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y Fo(2)1105 1343 y Fg(2)14 b Fl(NP)p Fg(g)468 1417 y Fl(co-D)568 1396 y Fo(P)652 1417 y Fl(=)41 b Fg(f)p 756 1378 V Ff(L)789 1424 y Fo(1)820 1417 y Fg([)11 b Ff(L)897 1424 y Fo(2)934 1417 y Fg(j)16 b Ff(L)997 1424 y Fo(1)1017 1417 y Ff(;)8 b(L)1072 1424 y Fo(2)1105 1417 y Fg(2)14 b Fl(NP)p Fg(g)382 1520 y Fl(SA)l(T)r Fg(^)p 515 1481 95 2 v 3 w Fl(SA)l(T)42 b(=)f Fg(f)p Fl(\()p Ff(F)807 1527 y Fo(1)826 1520 y Ff(;)8 b(F)880 1527 y Fo(2)900 1520 y Fl(\))16 b Fg(j)g Ff(F)997 1527 y Fo(1)1030 1520 y Fg(2)e Fl(SA)l(T)32 b(and)17 b Ff(F)1331 1527 y Fo(2)1364 1520 y Fg(2)p 1411 1481 V 14 w Fl(SA)l(T)f Fg(g)p 382 1554 V 382 1593 a Fl(SA)l(T)r Fg(_)s Fl(SA)l(T)42 b(=)f Fg(f)p Fl(\()p Ff(F)807 1600 y Fo(1)826 1593 y Ff(;)8 b(F)880 1600 y Fo(2)900 1593 y Fl(\))16 b Fg(j)g Ff(F)997 1600 y Fo(1)1030 1593 y Fg(2)p 1077 1554 V 14 w Fl(SA)l(T)32 b(or)17 b Ff(F)1296 1600 y Fo(2)1329 1593 y Fg(2)d Fl(SA)l(T)i Fg(g)p Ff(:)148 1748 y Fl(P)o(apadimitriou)k(and)j(Y)l(annak)m(akis)f(also)h 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b(and)h(since)f(the)h(conjunction)f(of)h(uniquely)e(satis\014able)i(form)o (ulas)e(is)h(also)75 2169 y(uniquely)h(satis\014able,)120 2269 y(USA)l(T)h(is)24 b Fg(\024)364 2251 y Fo(P)364 2282 y(m)403 2269 y Fl(-complete)15 b(for)h(D)737 2251 y Fo(P)791 2269 y Fg(\()-8 b(\))27 b Fl(SA)l(T)s Fg(^)p 1044 2230 V 3 w Fl(SA)l(T)8 b Fg(\024)1186 2251 y Fo(P)1186 2282 y(m)1225 2269 y Fl(USA)l(T)27 b Fg(\()-8 b(\))27 b Fl(SA)l(T)9 b Fg(\024)1646 2251 y Fo(P)1646 2282 y(m)1685 2269 y Fl(USA)l(T)o Ff(:)75 2369 y Fl(So,)20 b(the)g(question)f(of)h(whether)f(USA)l(T)g(can)h(b)q(e)28 b Fg(\024)1055 2351 y Fo(P)1055 2382 y(m)1094 2369 y Fl(-complete)18 b(for)i(D)1435 2351 y Fo(P)1481 2369 y Fl(hinges)f(on)h(whether)75 2430 y(there)k(is)f(a)33 b Fg(\024)360 2412 y Fo(P)360 2442 y(m)400 2430 y Fl(-reduction)24 b(from)f(SA)l(T)h(to)g(USA)l(T.)f(Then,)j (they)e(sho)o(w)o(ed)g(that)g(there)g(are)75 2490 y(oracle)15 b(w)o(orlds)g(where)g(no)h(suc)o(h)e(reduction)h(can)g(exist.)20 b(This)c(mean)o(t)d(a)j(non-relativizing)e(pro)q(of)75 2550 y(tec)o(hnique)g(w)o(ould)j(b)q(e)f(needed)g(to)g(answ)o(er)h(the)f (question|a)g(formidable)f(obstacle,)g(indeed.)p eop %%Page: 3 3 bop 148 42 a Fl(V)l(alian)o(t)13 b(and)i(V)l(azirani)e([VV86])g(did)h(not)h (surmoun)o(t)e(this)h(obstacle,)g(but)g(they)g(did)f(manage)75 102 y(to)21 b(circum)o(v)o(e)o(n)o(t)d(it.)33 b(Although,)22 b(it)e(ma)o(y)f(b)q(e)h(di\016cult)f(to)i(construct)g(a)29 b Fg(\024)1499 84 y Fo(P)1499 114 y(m)1539 102 y Fl(-reduction)20 b(from)75 162 y(SA)l(T)c(to)h(USA)l(T)f(\(assuming)g(one)h(ev)o(en)e (exists\),)h(they)g(w)o(ere)g(able)g(to)h(construct)f(a)h Fe(r)n(andomize)n (d)75 222 y(r)n(e)n(duction)p Fl(.)k(In)16 b(particular,)g(they)f (constructed)h(a)h(p)q(olynomial)e(function)h Ff(f)22 b Fl(suc)o(h)16 b(that)444 321 y Ff(x)e Fg(2)g Fl(SA)l(T)g(=)-8 b Fg(\))13 b Fl(Prob)839 328 y Fn(z)858 321 y Fl([)j Ff(f)5 b Fl(\()p Ff(x;)j(z)r Fl(\))14 b Fg(2)g Fl(USA)l(T)h(])f Fg(\025)f Fl(1)p Ff(=)p Fl(\(4)p 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Fg(\025)h Ff(\016)584 1304 y(x)g Fg(62)g Ff(A)f Fl(=)-8 b Fg(\))13 b Fl(Prob)920 1311 y Fn(z)940 1304 y Fl([)j Ff(f)5 b Fl(\()p Ff(x;)j(z)r Fl(\))13 b Fg(62)h Ff(B)19 b Fl(])13 b(=)h(1)p Ff(;)91 1412 y Fl(where)i Ff(z)i Fl(is)e(c)o(hosen)g(uniformly)f(o)o(v)o(er)g Fg(f)p Fl(0)p Ff(;)8 b Fl(1)p Fg(g)925 1393 y Fn(q)q Fo(\()p Fn(n)p Fo(\))993 1412 y Fl(.)148 1504 y(In)19 b(this)g(terminology)l(,)f(V)l (alian)o(t)g(and)i(V)l(azirani)e(sho)o(w)o(ed)h(that)h(SA)l(T)f(randomized)f (reduces)75 1564 y(to)d(USA)l(T)f(with)h(probabilit)o(y)f(1)p Ff(=)p Fl(\(4)p Ff(n)p Fl(\).)22 b(As)14 b(a)i(sp)q(ecial)e(case,)h(w)o(e)f (will)g(write)g Ff(A)8 b Fg(\024)1563 1546 y Fo(vv)1563 1576 y(m)1610 1564 y Ff(B)18 b Fl(if)c Ff(A)8 b Fg(\024)1792 1546 y Fo(rp)1792 1576 y(m)1835 1564 y Ff(B)75 1624 y Fl(with)15 b(probabilit)o(y)g(1)p Ff(=p)p Fl(\()p Ff(n)p Fl(\))i(for)f(some)f(p)q (olynomial)f(b)q(ound)j Ff(p)p Fl(.)k(W)l(e)16 b(will)e(reserv)o(e)h(the)g (term)f(\\the)75 1684 y(V)l(alian)o(t-V)l(azirani)i(reduction")h(to)h(name)f (the)g(randomized)f(reduction)h(from)g(SA)l(T)g(to)h(USA)l(T.)75 1745 y(Similarly)l(,)f(the)i(Adleman)f(and)i(Manders)g(de\014nition)f(of)h (randomized)e(reductions)h(w)o(ould)h(b)q(e)75 1805 y(randomized)g (reductions)i(with)g(probabilit)o(y)f(1)p Ff(=)p Fl(2.)39 b(Also,)23 b(in)e(statemen)o(ts)f(where)i(the)g(exact)75 1865 y(probabilit)o(y)10 b(b)q(ound)j(is)e(not)h(imp)q(ortan)o(t,)f(w)o(e)g(will)f(use)i(the)f(terms)f (1)p Ff(=)p Fl(p)q(oly)j(and)f(1)p Ff(=)p Fl(exp)f(to)h(indicate)75 1925 y(that)i(the)f(statemen)o(t)f(holds)i(for)g(an)o(y)f(in)o(v)o(erse)f(p)q (olynomial)g(and)i(in)o(v)o(erse)e(exp)q(onen)o(tial)h(function.)75 1985 y(As)18 b(w)o(e)h(men)o(tioned)d(b)q(efore,)j(under)g(certain)f (conditions,)27 b Fg(\024)1229 1967 y Fo(vv)1229 1998 y(m)1277 1985 y Fl(-reductions)18 b(and)h(randomized)75 2046 y(reductions)d(with)g (probabilit)o(y)f(1)p Ff(=)p Fl(2)j(are)e(equiv)m(alen)o(t.)75 2138 y Fh(De\014nition:)k Fl(F)l(or)c(an)o(y)h(language)g Ff(B)s Fl(,)e(w)o(e)h(de\014ne)g(the)g(follo)o(wing)g(classes.)351 2236 y(OR)425 2243 y Fo(2)444 2236 y Fl(\()p Ff(B)s Fl(\))41 b(=)h Fg(fh)p Ff(x;)8 b(y)r Fg(i)16 b(j)g Ff(x)e Fg(2)g Ff(B)k Fl(or)f Ff(y)e Fg(2)f Ff(B)s Fg(g)345 2309 y Fl(OR)419 2316 y Fn(!)444 2309 y Fl(\()p Ff(B)s Fl(\))41 b(=)h Fg(fh)p Ff(x)715 2316 y Fo(1)735 2309 y Ff(;)8 b(:)g(:)g(:)f(;)h(x)872 2316 y Fn(n)895 2309 y Fg(i)17 b(j)f Fl(for)g(some)f Ff(i;)24 b Fl(1)14 b Fg(\024)g Ff(i)f Fg(\024)h Ff(n;)24 b(x)1481 2316 y Fn(i)1509 2309 y Fg(2)14 b Ff(B)s Fg(g)314 2382 y Fl(AND)425 2389 y Fo(2)444 2382 y Fl(\()p Ff(B)s Fl(\))41 b(=)h Fg(fh)p Ff(x;)8 b(y)r Fg(i)16 b(j)g Ff(x)e Fg(2)g Ff(B)k Fl(and)f Ff(y)f Fg(2)e Ff(B)s Fg(g)309 2454 y Fl(AND)419 2461 y Fn(!)444 2454 y Fl(\()p Ff(B)s Fl(\))41 b(=)h Fg(fh)p Ff(x)715 2461 y Fo(1)735 2454 y Ff(;)8 b(:)g(:)g(:)f(;)h(x)872 2461 y Fn(n)895 2454 y Fg(i)17 b(j)f Fl(for)g(all)g Ff(i;)24 b Fl(1)14 b Fg(\024)g Ff(i)f Fg(\024)h Ff(n;)24 b(x)1427 2461 y Fn(i)1454 2454 y Fg(2)15 b Ff(B)s Fg(g)p Ff(:)p 75 2504 720 2 v 131 2535 a Fd(1)150 2550 y Fj(V)m(alian)o(t)d(and)i(V)m(azirani)e(credit)j(Alan)e(Selman)g(for)g (this)h(application)f(of)g(their)h(randomized)f(reduction.)p eop %%Page: 4 4 bop 75 42 a Fl(W)l(e)17 b(sa)o(y)h(that)g(the)f(set)g Ff(B)j Fl(has)e(OR)734 49 y Fo(2)771 42 y Fl(if)f(OR)891 49 y Fo(2)911 42 y Fl(\()p Ff(B)s Fl(\))8 b Fg(\024)1036 23 y Fo(P)1036 54 y(m)1075 42 y Ff(B)s Fl(,)17 b(and)h(similarly)d(for)i(OR)1592 49 y Fn(!)1618 42 y Fl(,)g(AND)1759 49 y Fo(2)1796 42 y Fl(and)75 102 y(AND)185 109 y Fn(!)211 102 y Fl(.)75 188 y Fh(Lemma)f(1)s(.)43 b Fl(If)16 b Ff(A)8 b Fg(\024)497 170 y Fo(vv)497 200 y(m)545 188 y Ff(B)19 b Fl(and)e Ff(B)h Fl(has)f(OR)912 195 y Fn(!)937 188 y Fl(,)f(then)g Ff(A)8 b Fg(\024)1162 170 y Fo(rp)1162 200 y(m)1206 188 y Ff(B)18 b Fl(with)f(probabilit)o(y)e(1)c Fg(\000)g Fl(1)p Ff(=)p Fl(exp.)75 274 y Fh(Pro)r(of)21 b(\(Sk)n(etc)n(h\):)j Fl(Let)18 b Ff(f)24 b Fl(b)q(e)18 b(the)26 b 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y(where)j(one)g(randomly)f(reduces)h(to)h(a)f(nice)f (language,)i(it)f(do)q(esn't)h(matter)d(whic)o(h)i(de\014nition)g(of)75 760 y(randomized)g(reduction)g(is)h(used.)21 b(Ho)o(w)o(ev)o(er,)14 b(USA)l(T)h(is)h(not)h(kno)o(wn)f(to)g(ha)o(v)o(e)f(OR)1625 767 y Fn(!)1650 760 y Fl(.)22 b(So,)16 b(there)75 820 y(w)o(asn't)e(an)h(ob)o (vious)f(w)o(a)o(y)g(to)g(amplify)e(the)i(V)l(alian)o(t-V)l(azirani)f (reduction)h(from)f(SA)l(T)h(to)g(USA)l(T.)148 880 y(Nev)o(ertheless,)f(the) 23 b Fg(\024)572 862 y Fo(vv)572 893 y(m)620 880 y Fl(-reduction)15 b(from)f(SA)l(T)i(to)f(USA)l(T)g(pro)o(v)o(ed)g(to)g(b)q(e)h(useful)f(in)f (man)o(y)75 941 y(areas)23 b(of)f(researc)o(h.)38 b(F)l(or)22 b(example,)f(Ric)o(hard)g(Beigel)f([Bei88])h(used)h(it)g(to)g(sho)o(w)h(that) f(SA)l(T)75 1001 y(is)e(sup)q(erterse)g(unless)f(RP)i(=)f(NP)o(.)33 b(Also,)20 b(T)l(o)q(da)h([T)l(o)q(d89)q(])f(used)g(a)g(similar)e(reduction)i (in)f(his)75 1061 y(pro)q(of)f(that)g(PH)d Fg(\022)g Fl(P)482 1043 y Fo(#P[1])575 1061 y Fl(.)24 b(This)18 b(result,)e(in)h(turn,)g(led)g (to)g(the)g(Lund,)h(F)l(ortno)o(w,)g(Karlo\013)f(and)75 1121 y(Nisan)j([LFKN90])f(result:)28 b(PH)20 b Fg(\022)f Fl(IP.)32 b(So,)21 b(there)e(should)h(b)q(e)g(little)e(doubt)i(in)g(the)f(reader's)75 1181 y(mind)14 b(regarding)i(the)f(usefulness)h(of)f(the)h(V)l(alian)o(t-V)l (azirani)e(reduction.)20 b(The)c(more)e(p)q(ertinen)o(t)75 1242 y(questions)22 b(are:)31 b(What)23 b(do)q(es)f(this)f(reduction)g(mean?) 37 b(Do)q(es)22 b(USA)l(T)f(b)q(eing)30 b Fg(\024)1621 1223 y Fo(vv)1621 1254 y(m)1669 1242 y Fl(-complete)75 1302 y(for)22 b(D)192 1284 y Fo(P)239 1302 y Fl(mean)f(that)g(it)g(is)g(someho)o(w)g (represen)o(tativ)o(e)f(of)h(the)g(whole)h(class?)37 b(Ho)o(w)21 b(do)q(es)h(the)75 1362 y(complexit)o(y)13 b(of)25 b Fg(\024)426 1344 y Fo(vv)426 1374 y(m)474 1362 y Fl(-complete)14 b(sets)i(compare)f(with) h(the)25 b Fg(\024)1229 1344 y Fo(P)1229 1374 y(m)1268 1362 y Fl(-complete)14 b(languages?)148 1422 y(Recen)o(t)f(results)g(ab)q(out)i (the)f(complexit)o(y)d(of)j(USA)l(T)f(and)h(of)h(D)1306 1404 y Fo(P)1346 1422 y Fl(shed)f(a)g(new)g(ligh)o(t)f(on)h(these)75 1482 y(questions.)22 b(Under)16 b(the)g(assumption)g(that)h(the)f(P)o (olynomial)f(Hierarc)o(h)o(y)f(has)k(in\014nitely)c(man)o(y)75 1543 y(lev)o(els,)19 b(m)o(uc)o(h)f(has)j(b)q(een)f(disco)o(v)o(ered)e(ab)q (out)k(SA)l(T)r Fg(^)p 1083 1503 V 3 w Fl(SA)l(T,)f(the)f(D)1338 1524 y Fo(P)1392 1543 y Fg(\024)1431 1524 y Fo(P)1431 1555 y(m)1471 1543 y Fl(-complete)e(set.)33 b(F)l(or)75 1603 y(example,)14 b(Kadin)i([Kad88)q(,)f(CK90a)r(])h(sho)o(w)o(ed)g(that)644 1688 y(D)681 1670 y Fo(P)721 1688 y Fl(=)e(co-D)872 1667 y Fo(P)912 1688 y Fl(=)-8 b Fg(\))30 b Fl(PH)16 b(collapses.)75 1774 y(So,)f(SA)l(T)s Fg(^)p 289 1735 V 3 w Fl(SA)l(T)e Fg(62)h Fl(co-D)544 1753 y Fo(P)585 1774 y Fl(unless)g(PH)h(collapses.)21 b(Similarly)l(,)11 b(one)k(can)g(sho)o(w)g(that)h(SA)l(T)r Fg(^)p 1780 1735 V 3 w Fl(SA)l(T)75 1834 y(cannot)i(ha)o(v)o(e)f(OR)424 1841 y Fn(!)467 1834 y Fl(unless)h(PH)f(collapses)g([CK90b)q(].)25 b(These)18 b(results)f(sho)o(w)h(that)h(SA)l(T)r Fg(^)p 1780 1795 V 3 w Fl(SA)l(T)75 1894 y(do)q(es)j(not)g(ha)o(v)o(e)f(the)g(same)f (robust)i(prop)q(erties)g(of)g(SA)l(T.)e(So,)j(ho)o(w)f(do)q(es)g(USA)l(T,)e (the)30 b Fg(\024)1811 1876 y Fo(vv)1811 1907 y(m)1859 1894 y Fl(-)75 1955 y(complete)18 b(set)i(for)g(D)480 1936 y Fo(P)506 1955 y Fl(,)h(compare)e(with)h(the)28 b Fg(\024)990 1936 y Fo(P)990 1967 y(m)1030 1955 y Fl(-complete)18 b(set?)33 b(As)20 b(it)f(turns)i(out,)g(Chang)75 2015 y(and)c(Kadin)f([CK90c])g(recen)o(tly)e (sho)o(w)o(ed)j(that)124 2096 y(USA)l(T)c Fg(62)h Fl(co-D)416 2075 y Fo(P)458 2096 y Fl(unless)i(PH)g(collapses,)124 2168 y(USA)l(T)f(do)q(es)i(not)g(ha)o(v)o(e)f(OR)654 2175 y Fn(!)696 2168 y Fl(unless)g(PH)g(collapses.)75 2249 y(F)l(rom)11 b(these)i(results)f (it)g(w)o(ould)h(app)q(ear)g(that)21 b Fg(\024)962 2231 y Fo(vv)962 2261 y(m)1010 2249 y Fl(-complete)11 b(sets)i(b)q(eha)o(v)o(e)e(m)o(uc)o(h)g (lik)o(e)g(the)20 b Fg(\024)1819 2231 y Fo(P)1819 2261 y(m)1859 2249 y Fl(-)75 2309 y(complete)13 b(sets)j(and)h(this)e(is)h(true)f Fe(in)j(spite)f(of)e Fl(the)h(fact)g(that)g(the)f(V)l(alian)o(t-V)l(azirani)f (reduction)75 2369 y(cannot)20 b(b)q(e)g(ampli\014ed)e(b)o(y)i(rep)q(eated)f (trials)h(\(since)f(USA)l(T)g(do)q(es)h(not)h(ha)o(v)o(e)e(OR)1613 2376 y Fn(!)1658 2369 y Fl(unless)g(PH)75 2430 y(collapses\).)33 b(Can)22 b(these)e(results)g(b)q(e)g(generalized)f(for)29 b Fg(\024)1174 2412 y Fo(vv)1174 2442 y(m)1222 2430 y Fl(-complete)19 b(sets)h(in)g(general?)34 b(Or)75 2490 y(do)19 b(these)e(results)h(only)g (apply)g(to)27 b Fg(\024)782 2472 y Fo(vv)782 2502 y(m)830 2490 y Fl(-complete)16 b(sets)i(for)h(D)1265 2472 y Fo(P)1291 2490 y Fl(?)28 b(W)l(e)18 b(will)f(in)o(v)o(estigate)f(these)75 2550 y(questions)g(in)g(the)g(next)g(section.)p eop %%Page: 5 5 bop 75 64 a Fm(2)83 b(A)28 b(Random)f(Complete)f(Set)h(for)h(co-D)1433 34 y Fh(P)75 174 y Fl(In)14 b(this)h(section)f(w)o(e)g(will)g(describ)q(e)g (the)g(b)q(eha)o(vior)h(of)g(a)g(set)f(that)h(is)g(complete)d(for)j(co-D)1712 153 y Fo(P)1753 174 y Fl(under)75 234 y(randomized)j(reductions.)28 b(W)l(e)19 b(will)f(sho)o(w)h(that)h(its)e(b)q(eha)o(vior)h(is)g(drastically) f(di\013eren)o(t)g(from)75 294 y(that)i(of)f(D)279 276 y Fo(P)334 294 y Fg(\024)373 276 y Fo(vv)373 306 y(m)420 294 y Fl(-complete)f(sets.)30 b(The)20 b(set)f(w)o(e)g(ha)o(v)o(e)g(in)g(mind)e(is)j(SA)l(T)1478 289 y Fg(\010)p 1516 255 95 2 v 1516 294 a Fl(SA)l(T)f(the)h(disjoin)o(t)75 354 y(union)c(of)h(SA)l(T)f(and)p 472 315 V 17 w(SA)l(T:)408 460 y(SA)l(T)503 455 y Fg(\010)p 542 421 V 5 x Fl(SA)l(T)d(=)h Fg(f)i Fl(0)p Ff(F)23 b Fg(j)16 b Ff(F)21 b Fg(2)14 b Fl(SA)l(T)i Fg(g)11 b([)g(f)16 b Fl(1)p Ff(F)23 b Fg(j)16 b Ff(F)k Fg(2)p 1392 421 V 14 w Fl(SA)l(T)c Fg(g)p Ff(:)75 569 y Fh(Lemma)g(2)s(.)87 b Fl(SA)l(T)503 564 y Fg(\010)p 542 530 V 5 x Fl(SA)l(T)21 b(is)g(complete)e(for)i(co-D)1102 548 y Fo(P)1149 569 y Fl(under)g (randomized)f(reductions.)36 b(In)75 629 y(particular,)p 313 590 V 15 w(SA)l(T)s Fg(_)s Fl(SA)l(T)8 b Fg(\024)589 611 y Fo(rp)589 641 y(m)633 629 y Fl(SA)l(T)728 624 y Fg(\010)p 766 590 V 766 629 a Fl(SA)l(T)16 b(with)g(probabilit)o(y)g(1)p Ff(=)p Fl(2)c(+)f(2)1395 611 y Fi(\000)p Fn(n)1443 599 y Fb(2)1464 629 y Fl(.)75 738 y Fh(Pro)r(of)21 b(\(Sk)n(etc)n(h\):)k Fl(Recall)17 b(that)p 735 699 V 19 w(SA)l(T)s Fg(_)s Fl(SA)l(T)h(is)27 b Fg(\024)1081 720 y Fo(P)1081 751 y(m)1120 738 y Fl(-complete)17 b(for)h(co-D)1521 717 y Fo(P)1547 738 y Fl(,)h(so)g(w)o(e)f(need)g(to)75 798 y(displa)o(y)g(a)g(randomized)f(reduction)h(from)p 886 759 V 17 w(SA)l(T)s Fg(_)s Fl(SA)l(T)g(to)g(SA)l(T)1289 793 y Fg(\010)p 1328 759 V 5 x Fl(SA)l(T.)27 b(It)18 b(is)g(simple)e(to)j(con-)75 859 y(struct)c(a)g(randomized)f(reduction)g(with)h(probabilit)o(y)f(greater)h (than)g(or)g(equal)g(to)g(1)p Ff(=)p Fl(2,)h(b)q(ecause)424 964 y Fg(h)p Ff(F)475 971 y Fo(1)494 964 y Ff(;)8 b(F)548 971 y Fo(2)568 964 y Fg(i)14 b(2)p 648 925 V 14 w Fl(SA)l(T)r Fg(_)s Fl(SA)l(T)28 b Fg(\()-8 b(\))27 b Ff(F)1055 971 y Fo(1)1088 964 y Fg(2)p 1135 925 V 14 w Fl(SA)l(T)16 b(or)h Ff(F)1338 971 y Fo(2)1371 964 y Fg(2)d Fl(SA)l(T)p Ff(:)75 1070 y Fl(Th)o(us,)h(a)f (randomized)f(function)h(can)h(c)o(ho)q(ose)g Ff(F)972 1077 y Fo(1)1005 1070 y Fl(or)g Ff(F)1095 1077 y Fo(2)1129 1070 y Fl(with)f(equal)g(probabilit)o(y)l(,)f(then)h(output)75 1130 y(1)p Ff(F)131 1137 y Fo(1)167 1130 y Fl(or)j(0)p Ff(F)283 1137 y Fo(2)302 1130 y Fl(.)k(If)16 b Fg(h)p Ff(F)437 1137 y Fo(1)457 1130 y Ff(;)8 b(F)511 1137 y Fo(2)530 1130 y Fg(i)17 b Fl(is)f(indeed)f(an)i(elemen)o(t)c(of)p 1073 1091 V 17 w(SA)l(T)r Fg(_)s Fl(SA)l(T,)j(then)559 1235 y(Prob)663 1243 y Fn(i)p Fi(2f)p Fo(0)p Fn(;)p Fo(1)p Fi(g)781 1235 y Fl([)g Ff(iF)860 1242 y Fo(2)p Fi(\000)p Fn(i)932 1235 y Fg(2)e Fl(SA)l(T)1074 1230 y Fg(\010)p 1113 1196 V 5 x Fl(SA)l(T)i(])d Fg(\025)h Fl(1)p Ff(=)p Fl(2)p Ff(:)75 1341 y Fl(On)i(the)g(other)h(hand,)f(if)g Fg(h)p Ff(F)599 1348 y Fo(1)619 1341 y Ff(;)8 b(F)673 1348 y Fo(2)692 1341 y Fg(i)14 b(62)p 772 1302 V 14 w Fl(SA)l(T)r Fg(_)s Fl(SA)l(T,)i(then)g(neither)f(case)i(holds.)k(Th)o(us,)584 1447 y(Prob)688 1454 y Fn(i)p Fi(2f)p Fo(0)p Fn(;)p Fo(1)p Fi(g)806 1447 y Fl([)16 b Ff(iF)885 1454 y Fo(2)p Fi(\000)p Fn(i)957 1447 y Fg(2)e Fl(SA)l(T)1099 1442 y Fg(\010)p 1138 1408 V 5 x Fl(SA)l(T)i(])d(=)h(0)p Ff(:)75 1552 y Fl(T)l(o)j(impro)o(v)o(e)c (the)k(probabilit)o(y)e(b)q(ey)o(ond)h(1)p Ff(=)p Fl(2,)h(simply)d(observ)o (e)i(that)h(if)f Ff(F)1451 1559 y Fo(2)1484 1552 y Fg(2)e Fl(SA)l(T,)h(then)i (there)75 1612 y(is)h(a)i(small)d(probabilit)o(y)g(of)i(guessing)h(a)f (satisfying)f(assignmen)o(t.)28 b(This)19 b(fact)f(can)h(b)q(e)g(used)g(to)75 1673 y(impro)o(v)o(e)14 b(the)i(proabilit)o(y)f(to)h(1)p Ff(=)p Fl(2)d(+)e(2)784 1655 y Fi(\000)p Fn(n)832 1643 y Fb(2)852 1673 y Fl(.)22 b(See)15 b([CR90)q(])g(for)i(details.)457 b Fc(2)148 1797 y Fl(Note)15 b(that)g(the)f(probabilit)o(y)g(b)q(ound)i(for)f (the)g(reduction)f(describ)q(ed)g(ab)q(o)o(v)o(e)h(is)g(m)o(uc)o(h)d(b)q (etter)75 1857 y(than)21 b(the)f(one)h(for)g(reducing)f(SA)l(T)r Fg(^)p 789 1818 V 3 w Fl(SA)l(T)h(to)f(USA)l(T.)g(Since)f(USA)l(T)h(b)q(eha)o (v)o(es)g(so)i(m)o(uc)o(h)c(lik)o(e)75 1918 y(the)i(D)200 1899 y Fo(P)255 1918 y Fg(\024)294 1899 y Fo(P)294 1930 y(m)334 1918 y Fl(-complete)e(sets,)j(one)g(w)o(ould)g(exp)q(ect)e(SA)l(T)1168 1913 y Fg(\010)p 1207 1878 V 5 x Fl(SA)l(T)h(to)h(b)q(eha)o(v)o(e)f(lik)o(e)e (co-D)1746 1897 y Fo(P)1780 1918 y Fg(\024)1819 1899 y Fo(P)1819 1930 y(m)1859 1918 y Fl(-)75 1978 y(complete)d(sets.)26 b(In)17 b(fact,)g(SA)l(T)669 1973 y Fg(\010)p 708 1939 V 5 x Fl(SA)l(T)g Fe(do)n(es)i(not)k Fl(resem)o(ble)p 1220 1939 V 14 w(SA)l(T)s Fg(_)s Fl(SA)l(T.)i(SA)l(T)1583 1973 y Fg(\010)p 1622 1939 V 5 x Fl(SA)l(T)17 b(can)h(b)q(e)75 2038 y(computed)i(in)g(P)399 2020 y Fo(SA)m(T)q([1])507 2038 y Fl(,)i(the)e(set)h(of)g(languages)i (recognized)d(b)o(y)g(a)h(p)q(olynomial)f(time)f(T)l(uring)75 2098 y(mac)o(hine)14 b(whic)o(h)i(asks)g(only)g(one)h(query)e(to)i(its)f(SA)l (T)g(oracle.)21 b(Moreo)o(v)o(er,)742 2204 y(P)775 2186 y Fo(SA)m(T)q([1])897 2204 y Fg(\022)14 b Fl(D)987 2186 y Fo(P)1024 2204 y Fg(\\)d Fl(co-D)1168 2183 y Fo(P)1194 2204 y Ff(:)75 2309 y Fl(So,)26 b(there)d(are)g(co-D)486 2288 y Fo(P)544 2309 y Fg(\024)583 2291 y Fo(vv)583 2322 y(m)631 2309 y Fl(-complete)f(sets)i(in)f(D)1065 2291 y Fo(P)1091 2309 y Fl(.)44 b(In)23 b(con)o(trast,)j(the)d(assumption)h (that)75 2369 y(USA)l(T)e Fg(2)g Fl(co-D)383 2349 y Fo(P)431 2369 y Fl(w)o(ould)f(collapse)g(the)f(P)o(olynomial)g(Hierarc)o(h)o(y)l(.)33 b(Th)o(us,)23 b(w)o(e)d(are)h(left)g(in)g(an)75 2430 y(asymmetric)16 b(p)q(osition:)27 b(D)587 2412 y Fo(P)622 2430 y Fg(\024)661 2412 y Fo(vv)661 2442 y(m)709 2430 y Fl(-complete)17 b(sets)i(b)q(eha)o(v)o (e)g(lik)o(e)e(the)28 b Fg(\024)1425 2412 y Fo(P)1425 2442 y(m)1464 2430 y Fl(-complete)17 b(sets,)j(but)75 2490 y(b)q(eing)28 b Fg(\024)257 2472 y Fo(vv)257 2502 y(m)305 2490 y Fl(-complete)18 b(for)j(co-D)709 2469 y Fo(P)755 2490 y Fl(is)f(next)g(to)g(meaningless.)31 b(A)o(t)19 b(this)h(p)q(oin)o(t,)h(the)e(sk)o(eptical)75 2550 y(reader)14 b(migh)o(t)e(suggest)j(that)f(this)g(asymmetry)d(is)j(due)g(to)g (some)f(\\sp)q(ecial)h(prop)q(ert)o(y")g(of)h(USA)l(T)p eop %%Page: 6 6 bop 75 42 a Fl(whic)o(h)11 b(SA)l(T)305 37 y Fg(\010)p 343 3 95 2 v 343 42 a Fl(SA)l(T)h(do)q(es)g(not)g(enjo)o(y)l(.)19 b(Ho)o(w)o(ev)o(er,)11 b(the)g(b)q(eha)o(vior)h(of)f(USA)l(T)g(w)o(e)h(men)o (tioned)d(ab)q(o)o(v)o(e)75 103 y(is)17 b(actually)g(a)h(prop)q(ert)o(y)f(of) g Fe(any)30 b Fg(\024)755 84 y Fo(vv)755 115 y(m)803 103 y Fl(-complete)16 b(set)h(for)g(D)1216 84 y Fo(P)1243 103 y Fl(,)g(not)h(just)f (of)h(USA)l(T.)e(Still,)g(this)75 163 y(asymmetry)11 b(seems)h(v)o(ery)h (unnatural)h(and)h(w)o(e)e(b)q(eliev)o(e)f(that)i(it)f(is)h(an)g(artifact)g (of)g(the)f(de\014nition)75 223 y(of)25 b Fg(\024)178 205 y Fo(vv)178 235 y(m)226 223 y Fl(-reductions.)c(The)16 b(follo)o(wing)g (theorem)f([CR90])h(illustrates)g(our)g(p)q(oin)o(t.)75 307 y Fh(Theorem)h(3)r(.)60 b Fl(If)p 469 268 V 17 w(SA)l(T)s Fg(_)s Fl(SA)l(T)8 b Fg(\024)745 289 y Fo(rp)745 319 y(m)788 307 y Fl(SA)l(T)s Fg(^)p 922 268 V 3 w Fl(SA)l(T)18 b(with)g(probabilit)o(y)f(1)p Ff(=)p Fl(2)c(+)f(1)p Ff(=p)p Fl(\()p Ff(n)p Fl(\))20 b(for)e(some)75 367 y(p)q(olynomial)d(b)q(ound)i Ff(p)p Fl(,)g(the)f(P)o(olynomial)e(Hierarc) o(h)o(y)g(collapses.)148 451 y(No)o(w,)28 b(consider)d(a)h(set)g Ff(G)31 b Fg(2)f Fl(co-D)851 431 y Fo(P)903 451 y Fl(suc)o(h)c(that)p 1138 412 V 26 w(SA)l(T)s Fg(_)s Fl(SA)l(T)8 b Fg(\024)1414 433 y Fo(rp)1414 464 y(m)1457 451 y Ff(G)26 b Fl(with)g(probabilit)o(y)75 512 y(1)p Ff(=)p Fl(2)10 b(+)f(1)p Ff(=)p Fl(p)q(oly)r(.)21 b(Then,)15 b(the)g(theorem)e(sho)o(ws)j(that)g Ff(G)g Fl(cannot)g(b)q(e)f(in) g(D)1407 494 y Fo(P)1449 512 y Fl(unless)g(PH)g(collapses.)75 572 y(Th)o(us,)24 b Ff(G)f Fl(is)g(m)o(uc)o(h)d(more)i(represen)o(tativ)o(e)e (of)j(co-D)1083 551 y Fo(P)1132 572 y Fl(than)g(SA)l(T)1347 567 y Fg(\010)p 1385 533 V 1385 572 a Fl(SA)l(T.)41 b(These)22 b(theorems)75 632 y(demonstrate)h(a)h(kind)f(of)h Fe(thr)n(eshold)f Fl(for)h(the)f(probabilit)o(y)g(b)q(ounds)i(of)f(randomized)e(reduc-)75 692 y(tions.)30 b(If)18 b(w)o(e)h(restrict)f(our)h(atten)o(tion)g(to)g (randomized)f(reductions)g(with)h(probabilit)o(y)f(ab)q(o)o(v)o(e)75 752 y(1)p Ff(=)p Fl(2)c(+)g(1)p Ff(=)p Fl(p)q(oly)q(,)20 b(then)f(the)h (languages)g(that)g(are)g(complete)d(for)j(co-D)1395 731 y Fo(P)1441 752 y Fl(under)f(this)h(notion)g(of)75 813 y(randomized)13 b(reducibilit)o(y)f(b)q(eha)o(v)o(e)h(lik)o(e)g(the)22 b Fg(\024)969 794 y Fo(P)969 825 y(m)1009 813 y Fl(-complete)12 b(sets.)21 b(Ho)o(w)o(ev)o(er,)12 b(if)i(w)o(e)g(allo)o(w)g(ran-)75 873 y(domized)f(reductions)h(with)h(probabilities)f(b)q(elo)o(w)h(this)f (threshold,)h(then)g(ev)o(en)e(trivial)h(sets)h(lik)o(e)75 933 y(SA)l(T)170 928 y Fg(\010)p 209 894 V 5 x Fl(SA)l(T)j(can)g(b)q(e)g (complete.)25 b(Th)o(us,)19 b(when)f(w)o(e)g(consider)26 b Fg(\024)1294 915 y Fo(rp)1294 945 y(m)1338 933 y Fl(-complete)16 b(sets)i(for)h(co-D)1835 912 y Fo(P)1861 933 y Fl(,)75 993 y(the)e(smallest)e(non-trivial)i(probabilit)o(y)f(b)q(ound)i(is)f(1)p Ff(=)p Fl(2)c(+)f(1)p Ff(=)p Fl(p)q(oly)q(.)24 b(In)17 b(con)o(trast,)g(the)g (smallest)75 1053 y(non-trivial)h(probabilit)o(y)g(b)q(ound)i(for)f(D)836 1035 y Fo(P)889 1053 y Fg(\024)928 1035 y Fo(rp)928 1066 y(m)972 1053 y Fl(-complete)d(sets)j(is)g(1)p Ff(=)p Fl(p)q(oly)q(.)29 b(It)18 b(w)o(ould)h(app)q(ear)75 1114 y(that)14 b(these)f(thresholds,)h(ab)q (o)o(v)o(e)g(whic)o(h)f(completeness)f(starts)i(making)f(sense,)g(are)h (di\013eren)o(t)f(for)75 1174 y(di\013eren)o(t)19 b(complexit)o(y)e(classes.) 32 b(This)20 b(still)f(lea)o(v)o(es)f(op)q(en)j(the)e(question)h(of)g (whether)g(V)l(alian)o(t)75 1234 y(and)e(V)l(azirani)e(made)g(the)h(correct)f (decision)h(when)g(they)f(c)o(hose)h(1)p Ff(=)p Fl(p)q(oly)i(as)f(the)f (threshold)g(for)75 1294 y(D)112 1276 y Fo(P)163 1294 y Fg(\024)202 1276 y Fo(rp)202 1306 y(m)245 1294 y Fl(-complete)e(sets.)148 1354 y(T)l(o)g(address)h(this)e(question,)g(w)o(e)g(return)h(to)f(USA)l(T.)g (Supp)q(ose)i(someone)d(w)o(ere)h(to)h(construct)75 1414 y(a)20 b(randomized)f(reduction)h(from)f(SA)l(T)h(to)g(USA)l(T)g(with)f(probabilit)o (y)h(1)p Ff(=)p Fl(2)14 b(+)g(1)p Ff(=)p Fl(p)q(oly)r(.)32 b(Then,)75 1475 y(USA)l(T)17 b(w)o(ould)h(b)q(e)g(complete)e(for)i(D)757 1457 y Fo(P)801 1475 y Fl(in)f(a)h(m)o(uc)o(h)e(stronger)j(sense.)26 b(In)17 b(fact,)h(suc)o(h)f(a)i(theorem)75 1535 y(w)o(ould)12 b(answ)o(er)g(the)g(frequen)o(tly)e(p)q(osed)j(question)f(of)g(whether)g(USA) l(T)f(has)i(OR)1533 1542 y Fo(2)1565 1535 y Fl([CH86,)e(GW86)q(,)75 1595 y(CGH)185 1577 y Fo(+)215 1595 y Fl(89,)23 b(GNW90)q(].)42 b(\(It)23 b(is)h(kno)o(wn)f(that)h(SA)l(T)s Fg(^)p 1088 1556 V 3 w Fl(SA)l(T)f(do)q(es)h(not)g(ha)o(v)o(e)f(OR)1611 1602 y Fo(2)1654 1595 y Fl(unless)g(PH)75 1655 y(collapses)16 b([CK90b)q(].\))75 1739 y Fh(Corollary)i(4)s(.)43 b Fl(If)16 b(SA)l(T)8 b Fg(\024)605 1721 y Fo(rp)605 1752 y(m)648 1739 y Fl(USA)l(T)16 b(with)f(probabilit)o(y)g (1)p Ff(=)p Fl(2)d(+)e(1)p Ff(=p)p Fl(\()p Ff(n)p Fl(\))17 b(for)f(some)f(p)q(olynomial)75 1800 y(b)q(ound)i Ff(p)p Fl(,)g(then)f(USA)l (T)f(do)q(es)i(not)g(ha)o(v)o(e)f(OR)921 1807 y Fo(2)957 1800 y Fl(unless)h(PH)f(collapses.)75 1884 y Fh(Pro)r(of:)k Fl(W)l(e)13 b(kno)o(w)g(that)p 555 1845 V 14 w(SA)l(T)8 b Fg(\024)697 1866 y Fo(P)697 1896 y(m)736 1884 y Fl(USA)l(T.)20 b(By)12 b(assumption,)h(SA)l(T) 8 b Fg(\024)1386 1866 y Fo(rp)1386 1896 y(m)1430 1884 y Fl(USA)l(T)13 b(with)g(probabil-)75 1944 y(it)o(y)f(1)p Ff(=)p Fl(2)5 b(+)g(1)p Ff(=)p Fl(p)q(oly)s(.)20 b(If)13 b(USA)l(T)f(has)i(OR)790 1951 y Fo(2)810 1944 y Fl(,)f(then)h(these)f(t)o(w)o(o)g(reductions)g(can)g(b)q(e) h(com)o(bined)d(in)o(to)i(a)75 2004 y(randomized)g(reduction)g(from)p 663 1965 V 13 w(SA)l(T)s Fg(_)s Fl(SA)l(T)g(to)i(USA)l(T)e(with)h(probabilit) o(y)f(1)p Ff(=)p Fl(2)6 b(+)g(1)p Ff(=)p Fl(p)q(oly)s(.)21 b(Since)75 2064 y(USA)l(T)c Fg(2)i Fl(D)313 2046 y Fo(P)339 2064 y Fl(,)g(this)g(also)g(giv)o(es)f(a)h(randomized)e(reduction)i(from)p 1337 2025 V 17 w(SA)l(T)s Fg(_)s Fl(SA)l(T)f(to)i(SA)l(T)r Fg(^)p 1780 2025 V 3 w Fl(SA)l(T)75 2125 y(with)c(probabilit)o(y)f(1)p Ff(=)p Fl(2)e(+)e(1)p Ff(=)p Fl(p)q(oly)q(.)21 b(Then,)16 b(b)o(y)g(Theorem)f (3,)h(PH)g(collapses.)323 b Fc(2)148 2249 y Fl(So,)20 b(if)e(w)o(e)g(w)o(ere) g(to)h(call)f(a)h(set)27 b Fg(\024)792 2231 y Fo(rp)792 2261 y(m)836 2249 y Fl(-complete)16 b(only)j(when)g(the)f(probabilit)o(y)g(b)q (ounds)i(are)75 2309 y(at)25 b(least)e(1)p Ff(=)p Fl(2)18 b(+)e(1)p Ff(=)p Fl(p)q(oly)r(,)26 b(then)e(w)o(e)f(w)o(ould)i(restore)f(symmetr)o(y)d (to)k(our)f(w)o(orld.)45 b(Ho)o(w)o(ev)o(er,)75 2369 y(considering)16 b(the)f(usefulness)h(of)g(the)g(V)l(alian)o(t-V)l(azirani)e(reduction)h(from) g(SA)l(T)h(to)g(USA)l(T,)f(one)75 2430 y(w)o(ould)j(hesitate)g(to)h(sa)o(y)f (that)h(they)f(made)f(the)h(wrong)i(de\014nition)e(of)27 b Fg(\024)1462 2412 y Fo(rp)1462 2442 y(m)1505 2430 y Fl(-completeness)17 b(for)75 2490 y(D)112 2472 y Fo(P)138 2490 y Fl(.)j(Ho)o(w)o(ev)o(er,)10 b(one)i(could)g(lamen)o(t)d(the)j(fact)f(that)h(they)g(did)f(not)h(pro)o (vide)f(a)h(stronger)g(reduction)75 2550 y(from)i(SA)l(T)h(to)g(USA)l(T.)f (Then,)h(our)g(w)o(orld)g(could)g(b)q(e)g(made)f(nice,)g(sane)h(and)h(symme)o (tric)c(again.)p eop %%Page: 7 7 bop 75 50 a Fm(3)83 b(A)28 b(Note)f(on)g(BPP)h(reductions)75 159 y Fl(So)18 b(far,)f(w)o(e)g(ha)o(v)o(e)f(b)q(een)h(discussing)h (randomized)e(reductions)h(with)g(one-sided)g(error.)24 b(In)17 b(this)75 220 y(section,)f(w)o(e)g(sho)o(w)i(that)f(similar)e(results)h (apply)h(to)g(randomized)e(reductions)i(with)f(t)o(w)o(o-sided)75 280 y(error.)32 b(T)l(raditionally)l(,)20 b(the)g(BP)o Fg(\001)g Fl(op)q(erator)h(on)g(a)f(class)g Fg(C)j Fl(has)e(b)q(een)f(de\014ned)g(as)g (follo)o(ws.)32 b(A)75 340 y(language)20 b Ff(A)e 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Fl(.)k(The)16 b(follo)o(wing)g(theorems)f([CR90])h (illustrate)f(our)i(p)q(oin)o(t.)75 1893 y Fh(Lemma)f(5)s(.)43 b Fl(There)16 b(exists)g(a)h(p)q(olynomial)e(time)f(function)i Ff(f)21 b Fl(suc)o(h)16 b(that)349 2000 y(Prob)453 2007 y Fn(z)473 2000 y Fl([)f Ff(x)f Fg(2)p 591 1961 95 2 v 14 w Fl(SA)l(T)s Fg(_)s Fl(SA)l(T)27 b Fg(\()-8 b(\))27 b Ff(f)5 b Fl(\()p Ff(x;)j(z)r Fl(\))14 b Fg(2)g Fl(SA)l(T)1264 1995 y Fg(\010)p 1302 1961 V 1302 2000 a Fl(SA)l(T)i(])e Fg(\025)f Fl(2)p Ff(=)p Fl(3)p Ff(:)75 2110 y Fh(Pro)r(of:)148 2170 y Fl(Let)19 b Ff(g)i Fl(b)q(e)d(the)27 b Fg(\024)484 2152 y Fo(rp)484 2183 y(m)528 2170 y Fl(-reduction)18 b(with)h(probabilit)o(y)e(1)p Ff(=)p Fl(2)j(from)p 1338 2131 V 18 w(SA)l(T)r Fg(_)s Fl(SA)l(T)f(to)g(SA)l(T)1742 2165 y Fg(\010)p 1780 2131 V 1780 2170 a Fl(SA)l(T)75 2230 y(describ)q(ed)25 b(in)f(Lemma)f(2.)48 b(Our)34 b Fg(\024)791 2212 y Fo(bpp)791 2243 y(m)860 2230 y Fl(-reduction)24 b Ff(f)31 b Fl(will)24 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