%!PS-Adobe-2.0 %%Creator: dvips 5.528 Copyright 1986, 1994 Radical Eye Software %%Title: col.dvi %%CreationDate: Wed Sep 23 13:08:40 1998 %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o col.ps col.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.09.23:1308 %%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 X}B /FMat[1 0 0 -1 0 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Fo(Righ)o(t)h({)g(except)e(that)i(all)g(the)f(problems)h(y)o (ou)f(men)o(tioned)g(are)h(in)g(the)f Fi(same)g Fo(class.)148 1924 y Fj(Q:)g Fo(Really?)23 b(Clique)17 b(is)g(an)f(enco)q(ding)i(of)f (SA)l(T?)148 1984 y Fj(A:)e Fo(Y)l(es.)20 b(Berman)13 b(and)j(Hartmanis)f([BH77)o(])f(ga)o(v)o(e)g(some)h(simple)g (su\016cien)o(t)f(conditions)j(for)75 2044 y(t)o(w)o(o)g(sets)f(to)i(b) q(e)e(isomorphic,)i(and)g(with)f(these)g(conditions,)h(it)f(is)h(easy)f (to)g(see)f(that)h(all)h(of)f(the)75 2104 y(NP-complete)f(problems)h (in)g([GJ79])f(are)h(isomorphic.)148 2165 y Fj(Q:)f Fo(What)h(ab)q(out) g(smaller)h(complexit)o(y)e(classes,)g(suc)o(h)g(as)h(P)g(and)g(NLOG?) 148 2225 y Fj(A:)j Fo(Hartmanis)g([Ha78)q(])f(studied)h(analogous)j (conjectures)c(for)h(these)f(classes,)i(using)g(log-)75 2285 y(space)15 b(reducibilit)o(y)h(\(and)f(logspace)h(isomorphisms\),) h(instead)f(of)f(p)q(oly-time.)22 b(Once)14 b(again,)j(all)75 2345 y(of)i(the)f(standard)i(complete)e(sets)g(are)h(easily)h(seen)e (to)h(b)q(e)f(isomorphic.)30 b(Of)18 b(course,)h(logspace)75 2405 y(reductions)f(are)f(to)q(o)i(p)q(o)o(w)o(erful)e(to)h(in)o(v)o (estigate)g(the)f(man)o(y)f(in)o(teresting)j(sub)q(classes)g(of)e(DLOG) 75 2466 y(\(suc)o(h)23 b(as)h(A)o(CC)384 2448 y Fp(0)404 2466 y Fo(,)h(TC)513 2448 y Fp(0)533 2466 y Fo(,)g(and)f(NC)746 2448 y Fp(1)765 2466 y Fo(\).)43 b(F)l(or)24 b(these)f(classes,)j(A)o (C)1317 2448 y Fp(0)1360 2466 y Fo(reducibilit)o(y)e(is)h(the)e(most)75 2526 y(natural)18 b(notion)f(of)g(reducibilit)o(y)g(to)g(use)f(\(see)g ([Ba89)q(,)f(Bu87)q(],)g(for)i(example\),)e(and)i(once)f(again)p eop %%Page: 4 4 4 3 bop 75 203 a Fo(the)16 b(\\standard")i(complete)e(sets)h(are)f(all) i(seen)e(to)g(b)q(e)h(isomorphic)g([ABI93)o(].)1519 185 y Fp(3)148 264 y Fj(Q:)k Fo(So,)h(I)f(guess)g(that,)h(with)g(so)g(m)o (uc)o(h)e(written)h(ab)q(out)i(the)d(isomorphism)j(conjectures,)75 324 y(most)16 b(complexit)o(y)g(theoreticians)i(b)q(eliev)o(e)e(that)h (the)f(conjectures)f(are)i(true?)148 384 y Fj(A:)d Fo(F)l(ar)h(from)f (it.)21 b(In)14 b(fact,)g(probably)h(most)g(complexit)o(y)e (theoreticians)j(b)q(eliev)o(e)e(that)h(there)75 444 y(is)k(a)g(one-w)o(a)o(y)g(function)g Fc(f)24 b Fo(\(one)18 b(that)h(is)g(easy)g(to)g(compute)f(but)g(v)o(ery)g(hard)h(to)g(in)o(v) o(ert\))e(suc)o(h)75 504 y(that)f Fc(f)5 b Fo(\(SA)l(T\))15 b(is)h Fi(not)g Fo(isomorphic)h(to)f(SA)l(T.)f(Some)g(really)h(in)o (teresting)g(w)o(ork)g(in)g(this)g(direction)75 565 y(has)h(b)q(een)f (done)h(b)o(y)e([JY85,)h(KLD86)r(,)f(KMR95)q(].)8 b(.)g(.)148 625 y Fj(Q:)15 b Fo(Let)g(me)f(in)o(terrupt.)20 b(Y)l(ou'll)c(nev)o(er) d(get)i(this)h(article)f(done)g(on)h(time)e(if)i(y)o(ou)e(are)h(going)i (to)75 685 y(surv)o(ey)e(the)h(en)o(tire)g(\014eld.)22 b(Are)15 b(there)h(an)o(y)g(surv)o(eys)g(y)o(ou)g(can)g(suggest?)148 745 y Fj(A:)h Fo(Y)l(es.)23 b(Tw)o(o)17 b(v)o(ery)f(nice)h(ones)h(that) f(spring)h(to)g(mind)f(are)g([KMR90])f(and)i([Y)l(o90].)23 b(Some)75 805 y(exciting)12 b(dev)o(elopmen)o(ts)f(\(suc)o(h)g(as)h ([FFK96,)f(JPY94,)g(Ro95)r(]\))g(are)g(to)q(o)i(recen)o(t)d(to)i(b)q(e) f(men)o(tioned)75 866 y(in)17 b(the)f(surv)o(eys.)148 926 y Fj(Q:)g Fo(What)h(do)q(es)g(this)g(ha)o(v)o(e)e(to)i(do)g(with)g (factorization?)148 986 y Fj(A:)h Fo(I'm)f(getting)i(to)f(that.)27 b(In)18 b(spite)g(of)g(the)g(general)h(feeling)f(that)h(the)f(original) i(Berman-)75 1046 y(Hartmanis)f(conjecture)f(is)h(probably)h(false,)f (it)g(w)o(as)g(sho)o(wn)g(in)g([AAR96)o(])f(that)h(the)f(Berman-)75 1106 y(Hartmanis)f(conjecture)e(for)i(A)o(C)695 1088 y Fp(0)730 1106 y Fo(reductions)g(is)g(actually)h Fi(true)p Fo(.)148 1166 y Fj(Q:)e Fo(Please)h(state)g(this)g(more)f(clearly)l(.) 148 1227 y Fj(A:)g Fo(F)l(or)f(an)o(y)h(\\reasonable")i(complexit)o(y)d (class,)i(all)g(of)f(the)f(sets)h(that)g(are)g(complete)f(under)75 1287 y(A)o(C)146 1269 y Fp(0)186 1287 y Fo(reductions)21 b(are)f(isomorphic)i(under)f(bijections)g(computable)g(and)g(in)o(v)o (ertible)g(b)o(y)f(A)o(C)1856 1269 y Fp(0)75 1347 y Fo(circuits.)i (\(In)16 b(fact,)g(they)f(are)i(computable)f(and)h(in)o(v)o(ertible)g (b)o(y)e Fi(depth)j(thr)n(e)n(e)e Fo(A)o(C)1606 1329 y Fp(0)1641 1347 y Fo(circuits!\))148 1407 y Fj(Q:)g Fo(Is)g(depth)g(three)g(optimal?)148 1467 y Fj(A:)g Fo(Y)l(es)g ([AAIPR97)o(].)148 1528 y Fj(Q:)g Fo(What)h(is)g(a)g(\\reasonable")h (complexit)o(y)e(class?)148 1588 y Fj(A:)k Fo(An)o(y)f(class)i(that)f (is)g(closed)h(under)e(man)o(y-one)h(reductions)h(computable)f(in)g(TC) 1764 1570 y Fp(0)1785 1588 y Fo(.)31 b(In)75 1648 y(particular,)26 b(NP)l(,)d(P)l(,)g(NC)563 1630 y Fp(1)583 1648 y Fo(,)h(DLOG,)g(BPP)l (,)f(and)h(just)f(ab)q(out)i(an)o(y)e(other)h(complexit)o(y)f(class)75 1708 y(y)o(ou'v)o(e)15 b(ev)o(er)f(heard)i(of)h(is)g(\\reasonable")h (in)f(this)f(sense)g(\(except)f(for)h(v)o(ery)f(small)j(classes,)e(suc) o(h)75 1768 y(as)h(A)o(C)206 1750 y Fp(0)241 1768 y Fo(and)g(A)o(CC)442 1750 y Fp(0)462 1768 y Fo(\).)148 1829 y Fj(Q:)22 b Fo(So,)j(in)e(ligh) o(t)h(of)f(y)o(our)f(earlier)i(commen)o(ts)d(ab)q(out)j Fc(f)5 b Fo(\(SA)l(T\),)23 b(when)f Fc(f)28 b Fo(is)c(a)f(one-w)o(a)o (y)75 1889 y(function,)17 b(do)q(es)g(this)g(mean)f(that)g(there)g(are) g(no)h(one-w)o(a)o(y)f(functions)i(computable)f(in)f(A)o(C)1783 1871 y Fp(0)1803 1889 y Fo(?)148 1949 y Fj(A:)k Fo(Quite)g(the)g(con)o (trary)l(.)32 b(It)19 b(w)o(as)i(sho)o(wn)g(in)f([Ni92)q(])f(that)h (there)g(are)g(functions)h Fc(f)k Fo(com-)75 2009 y(putable)16 b(in)g(A)o(C)377 1991 y Fp(0)412 2009 y Fo(that)f(are)h Fi(very)f Fo(one-w)o(a)o(y)g(with)i(resp)q(ect)e(to)g(depth-three)g(A)o (C)1556 1991 y Fp(0)1575 2009 y Fo(.)21 b(Nonetheless,)75 2069 y(ev)o(en)16 b(for)i(suc)o(h)e(a)i(\\random-lo)q(oking)q(")i (function)e Fc(f)5 b Fo(,)18 b Fc(f)5 b Fo(\(SA)l(T\))17 b(is)h(just)f(a)h(\\trivial")i(re-enco)q(ding)75 2130 y(of)d(SA)l(T)f(\(i.e.,)f(it)i(is)g(depth-three)e(A)o(C)784 2111 y Fp(0)804 2130 y Fo(-isomorphic)j(to)e(SA)l(T\).)148 2190 y Fj(Q:)11 b Fo(Are)g(there)g(some)h(anno)o(ying)h(details)g(ab)q (out)g(circuit)f(uniformit)o(y)g(conditions)i(that)e(y)o(ou're)75 2250 y(ignoring)19 b(in)e(this)g(article?)148 2310 y Fj(A:)f Fo(Y)l(es.)148 2370 y Fj(Q:)d Fo(This)h(seems)f(to)g(b)q(e)g (quite)h(a)f(strong)h(result!)21 b(Complexit)o(y)13 b(classes)h(are)g (the)e(fundamen)o(tal)75 2430 y(ob)s(jects)18 b(of)h(study)g(in)h (complexit)o(y)e(theory)l(,)h(b)q(ecause)g(the)f(natural)i (computational)h(problems)p 75 2474 720 2 v 131 2505 a Fg(3)150 2520 y Ff(Actually)13 b([ABI93])g(pro)o(v)o(es)h(something)f (m)o(uc)o(h)g(stronger,)h(but)g(that)g(w)o(ould)f(require)i(another)f (digression.)p eop %%Page: 5 5 5 4 bop 75 203 a Fo(that)18 b(w)o(e)f(are)g(in)o(terested)g(in)h(tend)f (to)h(cluster)f(in)o(to)h(classes)h(of)e(complete)g(sets)h(for)f (complexit)o(y)75 264 y(classes.)34 b(The)20 b(result)g(of)g([AAR96])f (sho)o(ws)i(that,)g(not)g(only)f(are)h(the)e(complete)h(sets)g(in)o (terre-)75 324 y(ducible,)d(but)f(in)h(fact)f(they)g(are)g(isomorphic)i (in)f(a)g Fi(very)f 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(reducibilit)o(y)l(,)75 805 y(but)i Fi(not)h Fo(under)f(A)o(C)458 787 y Fp(0)494 805 y Fo(reducibilit)o(y)l(.)148 866 y Fj(Q:)g Fo(What)h(do)q(es)g(this)g(set)f(lo)q(ok)i(lik)o(e?)148 926 y Fj(A:)13 b Fo(It)g(is)g(an)h(enco)q(ding)g(of)f(SA)l(T,)g(using)h (a)f(certain)h(kind)f(of)g(error-correcting)h(co)q(de.)20 b(This)14 b(set)75 986 y(\(let's)i(call)h(it)f(SA)l(T)429 968 y Fb(0)440 986 y Fo(\))g(is)h(clearly)f(p)q(oly-time)h(isomorphic)g (to)f(SA)l(T,)f(but)h(it)h(is)f(not)g(NP-complete)75 1046 y(under)e(A)o(C)282 1028 y Fp(0)316 1046 y Fo(reductions.)21 b(In)14 b(fact,)g(there)g(are)h(regular)g(sets)g(\(suc)o(h)f(as)h(the)f (P)l(ARITY)f(language\))75 1106 y(that)k(are)f(not)h(reducible)f(to)h (SA)l(T)712 1088 y Fb(0)740 1106 y Fo(under)f(A)o(C)949 1088 y Fp(0)984 1106 y Fo(reductions.)148 1166 y Fj(Q:)f Fo(Isn't)g(this)h(a)g(coun)o(terexample)f(to)h(y)o(our)f(claim)h(that)g (all)h(\\natural")h(NP-complete)d(sets)75 1227 y(are)h(complete)g (under)g(A)o(C)571 1209 y Fp(0)607 1227 y Fo(reducibilit)o(y?)148 1287 y Fj(A:)j Fo(This)h(dep)q(ends)f(on)g(y)o(our)g(de\014nition)h(of) f(\\natural".)32 b(I)18 b(think)h(that)h(not)f(man)o(y)f(p)q(eople)75 1347 y(w)o(ould)f(sa)o(y)g(that)f(this)h(is)g(the)f(\\natural")j(w)o(a) o(y)d(to)g(enco)q(de)h(SA)l(T.)148 1407 y Fj(Q:)i Fo(W)l(e'll)g(let)g (the)g(readers)g(of)g(this)h(column)g(decide)e(what)i(they)f(think)g (of)g(this)h(seman)o(tic)75 1467 y(hair-splitting)q(.)k(But)16 b(let's)g(get)g(bac)o(k)g(to)h(the)f(factorization)i(problem.)148 1528 y Fj(A:)e Fo(Thank)h(y)o(ou!)75 1694 y Fk(3)81 b(F)-7 b(actoring)27 b(and)g(Cylinders)75 1804 y Fj(A:)19 b Fo(First)h(let's)f(b)q(e)g(v)o(ery)f(precise)g(ab)q(out)j(what)f(w)o(e) e(mean)h(b)o(y)g(\\the)g(factorization)i(problem".)75 1864 y(Let)16 b(us)h(de\014ne)f(it)h(to)f(b)q(e)h(the)f(follo)o(wing)j (language:)265 1965 y(F)-5 b(A)o(CT)16 b(=)g Fh(f)p Fo(\()p Fc(x;)8 b(i;)g(b)p Fo(\))15 b(:)21 b(the)16 b Fc(i)p Fo(th)g(bit)h(of)f(the)g(prime)g(factorization)j(of)d Fc(x)g Fo(is)h Fc(b)p Fh(g)75 2067 y Fo(\(where)11 b(the)g(prime)g (factorization)i(is)f(presen)o(ted)e(as)i Fc(p)1056 2045 y Fe(e)1072 2050 y Fd(1)1056 2078 y Fp(1)1092 2067 y Fc(;)c(:)g(:)g(:)g(;)g(p)1226 2044 y Fe(e)1242 2050 y Fa(k)1226 2080 y Fe(k)1264 2067 y Fo(,)j(where)g(eac)o(h)g(exp)q(onen)o (t)g Fc(e)1758 2074 y Fe(i)1785 2067 y Fc(>)j Fo(0,)75 2127 y(and)j Fc(p)194 2134 y Fe(i)222 2127 y Fc(<)d(p)298 2134 y Fe(i)p Fp(+1)358 2127 y Fo(,)h(so)i(that)g(eac)o(h)f(n)o(um)o(b) q(er)f(has)i(a)g(unique)f(prime)h(factorization.)148 2187 y Fj(Q:)d Fo(What)h(are)f(the)g(complexit)o(y-theoretic)g(\\upp)q (er)h(and)f(lo)o(w)o(er)h(b)q(ounds")g(on)g(the)f(complex-)75 2248 y(it)o(y)i(of)h(F)-5 b(A)o(CT?)148 2308 y Fj(A:)14 b Fo(F)-5 b(A)o(CT)12 b(is)i(in)h(NP)p Fh(\\)p Fo(coNP)e(\(and)h(in)g (fact)f(it)h(is)h(ev)o(en)d(in)i(UP)p Fh(\\)p Fo(coUP)g([F)l(e92)o (]\).)20 b(Th)o(us)14 b(F)-5 b(A)o(CT)75 2368 y(is)21 b(not)g(NP-complete)e(unless)j(NP)d(=)i(coNP)f(=)g(UP)g(=)g(coUP)l(.)f 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b(:)g(:)g(x)722 331 y Fe(n)766 324 y Fo(is)22 b(a)f(m)o(ultiple)h(of)f(3)g(if)h(and)f(only)h(if)g(the) e(n)o(um)o(b)q(er)g(whose)75 384 y(binary)j(represen)o(tation)g(is)g Fc(x)642 391 y Fp(1)661 384 y Fo(0)p Fc(x)713 391 y Fp(2)733 384 y Fo(0)8 b Fc(:)g(:)g(:)g Fo(0)p Fc(x)883 391 y Fe(n)929 384 y Fo(is)23 b(a)g(m)o(ultiple)g(of)f(3.)39 b(Th)o(us,)24 b(if)f(F)-5 b(A)o(CT)21 b(w)o(ere)g(in)75 444 y(A)o(C)146 426 y Fp(0)165 444 y Fo(\()p Fh(\010)p Fo(\),)e(then)h(certainly)f(w)o (e)g(could)h(solv)o(e)g(the)f(Mo)q(d3)h(problem)g(with)g(A)o(C)1531 426 y Fp(0)1550 444 y Fo(\()p Fh(\010)p Fo(\))f(circuits,)h(in)75 504 y(con)o(tradiction)d(to)e([Smo87].)21 b(In)15 b(fact)g(this)h(sho)o (ws)g(that)f(F)-5 b(A)o(CT)14 b(is)i(not)g(in)g(A)o(C)1532 486 y Fp(0)1551 504 y Fo(\(Mo)q(d)p Fc(p)p Fo(\))g(for)f(an)o(y)75 565 y(prime)h Fc(p)f Fh(6)p Fo(=)e(3.)148 625 y Fj(Q:)j Fo(So,)g(is)h(F)-5 b(A)o(CT)16 b(in)h(A)o(C)629 607 y Fp(0)648 625 y Fo(\(Mo)q(d)g(3\)?)148 685 y Fj(A:)e Fo(No.)21 b(If)15 b(y)o(ou)g(lo)q(ok)h(at)g(the)f(\014nite)h(automaton)g(for)g (the)e(set)i(of)f(binary)h(strings)h(that)e(are)h(a)75 745 y(m)o(ultiple)e(of)h(5,)f(y)o(ou)f(will)j(see)d(that)h(the)g(n)o (um)o(b)q(er)f(of)h(1's)g(in)g(a)g(string)h Fc(x)f Fo(=)f Fc(x)1457 752 y Fp(1)1485 745 y Fc(:)8 b(:)g(:)g(x)1579 752 y Fe(n)1616 745 y Fo(is)14 b(a)g(m)o(ultiple)75 805 y(of)g(5)h(if)f(and)h(only)g(if)f(the)g(n)o(um)o(b)q(er)f(whose)i (binary)g(represen)o(tation)g(is)f(000)p Fc(x)1462 812 y Fp(1)1483 805 y Fo(000)p Fc(x)1583 812 y Fp(2)1604 805 y Fo(000)8 b Fc(:)g(:)g(:)h Fo(000)p Fc(x)1851 812 y Fe(n)75 866 y Fo(is)17 b(a)g(m)o(ultiple)h(of)f(5.)23 b(Th)o(us,)16 b(if)h(F)-5 b(A)o(CT)16 b(w)o(ere)g(in)h(A)o(C)1046 847 y Fp(0)1065 866 y Fo(\(Mo)q(d3\),)g(then)g(certainly)g(w)o(e)f (could)i(solv)o(e)75 926 y(the)13 b(Mo)q(d5)i(problem)f(with)g(A)o(C) 658 908 y Fp(0)677 926 y Fo(\(Mo)q(d3\))h(circuits,)f(again)h(in)f(con) o(tradiction)h(to)f([Smo87)q(].)20 b(Th)o(us)75 986 y(w)o(e)c(ha)o(v)o (e)f(that)i(F)-5 b(A)o(CT)15 b(is)i(not)g(in)g(A)o(C)777 968 y Fp(0)796 986 y Fo(\(Mo)q(d)p Fc(p)p Fo(\))g(for)g(an)o(y)f(prime) g Fc(p)p Fo(.)148 1046 y Fj(Q:)i Fo(So,)g(w)o(e)f(kno)o(w)h(that)h(F)-5 b(A)o(CT)16 b(is)j(not)f Fi(in)h Fo(A)o(C)1037 1028 y Fp(0)1056 1046 y Fo(\(Mo)q(d)p Fc(p)p Fo(\))g(for)f(an)o(y)f(prime)h Fc(p)p Fo(,)g(but)h(w)o(e)e(don't)75 1106 y(kno)o(w)f(if)h(F)-5 b(A)o(CT)15 b(is)i Fi(har)n(d)e Fo(for)i(an)o(y)f(suc)o(h)g(class.)22 b(Ho)o(w)17 b(ab)q(out)g(A)o(C)1285 1088 y Fp(0)1304 1106 y Fo(\(Mo)q(d6\)?)148 1166 y Fj(A:)g Fo(It)f(remains)h(an)g(op)q (en)g(question)g(if)g(there)f(is)i(an)o(ything)f(in)h(NTIME\(2)1530 1148 y Fe(n)1553 1166 y Fo(\))e(that)h(do)q(es)h(not)75 1227 y(ha)o(v)o(e)d(linear-size,)j(depth-three)e(circuits)h(of)f(Mo)q (d6)h(gates.)148 1287 y Fj(Q:)f Fo(Thanks)h(for)f(reminding)h(me)e (just)h(ho)o(w)h(little)g(w)o(e)f(kno)o(w.)21 b(What)16 b(do)q(es)h(this)g(ha)o(v)o(e)e(to)i(do)75 1347 y(with)g(the)f (question)h(of)g(whether)f(F)-5 b(A)o(CT)15 b(is)i(complete)f(for)g (some)h(complexit)o(y)f(class?)148 1407 y Fj(A:)22 b Fo(I'm)e(getting)i(to)g(that.)37 b(If)21 b(F)-5 b(A)o(CT)21 b(w)o(ere)g(complete)g(for)h(some)f(reasonable)i(complex-)75 1467 y(it)o(y)g(class,)i(then,)f(b)o(y)f([AAR96)o(],)h(F)-5 b(A)o(CT)22 b(w)o(ould)i(b)q(e)f(isomorphic)i(to)e(F)-5 b(A)o(CT)p Fh(\002f)p Fo(0)p Fc(;)8 b Fo(1)p Fh(g)1711 1449 y Fb(\003)1753 1467 y Fo(\(since)75 1528 y(F)-5 b(A)o(CT)p Fh(\002f)p Fo(0)p Fc(;)8 b Fo(1)p Fh(g)367 1510 y Fb(\003)402 1528 y Fo(w)o(ould)17 b(clearly)g(also)h(b)q(e)f (complete)e(for)i(the)f(same)g(complexit)o(y)g(class\).)148 1588 y Fj(Q:)f Fo(This)i(reminds)f(me)f(of)g(something)i(from)e(m)o(y)g (computabilit)o(y)i(textb)q(o)q(ok)f([Ro67)q(].)k(A)15 b(set)75 1648 y Fc(A)h Fo(that)h(is)g(recursiv)o(ely)e(isomorphic)j(to) f Fc(A)10 b Fh(\002)h(f)p Fo(0)p Fc(;)d Fo(1)p Fh(g)1049 1630 y Fb(\003)1086 1648 y Fo(is)17 b(called)g(a)g(\\cylinder".)148 1708 y Fj(A:)f Fo(Y)l(es.)21 b(This)c(same)g(terminology)g(w)o(as)g (imp)q(orted)g(to)g(complexit)o(y)e(theory)i(in)f([MY85].)148 1768 y Fj(Q:)g Fo(Are)g(w)o(e)f(no)o(w)i(ready)f(for)h(y)o(ou)f(to)g (state)h(y)o(our)f(conjecture?)148 1829 y Fj(A:)g Fo(Y)l(es.)197 1930 y Fj(Conjecture:)i Fo(There)11 b(is)g(no)h(isomorphism)g (computable)g(and)f(in)o(v)o(ertible)g(in)g(depth-)197 1990 y(three)16 b(A)o(C)393 1972 y Fp(0)428 1990 y Fo(mapping)i(F)-5 b(A)o(CT)p Fh(\002f)p Fo(0)p Fc(;)8 b Fo(1)p Fh(g)921 1972 y Fb(\003)956 1990 y Fo(on)o(to)16 b(F)-5 b(A)o(CT.)75 2092 y(That)19 b(is;)i(F)-5 b(A)o(CT)17 b(is)j(not)f(a)g (\(depth-three\))f(A)o(C)974 2074 y Fp(0)1012 2092 y Fo(cylinder.)29 b(If)18 b(this)h(conjecture)f(is)i(true,)e(then)75 2152 y(factoring)g(is)f(not)f(complete)g(for)h(an)o(y)f(reasonable)i (complexit)o(y)e(class)h(under)f(A)o(C)1598 2134 y Fp(0)1634 2152 y Fo(reductions.)148 2213 y Fj(Q:)i Fo(Isn't)g(this)h(conjecture)f (just)g(another)h(example)f(of)h(an)g(imp)q(ossibly)h(di\016cult)f (question)75 2273 y(in)e(complexit)o(y)f(theory?)21 b(W)l(ouldn't)c(it) g(imply)g(NLOG)g Fh(6)p Fo(=)f(P)l(,)g(or)g(TC)1363 2255 y Fp(0)1400 2273 y Fh(6)p Fo(=)g(NP)l(,)f(or)i(something?)148 2333 y Fj(A:)c Fo(I)f(don't)h(think)g(so.)20 b(If)12 b(I)h(could)g(sho)o(w)g(that)g Fc(A)g Fo(is)g(not)g(an)g(A)o(C)1309 2315 y Fp(0)1341 2333 y Fo(cylinder,)g(where)f Fc(A)h Fo(is)g(some)75 2393 y(set)f(in)h(NP)f(that)h(is)g(hard)g(for)g(TC)680 2375 y Fp(0)713 2393 y Fo(\(for)f(instance\),)i(then)e(I)g(w)o(ould)h (kno)o(w)g(that)g(TC)1593 2375 y Fp(0)1625 2393 y Fo(is)g(con)o(tained) 75 2453 y(in)18 b(the)f(class)h(of)f(languages)j(that)d(are)h (reducible)f(to)h Fc(A)p Fo(,)e(whic)o(h)i(is)g(prop)q(erly)g(con)o (tained)f(in)h(NP)75 2513 y(\(since)g(all)h(NP-complete)e(sets)h Fi(ar)n(e)f Fo(A)o(C)828 2495 y Fp(0)865 2513 y Fo(cylinders\),)h(and)g (th)o(us)g(I)f(could)i(conclude)f(that)g(TC)1855 2495 y Fp(0)p eop %%Page: 7 7 7 6 bop 75 203 a Fo(is)17 b(not)f(equal)g(to)h(NP)l(.)e(Ho)o(w)o(ev)o (er,)e(since)j(F)-5 b(A)o(CT)15 b(is)i Fi(not)f Fo(kno)o(wn)g(\(or)h (ev)o(en)d(b)q(eliev)o(ed\))i(to)g(b)q(e)g(hard)75 264 y(for)h(TC)220 246 y Fp(0)240 264 y Fo(,)e(this)j(obstacle)f(is)g(remo) o(v)o(ed.)148 324 y Fj(Q:)f Fo(What)h(in)o(tuition)h(do)f(y)o(ou)f(ha)o (v)o(e)f(ab)q(out)j(wh)o(y)e(the)g(conjecture)f(migh)o(t)i(b)q(e)f (true?)148 384 y Fj(A:)i Fo(First,)h(note)g(that)g(it)g(is)g(not)g(to)q (o)g(hard)g(to)g(sho)o(w)g(that)g(F)-5 b(A)o(CT)17 b(is)i(a)g (P-cylinder.)28 b(That)75 444 y(is,)21 b(there)f Fi(is)g Fo(an)h(isomorphism)h(computable)e(and)h(in)o(v)o(ertible)f(in)h(p)q (olynomial)i(time)d(b)q(et)o(w)o(een)75 504 y(F)-5 b(A)o(CT)23 b(and)h(F)-5 b(A)o(CT)p Fh(\002f)p Fo(0)p Fc(;)8 b Fo(1)p Fh(g)625 486 y Fb(\003)644 504 y Fo(.)43 b(\(Sk)o(etc)o(h)22 b(of)i(ho)o(w)g(to)g(compute)f(suc)o(h)h(an)g(isomorphism:)37 b(It)75 565 y(is)21 b(su\016cien)o(t)e(to)h(giv)o(e)g(a)h (length-increasing,)i(in)o(v)o(ertible)d(reduction)g(from)g(F)-5 b(A)o(CT)p Fh(\002f)p Fo(0)p Fc(;)8 b Fo(1)p Fh(g)1793 546 y Fb(\003)1832 565 y Fo(to)75 625 y(F)-5 b(A)o(CT;)13 b(a)i(result)g(in)g([BH77])f(can)h(b)q(e)f(used)h(to)g(construct)f(the) h(isomorphism)h(from)f(this.)21 b(Giv)o(en)75 685 y(\()p Fc(x;)8 b(i;)g(b)p Fo(\))19 b(and)h Fc(y)r Fo(,)h(w)o(e)e(need)h(to)g (\014nd)h(some)f(\()p Fc(x)932 667 y Fb(0)943 685 y Fc(;)8 b(i)982 667 y Fb(0)993 685 y Fc(;)g(b)1036 667 y Fb(0)1047 685 y Fo(\))20 b(suc)o(h)g(that)h(\()p Fc(x;)8 b(i;)g(b)p Fo(\))19 b Fh(2)h Fc(F)27 b Fo(if)20 b(and)h(only)g(if)75 745 y(\()p Fc(x)122 727 y Fb(0)133 745 y Fc(;)8 b(i)172 727 y Fb(0)183 745 y Fc(;)g(b)226 727 y Fb(0)238 745 y Fo(\))15 b Fh(2)h Fc(F)7 b Fo(.)24 b(W)l(e)17 b(can)g(directly)g (compute)g(the)g(largest)i(exp)q(onen)o(t)e Fc(e)1428 752 y Fe(j)1463 745 y Fo(for)g(eac)o(h)g(of)h(the)f(\014rst)75 805 y Fh(j)p Fc(y)r Fh(j)j Fo(primes,)h(suc)o(h)f(that)g Fc(p)572 778 y Fe(e)588 783 y Fa(j)572 817 y Fe(j)628 805 y Fo(to)g(that)h(exp)q(onen)o(t)f(divides)h Fc(x)p Fo(.)33 b(Then,)21 b(b)o(y)f(doing)i(some)e(simple)75 866 y(enco)q(ding)j(using)f(the)f(exp)q(onen)o(ts)h(of)f(the)g(\014rst) h Fh(j)p Fc(y)r Fh(j)f Fo(primes,)h(w)o(e)f(can)g(build)i(a)f(new)f(n)o (um)o(b)q(er,)75 926 y(whose)15 b(prime)g(factorization)i(\\enco)q (des")f(the)e(prime)h(factorization)i(of)e Fc(x)p Fo(,)f(along)j(with)f (the)e(bits)75 986 y(of)f Fc(y)r Fo(.)20 b(If)12 b(w)o(e)h(are)g (careful)g(ab)q(out)h(ho)o(w)g(w)o(e)e(do)i(this,)g(then)e(the)h(bits)h (of)f(the)g(prime)g(factorization)i(of)75 1046 y Fc(x)e Fo(that)h(deal)h(with)f(large)h(primes)f(are)f(essen)o(tially)i(unc)o (hanged)f(\(and)g(th)o(us)g Fc(i)f Fo(will)j(b)q(e)e(mapp)q(ed)g(to)75 1106 y Fc(i)92 1088 y Fb(0)117 1106 y Fo(=)g Fc(i)c Fo(+)h Fc(j)19 b Fo(for)e(some)f(easy-to-compute)g Fc(j)s Fo(\).)21 b(The)c(details)g(are)f(straigh)o(tforw)o(ard)i(but)e(tedious.\))148 1166 y(All)j(of)g(m)o(y)f(attempts)g(to)h(build)h(an)f(isomorphism)h(b) q(et)o(w)o(een)d(F)-5 b(A)o(CT)18 b(and)h(F)-5 b(A)o(CT)p Fh(\002f)p Fo(0)p Fc(;)8 b Fo(1)p Fh(g)1856 1148 y Fb(\003)75 1227 y Fo(in)o(v)o(olv)o(e)18 b(doing)i(quite)e(a)g(bit)h(of)g(m)o (ultiplication)h(and)f(division)i({)d(whic)o(h)h(is)g(imp)q(ossible)h (for)f(an)75 1287 y(A)o(C)146 1269 y Fp(0)187 1287 y Fo(circuit.)37 b(One)21 b(approac)o(h)h(to)g(try)f(to)h(pro)o(v)o(e)e (the)h(conjecture)g(w)o(ould)h(b)q(e)g(to)g(sho)o(w)g(that)75 1347 y Fi(any)15 b Fo(circuit)h(computing)g(suc)o(h)e(an)i(isomorphism) h(w)o(ould)f(b)q(e)f(usable)h(as)g(an)f(oracle)h(to)f(compute)75 1407 y(m)o(ultiplication,)k(or)d(at)h(least)g(to)f(compute)g(parit)o(y) h(in\014nitely)g(often.)148 1467 y Fj(Q:)f Fo(Still,)i(this)f(seems)e (lik)o(e)i(an)g(am)o(bitious)g(pro)s(ject.)148 1528 y Fj(A:)h Fo(P)o(erhaps.)27 b(Certainly)l(,)19 b(sho)o(wing)h(that)e(t)o (w)o(o)g(sets)g(are)g Fi(not)h Fo(isomorphic)g(often)f(seems)g(to)75 1588 y(b)q(e)f(quite)f(di\016cult!)23 b(\(Note,)16 b(ho)o(w)o(ev)o(er,) e(that)j([AAIPR97)o(])f(do)q(es)i(con)o(tain)f(results)g(of)g(this)h (form.)75 1648 y(The)e(sets)h(SA)l(T)f(and)h(SA)l(T)572 1630 y Fb(0)599 1648 y Fo(are)g(not)f(A)o(C)838 1630 y Fp(0)874 1648 y Fo(isomorphic.\))148 1708 y(Another)f(approac)o(h)h (w)o(ould)g(b)q(e)f(to)g(try)g(to)g(sho)o(w)h(that)f(there)g(is)g(no)h (length-increasing)i(one-)75 1768 y(one)g(function)i Fc(f)j Fo(in)c(NC)538 1750 y Fp(0)576 1768 y Fo(reducing)g(F)-5 b(A)o(CT)p Fh(\002f)p Fo(0)p Fc(;)8 b Fo(1)p Fh(g)1068 1750 y Fb(\003)1105 1768 y Fo(to)19 b(F)-5 b(A)o(CT.)17 b(\(Note)h(that)h(the)f(\\one-one")75 1829 y(condition)i(is)f(crucial)g (here.)26 b(Otherwise,)19 b(in)f(NC)1008 1810 y Fp(0)1046 1829 y Fo(one)g(could)h(map)g(\()p Fc(x;)8 b(i;)g(b;)g(z)r Fo(\))16 b(to)j(\()p Fc(x)p Fo(0)1724 1810 y Fb(j)p Fe(z)q Fb(j)1763 1829 y Fc(;)8 b(i)1802 1810 y Fb(0)1813 1829 y Fc(;)g(b)p Fo(\))75 1889 y(for)18 b(appropriate)i (\(easy-to-compute\))e Fc(i)837 1871 y Fb(0)849 1889 y Fo(.\))26 b(The)18 b(results)h(of)f([AAR96])f(can)h(again)i(b)q(e)e (used)h(to)75 1949 y(sho)o(w)e(that)g(this)g(implies)g(that)g(F)-5 b(A)o(CT)15 b(is)i(not)g(complete.)148 2009 y Fj(Q:)f Fo(Do)h(these)f(observ)m(ations)i(apply)f(to)g(problems)g(other)f(than) h(factorization?)148 2069 y Fj(A:)i Fo(They)g(certainly)g(do.)30 b(I)19 b(w)o(ould)h(guess)g(that)f(primalit)o(y)h(testing)g(and)f (computation)i(of)75 2130 y(the)d(greatest)h(common)g(divisor)h(also)g (fail)g(to)f(b)q(e)g(complete)f(for)h(an)o(y)f(reasonable)i(complexit)o (y)75 2190 y(class,)d(for)f(the)g(same)h(reason.)148 2250 y(It)e(w)o(ould)i(also)g(b)q(e)f(w)o(orth)g(trying)h(to)f(sa)o(y)f (something)i(concrete)e(ab)q(out)i(the)e(m)o(ultiplication)75 2310 y(problem.)31 b(That)20 b(is,)h(consider)f(the)f(set)g(MUL)l(T)h (=)f Fh(f)p Fo(\()p Fc(x;)8 b(y)r(;)g(i;)g(b)p Fo(\))17 b(:)28 b(the)19 b Fc(i)p Fo(th)g(bit)h(of)g(the)f(binary)75 2370 y(represen)o(tation)e(of)g Fc(x)11 b Fh(\001)h Fc(y)18 b Fo(is)g Fc(b)p Fh(g)p Fo(.)k(MUL)l(T)17 b(is)h(certainly)f(an)g(imp)q (ortan)o(t)h(set,)e(and)i(in)f(some)g(sense)75 2430 y(its)22 b(complexit)o(y)g(is)g(fairly)h(w)o(ell)f(understo)q(o)q(d.)39 b(It)21 b(is)i(in)f(TC)1228 2412 y Fp(0)1248 2430 y Fo(,)h(and)f(it)g (is)h(complete)e(for)h(TC)1855 2412 y Fp(0)75 2491 y Fo(under)17 b(A)o(C)285 2473 y Fp(0)304 2491 y Fo(-)p Fi(T)l(uring)h Fo(reducibilit)o(y)l(.)24 b(Since)17 b(it)h(is)f(still)i (an)f(imp)q(ortan)o(t)g(op)q(en)f(question)h(if)f(TC)1805 2473 y Fp(0)1842 2491 y Fo(is)75 2551 y(equal)k(to)g(NC)344 2533 y Fp(1)363 2551 y Fo(,)g(I)f(b)q(eliev)o(e)h(it)f(is)i(still)g(an) f(op)q(en)g(question)g(whether)f(MUL)l(T)g(is)i(complete)e(for)p eop %%Page: 8 8 8 7 bop 75 203 a Fo(NC)147 185 y Fp(1)183 203 y Fo(under)16 b(A)o(C)392 185 y Fp(0)428 203 y Fo(man)o(y-one)g(reducibilit)o(y)l(.) 22 b(I)16 b(think)h(it)g(is)g(quite)f(p)q(ossible)j(that)e(one)f(migh)o (t)h(b)q(e)75 264 y(able)g(to)g(sho)o(w)g(that)g(MUL)l(T)g(is)g Fi(not)g Fo(complete)f(for)h(an)o(y)f(reasonable)i(complexit)o(y)e (class)i(\(under)75 324 y(A)o(C)146 306 y Fp(0)181 324 y Fo(man)o(y-one)f(reducibilit)o(y\).)148 384 y Fj(Q:)f Fo(W)l(ouldn't)h(this)g(allo)o(w)h(one)f(to)f(pro)o(v)o(e)g(that)g(TC) 1113 366 y Fp(0)1150 384 y Fo(is)h(not)g(equal)f(to)h(NC)1546 366 y Fp(1)1565 384 y Fo(?)148 444 y Fj(A:)k Fo(Not)f(that)i(I)e(see.) 34 b(Note)20 b(in)i(this)f(regard)h(that)f(it)g(is)g(already)h(kno)o (wn)f(that)g(the)f(MA-)75 504 y(JORITY)d(function)h(is)g(not)g (complete)g(for)f(NC)966 486 y Fp(1)1004 504 y Fo(under)g(A)o(C)1214 486 y Fp(0)1251 504 y Fo(man)o(y-one)g(reducibilit)o(y)l(.)26 b(\(That)75 565 y(is)20 b(b)q(ecause)f(it)h(is)g(sho)o(wn)g(in)g([Be94) o(])f(that)h(there)e(is)i(no)g(A)o(C)1206 546 y Fp(0)1244 565 y Fo(man)o(y-one)f(reduction)h(from)f(the)75 625 y(P)l(ARITY)13 b(language)i(to)f(MAJORITY.\))e(MAJORITY)g(and)i(MUL)l (T)f(ha)o(v)o(e)g(quite)h(similar)h(com-)75 685 y(plexit)o(y)l(,)h(in)h (that)f(b)q(oth)h(are)g(complete)f(for)g(TC)955 667 y Fp(0)992 685 y Fo(under)g(A)o(C)1201 667 y Fp(0)1220 685 y Fo(-T)l(uring)i(reducibilit)o(y)l(.)148 745 y Fj(Q:)e Fo(Is)g(this)h(the)f(end)g(of)h(the)f(article?)148 805 y Fj(A:)g Fo(Almost.)22 b(I)16 b(just)g(w)o(an)o(t)g(to)h(commen)o(t)e (that)i(w)o(e)f(really)h(do)f(agree)h(on)g(one)f(p)q(oin)o(t.)148 866 y Fj(Q:)g Fo(What)h(is)g(that?)148 926 y Fj(A:)f Fo(Curren)o(tly)l(,)g(complexit)o(y)g(theory)h(has)g(no)g(credible)g (approac)o(h)g(to)o(w)o(ard)f(trying)i(to)e(sho)o(w)75 986 y(that)22 b(factorization)i(is)f(di\016cult)g(to)f(compute.)38 b(The)22 b(only)h(real)g(evidence)e(of)h(in)o(tractabilit)o(y)75 1046 y(that)h(complexit)o(y)f(theory)g(o\013ers)h(curren)o(tly)f(is)h (based)f(on)h(reducibilit)o(y)h(and)f(completeness.)75 1106 y(F)l(actorization)c(seems)d(to)h(fall)h(outside)g(that)g(framew)o (ork.)k(With)c(luc)o(k,)e(ho)o(w)o(ev)o(er,)f(it)j(migh)o(t)f(b)q(e)75 1166 y(p)q(ossible)h(to)f Fi(pr)n(ove)f Fo(that)h(this)g(is)g(the)f (case.)75 1333 y Fk(References)75 1442 y Fo([AA96])23 b(M.)14 b(Agra)o(w)o(al)g(and)h(E.)e(Allender,)i Fi(A)o(n)g (Isomorphism)f(The)n(or)n(em)g(for)h(Cir)n(cuit)g(Complex-)245 1503 y(ity)p Fo(,)22 b(in)f(Pro)q(c.)g(11th)h(Ann)o(ual)g(IEEE)f (Conference)f(on)i(Computational)i(Complexit)o(y)245 1563 y(\(1996\))18 b(pp.)e(2{11.)75 1664 y([AAIPR97])23 b(M.)g(Agra)o(w)o(al,)k(E.)c(Allender,)j(R.)e(Impagliazzo,)j(T.)c (Pitassi,)28 b(and)d(S.)e(Rudic)o(h,)245 1725 y Fi(R)n(e)n(ducing)16 b(the)h(c)n(omplexity)g(of)f(r)n(e)n(ductions)p Fo(,)e(in)i(Pro)q(c.)f (29th)h(A)o(CM)e(STOC)h(\(1997\),)h(pp.)245 1785 y(730{738.)75 1887 y([AAR96])23 b(M.)15 b(Agra)o(w)o(al,)i(E.)e(Allender)h(and)h(S.)e (Rudic)o(h,)h Fi(R)n(e)n(ductions)g(in)i(Cir)n(cuit)e(Complexity:)245 1947 y(A)o(n)k(Isomorphism)e(The)n(or)n(em)g(and)h(a)h(Gap)f(The)n(or)n (em)p Fo(,)e(to)i(app)q(ear)g(in)g(J.)f(Comp.)g(Sys.)245 2007 y(Sci.)i(Some)g(of)g(the)g(results)h(in)g(this)g(pap)q(er)g(w)o (ere)e(originall)q(y)j(announced)f(in)g([AA96)o(])245 2067 y(and)c([AAIPR97)o(].)75 2169 y([ABI93])23 b(E.)15 b(Allender,)h(N.)f(Immerman,)f(and)i(J.)g(Balc\023)-24 b(azar,)16 b Fi(A)h(\014rst-or)n(der)f(isomorphism)g(the-)245 2229 y(or)n(em)p Fo(,)f(SIAM)g(Journal)j(on)f(Computing)h Fj(26)e Fo(\(1997\))i(557{567.)75 2331 y([Al97])44 b(E.)25 b(Allender,)i Fi(Some)f(p)n(ointe)n(d)f(questions)i(c)n(onc)n(erning)g (asymptotic)e(lower)h(b)n(ounds)p Fo(,)245 2391 y(Computational)19 b(Complexit)o(y)e(Column,)f(Bulletin)i(of)e(the)g(EA)l(TCS)h(62)g (\(1997\).)p eop %%Page: 9 9 9 8 bop 75 203 a Fo([AR98])24 b(E.)18 b(Allender)h(and)g(K.)f (Reinhardt,)i Fi(Isolation,)h(Matching,)g(and)e(Counting)p Fo(,)i(in)e(Pro)q(c.)245 264 y(13th)f(Ann)o(ual)f(IEEE)g(Conference)f (on)h(Computational)j(Complexit)o(y)l(,)c(1998,)i(pp.)f(92{)245 324 y(100)75 425 y([Ba89])35 b(D.)18 b(Barrington,)h Fi(Bounde)n(d-width)i(p)n(olynomial)e(size)g(br)n(anching)h(pr)n(o)n (gr)n(ams)d(r)n(e)n(c)n(o)n(gnize)245 486 y(exactly)g(those)e (languages)j(in)d(NC)875 468 y Fp(1)895 486 y Fo(,)f(Journal)i(of)e (Computer)g(and)g(System)g(Sciences)f(38)245 546 y(\(1989\),)k (150{164.)75 648 y([Be94])37 b(R.)16 b(Beigel,)g Fi(When)i(do)f(extr)n (a)h(majority)e(gates)j(help?)k(Polylo)n(g)p Fo(\()p Fc(n)p Fo(\))18 b Fi(majority)e(gates)j(ar)n(e)245 708 y(e)n(quivalent)h(to)d(one)p Fo(,)g(Computational)i(Complexit)o(y)d Fj(4)h Fo(\(1984\))g(297{388.)75 809 y([BH77])23 b(L.)17 b(Berman)f(and)h(J.)f(Hartmanis,)h Fi(On)h(isomorphism)f(and)h(density) g(of)f(NP)h(and)g(other)245 870 y(c)n(omplete)h(sets)p Fo(,)d(SIAM)f(J.)h(Comput.)g Fj(6)g Fo(\(1977\))i(305{322.)75 971 y([BL87])29 b(Ra)o(vi)12 b(Boppana)g(and)h(Je\013)e(Lagarias,)k Fi(One-Way)f(F)l(unctions)h(and)e(Cir)n(cuit)g(Complexity)p Fo(,)245 1032 y(Information)k(and)g(Computation)h Fj(74)e Fo(\(1987\))i(226-240.)75 1133 y([Bu87])32 b(S.)21 b(Buss,)h Fi(The)h(Bo)n(ole)n(an)f(F)l(ormula)g(V)l(alue)i(Pr)n(oblem)e(is)g(in)h (ALOGTIME)p Fo(,)d(in)i(Pro)q(c.)245 1193 y(19th)17 b(A)o(CM)f(Symp)q (osium)h(on)f(Theory)h(of)f(Computing,)i(pp.)e(123-131,)i(1987.)75 1295 y([F)l(e92])44 b(M.)20 b(F)l(ello)o(ws)i(and)f(N.)f(Koblitz.)h Fi(Self-witnessing)k(p)n(olynomial-time)e(c)n(omplexity)f(and)245 1355 y(prime)e(factorization)g Fo(in)g Fi(Pr)n(o)n(c.)f(of)h(the)h(7th) g(IEEE)f(Structur)n(e)h(in)f(Complexity)h(Con-)245 1416 y(fer)n(enc)n(e)p Fo(,)16 b(pages)h(107{110,)i(1992.)75 1517 y([FFK96])24 b(S.)f(F)l(enner,)g(L.)g(F)l(ortno)o(w,)i(and)f(S.)f (Kurtz,)h Fi(The)g(isomorphism)e(c)n(onje)n(ctur)n(e)i(holds)245 1577 y(r)n(elative)18 b(to)g(an)g(or)n(acle)p Fo(,)d(SIAM)h(Journal)i (on)e(Computing)i Fj(25)e Fo(\(1996\))i(193-206.)75 1679 y([GHR95])24 b(R.)15 b(Greenla)o(w,)g(J.)g(Ho)q(o)o(v)o(er,)g(and)h (W.L.)f(Ruzzo.)k Fi(Limits)d(to)h(Par)n(al)r(lel)h(Computation:)245 1739 y(P-Completeness)i(The)n(ory)p Fo(.)g(Oxford)c(Univ)o(ersit)o(y)g (Press,)g(1995.)75 1841 y([GJ79])31 b(Mic)o(hael)43 b(R.)e(Garey)22 b(and)f(Da)o(vid)h(S.)42 b(Johnson,)24 b Fi(Computers)d(and)i(Intr)n (actability,)245 1901 y Fo(F)l(reeman,)15 b(1979.)75 2003 y([Ha78])33 b(Juris)21 b(Hartmanis,)h Fi(On)g(the)g(lo)n(gtap)n(e) f(isomorphism)f(of)h(c)n(omplete)h(sets,)g Fo(Theoretical)245 2063 y(Computer)16 b(Science)g Fj(7)g Fo(\(1978\))i(273-286.)75 2165 y([JPY94])24 b(D.)d(Joseph,)h(R.)e(Pruim,)i(and)f(P)l(.)g(Y)l (oung,)h Fi(Col)r(lapsing)h(de)n(gr)n(e)n(es)e(in)h(sub)n(exp)n (onential)245 2225 y(time)p Fo(,)17 b(Pro)q(c.)f(9th)h(Structure)f(in)h (Complexit)o(y)g(Theory)f(Conference)g(\(1994\))i(pp.)e(367{)245 2285 y(382.)75 2387 y([JY85])32 b(D.)d(Joseph)h(and)f(P)l(.)g(Y)l (oung,)j Fi(Some)e(r)n(emarks)f(on)g(witness)i(functions)f(for)f(non-) 245 2447 y(p)n(olynomial)20 b(and)g(non-c)n(omplete)i(sets)e(in)h(NP)p Fo(,)d(Theoretical)j(Computer)e(Science)f Fj(39)245 2507 y Fo(\(1985\))g(225{237.)p eop %%Page: 10 10 10 9 bop 75 203 a Fo([KLD86])25 b(Ker-I)17 b(Ko,)g(Timoth)o(y)h(J.)f (Long,)i(and)f(Ding-Zh)o(u)g(Du,)g Fi(On)h(one-way)h(functions)g(and) 245 264 y(p)n(olynomial-time)15 b(isomorphisms)p Fo(,)c(Theoretical)j (Computer)e(Science)g Fj(47)h Fo(\(1986\))g(263{)245 324 y(276.)75 425 y([KMR90])24 b(S.)16 b(Kurtz,)g(S.)h(Mahaney)l(,)g (and)g(J.)g(Ro)o(y)o(er,)e Fi(The)j(structur)n(e)g(of)g(c)n(omplete)h (de)n(gr)n(e)n(es)p Fo(,)d(in)245 486 y(A.)j(Selman,)i(editor,)g Fi(Complexity)g(The)n(ory)f(R)n(etr)n(osp)n(e)n(ctive)p Fo(,)f(Springer-V)l(erlag,)j(1990,)245 546 y(pp.)16 b(108{146.)75 648 y([KMR95])24 b(S.)g(Kurtz,)h(S.)f(Mahaney)l(,)i(and)f(J.)f(Ro)o(y)o (er,)h Fi(The)g(isomorphism)f(c)n(onje)n(ctur)n(e)h(fails)245 708 y(r)n(elative)18 b(to)g(a)f(r)n(andom)g(or)n(acle)p Fo(,)f(J.)f(A)o(CM)h Fj(42)g Fo(\(1995\),)h(401{420.)75 809 y([KST93])25 b(J.)20 b(K\177)-24 b(obler,)23 b(U.)d(Sc)o(h\177)-24 b(oning,)23 b(and)f(J.)f(T)l(or\023)-24 b(an,)23 b Fi(The)f(Gr)n(aph)e (Isomorphism)h(Pr)n(oblem;)245 870 y(It's)c(Structur)n(al)h(Complexity) p Fo(,)f(Birkh\177)-24 b(auser,)16 b(1993.)75 971 y([KUW86])24 b(R.)15 b(Karp)h(and)g(E.)g(Upfal)g(and)g(A.)f(Wigderson,)i Fi(Constructing)h(a)f(p)n(erfe)n(ct)g(matching)245 1032 y(is)g(in)h(r)n(andom)f(NC)p Fo(,)e(Com)o(binatorica)k Fj(6)d Fo(\(1986\))i(35{48.)75 1133 y([La75])40 b(R.)17 b(Ladner.)h Fi(On)i(the)f(structur)n(e)g(of)f(p)n(olynomial-time)i(r)n (e)n(ducibility)p Fo(,)d(J.)h(Assn.)f(Comp.)245 1193 y(Mac)o(h.)e Fj(22)i Fo(\(1975\))g(155{171.)75 1295 y([MY85])24 b(S.)13 b(Mahaney)g(and)h(P)l(.)f(Y)l(oung,)h Fi(R)n(e)n(ductions)h (among)h(p)n(olynomial)e(isomorphism)g(typ)n(es)p Fo(,)245 1355 y(Theoretical)j(Computer)g(Science)e Fj(39)i Fo(\(1985\))h (207-224.)75 1457 y([Ni92])44 b(Noam)12 b(Nisan,)h Fi(Using)h(Har)n(d)f (Pr)n(oblems)g(to)g(Cr)n(e)n(ate)g(Pseudor)n(andom)g(Gener)n(ators)p Fo(,)e(MIT)245 1517 y(Press)17 b(\(1992\).)75 1619 y([Ro67])34 b(H.)d(Rogers,)k Fi(The)n(ory)30 b(of)i(R)n(e)n(cursive)f(F)l(unctions) i(and)e(E\013e)n(ctive)i(Computability)p Fo(,)245 1679 y(McGra)o(w-Hill,)17 b(1967.)75 1781 y([Ro95])34 b(J.)24 b(Rogers,)i Fi(The)e(isomorphism)f(c)n(onje)n(ctur)n(e)i(holds)g(and)g (one-way)g(functions)h(exist)245 1841 y(r)n(elative)18 b(to)g(an)g(or)n(acle)p Fo(,)d(J.)h(Computer)h(Sys.)f(Sci.)g Fj(54)g Fo(\(1997\))i(412{423.)75 1943 y([Smo87])24 b(R.)17 b(Smolensky)l(.)g Fi(A)o(lgebr)n(aic)i(metho)n(ds)f(in)h(the)g(the)n (ory)e(of)h(lower)h(b)n(ounds)g(for)f(Bo)n(ole)n(an)245 2003 y(cir)n(cuit)e(c)n(omplexity)p Fo(,)g(in)f Fi(Pr)n(o)n(c)n(e)n(e)n (dings,)g(19th)i(A)o(CM)e(Symp)n(osium)g(on)i(The)n(ory)e(of)h(Com-)245 2063 y(puting)p Fo(,)h(pages)g(77{82,)h(1987.)75 2165 y([T)l(o98])39 b(J.)23 b(T)l(or\023)-24 b(an,)26 b(talk)e(presen)o(ted) e(at)i(the)f(1998)i(IEEE)f(Conference)e(on)i(Computational)245 2225 y(Complexit)o(y)l(,)16 b(Bu\013alo,)h(NY.)75 2327 y([Y)l(o90])37 b(P)o(aul)21 b(Y)l(oung,)h Fi(Juris)e(Hartmanis:)30 b(F)l(undamental)23 b(Contributions)f(to)g(Isomorphism)245 2387 y(Pr)n(oblems,)f Fo(in)g Fi(Complexity)g(The)n(ory)f(R)n(etr)n (osp)n(e)n(ctive,)g Fo(Alan)g(Selman,)h(ed.,)f(Springer-)245 2447 y(V)l(erlag)d(\(1990\),)g(28-58.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF