(f)32 b Fy(to)23 b(the)h(closest)h(singleton.)31 b(W)-7 b(e)23 b(shall)h(think)h(of)f Fv(\015)2842 2677 y Fp(0)2904 2663 y Fy(as)f(a)g(constant.)0 2896 y Fj(Algorithm)h(4)46 b Fy(T)-6 b(est)23 b(for)g(Singleton)j(with)d(lo)n(wer)g(bound)i Fv(\015)1899 2910 y Fp(0)114 3103 y Fw(1.)45 b Fx(Siz)o(e)26 b(T)-11 b(est:)54 b Fw(Uniformly)25 b(select)h(a)e(sample)h(of)f Fv(m)j Fu(=)g(\002\(1)p Fv(=\017)2184 3070 y Fp(2)2224 3103 y Fu(\))d Fw(strings)i(in)f Ft(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)2871 3061 y Fs(n)2918 3103 y Fw(.)31 b(F)-10 b(or)24 b(eac)o(h)h Fv(x)f Fw(in)g(the)h(sample)o(,)227 3216 y(obtain)31 b Fv(f)10 b Fu(\()p Fv(x)p Fu(\))p Fw(.)45 b(Let)28 b Fv(\013)h Fw(be)g(the)h(fr)o(action)h(of)e(sample)g(strings) i Fv(x)e Fw(suc)o(h)g(that)h Fv(f)10 b Fu(\()p Fv(x)p Fu(\))36 b(=)f(1)p Fw(.)45 b(If)29 b Ft(j)p Fv(\013)c Ft(\000)f Fu(1)p Fv(=)p Fu(2)p Ft(j)38 b Fv(>)3673 3180 y Fs(\017)p 3670 3195 V 3670 3247 a Fp(4)3743 3216 y Fw(then)227 3329 y Fx(reject)p Fw(,)23 b(otherwise)i(continue)o(.)114 3516 y(2.)45 b Fx(Closure-Under-Intersection)23 b(T)-11 b(est:)50 b Fw(Repeat)24 b(the)f(following)i Fu(\002\()p Fv(\017)2501 3483 y FH(\000)p Fp(1)2595 3516 y Fv(\015)2647 3478 y FH(\000)p Fp(1)2642 3541 y(0)2741 3516 y Fu(\))e Fw(times:)29 b(Uniformly)24 b(select)g Fv(x;)15 b(y)28 b Ft(2)227 3629 y(f)p Fu(0)p Fv(;)15 b Fu(1)p Ft(g)447 3596 y Fs(n)496 3629 y Fw(.)28 b(If)c Fv(x)e Fw(and)i Fv(y)i Fw(ar)m(e)e(a)f(violating)j(pair)-10 b(,)24 b(then)g(r)m(eject.) 114 3817 y(3.)45 b(If)24 b(no)f(step)h(caused)i(r)m(ejection,)f(then)f (accept.)0 4049 y Fj(Theor)n(em)g(4)45 b Fw(If)23 b Fv(f)31 b Fw(is)22 b(a)g(singleton,)j(then)e(Algorithm)h(4)e(accepts)i(with)f (pr)l(obability)i(at)e(least)g Fu(2)p Fv(=)p Fu(3)p Fw(.)29 b(If)22 b Fv(f)31 b Fw(is)22 b Fv(\017)p Fw(-far)h(fr)l(om)g(any)0 4162 y(singleton)30 b(wher)m(e)d Fv(\017)f Fw(is)h(bounded)i(away)e(fr) l(om)g Fu(1)p Fv(=)p Fu(2)p Fw(,)h(then)g(the)f(algorithm)i(r)m(ejects) f(with)f(pr)l(obability)j(at)d(least)h Fu(2)p Fv(=)p Fu(3)p Fw(.)40 b(The)0 4275 y(query)25 b(comple)n(xity)g(of)f(the)g (algorithm)h(is)e Fv(O)s Fu(\(1)p Fv(=\017)1570 4242 y Fp(2)1611 4275 y Fu(\))p Fw(.)0 4507 y Fj(Pr)n(oof:)54 b Fy(If)26 b Fv(f)36 b Fy(is)26 b(a)h(singleton)i(then)f Fu(Pr)o([)p Fv(f)41 b Fu(=)31 b(1])h(=)f(1)p Fv(=)p Fu(2)p Fy(.)39 b(By)27 b(an)f(additi)n(v)o(e)j(Chernof)n(f)f(bound,)h(and)e (for)g(the)g(appropriate)0 4620 y(constant)k(in)e(the)f Fu(\002\()p Ft(\001)p Fu(\))h Fy(notation,)j(the)d(probability)j(that)d (it)g(is)f(rejected)j(in)d(the)h(\002rst)g(step)g(of)g(Algorithm)g(4)g (is)f(less)h(than)0 4733 y Fu(1)p Fv(=)p Fu(3)p Fy(.)g(By)23 b(the)h(de\002nition)h(of)f(singletons,)i Fv(f)32 b Fy(al)o(w)o(ays)24 b(passes)h(the)f(closure-under)n(-inte)q(rse)q(cti)q(on)29 b(test.)141 4865 y(Suppose)h(that)f Fv(f)38 b Fy(is)28 b Fv(\017)p Fy(-f)o(ar)h(from)g(an)o(y)g(singleton)i(and)f(let)e Fv(\016)k Fy(be)d(its)g(distance)i(to)d(the)h(closest)i(singleton.)47 b(Thus)28 b Fv(\017)35 b(<)0 4978 y(\016)29 b Ft(\024)c Fu(1)p Fv(=)p Fu(2)d Ft(\000)e Fv(\015)460 4992 y Fp(0)499 4978 y Fy(.)28 b(W)-7 b(e)23 b(sho)n(w)g(that)h Fv(f)32 b Fy(is)23 b(rejected)j(with)d(probability)k(greater)e(than)f Fu(2)p Fv(=)p Fu(3)p Fy(.)114 5185 y(1.)45 b(If)24 b Ft(j)p Fu(Pr[)p Fv(f)36 b Fu(=)27 b(1])21 b Ft(\000)g Fu(1)p Fv(=)p Fu(2)p Ft(j)28 b Fv(>)1120 5149 y Fs(\017)p 1117 5164 V 1117 5217 a Fp(2)1162 5185 y Fy(,)23 b(then)i Fv(f)33 b Fy(is)24 b(rejected)j(in)d(the)h(\002rst)f(step)h(of)f(the)g (algorithm)i(with)f(probability)i(at)d(least)227 5298 y Fu(5)p Fv(=)p Fu(6)p Fy(.)1905 5589 y(24)p eop %%Page: 25 25 25 24 bop 114 91 a Fy(2.)45 b(Otherwise,)22 b Ft(j)p Fu(Pr[)p Fv(f)35 b Fu(=)25 b(1])11 b Ft(\000)g Fu(1)p Fv(=)p Fu(2)p Ft(j)26 b(\024)1427 56 y Fs(\017)p 1423 71 36 4 v 1423 123 a Fp(2)1494 91 y Fv(<)1600 56 y Fs(\016)p 1600 71 V 1600 123 a Fp(2)1645 91 y Fy(.)h(In)21 b(this)g(case,)h(as)f (we)f(sho)n(w)h(shortly)i(in)e(Lemma)e(27,)j(the)f(probability)227 213 y(of)g(obtaining)i(a)d(violating)j(pair)e(is)f(at)g(least)1612 177 y Fs(\016)p 1612 192 V 1612 244 a Fp(4)1657 213 y Fu(\()1702 177 y Fp(1)p 1702 192 V 1702 244 a(2)1756 213 y Ft(\000)9 b Fv(\016)s Fu(\))26 b Ft(\025)2049 177 y Fs(\017)p 2046 192 V 2046 244 a Fp(4)2100 213 y Ft(\001)9 b Fv(\015)2181 227 y Fp(0)2221 213 y Fy(.)26 b(Therefore,)d Fv(f)29 b Fy(will)20 b(be)g(rejected)j(with)d(probability)227 326 y(of)k(at)f(least)h Fu(5)p Fv(=)p Fu(6)h Fy(in)e(the)h(second)h (step)f(of)g(the)f(algorithm)j(\(the)e(closure-under)n(-inter)q(sec)q (tio)q(n)29 b(test\).)0 526 y(Thus,)23 b(the)h(probability)j(that)d Fv(f)32 b Fy(is)23 b(accepted)j(by)e(the)f(algorithm)j(is)d(at)g(most)h Fu(1)p Fv(=)p Fu(3)p Fy(,)g(as)f(required.)p 3148 526 67 67 v 0 749 a Fj(Lemma)g(27)46 b Fw(Let)20 b Fv(\016)k Fw(be)e(the)f(distance)i(of)e Fv(f)30 b Fw(to)20 b(the)i(closest)g (singleton.)31 b(If)21 b Fu(Pr)o([)p Fv(f)10 b Fu(\()p Fv(x)p Fu(\))26 b(=)f(1])h Ft(\025)2955 713 y Fp(1)p 2955 728 36 4 v 2955 780 a(2)3012 749 y Ft(\000)3104 713 y Fs(\016)p 3103 728 V 3103 780 a Fp(2)3148 749 y Fw(,)21 b(then)h(the)f(pr)l(obability)0 871 y(of)i(obtaining)k(a)c (violating)j(pair)e(is)g(at)f(least)1435 835 y Fs(\016)p 1434 850 V 1434 902 a Fp(4)1480 871 y Fu(\()1525 835 y Fp(1)p 1525 850 V 1525 902 a(2)1590 871 y Ft(\000)d Fv(\016)s Fu(\))p Fw(.)0 1094 y Fj(Pr)n(oof:)47 b Fy(Let)23 b Fv(x)487 1108 y Fs(i)538 1094 y Fy(be)g(the)h(closest)h(singleton)i (to)c Fv(f)10 b Fy(,)22 b(so)h(that)h Fu(Pr[)p Fv(f)10 b Fu(\()p Fv(x)p Fu(\))25 b Ft(6)p Fu(=)g Fv(x)2335 1108 y Fs(i)2363 1094 y Fu(])h(=)f Fv(\016)s Fy(.)k(De\002ne)1118 1290 y Fv(G)1189 1304 y Fp(1)1254 1290 y Fu(=)c Ft(f)p Fv(x)p Ft(j)p Fv(f)10 b Fu(\()p Fv(x)p Fu(\))26 b(=)f(1)p Fv(;)15 b(x)1908 1304 y Fs(i)1962 1290 y Fu(=)25 b(1)p Ft(g)p Fv(;)85 b(B)2327 1304 y Fp(1)2391 1290 y Fu(=)25 b Fv(F)2545 1304 y Fp(1)2605 1290 y Ft(n)c Fv(G)2742 1304 y Fp(1)1118 1485 y Fv(G)1189 1499 y Fp(0)1254 1485 y Fu(=)k Ft(f)p Fv(x)p Ft(j)p Fv(f)10 b Fu(\()p Fv(x)p Fu(\))26 b(=)f(0)p Fv(;)15 b(x)1908 1499 y Fs(i)1962 1485 y Fu(=)25 b(0)p Ft(g)p Fv(;)85 b(B)2327 1499 y Fp(0)2391 1485 y Fu(=)25 b Fv(F)2545 1499 y Fp(0)2605 1485 y Ft(n)c Fv(G)2742 1499 y Fp(0)141 1667 y Fy(A)k(simple)h(counting)i(ar)n (gument)g(sho)n(ws)e(that)g(there)g(are)g Fu(\()2007 1631 y Fp(1)p 2007 1646 V 2007 1699 a(2)2075 1667 y Ft(\000)21 b Fv(\016)s Fu(\)2)2290 1634 y Fs(n)2364 1667 y Fy(disjoint)27 b(pairs)g Fv(x;)15 b(x)3007 1634 y FH(0)3031 1667 y Fy(,)25 b(such)h(that:)34 b(\(1\))26 b Fv(x)k Ft(2)e Fv(G)3837 1681 y Fp(1)3877 1667 y Fy(,)0 1780 y Fv(x)52 1747 y FH(0)113 1780 y Ft(2)38 b Fv(G)283 1794 y Fp(0)323 1780 y Fy(;)33 b(\(2\))e Fv(x)e Fy(and)i Fv(x)811 1747 y FH(0)864 1780 y Fy(dif)n(fer)g(only)h(on)e(the)h Fv(i)p Fy(')n(th)g(bit.)50 b(T)-7 b(o)29 b(see)i(why)e(this)i(is)g(true,)h(simply)f(match)g(each)g Fv(x)38 b Ft(2)g Fv(G)3760 1794 y Fp(1)3829 1780 y Fy(to)0 1893 y(a)30 b(point)i Fv(x)339 1860 y FH(0)363 1893 y Fy(,)f(which)h(dif)n(fers)g(with)e Fv(x)g Fy(only)i(on)f(the)g Fv(i)p Fy(')n(th)h(bit.)51 b(Thus,)33 b(there)e(are)h(at)e(least)i Ft(j)p Fv(G)2970 1907 y Fp(1)3010 1893 y Ft(j)26 b(\000)f(j)p Fv(B)3251 1907 y Fp(1)3291 1893 y Ft(j)30 b Fy(points)j Fv(x)39 b Ft(2)f Fv(G)3860 1907 y Fp(1)0 2006 y Fy(that)29 b(must)f(be)h(matched)g(to)f(points)i Fv(x)1217 1973 y FH(0)1274 2006 y Ft(2)k Fv(G)1440 2020 y Fp(0)1480 2006 y Fy(.)42 b(But)28 b Ft(j)p Fv(G)1800 2020 y Fp(1)1840 2006 y Ft(j)c Fu(+)g Ft(j)p Fv(B)2078 2020 y Fp(0)2117 2006 y Ft(j)35 b Fu(=)f(2)2327 1973 y Fs(n)p FH(\000)p Fp(1)2464 2006 y Fy(,)29 b(and)f Ft(j)p Fv(B)2768 2020 y Fp(1)2808 2006 y Ft(j)c Fu(+)g Ft(j)p Fv(B)3046 2020 y Fp(0)3085 2006 y Ft(j)35 b Fu(=)e Fv(\016)s Fu(2)3337 1973 y Fs(n)3413 2006 y Fy(and)c(therefore)0 2119 y Ft(j)p Fv(G)96 2133 y Fp(1)136 2119 y Ft(j)21 b(\000)f(j)p Fv(B)367 2133 y Fp(1)406 2119 y Ft(j)26 b Fu(=)f(\()598 2083 y Fp(1)p 598 2098 V 598 2150 a(2)663 2119 y Ft(\000)20 b Fv(\016)s Fu(\)2)877 2086 y Fs(n)926 2119 y Fy(.)141 2251 y(No)n(w)27 b(consider)k(an)o(y)d(point)h Fv(y)37 b Ft(2)d Fv(B)1296 2265 y Fp(1)1335 2251 y Fy(,)29 b(and)g(let)f Fv(x)34 b Ft(2)g Fv(G)1916 2265 y Fp(1)1956 2251 y Fy(,)28 b Fv(x)2059 2218 y FH(0)2117 2251 y Ft(2)33 b Fv(G)2282 2265 y Fp(0)2350 2251 y Fy(be)28 b(a)g(matched)h(pair)g(as)f(de\002ned) i(abo)o(v)o(e.)43 b(Then)0 2364 y Fv(x)19 b Ft(^)g Fv(y)28 b Fu(=)d Fv(x)372 2331 y FH(0)414 2364 y Ft(^)19 b Fv(y)s Fy(,)j(b)n(ut)i Fv(f)10 b Fu(\()p Fv(x)p Fu(\))19 b Ft(^)g Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b(=)g(1)e Fy(while)g Fv(f)10 b Fu(\()p Fv(x)1728 2331 y FH(0)1751 2364 y Fu(\))19 b Ft(^)g Fv(f)10 b Fu(\()p Fv(y)s Fu(\))25 b(=)g(0)p Fy(.)j(Therefore,)d(either)f Fv(f)10 b Fu(\()p Fv(x)19 b Ft(^)g Fv(y)s Fu(\))25 b Ft(6)p Fu(=)g Fv(f)10 b Fu(\()p Fv(x)p Fu(\))19 b Ft(^)g Fv(f)10 b Fu(\()p Fv(y)s Fu(\))22 b Fy(or)0 2477 y Fv(f)10 b Fu(\()p 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2624 y Fp(1)3776 2610 y Ft(j)28 b Fu(=)0 2722 y(\()45 2687 y Fp(1)p 45 2702 V 45 2754 a(2)111 2722 y Ft(\000)21 b Fv(\016)s Fu(\)2)326 2689 y Fs(n)375 2722 y Fy(,)i(we)h(get)g(that)h Ft(j)p Fv(B)939 2736 y Fp(1)979 2722 y Ft(j)i(\025)f Fv(\016)s Fu(2)1216 2689 y Fs(n)p FH(\000)p Fp(2)1355 2722 y Fy(.)k(It)24 b(follo)n(ws)h(that)g(the)g(probability)j(of)c(obtaining)j(a)d (violating)j(pair)l(,)e(is)f(at)g(least)11 2808 y Fs(\016)p 10 2823 V 10 2875 a Fp(4)55 2844 y Fu(\()100 2808 y Fp(1)p 100 2823 V 100 2875 a(2)166 2844 y Ft(\000)c Fv(\016)s Fu(\))p Fy(.)p 463 2844 67 67 v 141 2977 a(The)j(abo)o(v)o(e)h (analysis)i(breaks)f(when)e Fv(f)32 b Fy(is)23 b(actually)j(almost)e Fu(1)p Fv(=)p Fu(2)d Ft(\000)e Fv(f)10 b(ar)25 b Fy(from)e(e)n(v)o(ery) h(singleton,)i(since)e(in)g(this)g(case)0 3089 y Fv(\016)35 b Fy(is)d(close)h(to)e Fu(1)p Fv(=)p Fu(2)p Fy(,)k(and)d(the)g (probability)1424 3054 y Fs(\016)p 1423 3069 36 4 v 1423 3121 a Fp(4)1468 3089 y Fu(\()1513 3054 y Fp(1)p 1513 3069 V 1513 3121 a(2)1585 3089 y Ft(\000)26 b Fv(\016)s Fu(\))33 b Fy(of)f(obtaining)i(a)e(violating)i(pair)e(is)g(not)g (bounded)i(from)e(belo)n(w)-6 b(.)0 3202 y(Another)31 b(disadv)n(antage)i(of)d(Algorithm)h(4)e(is)h(the)g(tw)o(o)f(sided)i (error)g(probability)i(for)d(testing)h(singletons,)j(as)c(opposed)0 3315 y(to)f(the)h(one)g(sided)h(error)f(we)f(achie)n(v)o(ed)i(in)e (Algorithm)i(1)e(when)g(we)g(added)i(the)f(parity)g(test.)48 b(Ho)n(we)n(v)o(er)l(,)30 b(Algorithm)h(4)0 3428 y(can)25 b(be)g(generalized)j(to)d(testing)h Fv(k)s Fy(-monomials,)h(with)e(a)f (query)i(comple)o(xity)h(of)e(only)g Fv(O)s Fu(\(1)p Fv(=\017)3028 3395 y Fp(2)3069 3428 y Fu(\))p Fy(,)f(in)h(comparison)i (to)e(the)21 3518 y Fu(~)0 3541 y Fv(O)s Fu(\(1)p Fv(=\017)234 3508 y Fp(3)274 3541 y Fu(\))f Fy(query)h(comple)o(xity)h(of)e (Algorithm)h(2.)k(The)24 b(probability)j(of)d(choosing)i(a)e(violating) i(pair)f(can)f(be)g(sho)n(wn)g(to)g(be)0 3654 y(at)f(least)287 3618 y Fs(\016)p 287 3633 V 287 3685 a Fp(4)332 3654 y 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