%!PS-Adobe-2.0 %%Creator: dvips 5.528 Copyright 1986, 1994 Radical Eye Software %%Title: rolim.final.dvi %%CreationDate: Tue Jan 13 10:17:19 1998 %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o rolim.final.ps rolim.final.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.01.13:1012 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies 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y(whic)o(h)c(one-sided)h(randomness)f(seems)f(to)h(b)q(e)h(easier)f(to) g(ac)o(hiev)o(e)h(is)f(in)g(the)h(case)f(of)g Fp(\\smal)r(l")i(line)n (ar)f(subsp)n(ac)n(es)-35 1149 y FB(of)f Fk(f)p FB(0)p Fo(;)8 b FB(1)p Fk(g)130 1128 y Fj(n)172 1149 y FB([4)o(].)28 b(It)18 b(is)g(indeed)h(p)q(ossible)e(to)h(construct)f(small)f(hitting) h(sets)h(for)f(this)h(class)f(of)h(subsets)g(\(and,)g(so,)-35 1205 y(for)13 b(the)h(corresp)q(onding)g(c)o(haracteristic)f (functions\))g(that)h(imply)e(some)h(explicit,)g(exp)q(onen)o(tial)g (lo)o(w)o(er)g(b)q(ounds)i(for)-35 1262 y(the)h(branc)o(hing)f(program) f(mo)q(del)h([4],)g(but)h(no)g(construction)f(of)h(non-trivial)f (discrepancy)h(sets)g(for)f(this)g(class)h(is)-35 1318 y(kno)o(wn.)35 1375 y(One)d(general)e(reason)h(for)f(the)h(fact)g(that) f(one-sided)h(random)f(ob)s(jects)g(seem)g(to)h(b)q(e)g(easier)g(to)f (construct)h(is)f(that)-35 1431 y(they)16 b(ha)o(v)o(e)g(a)g Fp(monotonicity)h FB(prop)q(ert)o(y)f(not)g(satis\014ed)g(b)o(y)g (discrepancy)h(sets:)22 b(if)16 b Fo(H)i FB(=)d Fk(f)p Fo(H)1565 1438 y Fj(n)1588 1431 y Fo(;)8 b(n)15 b(>)g FB(0)p Fk(g)h FB(is)g(a)g(family)-35 1488 y(of)h(hitting)f(ob)s(jects)h (then)h(an)o(y)f(other)g(family)f Fo(H)827 1471 y Fc(0)855 1488 y FB(=)h Fk(f)p Fo(H)972 1471 y Fc(0)968 1499 y Fj(n)991 1488 y Fo(;)8 b(n)16 b(>)h FB(0)p Fk(g)p FB(,)g(suc)o(h)h (that)f(for)g(an)o(y)g Fo(n)g(>)g FB(0)g Fo(H)1727 1471 y Fc(0)1723 1499 y Fj(n)1764 1488 y FB(\\con)o(tains")-35 1544 y Fo(H)3 1551 y Fj(n)26 1544 y FB(,)e(has)g(\(at)f(least\))g(the)h (same)f(same)g(hitting)g(prop)q(erties.)35 1601 y(In)e(a)e(rather)h (surprising)f(w)o(a)o(y)l(,)g(Andreev)i Fp(et)g(al)f FB(sho)o(w)o(ed)g(that)f(the)h(ab)q(o)o(v)o(e)f(in)o(tuition)g(ab)q (out)g(the)h(real)g(relationship)-35 1657 y(b)q(et)o(w)o(een)k(PR)o (G's)g(and)g(HSG's)g(is)f(somewhat)g(false.)-35 1758 y Fi(Theorem)g(3.1)j(\([1]\))22 b Fp(L)n(et)16 b Fo(k)q FB(\()p Fo(n)p FB(\))c(=)h Fo(O)q FB(\(log)7 b Fo(n)p FB(\))p Fp(.)21 b(If)16 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1726 y Fo(n)p FB(\))p Fo(=)p FB(\()p Fo(k)g FB(+)f(log)d Fo(n)p FB(\))p Fo(:)-35 1819 y FB(F)l(urthermore,) 13 b(for)h(almost)f(ev)o(ery)i(Bo)q(olean)g(op)q(erator)f Fo(H)j FB(:)12 b Fo(k)h Fk(!)h Fo(n)p FB(,)h(w)o(e)g(ha)o(v)o(e)78 1912 y Fo(L)109 1893 y Fj(op)146 1912 y FB(\()p Fo(H)t FB(\))c(=)i(\002\(\(2)377 1893 y Fj(k)398 1912 y Fo(n)p FB(\))p Fo(=)p FB(\()p Fo(k)e FB(+)f(log)e Fo(n)p FB(\)\))14 b Fo(:)35 2005 y FB(The)j(follo)o(wing)d(theorem)g(giv)o(es)i(a)g (su\016cien)o(t)g(condition)f(for)h FC(P)e FB(=)h FC(BPP)h FB(in)g(terms)f(of)h(the)g(w)o(orst-case)f(circuit)-35 2062 y(complexit)o(y)e(of)h(c)o(haracteristic)g(functions)h(of)g(sets)g (generated)g(b)o(y)g(Bo)q(olean)g(op)q(erators)f(\(as)h(de\014ned)h(ab) q(o)o(v)o(e\).)-35 2167 y Fi(Theorem)e(3.2)j(\([3]\))22 b Fp(L)n(et)14 b Fo(k)q FB(\()p Fo(n)p FB(\))e(=)h(\(2)6 b(+)g Fo(O)q FB(\(1\)\))i(log)d Fo(n)p Fp(.)20 b(A)14 b(c)n(onstant)g FB(0)e Fo(<)h(c)1279 2174 y Fw(0)1311 2167 y Fo(<)g FB(1)h 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y(Theorem)f(2.1)h(since)h(it) f(has)h Fo(O)q FB(\()p Fo(n)557 1515 y Fw(2)576 1531 y FB(\))g(inputs)g(and)g(the)f(hardness)i(is)e(\\only")g(2)1328 1515 y Fw(\012\()p Fj(n)p Fw(\))1404 1531 y FB(.)25 b(F)l(rom)15 b(this)h(assumption)f(it)h(is)-35 1588 y(only)h(p)q(ossible)h(to)f (infer)h(from)e(the)i(tec)o(hniques)g(of)g([14)o(,)f(6])h(that)f FC(BPP)h FB(can)g(b)q(e)h(deterministically)c(sim)o(ulated)h(in)-35 1644 y(time)d Fo(n)95 1628 y Fw(p)q(oly)5 b(log)h Fj(n)243 1644 y FB(.)35 1701 y(Impagliazzo)16 b(and)i(Wigderson)f([8])g(recen)o (tly)h(made)f(further)g(progress)h(b)o(y)g(sho)o(wing)e(ho)o(w)i(to)f (de-randomize)-35 1757 y(the)c(Xor)g(Lemma.)18 b(Their)13 b(main)f(result)h(is)g(a)g(pro)q(cedure)h(that,)f(giv)o(en)g(a)g (function)g Fo(g)i FB(with)e Fo(n)h FB(inputs)f(and)h(a)f(certain)-35 1814 y(unpredictabilit)o(y)l(,)f(constructs)h(another)h(function)f Fo(h)h FB(with)f Fo(O)q FB(\()p Fo(n)7 b FB(+)g Fo(k)q FB(\))14 b(inputs)g(whose)g(unpredictabilit)o(y)e(decreases)-35 1870 y(exp)q(onen)o(tially)h(in)h Fo(k)q FB(.)19 b(Th)o(us,)14 b(assuming)e(that)i(a)g(function)f Fo(f)20 b FB(in)14 b FC(EXP)g FB(exists)f(with)h(circuit)f(complexit)o(y)f(2)1801 1853 y Fw(\012\()p Fj(n)p Fw(\))1877 1870 y FB(,)i(one)-35 1926 y(can)e(deduce)i(that)e(a)g(function)g Fo(g)i FB(exists)e(in)g FC(EXP)h FB(suc)o(h)g(that)e(circuits)h(of)g(size)g(2)1315 1910 y Fw(\012\()p Fj(n)p Fw(\))1404 1926 y FB(only)g(ha)o(v)o(e)g (success)h(probabilit)o(y)-35 1983 y(1)f Fk(\000)g FB(1)p Fo(=n)120 1966 y Fw(2)159 1983 y FB(on)18 b Fo(g)r FB(.)30 b(Then,)19 b(using)g(a)f(result)g(of)g(Impagliazzo)f([11)o(])h(it)g (follo)o(ws)f(that)h(a)g(function)h Fo(g)1671 1966 y Fc(0)1700 1983 y FB(in)g FC(EXP)g FB(exists)-35 2039 y(suc)o(h)d(that)f(circuits)h(of)f(size)h(2)490 2023 y Fw(\012\()p Fj(n)p Fw(\))582 2039 y FB(ha)o(v)o(e)g(success)h (probabilit)o(y)d(at)h(most)f(2)p Fo(=)p FB(3,)h(and,)h(ev)o(en)o (tually)l(,)g(that)f(a)h(function)-35 2096 y Fo(h)h FB(exists)g(in)g FC(EXP)g FB(suc)o(h)h(that)f(circuits)f(of)h(size)g(2)826 2079 y Fj(\017n)881 2096 y FB(only)g(ha)o(v)o(e)g(success)h(probabilit) o(y)d(2)1504 2079 y Fc(\000)p Fj(\017n)1569 2096 y FB(.)26 b(In)18 b(turn,)f(the)h(latter)-35 2152 y(statemen)o(t)13 b(implies)g(the)i(existence)h(of)e(PR)o(G)h(of)g(logarithmic)d(price)j (and)h(th)o(us)f FC(P)d FB(=)h FC(BPP)q FB(.)-35 2246 y Fi(Theorem)h(3.3)22 b Fp([8])16 b(If)g(a)h(function)f(in)f Fo(f)22 b Fp(exists)15 b(having)h(cir)n(cuit)h(c)n(omplexity)f FB(2)1366 2229 y Fw(\012\()p Fj(n)p Fw(\))1458 2246 y Fp(then)g FC(P)d FB(=)g FC(BPP)p Fp(.)-35 2387 y Fq(4)66 b(Related)23 b(Results)g(and)g(Conclusion)-35 2488 y FB(De-randomization)7 b(is)j(not)g(the)g(only)g(researc)o(h)g (direction)g(that)f(has)h(b)q(een)i(explored)e(ab)q(out)g(the)g(use)h (of)e(randomness)-35 2545 y(in)14 b(computation.)j(An)e(alternativ)o(e) d(approac)o(h)i(deals)h(with)e(the)i(use)f(of)g Fp(we)n(ak)i(sour)n(c)n (es)e(of)i(r)n(andomness)d FB(\(see)h([13]\).)-35 2601 y(Ev)o(en)21 b(in)h(this)f(case)h(there)g(is)f(a)h(di\013erence)g(b)q (et)o(w)o(een)g(one-sided)g(pseudorandom)f(structures)g(and)h(t)o(w)o (o-sided)-35 2658 y(pseudorandom)14 b(structures)h(and)g(Theorem)f(3.1) h(is)f(a)h(useful)g(to)q(ol)g([5)o(].)p eop %%Page: 7 7 7 6 bop 35 140 a FB(A)21 b(natural)f(question)h(to)f(ask)g(is)h (whether)g(the)g(results)f(describ)q(ed)i(in)f(this)f(column)g(\(or)g (at)g(least)g(part)h(of)-35 197 y(them\))15 b(extend)i(to)e(parallel)g (and)i(space-b)q(ounded)h(classes.)23 b(The)17 b(issue)f(of)g (parallelization)e(is)i(not)g(addressed)g(b)o(y)-35 253 y(Impagliazzo)c(and)i(Wigderson)g([8)o(],)g(and)g(some)f(steps)h(in)g (their)g(construction)f(app)q(ear)i(to)e(b)q(e)i(hard)g(to)e (parallelize.)-35 310 y(On)i(the)g(other)f(hand,)h(a)f(b)q(ottlenec)o (k)g(for)g(the)h(parallelization)d(of)i(the)h(tec)o(hniques)g(of)f (Andreev)i(et)e(al.)g([3)o(])g(w)o(as)g(the)-35 366 y(use)d(of)g (Theorem)g(3.1,)f(whose)i(pro)q(of)f(in)g([1)o(])g(w)o(as)g(inheren)o 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y(more)f(p)q(o)o(w)o(erful)g (than)i FC(RP)p FB(.)20 b(A)14 b(recen)o(t)h(result)f(due)h(to)f(F)l (ortno)o(w)f(establishes)h(that)g(an)o(y)g FC(BPP)h FB(problem)e(is)h (solv)m(able)-35 818 y(b)o(y)19 b(an)g FC(RP)h FB(algorithm)c(making)h (one)j(oracle)e(query)h(to)g(a)g(promise-)p FC(RP)f FB(oracle)h (\(promise-)p FC(RP)e FB(is)i(the)h(extension)-35 874 y(of)d FC(RP)h FB(to)f Fp(pr)n(omise)h(pr)n(oblems)p FB(\).)26 b(It)18 b(is)f(an)h(op)q(en)g(question)f(to)g(pro)o(v)o(e)g (the)h(same)e(result)h(without)g(using)g(promise)-35 931 y(problems.)j(Solving)15 b(this)h(question)f(ma)o(y)g(b)q(e)h(an)g (imp)q(ortan)o(t)e(in)o(termediate)g(step)i(to)o(w)o(ards)e(pro)o(ving) h FC(BPP)f FB(=)g FC(RP)q FB(,)-35 987 y(a)j(result)h(that)f(w)o(ould)g (ha)o(v)o(e)h(sev)o(eral)f(complexit)o(y-theoretic)e(consequences)k (\(e.g.)28 b FC(RP)18 b FB(w)o(ould)f(b)q(e)i(closed)f(under)-35 1043 y(complemen)o(t,)12 b FC(BPP)k FB(w)o(ould)e(b)q(e)i(con)o(tained) f(in)g FC(NP)g FB(and)h(so)e(on\).)-35 1187 y Fq(Ac)n(kno)n(wledgmen)n (ts)-35 1288 y FB(W)l(e)h(are)g(grateful)f(to)g(Eric)h(Allender)g(for)g (his)g(helpful)g(commen)o(ts)e(and)i(suggestions.)-35 1431 y Fq(References)-13 1533 y FB([1])22 b(Andreev)d(A.,)g(Clemen)o (ti)d(A.,)j(and)g(Rolim)e(J.)h(\(1996\),)f(\\A)i(New)f(General)g (De-Randomization)f(Metho)q(d",)58 1589 y Fp(J.)23 b(of)g(the)h(A)o(CM) p FB(.)d(to)h(app)q(ear.)g(Extended)i(Abstract)e(in)g Fp(23-th)i(A)o(nnual)f(International)e(Col)r(lo)n(quium)j(on)58 1646 y(A)o(lgorithms,)16 b(L)n(o)n(gic)f(and)h(Pr)n(o)n(gr)n(amming)g (\(ICALP'96\))p FB(,)c(LNCS,)k(1099,)d(pp.)j(357-368.)-13 1740 y([2])22 b(Andreev)c(A.,)g(Clemen)o(ti)e(A.,)i(and)g(Rolim)e(J.)h (\(1996\))f(\\Hitting)g(Prop)q(erties)h(of)h(Hard)f(Bo)q(olean)h(Op)q (erators)58 1796 y(and)d(Their)g(Consequences)h(on)g FC(BPP)p FB(,)f Fp(ECCC)f(R)n(ep)n(ort)j(TR96-055)p FB(.)-13 1890 y([3])22 b(Andreev)f(A.,)f(Clemen)o(ti)e(A.,)i(and)g(Rolim)e(J.)i (\(1997\),)f(\\W)l(orst-case)g(Hardness)h(Su\016ces)g(for)g(Derandom-) 58 1946 y(ization:)30 b(a)21 b(New)f(metho)q(d)h(for)f (Hardness-Randomness)h(T)l(rade-O\013s",)g(in)g Fp(24-th)h(A)o(nnual)f (International)58 2003 y(Col)r(lo)n(quium)16 b(on)g(A)o(lgorithms,)g(L) n(o)n(gic)f(and)i(Pr)n(o)n(gr)n(amming)e(\(ICALP'97\))p FB(,)e(LNCS,)i(1256,)f(pp.)h(177-187.)-13 2097 y([4])22 b(Andreev)16 b(A.,)g(Bask)m(ak)o(o)o(v)f(J.,)g(Clemen)o(ti)f(A.,)h(and) h(Rolim)e(J.)h(\(1997\),)f(\\E\016cien)o(t)h(Construction)g(of)g Fo(\017)p FB(-Biased)58 2153 y(Sample)f(Spaces)i(for)f(Systems)f(of)h (Linear)g(T)l(ests)g(and)g(Applications",)f Fp(T)m(e)n(chnic)n(al)g(R)n (ep)n(ort)i(in)g(ECCC)f(-)h(TR-)58 2209 y(97-053)p FB(.)-13 2303 y([5])22 b(Andreev)13 b(A.,)f(Clemen)o(ti)f(A.,)h(Rolim)f(J.)h (and)g(T)l(revisan)g(L.)h(\(1997\),)d(\\W)l(eak)i(Random)g(Sources,)h (Hitting)e(Sets,)58 2360 y(and)j FC(BPP)g FB(Sim)o(ulation",)d Fp(SIAM)i(J.)i(of)g(Comput.)p FB(,)f(to)f(app)q(ear.)h(Extended)g (abstract)f(in)h Fp(38-th)i(A)o(nnual)d(IEEE)58 2416 y(Symp)n(osium)j(on)g(F)m(oundations)g(on)g(Computer)h(Scienc)n(e)p FB(,)c(pp.)i(264-273.)-13 2510 y([6])22 b(Babai)14 b(L.,)g(F)l(ortno)o (w)g(L.,)g(Nisan)g(N.)h(and)f(Wigderson)g(A.)h(\(1993\))d(\\BPP)j(has)f (Sub)q(exp)q(onen)o(tial)h(Time)f(Sim)o(u-)58 2566 y(lations)g(unless)h (EXPTIME)g(has)g(Publishable)g(Pro)q(ofs")f Fp(Computational)j (Complexity)p FB(,)e(3,)f(pp.)h(307-318.)p eop %%Page: 8 8 8 7 bop -13 140 a FB([7])22 b(Blum)14 b(M.,)h(and)g(Micali)f(S.)h (\(1984\),)e(\\Ho)o(w)i(to)f(generate)h(cryptographically)f(strong)g (sequences)j(of)e(pseudo-)58 197 y(random)f(bits",)g Fp(SIAM)h(J.)h(of)g(Computing)p FB(,)f(13\(4\),)e(pp.)j(850-864.)-13 290 y([8])22 b(Impagliazzo)16 b(R.,)i(and)g(Wigderson)f(A.)g(\(1997\),) g(\\)p FC(P)o FB(=)h FC(BPP)h FB(if)e(E)g(requires)h(exp)q(onen)o(tial) f(circuits:)25 b(Deran-)58 347 y(domizing)20 b(the)h(X)o(OR)i(lemma")18 b(In)k Fp(29-th)h(A)o(nnual)e(A)o(CM)g(Symp)n(osium)i(on)e(The)n(ory)h (of)h(Computing)p FB(,)f(pp.)58 403 y(220-229.)-13 497 y([9])g(Mot)o(w)o(ani)13 b(R.,)i(and)g(Ragha)o(v)m(an)h(P)l(.)f (\(1995\),)e Fp(R)n(andomize)n(d)i(A)o(lgorithms)p FB(,)g(Cam)o(bridge) e(Univ)o(ersit)o(y)h(Press.)-35 591 y([10])21 b(Goldreic)o(h)10 b(O.)h(\(1997\),)f(\\A)g(Sample)g(of)h(Samplers:)16 b(A)11 b(Computational)d(P)o(ersp)q(ectiv)o(e)j(on)g(Sampling",)e Fp(ECCC)p FB(,)58 647 y(TR97-20.)-35 741 y([11])21 b(Impagliazzo,)12 b(R.)h(\(1995\),)f(\\Hard-core)i(distributions)e(for)h(somewhat)f(hard) h(problems",)f(in)i Fp(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)58 798 y(the)17 b(36th)g(IEEE)e(Symp)n(osium)h(on)g(F)m(oundations)g(of)g (Computer)h(Scienc)n(e)p FB(,)c(pp.)j(538{545.)-35 892 y([12])21 b(Nisan)14 b(N.)h(\(1990\),)d Fp(Using)j(Har)n(d)g(Pr)n (oblems)g(to)h(Cr)n(e)n(ate)f(Pseudor)n(andom)h(Gener)n(ators)p FB(,)e Fp(A)o(CM)h(Distinguishe)n(d)58 948 y(Dissertation)p FB(,)f(MIT)i(Press.)-35 1042 y([13])21 b(Nisan)e(N.)g(\(1996\),)f (\\Extracting)g(randomness:)27 b(Ho)o(w)19 b(and)h(wh)o(y",)f(In)h Fp(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)i(the)h(11th)f(A)o(nnual)58 1098 y(IEEE)15 b(Confer)n(enc)n(e)g(on)h(Computational)g(Complexity)p FB(,)f(pp.)g(44-58.)-35 1192 y([14])21 b(Nisan)14 b(N.,)f(and)h (Wigderson)g(A.)f(\(1994\),)f(\\Hardness)i(vs)g(Randomness",)f Fp(J.)i(Comput.)h(System)f(Sci.)e FB(49,)g(pp.)58 1249 y(149-167.)-35 1342 y([15])21 b(P)o(apadimitriou)12 b(C.H.)i(\(1993\),) f Fp(Computational)k(Complexity,)31 b FB(Addison-W)l(esley)l(.)-35 1436 y([16])21 b(Saks)c(M.,)h(Sriniv)m(asan)f(A.,)g(and)h(Zhou)g(S.)f (\(1995\).)f(\\Explicit)g(disp)q(ersers)i(with)f(p)q(olylog)f(degree",) i(In)g Fp(Pr)n(o-)58 1493 y(c)n(e)n(e)n(dings)c(of)j(the)f(27th)h(A)o (nnual)e(A)o(CM)h(Symp)n(osium)g(on)g(The)n(ory)g(of)h(Computing)p FB(,)d(pp.)i(479-488,)d(1995.)-35 1586 y([17])21 b(W)l(egener,)12 b(I.)f(\(1987\),)f Fp(The)i(c)n(omplexity)h(of)g(\014nite)f(Bo)n(ole)n (an)f(functions)p FB(,)g Fp(Wiley-T)m(eubner)h(Series)g(in)g(Computer) 58 1643 y(Scienc)n(e)p FB(.)-35 1737 y([18])21 b(T)l(a-Shma)e(A.)h (\(1996\),)f(\\On)i(extracting)e(randomness)g(from)f(w)o(eak)i(random)e (sources",)j(In)g Fp(28th)g(A)o(nnual)58 1793 y(A)o(CM)15 b(Symp)n(osium)h(on)h(The)n(ory)f(of)g(Computing)p FB(,)f(pp.)g (276-285.)-35 1887 y([19])21 b(Y)l(ao)11 b(A.)g(\(1982\),)f(\\Theory)g (and)i(applications)d(of)i(trap)q(do)q(or)g(functions",)g(in)g Fp(23th)i(A)o(nnual)f(IEEE)f(Symp)n(osium)58 1943 y(on)16 b(F)m(oundations)g(on)g(Computer)h(Scienc)n(e)p FB(,)c(pp.)i(80-91.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF