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b(\(again)j(using)d(magic\)|a)448 3692 y(problem)31 b(that)i(Sto)s(c)m (kmey)m(er)g(and)f(Mey)m(er,)i(in)d(a)i(STOC)c('73)k(pap)s(er,)f(pro)m (v)m(ed)h(NP)3302 3659 y Fu(NP)3407 3692 y FA(-)448 3810 y(complete.)40 b(I'll)24 b(ask)h(y)m(ou)h(to)g(write)e(a)i(p)s (olynomial-time)c(algorithm)i(that)i(accepts)h(the)448 3929 y(language)k(F)m(OO,)f(and)f(that)i(on)f(eac)m(h)h(input)d(mak)m (es)j(at)f(most)h(one)f(query)f(to)i(eac)m(h)g(of)448 4047 y(the)36 b(blac)m(k)f(b)s(o)m(xes.)55 b(Actually)-8 b(,)36 b(at)g(this)e(momen)m(t)i(I)f(don't)g(demand)g(that)g(y)m(ou)h (write)448 4166 y(the)c(algorithm.)43 b(I)31 b(just)g(w)m(an)m(t)h(y)m (ou)g(to)g(decide)f(on)g(a)h(\014xed)f(order)g(in)f(whic)m(h)g(y)m(ou)i (will)448 4285 y(access)39 b(the)e(blac)m(k)g(b)s(o)m(xes.)60 b(Either)36 b(y)m(ou)h(m)m(ust)g(\(righ)m(t)g(no)m(w\))g(commit)g(to)h (accessing)448 4403 y(SA)-8 b(T)34 b(\014rst)g(on)g(eac)m(h)h(input,)e (or)i(y)m(ou)f(m)m(ust)g(\(righ)m(t)g(no)m(w\))h(commit)f(to)h (accessing)g(IEI)448 4522 y(\014rst)30 b(on)g(eac)m(h)i(input.)448 4641 y Ft(Y)-9 b(ou:)92 b FA(W)-8 b(ell,)39 b(b)s(efore)f(I)f(commit,)j (will)35 b(y)m(ou)j(tell)f(me)h(a)g(bit)f(ab)s(out)g(what)h(problem)448 4760 y(F)m(OO)31 b(is?)448 4880 y Ft(Stranger:)91 b FA(No.)448 5000 y Ft(Y)-9 b(ou:)92 b FA(Oh.)49 b(W)-8 b(ait)34 b(a)f(second.)50 b(Y)-8 b(ou're)34 b(trying)e(to)i(tric)m(k)g(me.)49 b(NP)2800 4967 y Fu(NP)2938 5000 y FA(\(as)34 b(captured)448 5118 y(b)m(y)h(its)g(complete)g(set,)h(IEI\))f(is)f Fy(much)42 b FA(more)35 b(p)s(o)m(w)m(erful)f(than)h(NP)g(\(as)g(captured)g(b)m(y) 448 5237 y(its)i(complete)h(set,)h(SA)-8 b(T\).)38 b(So)f(for)g(b)s (oth)g(classes)h(y)m(ou)g(are)g(sp)s(eaking)e(of)i(the)g(query)448 5355 y(to)e(SA)-8 b(T)35 b(is)e(sup)s(er\015uous.)52 b(That)35 b(is:)49 b(P)1829 5322 y Fu(IEI[1])2039 5355 y FA(=)32 b(P)2204 5322 y Fu(IEI:SA)-6 b(T)2497 5355 y FA(=)32 b(P)2662 5322 y Fu(SA)-6 b(T:IEI)2922 5355 y FA(,)36 b(where)e(\\[1]")448 5474 y(indicates)j(one)i(query)f(and)f (P)1517 5441 y Fp(A)p Fu(:)p Fp(B)1688 5474 y FA(indicates)g(a)i(P)f (mac)m(hine)g(making)f(at)i(most)g(one)448 5592 y(query)30 b(to)h(\(the)g(set\))g Fo(A)g FA(set)g(follo)m(w)m(ed)f(b)m(y)g(at)h (most)g(one)g(query)e(to)i(\(the)g(set\))h Fo(B)5 b FA(.)p eop %%Page: 3 3 3 2 bop 448 399 a Ft(Stranger:)91 b FA(F)-8 b(alse!)66 b(Y)-8 b(our)38 b(reasoning)g(is)g(tempting,)i(but)e(is)f(not)i(v)-5 b(alid.)63 b(In)38 b(fact,)448 517 y(that)31 b(w)m(eak)g(SA)-8 b(T)30 b(query)g(pro)m(v)-5 b(ably)29 b(giv)m(es)i(strictly)e(more)h (computational)g(p)s(o)m(w)m(er,)h(as)448 636 y(y)m(ou)k(w)m(ould)e (kno)m(w)h(if)f(y)m(ou)i(had)f(read)g(a)g(certain)h Fy(BEA)-7 b(TCS)44 b FA(article)34 b([HHH97a)r(])h(and)448 754 y(had)30 b(applied)e(it)i(in)f(ligh)m(t)h(of)g(the)h(Sipser-Y)-8 b(ao)30 b(b)s(o)s(ok)g(under)f(m)m(y)h(arm.)448 873 y Ft(Y)-9 b(ou:)92 b FA(I'v)m(e)23 b(got)g(y)m(ou)f(no)m(w,)i(as)e(I)g Fy(did)32 b FA(read)22 b(that)g(obscure)g(little)f(article!)37 b(I)22 b(retract)h(the)448 991 y(P)510 958 y Fu(IEI[1])717 991 y FA(=)29 b(P)879 958 y Fu(IEI:SA)-6 b(T)1171 991 y FA(claim)32 b(I)g(just)g(made,)i(but)e(I)h(reassert)g(the)g(P)2727 958 y Fu(IEI:SA)-6 b(T)3016 991 y FA(=)29 b(P)3178 958 y Fu(SA)-6 b(T:IEI)448 1110 y FA(claim!)39 b(Order)24 b(do)s(es)i(not)g(matter)h(here,)g(and)f(so)g(either)g(order)f(I)h (commit)g(to)h(no)m(w)f(will)448 1228 y(lea)m(v)m(e)32 b(it)e(equally)f(lik)m(ely)g(that)i(F)m(OO)g(can)f(b)s(e)g(solv)m(ed)g (via)g(the)h(giv)m(en)f(access)i(order!)448 1347 y Ft(Stranger:)91 b FA(Sheesh...)37 b(I)g(though)m(t)h(no)e(one)i(except)g(the)f(authors) g(and)f(the)i(editor)448 1466 y(had)30 b(read)g(that)h(column.)40 b(W)-8 b(ell,)30 b(y)m(ou'v)m(e)i(b)s(eaten)e(me)h(so)g(here)f(are)h(y) m(our)f(b)s(o)s(oks.)448 1584 y Ft(Y)-9 b(ou:)92 b FA(But)31 b(this)g(Sipser-Y)-8 b(ao)31 b(b)s(o)s(ok)g(is)f(from)h(the)g(y)m(ear)h (2010.)46 b(If)31 b(I)g(just)f(steal)i(their)448 1703 y(result)g(and)h(pro)s(of,)g(and)f(publish)d(them)k(no)m(w,)h(w)m(on't) g(that)f(c)m(hange)h(the)g(future)e(and)448 1821 y(cause)c(the)g(b)s(o) s(ok)f(nev)m(er)h(to)g(b)s(e)f(written,)g(in)f(whic)m(h)g(case)j(ho)m (w)e(can)h(I)f(b)s(e)g(meeting)h(y)m(ou)448 1940 y(and)i(receiving)g (this)f(b)s(o)s(ok)h(in)f(the)h(\014rst)g(place?)448 2059 y Ft(Stranger:)91 b FA(\(Suddenly)42 b(starts)k(fading)d(a)m(w)m (a)m(y)-8 b(,)51 b(but)44 b(b)s(efore)h(he)f(disapp)s(ears)f(y)m(ou)448 2177 y(hear\))100 b(I)32 b(commend)h(to)h(y)m(ou)f(the)g Fx(Nebula-Aw)n(ard-winning)c(b)r(o)r(ok)g Fn(Timesc)l(ap)l(e)35 b Fr([Ben80)q(],)448 2296 y(whic)n(h)26 b(also)h(has)f(quite)f(a)h(bit) g(to)f(sa)n(y)h(ab)r(out)31 b Fu(curiosit)n(y-driv)n(en)23 b(researc)n(h,)54 b Fs(scien)n(ti\014c)21 b(ethics,)f(and...)589 2514 y FA(Y)-8 b(ou)32 b(suddenly)d(feel)i(a)h(bit)e(disorien)m(ted)g (and|distractedly)f(tossing)i(the)h(b)s(o)s(oks)448 2633 y(in)m(to)27 b(a)g(trash)g(bin)e(and)h(m)m(um)m(bling)f(ab)s(out)i(sw)m (earing)f(o\013)h(that)h(nin)m(th)d(cup)h(of)i(co\013ee|)448 2751 y(y)m(ou)36 b(go)g(bac)m(k)h(in)m(to)e(the)h(lecture)f(hall)f(and) h(listen)f(with)g(half-hearted)h(atten)m(tion)h(to)448 2870 y(an)31 b(in-progress)e(lecture)h(on)g(query)g(order.)448 3164 y FC(2)135 b(Query)45 b(Order)g(in)g(the)g(P)l(olynomial)i (Hierarc)l(h)l(y)448 3376 y Fm(2.1)113 b(Results)448 3554 y FA(Recall)32 b(from)g(the)h(in)m(tro)s(duction)d(that,)k(for)e (an)m(y)h(sets)g Fo(A)f FA(and)g Fo(B)5 b FA(,)33 b(P)2815 3521 y Fp(A)p Fu(:)p Fp(B)2980 3554 y FA(denotes)g(the)448 3672 y(class)28 b(of)h(languages)f(that)h(can)g(b)s(e)e(accepted)j(via) e(P)g(mac)m(hines)g(making)f(at)i(most)g(one)448 3791 y(query)35 b(to)h Fo(A)f Fy(fol)5 b(lowe)-5 b(d)39 b(by)k FA(at)35 b(most)h(one)f(query)g(to)g Fo(B)5 b FA(.)55 b(Similarly)-8 b(,)33 b(for)i(an)m(y)g(classes)448 3910 y Fl(C)43 b FA(and)36 b Fl(D)s FA(,)j(P)921 3877 y Fk(C)t Fu(:)p Fk(D)1079 3910 y FA(denotes)f(the)g(class)f(of)g(languages)h (that)g(can)f(b)s(e)g(accepted)h(via)f(P)448 4028 y(mac)m(hines)27 b(making)g(at)h(most)f(one)h(query)e(to)i(a)g(set)g(from)e Fl(C)33 b Fy(fol)5 b(lowe)-5 b(d)31 b(by)k FA(at)28 b(most)g(one)448 4147 y(query)i(to)h(a)g(set)g(from)f Fl(D)s FA(.)40 b(That)30 b(is,)1495 4357 y(P)1557 4319 y Fk(C)t Fu(:)p Fk(D)1704 4357 y FA(=)1928 4276 y Fj([)1800 4460 y Fp(C)5 b Fk(2C)t Fp(;)11 b(D)r Fk(2D)2164 4357 y FA(P)2226 4319 y Fp(C)5 b Fu(:)p Fp(D)2365 4357 y Fo(:)448 4644 y FA(These)39 b(notions)e(and)h(notations)h(w)m(ere)g(in)m(tro)s(duced)e(b)m(y)h (Hemaspaandra,)j(Hemp)s(el,)448 4762 y(and)32 b(W)-8 b(ec)m(hsung)34 b([HHW95)r(],)f(who)f(studied)f(them)h(for)g(the)h (case)h(in)d(whic)m(h)g Fl(C)38 b FA(and)31 b Fl(D)448 4881 y FA(are)c(lev)m(els)g(of)f(the)h(b)s(o)s(olean)f(hierarc)m(h)m(y) g(\(see)i(Section)e(3\);)j(in)c(brief,)i(they)f(pro)m(v)m(ed)h(that)448 5000 y(query)j(order)g(usually)e(do)s(es)i(matter)h(in)e(the)i(b)s(o)s (olean)e(hierarc)m(h)m(y)-8 b(.)589 5118 y(Hemaspaandra,)102 b(Hemaspaandra,)g(and)86 b(Hemp)s(el)g([HHH96b)q(,HHH97b)r(],)448 5237 y(follo)m(wing)29 b(up)g(on)h(the)h(concept)g(of)f(query)g(order,) g(ask)m(ed)h(whether)f(query)f(order)h(also)448 5355 y(matters)d(in)d(the)i(p)s(olynomial)d(hierarc)m(h)m(y)-8 b(.)39 b(What)26 b(they)g(found)e(w)m(as)i(that)h(query)e(order)448 5474 y Fy(never)38 b FA(matters)28 b(in)f(the)g(p)s(olynomial)f (hierarc)m(h)m(y)-8 b(.)39 b(This)26 b(is)h(stated)i(formally)d(b)s (elo)m(w)h(as)448 5592 y(Theorem)j(2.2.)p eop %%Page: 4 4 4 3 bop 448 399 a Ft(De\014nition)36 b(2.1)76 b FA([Sto77)q(])559 593 y(1.)47 b(\006)742 548 y Fu(p)742 617 y(0)810 593 y FA(=)25 b(P.)559 797 y(2.)47 b(F)-8 b(or)38 b(eac)m(h)i Fo(k)h(>)d FA(0,)j(\006)1433 753 y Fu(p)1433 825 y Fp(k)1514 797 y FA(=)d(NP)1753 755 y Fu(\006)1804 723 y Fs(p)1804 780 y Fi(k)q Fh(\000)p Fs(1)1924 797 y FA(.)64 b(\(F)-8 b(or)40 b(example,)g(\006)2677 753 y Fu(p)2677 822 y(1)2758 797 y FA(=)e(NP)g(and)f(\006)3285 753 y Fu(p)3285 822 y(2)3366 797 y FA(=)676 916 y(NP)806 880 y Fu(NP)911 916 y FA(.\))559 1110 y(3.)47 b(PH)25 b(=)957 1046 y Fj(S)927 1189 y Fp(k)r Fk(\025)p Fu(0)1071 1110 y FA(\006)1137 1066 y Fu(p)1137 1138 y Fp(k)1180 1110 y FA(.)448 1377 y Ft(Theorem)34 b(2.2)77 b FA([HHH97b)q(])121 b(F)-8 b(or)31 b(eac)m(h)h Fo(i;)15 b(j)32 b Fl(\025)25 b FA(0,)1606 1597 y(P)1668 1554 y Fu(\006)1719 1521 y Fs(p)1719 1577 y Fi(i)1757 1554 y Fu(:\006)1828 1521 y Fs(p)1828 1577 y Fi(j)1895 1597 y FA(=)g(P)2053 1554 y Fu(\006)2104 1521 y Fs(p)2104 1577 y Fi(j)2141 1554 y Fu(:\006)2212 1521 y Fs(p)2212 1577 y Fi(i)2254 1597 y Fo(:)589 1807 y FA(In)42 b(fact,)47 b(in)42 b(all)f(but)h(the)h(\\diagonal")g(cases)h (of)e(this)g(theorem)h(\(where)g(order)448 1926 y(elimination)f(is)i (imp)s(ossible)d(unless)h(the)j(p)s(olynomial)d(hierarc)m(h)m(y)i (collapses\),)k(one)448 2045 y(can)31 b(eliminate)e(order)h(en)m (tirely:)448 2256 y Ft(Theorem)k(2.3)77 b FA([HHH97b)q(])121 b(F)-8 b(or)31 b(eac)m(h)h Fo(i;)15 b(j)32 b Fl(\025)25 b FA(0)30 b(with)f Fo(i)d Fl(6)p FA(=)f Fo(j)5 b FA(,)1387 2497 y(P)1449 2433 y Fu(\(\006)1527 2400 y Fs(p)1527 2456 y Fi(i)1565 2433 y Fp(;)p Fu(\006)1636 2400 y Fs(p)1636 2456 y Fi(j)1673 2433 y Fu(\))1449 2522 y(1)p Fp(;)p Fu(1-tt)1730 2497 y FA(=)25 b(P)1888 2454 y Fu(\006)1939 2422 y Fs(p)1939 2478 y Fi(i)1977 2454 y Fu(:\006)2048 2422 y Fs(p)2048 2478 y Fi(j)2114 2497 y FA(=)g(P)2272 2454 y Fu(\006)2323 2422 y Fs(p)2323 2478 y Fi(j)2361 2454 y Fu(:\006)2432 2422 y Fs(p)2432 2478 y Fi(i)2474 2497 y Fo(;)448 2753 y FA(where)j(P)771 2689 y Fu(\(\006)849 2656 y Fs(p)849 2712 y Fi(i)887 2689 y Fp(;)p Fu(\006)958 2656 y Fs(p)958 2712 y Fi(j)995 2689 y Fu(\))771 2778 y(1)p Fp(;)p Fu(1-tt)1054 2753 y FA(denotes)g(the)g(class)g(of)f (languages)h(accepted)h(b)m(y)f(mac)m(hines)f(that,)i(in)448 2872 y(parallel)i(\(i.e.,)j(sim)m(ultaneously\),)e(ask)h(at)h(most)f (one)g(question)f(to)i(a)f(\006)2957 2828 y Fu(p)2957 2898 y Fp(i)3033 2872 y FA(set)g(and)f(at)448 2991 y(most)f(one)g (question)e(to)i(a)g(\006)1452 2946 y Fu(p)1452 3016 y Fp(j)1526 2991 y FA(set.)589 3201 y(Ho)m(w)m(ev)m(er,)i(w)m(e)e (should)d(no)m(w)i(address)g(the)g(\\tempting")h(w)m(orry)f(y)m(ou,)g (the)h(reader,)448 3320 y(raised)j(during)f(the)i(in)m(tro)s(duction.) 52 b(Let)36 b Fo(i)d(<)f(j)5 b FA(.)55 b(Clearly)-8 b(,)35 b(\006)2601 3276 y Fu(p)2601 3346 y Fp(j)2679 3320 y FA(can)g(b)s(eat)g(\006)3125 3276 y Fu(p)3125 3346 y Fp(i)3203 3320 y FA(in)m(to)g(a)448 3454 y(pulp.)57 b(F)-8 b(or)37 b(example,)h(it)e(is)g(w)m(ell)f(kno)m(wn)h(that)i(P)2212 3421 y Fu(\006)2263 3388 y Fs(p)2263 3444 y Fi(i)2340 3454 y Fl(\022)d FA(\006)2512 3410 y Fu(p)2512 3480 y Fp(j)2555 3454 y FA(.)59 b(That)37 b(is,)g(\006)3063 3410 y Fu(p)3063 3480 y Fp(j)3142 3454 y FA(is)f(more)448 3573 y(p)s(o)m(w)m(erful)k(than)h(ev)m(en)g(a)h(p)s(olynomially)c(long) i(series)h(of)g(adaptiv)m(e)g(queries)f(to)i(\006)3369 3528 y Fu(p)3369 3599 y Fp(i)3412 3573 y FA(.)448 3704 y(So)37 b(it)f(w)m(ould)g(indeed)f(b)s(e)h(tempting)g(to)i(assert:)54 b(P)2256 3661 y Fu(\006)2307 3628 y Fs(p)2307 3684 y Fi(i)2344 3661 y Fu(:\006)2415 3628 y Fs(p)2415 3684 y Fi(j)2493 3704 y FA(=)35 b(P)2661 3661 y Fu(\006)2712 3628 y Fs(p)2712 3684 y Fi(j)2750 3661 y Fu([1])2865 3704 y FA(for)i Fo(i)f(<)f(j)5 b FA(,)40 b(and)448 3822 y(indeed)f(there)i(is)e(an)i(ob)m(vious)f(\(but)g(\015a)m(w)m(ed\))h (\\pro)s(of)7 b(")41 b(of)f(this,)j(in)m(v)m(olving)38 b(ha)m(ving)448 3950 y(\006)514 3906 y Fu(p)514 3976 y Fp(j)588 3950 y FA(sim)m(ulate)31 b(the)g(\006)1174 3906 y Fu(p)1174 3976 y Fp(i)1248 3950 y FA(query)g(of)g(P)1668 3907 y Fu(\006)1719 3875 y Fs(p)1719 3931 y Fi(i)1757 3907 y Fu(:\006)1828 3875 y Fs(p)1828 3931 y Fi(j)1869 3950 y FA(,)h(get)g(the)g(answ)m(er,)f(and)g(then)f(sim)m(ulate)h(the) 448 4094 y(\006)514 4050 y Fu(p)514 4120 y Fp(j)600 4094 y FA(query)42 b(of)h(P)1043 4051 y Fu(\006)1094 4019 y Fs(p)1094 4075 y Fi(i)1131 4051 y Fu(:\006)1202 4019 y Fs(p)1202 4075 y Fi(j)1244 4094 y FA(.)77 b(The)42 b(\015a)m(w)h(is)e(that)i(though)g(\006)2446 4050 y Fu(p)2446 4120 y Fp(j)2531 4094 y FA(can)g(do)g(this,)h(it)e(cannot)448 4213 y(pass)f(to)i(the)e(base)h(P)f(mac)m(hine)h(the)f(information)f (on)i(whic)m(h)e(truth-table)h(to)h(use)448 4331 y(to)37 b(pro)s(cess)e(its)g(answ)m(er;)k(there)d(is)e(a)j(1-bit)e(information) f(b)s(ottlenec)m(k!)57 b(Indeed,)37 b(the)448 4450 y(tempting)f (equalit)m(y|P)1312 4407 y Fu(\006)1363 4374 y Fs(p)1363 4430 y Fi(i)1400 4407 y Fu(:\006)1471 4374 y Fs(p)1471 4430 y Fi(j)1548 4450 y FA(=)f(P)1716 4407 y Fu(\006)1767 4374 y Fs(p)1767 4430 y Fi(j)1805 4407 y Fu([1])1920 4450 y FA(for)h Fo(i)f(<)g(j)5 b FA(|is)36 b(outrigh)m(t)g(false)g (unless)f(the)448 4568 y(p)s(olynomial)40 b(hierarc)m(h)m(y)i (collapses.)76 b(This)40 b(follo)m(ws)i(immediately)f(from)g(the)i (more)448 4687 y(general)25 b(fact)h(that)f(all)f(\\ordered)h(access)h (to)g(the)f(p)s(olynomial)d(hierarc)m(h)m(y")j(classes)g(are)448 4806 y(either)30 b(trivially)f(equal)h(or)g(are)i(truly)d(di\013eren)m (t)h(\(unless)f(the)i(p)s(olynomial)d(hierarc)m(h)m(y)448 4924 y(collapses\).)448 5144 y Ft(Theorem)34 b(2.4)85 b FA([HHH97b)q(])130 b(Let)39 b Fo(i;)15 b(j;)g(`;)g(m)41 b Fl(\025)d FA(0.)66 b(If)39 b(P)2561 5102 y Fu(\006)2612 5069 y Fs(p)2612 5125 y Fi(i)2649 5102 y Fu(:\006)2720 5069 y Fs(p)2720 5125 y Fi(j)2801 5144 y FA(=)f(P)2972 5110 y Fu(\006)3023 5077 y Fs(p)3023 5135 y Fi(`)3061 5110 y Fu(:\006)3132 5077 y Fs(p)3132 5122 y Fi(m)3194 5144 y FA(,)j(then)448 5263 y(either)30 b Fl(f)p Fo(i;)15 b(j)5 b Fl(g)28 b FA(=)d Fl(f)p Fo(`;)15 b(m)p Fl(g)31 b FA(or)f(the)h(p)s(olynomial)d(hierarc)m(h)m(y)i(collapses.)589 5474 y(The)25 b(just-stated)h(theorem)g(merely)e(concludes)h(that)g (the)h(p)s(olynomial)c(hierarc)m(h)m(y)448 5592 y(collapses)h(\(unless) g Fl(f)p Fo(i;)15 b(j)5 b Fl(g)27 b FA(=)e Fl(f)p Fo(`;)15 b(m)p Fl(g)p FA(\).)40 b(In)23 b(fact,)j(in)c(almost)i(all)f(cases,)j (the)e(p)s(olynomial)p eop %%Page: 5 5 5 4 bop 448 399 a FA(hierarc)m(h)m(y)23 b(collapses)g(to)h(an)f (alarmingly)e(lo)m(w)i(lev)m(el|one)g(that)h Fy(a)j(priori)33 b FA(seems)24 b(lo)m(w)m(er)448 517 y(than)30 b(either)f(of)h(the)f (classes)h(men)m(tioned)f(in)g(the)g(theorem)h(\(this)f(can)h(b)s(e)f (seen)h(easily)448 636 y(from)c([HHH)q(,BF96)r(,HHH97d)q(,HHH97b)q(],)i (see)f(esp)s(ecially)d([HHH97b)r(,)j(Section)f(3.2]\).)448 754 y(F)-8 b(or)31 b(example,)1101 966 y(P)1163 928 y Fu(\006)1214 896 y Fs(p)1214 951 y(1997)1339 928 y Fu(:\006)1410 896 y Fs(p)1410 951 y(1999)1565 966 y FA(=)25 b(P)1723 928 y Fu(\006)1774 896 y Fs(p)1774 951 y(1998)1900 928 y Fu(:\006)1971 896 y Fs(p)1971 951 y(1999)2126 966 y FA(=)-15 b Fl(\))24 b FA(PH)i(=)f(\006)2615 922 y Fu(p)2615 991 y(1999)2760 966 y Fo(;)448 1178 y FA(ev)m(en)35 b(though)f Fy(a)j(priori)45 b FA(one)35 b(w)m(ould)e(susp)s(ect)h(that)h(\006)2356 1134 y Fu(p)2356 1203 y(1999)2535 1178 y FA(is)e(strictly)h(con)m (tained)g(in)448 1296 y(P)510 1263 y Fu(\006)561 1231 y Fs(p)561 1286 y(1997)687 1263 y Fu(:\006)758 1231 y Fs(p)758 1286 y(1999)888 1296 y FA(.)589 1416 y(Note)25 b(that)g(in)d(all)h(the)h(results)e(w)m(e)i(ha)m(v)m(e)h(discussed)d (so)i(far,)h(w)m(e)f(ha)m(v)m(e)h(a)f(P)f(mac)m(hine)448 1534 y(doing)42 b(the)h(querying,)h(i.e.,)i(P)1540 1501 y Fu(SA)-6 b(T:IEI)1845 1534 y FA(=)45 b(P)2023 1501 y Fu(IEI:SA)-6 b(T)2282 1534 y FA(.)77 b(In)42 b(fact,)47 b(Hemaspaandra)42 b(et)448 1653 y(al.)25 b([HHH96b)q(])f(ha)m(v)m(e)i (sho)m(wn)e(that)h(all)e(standard)h(complexit)m(y)g(classes)g(\(in)f (particular,)448 1771 y(all)j(leaf-de\014nable)g(classes\))h (automatically)g(inherit)d(all)i(query-order)g(con)m(tainmen)m(ts)448 1890 y(that)33 b(hold)f(for)g(P)g(mac)m(hines.)47 b(Th)m(us,)32 b(for)h(example,)g(since)f(P)2592 1857 y Fu(NP)o(:NP)2813 1833 y Fs(NP)2935 1890 y FA(=)d(P)3097 1857 y Fu(NP)3198 1833 y Fs(NP)3287 1857 y Fu(:NP)3412 1890 y FA(,)448 2008 y(w)m(e)i(ma)m(y)g(conclude)f(immediately)f(that)i(PP)1991 1973 y Fu(NP:NP)2213 1949 y Fs(NP)2332 2008 y FA(=)24 b(PP)2551 1973 y Fu(NP)2652 1949 y Fs(NP)2742 1973 y Fu(:NP)2866 2008 y FA(.)448 2261 y Fm(2.2)113 b(Pro)s(of)37 b(b)m(y)g(Example)448 2440 y FA(Our)d(goal)i(here)e(is)g(just)h(to)h (giv)m(e)f(the)g(general)g(\015a)m(v)m(or)h(of)f(a)h(pro)s(of)e (related)h(to)h(query)448 2558 y(order)42 b(in)f(the)h(p)s(olynomial)e (hierarc)m(h)m(y)-8 b(.)76 b(W)-8 b(e)44 b(will)39 b(pro)m(v)m(e)k (part)f(of)h(an)f(instance)g(of)448 2677 y(Theorem)30 b(2.2.)42 b(That)31 b(is,)e(w)m(e)i(will)d(partially)g(pro)m(v)m(e:) 1594 2889 y(P)1656 2851 y Fu(\006)1707 2819 y Fs(p)1707 2874 y(2)1745 2851 y Fu(:NP)1895 2889 y Fl(\022)d FA(P)2053 2851 y Fu(NP)o(:\006)2224 2819 y Fs(p)2224 2874 y(2)2266 2889 y Fo(:)448 3101 y FA(In)45 b(particular,)j(w)m(e)e(will)c(pro)m(v) m(e)47 b(that)f Fo(X)57 b Fl(\022)50 b FA(P)2164 3068 y Fu(NP:\006)2336 3035 y Fs(p)2336 3090 y(2)2378 3101 y FA(,)f(where)c Fo(X)53 b FA(is)44 b(the)i(class)f(of)448 3219 y(languages)30 b(that)g(are)g(in)e(P)1376 3186 y Fu(\006)1427 3154 y Fs(p)1427 3209 y(2)1464 3186 y Fu(:NP)1618 3219 y FA(via)h(a)h(P)1903 3186 y Fu(\006)1954 3154 y Fs(p)1954 3209 y(2)1992 3186 y Fu(:NP)2145 3219 y FA(mac)m(hine)g(in)e (whic)m(h)g(the)h(P)g(mac)m(hine)448 3338 y(on)i(eac)m(h)g(input)e (asks)h Fy(exactly)40 b FA(one)30 b(question)g(to)h(eac)m(h)h(of)e(its) g(oracles,)h(and)f(in)f(whic)m(h)448 3456 y(the)k(P)f(mac)m(hine)h (accepts)h(if)d(and)h(only)g(if)f Fy(exactly)36 b(one)f(of)f(its)h(two) h(queries)e(gets)h(the)448 3575 y(answer)f(\\yes.")60 b FA(That)31 b(is,)e(w)m(e)i(will)d(do)i(the)h(parit)m(y)f(case.)589 3694 y(The)41 b(pro)s(of)f(is)g(not)h(hard,)h(and)f(\014nding)d(it)j (for)f(oneself)h(will)d(help)h(one)i(gain)g(a)448 3813 y(feeling)29 b(for)h(what)g(it)f(is)g(lik)m(e)g(to)i(study)e(query)g (order.)40 b(Th)m(us,)29 b(w)m(e)i(urge)e(the)i(reader)e(to)448 3931 y(try)i(to)g(pro)m(v)m(e)g(this)e(him-)g(or)i(herself)e(b)s(efore) h(reading)f(the)i(pro)s(of)e(w)m(e)i(include)d(b)s(elo)m(w.)448 4050 y Ft(Pro)s(of:)47 b FA(Let)33 b Fo(L)d Fl(2)f FA(P)1186 4017 y Fu(\006)1237 3985 y Fs(p)1237 4040 y(2)1274 4017 y Fu(:NP)1399 4050 y FA(.)48 b(Let)34 b Fo(M)43 b FA(b)s(e)32 b(a)h(P)g(mac)m(hine,)h Fo(A)29 b Fl(2)g FA(\006)2706 4006 y Fu(p)2706 4075 y(2)2750 4050 y FA(,)k(and)g Fo(B)h Fl(2)29 b FA(NP)k(b)s(e)448 4169 y(suc)m(h)26 b(that)g Fo(M)36 b FA(accepts)27 b Fo(L)f FA(and,)g(on)g(eac)m(h)h(input,)e Fo(M)36 b FA(mak)m(es)26 b(exactly)h(one)f(query)f(to)i Fo(A)448 4288 y FA(follo)m(w)m(ed)k(b)m(y)h(exactly)g(one)g(query)f(to) h Fo(B)5 b FA(.)43 b(\(This)30 b(do)s(es)h(not)h(rule)e(out)i(the)g(p)s (ossibilit)m(y)448 4406 y(that)24 b(w)m(ere)f Fo(M)10 b FA(\()p Fo(x)p FA(\))23 b(to)h(b)s(e)e(giv)m(en)h(an)g(incorrect)f (answ)m(er)h(to)g(its)g(\014rst)f(query)g(it)g(w)m(ould)g(not)448 4525 y(ask)28 b(a)h(second)f(query)-8 b(.)39 b(Ho)m(w)m(ev)m(er,)31 b(without)c(loss)g(of)h(generalit)m(y)g(w)m(e)h(can)f(assume)f(that)448 4643 y(it)k(alw)m(a)m(ys)h(asks)f(exactly)h(one)g(query)f(to)h(eac)m(h) g(oracle,)h(regardless)d(of)i(the)f(answ)m(er)g(to)448 4762 y(the)24 b(\014rst)f(query)-8 b(.)38 b(W)-8 b(e)25 b(do)e(assume)h(this)e(b)s(oth)h(here)g(and)g(in)g(the)g(pro)s(of)g(of) h(Section)f(3.2.\))448 4880 y(W)-8 b(e)37 b(will)32 b(partially)h (describ)s(e)h(a)i(P)f(mac)m(hine)f Fo(M)2150 4847 y Fk(0)2209 4880 y FA(that)i(accepts)g Fo(L)f FA(with)f(one)i(query)448 4999 y(to)31 b(an)g(NP)f(set)h(follo)m(w)m(ed)f(b)m(y)g(one)h(query)f (to)h(a)g(\006)2138 4955 y Fu(p)2138 5024 y(2)2211 4999 y FA(set.)589 5118 y(On)24 b(input)f Fo(x)p FA(,)i Fo(M)1172 5085 y Fk(0)1220 5118 y FA(determines)e(the)i(\014rst)f(query)g(of)g Fo(M)10 b FA(\()p Fo(x)p FA(\))25 b(and)f(the)h(t)m(w)m(o)h(p)s(oten)m (tial)448 5237 y(second)50 b(queries)e(of)h Fo(M)10 b FA(\()p Fo(x)p FA(\).)99 b(W)-8 b(e)50 b(will)d(write)h Fo(q)k FA(for)d(the)h(\014rst)f(query)f(ask)m(ed)i(b)m(y)448 5355 y Fo(M)10 b FA(\()p Fo(x)p FA(\),)44 b Fo(q)778 5369 y Fp(Y)879 5355 y FA(for)d(the)g(second)f(query)g(that)i(w)m(ould) d(b)s(e)h(ask)m(ed)h(b)m(y)g Fo(M)10 b FA(\()p Fo(x)p FA(\))41 b(w)m(ere)g(it)f(to)448 5474 y(receiv)m(e)48 b(a)f(\\y)m(es")i(answ)m(er)d(to)i(the)f(\014rst)f(query)-8 b(,)51 b(and)46 b Fo(q)2462 5488 y Fp(N)2576 5474 y FA(for)h(the)g (second)g(query)448 5592 y(that)35 b(w)m(ould)d(b)s(e)h(ask)m(ed)h(b)m (y)g Fo(M)10 b FA(\()p Fo(x)p FA(\))34 b(w)m(ere)g(it)f(to)h(receiv)m (e)h(a)f(\\no")g(answ)m(er)g(to)g(the)g(\014rst)p eop %%Page: 6 6 6 5 bop 3 setlinewidth np 1307 929 a 1806 1247 li st 3 setlinewidth np 2304 454 a 2712 340 li st 3 setlinewidth np 2304 567 a 2712 703 li st 3 setlinewidth np 2304 1178 a 2712 1066 li st 3 setlinewidth np 2304 1292 a 2712 1428 li st 973 907 a Fg(q)26 b Ff(2)d Fg(A)28 b Fx(?)1879 544 y Fg(q)1916 556 y Fe(Y)1996 544 y Ff(2)c Fg(B)31 b Fx(?)1879 1269 y Fg(q)1916 1281 y Fe(N)2002 1269 y Ff(2)23 b Fg(B)32 b Fx(?)p 3 setlinewidth np 1307 816 a 1806 521 li st 2757 363 a(Reject)2461 1428 y Fd(no)2757 725 y Fx(Accept)2757 1088 y(Accept)2757 1450 y(Reject)1445 658 y Fd(y)n(es)1464 1133 y(no)2442 363 y(y)n(es)2438 703 y(no)2442 1088 y(y)n(es)448 1756 y FA(Figure)60 b(1:)102 b(A)61 b(p)s(ossible)d(b)s(eha)m(vior)i(of)h Fo(M)10 b FA(\()p Fo(x)p FA(\))61 b(\(applies)e(to)i(Section)g(2.2)h(and)448 1875 y(Section)31 b(3.2\).)p 3 setlinewidth np 1129 2648 a 1718 2966 li st 3 setlinewidth np 2488 2897 a 2987 2785 li st 3 setlinewidth np 2488 3011 a 2987 3147 li st 3 setlinewidth np 2488 2286 a 2987 2422 li st 3 setlinewidth np 2488 2173 a 2987 2059 li st 1792 2263 a Fg(q)26 b Ff(2)d Fg(A)c Ff(^)g Fg(q)2125 2275 y Fe(Y)2205 2263 y Ff(2)24 b Fg(B)31 b Fx(?)1792 2987 y Fg(q)26 b Ff(2)d Fg(A)c Ff(^)g Fg(q)2125 2999 y Fe(Y)2205 2987 y Ff(62)24 b Fg(B)31 b Fx(?)1358 2353 y Fd(y)n(es)p 3 setlinewidth np 1129 2535 a 1718 2240 li st 1377 2874 a(no)2668 3123 y(no)767 2625 y Fg(q)804 2637 y Fe(N)890 2625 y Ff(2)24 b Fg(B)32 b Fx(?)3032 2081 y(Reject)3032 2444 y(Accept)3032 2806 y(Accept)3032 3169 y(Reject)2649 2104 y Fd(y)n(es)2649 2829 y(y)n(es)2668 2399 y(no)448 3475 y FA(Figure)25 b(2:)39 b(Query)25 b(structure)g(of)h(the)g(P)1808 3442 y Fu(NP:\006)1980 3409 y Fs(p)1980 3464 y(2)2047 3475 y FA(mac)m(hine)f(corresp)s(onding)f(to)i(Figure)f(1.)448 3793 y(query)-8 b(.)40 b Fo(M)826 3760 y Fk(0)850 3793 y FA(\()p Fo(x)p FA(\))30 b(then)e(determines)h(for)f(whic)m(h)g(of)i (the)f(four)f(p)s(ossible)f(answ)m(ers)i(to)h(t)m(w)m(o)448 3911 y(sequen)m(tial)42 b(queries)f(\(namely)-8 b(,)45 b(\\no/no,")i(\\no/y)m(es,")g(\\y)m(es/no,")g(and)42 b(\\y)m(es/y)m(es"\))448 4030 y Fo(M)10 b FA(\()p Fo(x)p FA(\))24 b(accepts.)39 b(All)22 b(this)f(can)i(b)s(e)f(done)h(in)e(p)s (olynomial)f(time)j(without)e(querying)h(an)m(y)448 4148 y(strings.)39 b(Since)25 b Fo(M)10 b FA(\()p Fo(x)p FA(\))27 b(asks)g(t)m(w)m(o)h(queries,)e(there)h(are)f(sixteen)h(di\013eren)m(t) f(p)s(ossibilities)448 4267 y(for)37 b(Accept/Reject)k(b)s(eha)m(vior) 36 b(of)h Fo(M)10 b FA(\()p Fo(x)p FA(\).)62 b(As)38 b(an)f(example,)i(supp)s(ose)c(that)j Fo(M)10 b FA(\()p Fo(x)p FA(\))448 4385 y(accepts)35 b(if)e(and)g(only)g(if)g(the)h(answ) 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Fu(p)738 5143 y(2)820 5118 y FA(predicates.)68 b(Since)38 b(\006)1625 5074 y Fu(p)1625 5143 y(2)1708 5118 y FA(is)g(closed)h(under)f(disjoin)m(t)g(union)g(w)m(e)h(are)h (done)448 5237 y(for)g(the)f(case)i(corresp)s(onding)c(to)k(Figure)e (1.)68 b(There)39 b(are)h(of)g(course)f(\014fteen)h(other)448 5355 y(cases)35 b(to)f(consider,)g(but)f(all)f(of)i(these)g(are)g (similar)d(to)k(or)e(easier)h(than)f(the)h(case)h(w)m(e)448 5474 y(just)43 b(treated.)78 b(W)-8 b(e)44 b(men)m(tion)f(in)e(passing) h(that)h(the)g(full)e(pro)s(of)h(of)g(Theorem)h(2.2)448 5592 y(in)g([HHH97b)q(])h(is)f(more)h(elegan)m(t)i(than)d(w)m(orking)h (through)f(the)h(sixteen)g(di\013eren)m(t)p eop %%Page: 7 7 7 6 bop 448 399 a FA(sub)s(cases)21 b(and)g(splicing)e(them)j(together) h(dynamically;)g(the)f(presen)m(t)f(pro)s(of)g(fragmen)m(t)448 517 y(is)28 b(merely)f(in)m(tended)g(to)i(con)m(v)m(ey)h(some)e(of)g (the)h(\015a)m(v)m(or)g(of)f(ho)m(w)g(one)h(can)f(pro)m(v)m(e)h(things) 448 636 y(ab)s(out)h(query)g(order.)p 3305 648 42 88 v 448 928 a FC(3)135 b(Query)45 b(Order)g(in)g(the)g(Bo)t(olean)g (Hierarc)l(h)l(y)448 1140 y Fm(3.1)113 b(Results)448 1318 y FA(The)45 b(b)s(o)s(olean)e(hierarc)m(h)m(y)i([CGH)1644 1285 y Fu(+)1703 1318 y FA(88)q(,CGH)2024 1285 y Fu(+)2084 1318 y FA(89])g(w)m(as)h(in)m(tro)s(duced)d(in)g(the)i(1980s,)448 1436 y(and)30 b(captures)g(and)f(classi\014es)g(those)h(languages)h (that)f(can)g(b)s(e)g(computed)g(via)f(\014nite)448 1555 y(hardw)m(are)d(op)s(erating)g(o)m(v)m(er)i(NP)f(predicates)f(\(equiv) -5 b(alen)m(tly)d(,)27 b(that)g(can)f(b)s(e)g(computed)448 1673 y(via)k(b)s(ounded)f(access)i(to)g(SA)-8 b(T\).)448 1876 y Ft(De\014nition)36 b(3.1)157 b FA(1.)46 b(Let)31 b Fl(C)25 b(\011)20 b(D)28 b FA(=)d Fl(f)p Fo(L)1971 1890 y Fu(1)2031 1876 y Fl(\000)20 b Fo(L)2184 1890 y Fu(2)2238 1876 y Fl(j)h Fo(L)2346 1890 y Fu(1)2410 1876 y Fl(2)k(C)36 b(^)24 b Fo(L)2727 1890 y Fu(2)2792 1876 y Fl(2)h(D)s(g)p FA(.)559 2067 y(2.)47 b([CGH)906 2034 y Fu(+)965 2067 y FA(88)q(])714 2258 y(\(a\))g(BH)1008 2272 y Fu(0)1073 2258 y FA(=)25 b(P.)709 2407 y(\(b\))46 b(F)-8 b(or)31 b(eac)m(h)g Fo(k)e(>)c FA(0,)31 b(BH)1648 2422 y Fp(k)1716 2407 y FA(=)25 b(NP)20 b Fl(\011)g FA(BH)2186 2422 y Fp(k)r Fk(\000)p Fu(1)2319 2407 y FA(.)719 2556 y(\(c\))47 b(BH,)31 b(the)f(b)s(o)s(olean)g(hierarc)m(h)m(y)-8 b(,)30 b(is)g(de\014ned)f(as)2519 2491 y Fj(S)2489 2634 y Fp(k)r Fk(\025)p Fu(0)2633 2556 y FA(BH)2766 2571 y Fp(k)2809 2556 y FA(.)589 2815 y(So)h(BH)848 2829 y Fu(1)913 2815 y FA(=)25 b(NP,)30 b(and)g(BH)1503 2829 y Fu(2)1573 2815 y FA(equals)f(P)m(apadimitriou)f(and)h(Y)-8 b(annak)j(akis's)31 b([PY84)q(])448 2934 y(class)22 b(DP)-8 b(,)22 b(namely)f(the)h(class)f (of)h(all)e(sets)i(that)g(can)g(b)s(e)f(written)g(as)h(the)g(in)m (tersection)f(of)448 3052 y(some)g(NP)g(set)g(with)f(some)h(coNP)g (set.)38 b(NP)-8 b(,)22 b(DP)-8 b(,)21 b(and)f(the)h(other)g(lev)m(els) f(of)h(the)g(b)s(o)s(olean)448 3171 y(hierarc)m(h)m(y)44 b(con)m(tain)g(a)g(large)f(v)-5 b(ariet)m(y)44 b(of)g(complete)g (problems)e(\(see,)48 b(e.g.,)e([GJ79)q(,)448 3289 y(CM87)q(,CGH)918 3256 y Fu(+)978 3289 y FA(88)q(,Bor95)q(]\).)589 3408 y(Hemaspaandra,)j(Hemp)s(el,)e(and)c(W)-8 b(ec)m(hsung)45 b([HHW95)r(])g(raised)e(the)i(topic)f(of)448 3526 y(query)55 b(order)f(b)m(y)h(asking)f(whether)g(P)1879 3493 y Fu(BH)1982 3503 y Fi(i)2008 3493 y Fu(:BH)2130 3503 y Fi(j)2233 3526 y FA(=)66 b(P)2432 3493 y Fu(BH)2534 3503 y Fi(j)2567 3493 y Fu(:BH)2689 3503 y Fi(i)2720 3526 y FA(.)114 b(They)54 b(resolv)m(ed)448 3645 y(this)42 b(question)g(as)h(follo)m(ws.)77 b(They)42 b(noted)h(that)h(equalit)m(y)e(trivially)e(holds)h(if)h Fo(i)k FA(=)448 3763 y Fo(j)60 b Fl(_)53 b Fo(i)34 b FA(=)f(0)54 b Fl(_)f Fo(j)39 b FA(=)33 b(0.)56 b(They)34 b(pro)m(v)m(ed)i(that)g(equalit)m(y)e(\(not)i(so)g(trivially\))c(holds) i(if)448 3882 y Fo(i)d FA(is)e(ev)m(en)62 b Fl(^)55 b Fo(j)31 b FA(=)25 b Fo(i)18 b FA(+)f(1,)31 b(or)e(if)f Fo(j)36 b FA(is)29 b(ev)m(en)62 b Fl(^)55 b Fo(i)25 b FA(=)g Fo(j)f FA(+)17 b(1.)41 b(They)29 b(pro)m(v)m(ed)g(that)h(for)f (all)448 4001 y(other)i(cases)g(inequalit)m(y)e(holds)g(unless)g(the)h (p)s(olynomial)e(hierarc)m(h)m(y)i(collapses.)448 4204 y Ft(Theorem)k(3.2)133 b FA([HHW95)r(])i(F)-8 b(or)44 b(eac)m(h)h Fo(i)f FA(and)f Fo(j)5 b FA(,)48 b(the)43 b(relationship)e(b)s(et)m(w)m(een)448 4322 y(P)510 4289 y Fu(BH)613 4299 y Fi(i)639 4289 y Fu(:BH)762 4299 y Fi(j)829 4322 y FA(and)30 b(P)1068 4289 y Fu(BH)1170 4299 y Fi(j)1203 4289 y Fu(:BH)1325 4299 y Fi(i)1386 4322 y FA(is)f(as)i(sho)m(wn)f(in)f(Figure)h(3.)589 4525 y(The)45 b(most)g(strikingly)d(o)s(dd)i(feature)i(of)f(this)f(theorem)h (is)f(that)h(the)h(just-o\013-)448 4644 y(diagonal)26 b(en)m(tries)h(alternate)g(b)s(et)m(w)m(een)g(equalit)m(y)f(and)g (inequalit)m(y)f(\(e.g.,)30 b(P)3043 4611 y Fu(BH)3145 4620 y Fs(2)3180 4611 y Fu(:BH)3302 4620 y Fs(3)3366 4644 y FA(=)448 4762 y(P)510 4729 y Fu(BH)613 4739 y Fs(3)648 4729 y Fu(:BH)770 4738 y Fs(2)809 4762 y FA(,)72 b(y)m(et)65 b(unless)d(the)i(p)s(olynomial)d(hierarc)m(h)m(y)j (collapses)f(P)2987 4729 y Fu(BH)3090 4739 y Fs(3)3124 4729 y Fu(:BH)3247 4738 y Fs(4)3366 4762 y Fl(6)p FA(=)448 4881 y(P)510 4848 y Fu(BH)613 4857 y Fs(4)648 4848 y Fu(:BH)770 4857 y Fs(3)809 4881 y FA(\).)111 b(The)53 b(curious)f(asymmetry)i(b)s(ecomes)g(a)g(bit)e(less)h(opaque)h(if)f (one)448 5000 y(lo)s(oks)42 b(at)i(what)e(is)g(actually)g(underpinning) c(Theorem)k(3.2.)78 b(The)42 b(k)m(ey)i(result)d(on)448 5118 y(whic)m(h)g(Theorem)g(3.2)h(rests)g(is)f(Lemma)g(3.3)i(b)s(elo)m (w,)h(whic)m(h)c(states)j(that)f(ordered)448 5237 y(access)h(to)f(the)f (b)s(o)s(olean)f(hierarc)m(h)m(y's)h(lev)m(els)g(can)g(without)g(loss)f (of)h(generalit)m(y)h(b)s(e)448 5355 y(restructured)32 b(as)h(parallel)d(access)k(to)f(NP.)47 b(As)33 b(is)e(standard,)i(let)f (R)2830 5311 y Fu(p)2830 5383 y Fp(`)p Fu(-tt)2941 5355 y FA(\(NP)q(\))g(denote)448 5474 y Fl(f)p Fo(L)15 b Fl(j)23 b FA(\()p Fl(9)p Fo(A)30 b Fl(2)g FA(NP\)[)p Fo(L)h Fl(\024)1247 5430 y Fu(p)1247 5502 y Fp(`)p Fu(-tt)1388 5474 y Fo(A)p FA(])p Fl(g)p FA(,)k(where)e Fl(\024)1923 5430 y Fu(p)1923 5502 y Fp(`)p Fu(-tt)2068 5474 y FA(is)f(the)i(standard)e(reduction)g (allo)m(wing)448 5592 y Fo(`)f FA(parallel)d(queries)h([LLS75)q(].)p eop %%Page: 8 8 8 7 bop 683 1136 a Fo(i)p 829 525 4 119 v 890 490 a FA(:)p 974 525 V 991 525 V 127 w(etc.)p 1231 525 V 101 w(etc.)p 1472 525 V 101 w(etc.)p 1713 525 V 101 w(etc.)p 1954 525 V 101 w(etc.)p 2196 525 V 101 w(etc.)p 2437 525 V 101 w(etc.)p 2678 525 V 101 w(etc.)p 2919 525 V 101 w(etc.)p 830 529 2332 4 v 828 647 4 119 v 880 612 a(7)p 974 647 V 991 647 V 135 w FC(=)p 1231 647 V 165 w Fi(<)p 1472 647 V 193 w(<)p 1713 647 V 193 w(<)p 1954 647 V 193 w(<)p 2196 647 V 193 w(<)p 2437 647 V 164 w FC(=)p 2678 647 V 136 w(=)p 2919 647 V 118 w FA(etc.)p 830 651 2332 4 v 828 769 4 119 v 880 734 a(6)p 974 769 V 991 769 V 135 w FC(=)p 1231 769 V 165 w Fi(<)p 1472 769 V 193 w(<)p 1713 769 V 193 w(<)p 1954 769 V 193 w(<)p 2196 769 V 193 w(<)p 2437 769 V 164 w FC(=)p 2678 769 V 136 w(=)p 2919 769 V 118 w FA(etc.)p 830 772 2332 4 v 828 891 4 119 v 880 855 a(5)p 974 891 V 991 891 V 135 w FC(=)p 1231 891 V 165 w Fi(<)p 1472 891 V 193 w(<)p 1713 891 V 193 w(<)p 1954 891 V 164 w FC(=)p 2196 891 V 136 w(=)p 2437 891 V 165 w Fi(>)p 2678 891 V 193 w(>)p 2919 891 V 146 w FA(etc.)p 830 894 2332 4 v 828 1013 4 119 v 880 977 a(4)p 974 1013 V 991 1013 V 135 w FC(=)p 1231 1013 V 165 w Fi(<)p 1472 1013 V 193 w(<)p 1713 1013 V 193 w(<)p 1954 1013 V 164 w FC(=)p 2196 1013 V 136 w(=)p 2437 1013 V 165 w Fi(>)p 2678 1013 V 193 w(>)p 2919 1013 V 146 w FA(etc.)p 830 1016 2332 4 v 828 1135 4 119 v 880 1099 a(3)p 974 1135 V 991 1135 V 135 w FC(=)p 1231 1135 V 165 w Fi(<)p 1472 1135 V 164 w FC(=)p 1713 1135 V 136 w(=)p 1954 1135 V 165 w Fi(>)p 2196 1135 V 193 w(>)p 2437 1135 V 193 w(>)p 2678 1135 V 193 w(>)p 2919 1135 V 146 w FA(etc.)p 830 1138 2332 4 v 828 1257 4 119 v 880 1221 a(2)p 974 1257 V 991 1257 V 135 w FC(=)p 1231 1257 V 165 w Fi(<)p 1472 1257 V 164 w FC(=)p 1713 1257 V 136 w(=)p 1954 1257 V 165 w Fi(>)p 2196 1257 V 193 w(>)p 2437 1257 V 193 w(>)p 2678 1257 V 193 w(>)p 2919 1257 V 146 w FA(etc.)p 830 1260 2332 4 v 828 1378 4 119 v 880 1343 a(1)p 974 1378 V 991 1378 V 135 w FC(=)p 1231 1378 V 136 w(=)p 1472 1378 V 165 w Fi(>)p 1713 1378 V 193 w(>)p 1954 1378 V 193 w(>)p 2196 1378 V 193 w(>)p 2437 1378 V 193 w(>)p 2678 1378 V 193 w(>)p 2919 1378 V 146 w FA(etc.)p 830 1382 2332 4 v 828 1500 4 119 v 880 1465 a(0)p 974 1500 V 991 1500 V 135 w FC(=)p 1231 1500 V 136 w(=)p 1472 1500 V 136 w(=)p 1713 1500 V 136 w(=)p 1954 1500 V 136 w(=)p 2196 1500 V 136 w(=)p 2437 1500 V 136 w(=)p 2678 1500 V 136 w(=)p 2919 1500 V 118 w FA(etc.)p 830 1504 2332 4 v 830 1520 V 828 1639 4 119 v 974 1639 V 991 1639 V 1090 1603 a(0)p 1231 1639 V 196 w(1)p 1472 1639 V 196 w(2)p 1713 1639 V 196 w(3)p 1954 1639 V 196 w(4)p 2196 1639 V 196 w(5)p 2437 1639 V 196 w(6)p 2678 1639 V 196 w(7)p 2919 1639 V 194 w Fl(\001\001)p 830 1642 2332 4 v 1990 1802 a Fo(j)838 2080 y FA(Figure)30 b(3:)41 b(Relationship)28 b(b)s(et)m(w)m(een)j(P)2176 2047 y Fu(BH)2279 2058 y Fi(i)2305 2047 y Fu(:BH)2428 2057 y Fi(j)2495 2080 y FA(and)f(P)2734 2047 y Fu(BH)2836 2058 y Fi(j)2869 2047 y Fu(:BH)2991 2057 y Fi(i)3022 2080 y FA(.)498 2221 y Fc(Key:)104 b FC(=)51 b Fr(in)25 b(ro)n(w)i Fb(i)f Fr(and)f(column)g Fb(j)104 b Fr(means)25 b(P)2042 2189 y Fs(BH)2129 2200 y Fa(i)2156 2189 y Fs(:BH)2260 2200 y Fa(j)2317 2221 y Fr(=)c(P)2450 2189 y Fs(BH)2537 2200 y Fa(j)2568 2189 y Fs(:BH)2673 2200 y Fa(i)2703 2221 y Fr(.)785 2317 y Fi(<)52 b Fr(in)25 b(ro)n(w)h Fb(i)g Fr(and)g(column)e Fb(j)133 b Fr(means)42 b(P)2059 2285 y Fs(BH)2146 2295 y Fa(i)2172 2285 y Fs(:BH)2277 2296 y Fa(j)2354 2317 y Fr(is)h(a)g(strict)g(subset)f(of)h(P)3155 2285 y Fs(BH)3243 2295 y Fa(j)3274 2285 y Fs(:BH)3378 2296 y Fa(i)1755 2413 y Fr(unless)26 b(the)f(p)r(olynomial)h(hierarc)n (h)n(y)f(collapses.)785 2508 y Fi(>)52 b Fr(in)25 b(ro)n(w)h Fb(i)g Fr(and)g(column)e Fb(j)133 b Fr(means)32 b(P)2049 2477 y Fs(BH)2137 2487 y Fa(i)2163 2477 y Fs(:BH)2268 2488 y Fa(j)2336 2508 y Fr(is)h(a)g(strict)h(sup)r(erset)f(of)g(P)3155 2477 y Fs(BH)3243 2487 y Fa(j)3274 2477 y Fs(:BH)3378 2488 y Fa(i)1755 2604 y Fr(unless)26 b(the)f(p)r(olynomial)h(hierarc)n (h)n(y)f(collapses.)448 2985 y Ft(Lemma)33 b(3.3)46 b FA(F)-8 b(or)31 b(eac)m(h)h Fo(i;)15 b(j)31 b Fl(\025)25 b FA(1,)507 3261 y(P)569 3224 y Fu(BH)671 3234 y Fi(i)698 3224 y Fu(:BH)820 3234 y Fi(j)882 3261 y FA(=)978 3117 y Fj(\()1086 3190 y FA(R)1153 3146 y Fu(p)1153 3222 y(\()p Fp(i)p Fu(+2)p Fp(j)t Fk(\000)p Fu(1\)-tt)1527 3190 y FA(\(NP)q(\))182 b(if)29 b Fo(i)i FA(is)e(ev)m(en)i(and)f Fo(j)36 b FA(is)30 b(o)s(dd)1086 3321 y(R)1153 3277 y Fu(p)1153 3353 y(\()p Fp(i)p Fu(+2)p Fp(j)t Fu(\)-tt)1437 3321 y FA(\(NP)q(\))272 b(otherwise.)589 3539 y(This)39 b(lemma)h(is)g(the)h(source)g(of)f(the)h(asymmetry)g(b)s(et)m(w)m(een,) j(for)c(example,)j(2-)448 3657 y(v)m(ersus-3)34 b(and)f(3-v)m(ersus-4.) 51 b(Of)33 b(course,)h(this)e(in)g(some)i(w)m(a)m(y)h(b)s(egs)e(the)g (question,)h(as)448 3776 y(the)c(reader)g(ma)m(y)g(w)m(ell)e(ask)i(ab)s (out)g(the)f(asymmetry)h(b)s(et)m(w)m(een)g(the)g(\014rst)f(and)g (second)448 3895 y(cases)e(of)g(Lemma)f(3.3.)40 b(Brie\015y)26 b(and)f(informally)e(put,)k(when)e Fo(i)h FA(is)g(ev)m(en)g(and)g Fo(j)32 b FA(is)25 b(o)s(dd,)448 4013 y(a)32 b(certain)f(underlying)d (graph)i(mo)s(deling)f(the)j(computation)e(of)i(P)2787 3980 y Fu(BH)2889 3990 y Fi(i)2916 3980 y Fu(:BH)3038 3990 y Fi(j)3106 4013 y FA(b)s(ecomes)448 4132 y(non-bipartite,)45 b(and)e(b)m(y)g(doing)f(so)h(allo)m(ws)f(one)i(to)g(guaran)m(tee)g(a)g (sa)m(vings)f(of)g(one)448 4250 y(parallel)29 b(query)h(to)h(NP)f (\(see)h([HHW95)r(])g(for)f(full)e(details\).)448 4509 y Fm(3.2)113 b(Pro)s(of)37 b(b)m(y)g(Example)448 4689 y FA(In)28 b(this)f(subsection)g(w)m(e)i(will)c(giv)m(e)k(a)g(partial)e (pro)s(of)g(for)h(an)g(instance)g(of)g(Lemma)h(3.3,)448 4807 y(namely)-8 b(,)31 b(w)m(e)f(will)e(giv)m(e)j(a)g(partial)e(pro)s (of)h(for)1533 5022 y(P)1595 4985 y Fu(DP)o(:NP)1846 5022 y Fl(\022)25 b FA(R)2009 4978 y Fu(p)2009 5047 y(3-tt)2127 5022 y FA(\(NP\))p Fo(:)448 5237 y FA(In)39 b(particular,)i(w)m(e)f (will)d(sho)m(w)j(that)g Fo(X)48 b Fl(\022)41 b FA(R)2095 5192 y Fu(p)2095 5261 y(3-tt)2213 5237 y FA(\(NP\),)h(where)e Fo(X)47 b FA(is)39 b(the)g(class)h(of)448 5355 y(languages)28 b(that)g(are)g(in)e(P)1368 5322 y Fu(DP)o(:NP)1621 5355 y FA(via)h(a)h(P)1902 5322 y Fu(DP:NP)2156 5355 y FA(mac)m(hine)f(in)f (whic)m(h)g(the)h(P)g(mac)m(hine)448 5474 y(on)k(eac)m(h)g(input)e (asks)h Fy(exactly)40 b FA(one)30 b(question)g(to)h(eac)m(h)h(of)e(its) g(oracles,)h(and)f(in)f(whic)m(h)448 5592 y(the)k(P)f(mac)m(hine)h (accepts)h(if)d(and)h(only)g(if)f Fy(exactly)36 b(one)f(of)f(its)h(two) h(queries)e(gets)h(the)p eop %%Page: 9 9 9 8 bop 448 399 a Fy(answer)38 b(\\yes.")54 b FA(That)35 b(is,)g(as)h(w)m(as)f(the)g(case)h(also)f(in)f(Section)h(2.2,)i(w)m(e)f (will)c(do)j(the)448 517 y(parit)m(y)30 b(case.)589 649 y(W)-8 b(e)67 b(w)m(arn)d(the)h(reader)g(that)h(the)f(pro)s(of)f (approac)m(h)i(tak)m(en)g(here)f(is)f(not)448 767 y(suited)41 b(to)i(b)s(e)e(elegan)m(tly)i(generalized)f(to)g(ev)m(en)m(tually)g (yield)f(Lemma)h(3.3.)77 b(F)-8 b(or)43 b(a)448 886 y(complete)j(and)f (uni\014ed)e(pro)s(of)i(of)h(P)1786 853 y Fu(DP)o(:NP)2062 886 y Fl(\022)k FA(R)2250 842 y Fu(p)2250 911 y(3-tt)2368 886 y FA(\(NP)q(\),)f(w)m(e)d(refer)f(the)h(reader)448 1004 y(to)h([HHW95)r(].)89 b(Ho)m(w)m(ev)m(er,)52 b(the)46 b(pro)s(of)g(giv)m(en)g(b)s(elo)m(w)f(pro)m(vides)h(a)g(go)s(o)s(d)g (starting)448 1123 y(p)s(oin)m(t)25 b(for)g(understanding)e(ho)m(w)j (these)g(pro)s(ofs)f(w)m(ork)h(in)e(the)i(con)m(text)h(of)f(the)g(b)s (o)s(olean)448 1242 y(hierarc)m(h)m(y)-8 b(.)38 b(Note)22 b(that)f(the)f(pro)s(of)g(pic)m(ks)g(one)h(of)g(the)f(in)m(teresting)g (cases)i(of)e(Lemma)h(3.3)448 1360 y(\(\\)p Fo(i)33 b FA(ev)m(en)g(and)e Fo(j)38 b FA(o)s(dd"\).)45 b(As)32 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(and)g(generalized)g(settings.)448 774 y Ft(T)-9 b(ranslating)35 b(Equalities)g(Do)m(wn)m(w)m(ards)448 954 y FA(Supp)s(ose)c(t)m(w)m(o)i (questions)f(to)h(\006)1559 910 y Fu(p)1559 983 y Fp(k)1634 954 y FA(yield)e(no)h(new)g(languages)g(b)s(ey)m(ond)g(those)g(already) 448 1073 y(solv)-5 b(able)30 b(via)g(one)g(query)g(to)h(\006)1535 1029 y Fu(p)1535 1101 y Fp(k)1578 1073 y FA(.)41 b(What)31 b(follo)m(ws?)589 1193 y(Un)m(til)23 b(v)m(ery)i(recen)m(tly)-8 b(,)26 b(all)d(one)h(could)f(conclude)g(from)h(this)f(assumption)f(w)m (as)i(that)448 1312 y(the)42 b(p)s(olynomial)c(hierarc)m(h)m(y)j (collapses)g(to)h(a)g(lev)m(el)f(sligh)m(tly)e(ab)s(o)m(v)m(e)k(one)f (query)e(to)448 1431 y(\006)514 1386 y Fu(p)514 1459 y Fp(k)r Fu(+1)678 1431 y FA(\(note)32 b(the)f(\\+1")h(here\))g ([Kad88,W)-8 b(ag87)s(,W)g(ag89)s(,CK96)q(,BCO93)q(].)43 b(Ho)m(w)m(ev)m(er,)448 1549 y(w)m(ork)50 b(gro)m(wing)g(directly)e (out)i(of)g(the)g(study)f(of)h(query-order)f(classes|namely)-8 b(,)448 1668 y(out)45 b(of)g(the)g(goal)h(of)f(sho)m(wing)f(that)h (di\013eren)m(t)f(ordered)h(access)h(to)f(lev)m(els)g(of)g(the)448 1786 y(p)s(olynomial)32 b(hierarc)m(h)m(y)i(yields)e(di\013eren)m(t)i (language)g(classes)h(\(see)g(Theorem)f(2.4\)|)448 1905 y(led)26 b(to)h(a)g(collapse)f(a)h(full)d(lev)m(el)i(lo)m(w)m(er)h(in)e (the)i(p)s(olynomial)c(hierarc)m(h)m(y)-8 b(.)40 b(In)25 b(particular,)448 2023 y(the)h(one-t)m(w)m(o)h(punc)m(h)d(of)h([HHH96a) r(,HHH)q(]/[BF96)s(])g(yielded)e(the)j(follo)m(wing)d(theorem.)448 2243 y Ft(Theorem)34 b(4.1)46 b FA(Let)31 b Fo(k)e(>)c FA(1.)41 b(If)30 b(P)1670 2208 y Fu(\006)1721 2175 y Fs(p)1721 2233 y Fi(k)1759 2208 y Fu([1])1862 2243 y FA(=)25 b(P)2020 2208 y Fu(\006)2071 2175 y Fs(p)2071 2233 y Fi(k)2109 2208 y Fu([2])2188 2243 y FA(,)31 b(then)f(PH)25 b(=)g(\006)2768 2198 y Fu(p)2768 2271 y Fp(k)2811 2243 y FA(.)448 2460 y(In)f(other)g(w)m(ords,)h(if)e(P)1207 2425 y Fu(\006)1258 2393 y Fs(p)1258 2450 y Fi(k)1296 2425 y Fu([1])1400 2460 y FA(=)i(P)1558 2425 y Fu(\006)1609 2393 y Fs(p)1609 2450 y Fi(k)1647 2425 y Fu([2])1726 2460 y FA(,)g Fo(k)k(>)c FA(1,)h(the)e(p)s(olynomial)d(hierarc)m(h)m(y) j(crashes)g(to)448 2579 y(a)29 b(class)g(that)g(\(b)s(efore)g(the)g (crash\))f(w)m(as)h(seemingly)f(ev)m(en)h(lo)m(w)m(er)g(than)g(that)g (at)g(whic)m(h)448 2697 y(the)i(h)m(yp)s(othesis's)e(equalit)m(y)h (stands.)589 2818 y(It)25 b(has)g(b)s(een)f(sho)m(wn)g([HHH)q(,HHH97d)q (])h(that)g(the)g(ab)s(o)m(v)m(e)h(theorem)f(in)e(fact)j(has)e(an)448 2936 y(analog)e(for)g(the)f Fo(j)27 b FA(v)m(ersus)22 b Fo(j)8 b FA(+)s(1)21 b(queries)f(case.)39 b(In)21 b(particular,)h(w)m (e)g(ha)m(v)m(e)h(the)e(follo)m(wing,)448 3055 y(whic)m(h)28 b(lik)m(e)f(Theorem)i(4.1)g(w)m(as)g(established)e(in)h(the)g (literature)g(via)g(pro)m(ving)g(a)h(more)448 3173 y(general)44 b(theorem)g(ab)s(out)f(query-order)g(classes,)k(and)c(then)g(deriving)e (the)j(stated)448 3292 y(result)30 b(as)g(a)h(corollary)f(to)h(the)f (more)h(general)f(theorem)h([HHH97d)q(].)448 3511 y Ft(Theorem)j(4.2)46 b FA(Let)54 b Fo(k)67 b(>)c FA(1)54 b(and)e Fo(m)64 b Fl(\025)f FA(1.)110 b(If)53 b(P)2437 3476 y Fu(\006)2488 3444 y Fs(p)2488 3501 y Fi(k)2526 3476 y Fu([)p Fp(m)p Fu(])2695 3511 y FA(=)63 b(P)2891 3476 y Fu(\006)2942 3444 y Fs(p)2942 3501 y Fi(k)2980 3476 y Fu([)p Fp(m)p Fu(+1])3176 3511 y FA(,)c(then)448 3630 y(DIFF)669 3644 y Fp(m)736 3630 y FA(\(\006)837 3585 y Fu(p)837 3658 y Fp(k)880 3630 y FA(\))26 b(=)f(coDIFF)1344 3644 y Fp(m)1410 3630 y FA(\(\006)1511 3585 y Fu(p)1511 3658 y Fp(k)1555 3630 y FA(\).)448 3847 y(Again,)g(this)e(sa)m(ys)h(that,)h(under)d(the) i(stated)g(assumption,)g(there)f(is)g(a)h(collapse)f(within)448 3966 y(the)31 b(b)s(o)s(olean)e(hierarc)m(h)m(y)h(to)h(a)g(lev)m(el)f (that,)h Fy(a)i(priori)p FA(,)f(w)m(as)f(just)f Fy(b)-5 b(elow)41 b FA(P)2982 3931 y Fu(\006)3033 3898 y Fs(p)3033 3956 y Fi(k)3071 3931 y Fu([)p Fp(m)p Fu(])3177 3966 y FA(.)589 4086 y(In)51 b(a)g(n)m(utshell,)k(in)50 b(this)g(setting)h (smaller)f(classes)h(collapse)g(if)f(and)h(only)f(if)448 4205 y(larger)38 b(classes)g(collapse|a)f(t)m(yp)s(e)h(of)g(b)s(eha)m (vior)f(p)s(eople)f(ha)m(v)m(e)j(b)s(een)e(stalking)g(ev)m(er)448 4323 y(since)j(in\015uen)m(tial)e(pap)s(ers)h(of)i(Bo)s(ok)g([Bo)s(o74) r(])f(and)g(Hartmanis,)j(Immerman,)f(and)448 4442 y(Sew)m(elson)50 b([Har83)r(,HIS85])h(raised)f(the)g(issue)g(of)g(when)g(classes)g (stand)g(and)g(fall)448 4560 y(together.)448 4817 y Ft(Multiple)35 b(Queries)g(and)g(Bottlenec)m(k)g(Computations)448 4998 y FA(In)42 b(this)g(surv)m(ey)-8 b(,)46 b(w)m(e)d(ha)m(v)m(e)h(fo)s (cused)e(on)g(the)h(most)g(natural)f(case:)66 b(one)43 b(query)f(to)448 5116 y(eac)m(h)32 b(of)e(the)h(t)m(w)m(o)g (information)e(sources.)40 b(A)31 b(n)m(um)m(b)s(er)e(of)h(pap)s(ers)f (building)d(on)31 b(those)448 5235 y(men)m(tioned)f(here)g(ha)m(v)m(e)i (studied)d(more)h(elab)s(orate)h(settings.)589 5355 y(In)99 b(fact,)117 b(the)100 b(initial)c(pap)s(er)i(of)h(Hemaspaandra,)117 b(Hemp)s(el,)f(and)448 5474 y(W)-8 b(ec)m(hsung)50 b([HHW95)r(])f (already)g(studied)e(the)i(case)h(of)f(general)g(tree-lik)m(e)h(access) 448 5592 y(to)f(lev)m(els)f(of)h(the)g(b)s(o)s(olean)e(hierarc)m(h)m(y) -8 b(,)53 b(and)48 b(in)f(doing)g(so)i(studied)e(the)h(case)i(of)p eop %%Page: 12 12 12 11 bop 448 399 a FA(m)m(ultiple)31 b(rounds)h(of)i(single)e (queries;)i(Beigel)f(and)g(Chang)g([BC97)q(])h(study)e(m)m(ultiple)448 517 y(rounds)49 b(of)h(m)m(ultiple)e(queries)h(to)i(the)g(p)s (olynomial)c(hierarc)m(h)m(y)-8 b(,)56 b(and)49 b(sho)m(w)h(that)448 636 y(here)39 b(the)g(order)g(do)s(es)g(not)g(matter,)j(and)d(they)g (also)g(study)f(the)h(case)h(of)f(function)448 754 y(classes;)h(W)-8 b(agner)38 b([W)-8 b(ag97)s(])36 b(studies)f(parallel)g(rounds)g(of)h (one)h(or)f(more)h(queries)e(to)448 873 y(the)46 b(p)s(olynomial)c (hierarc)m(h)m(y)i(and)h(other)g(classes)g(and)g(also)g(tigh)m(tly)f (relates)h(suc)m(h)448 991 y(classes)31 b(to)f(the)h(re\014ned)e (hierarc)m(h)m(y)h(w)m(ork)g(of)g(Seliv)-5 b(ano)m(v)30 b([Sel94,Sel95])h(\(see)g(also)f(the)448 1110 y(discussion)e(in)h(the)i (\014nal)e(paragraph)h(of)g(Section)g(2)h(of)g([HHH97c)r(]\).)589 1231 y(In)j(a)h(quite)f(di\013eren)m(t)g(direction,)g(b)s(ottlenec)m(k) h(mac)m(hines)f(are)h(a)f(mo)s(del)g(used)f(to)448 1350 y(study)41 b(whether)f(a)i(computational)f(problem)e(can)j(b)s(e)e (decomp)s(osed)h(in)m(to)g(a)h(large)448 1468 y(n)m(um)m(b)s(er)d(of)i (simple,)g(sequen)m(tial,)h(tasks,)h(eac)m(h)f(of)e(whic)m(h)f(passes)h (on)h(only)e(a)i(v)m(ery)448 1587 y(limited)j(amoun)m(t)i(of)g (information)e(to)j(the)f(next)g(task,)51 b(and)45 b(all)g(of)h(whic)m (h)e(di\013er)448 1705 y(only)g(in)g(that)i(input)d(and)h(in)g(a)h (\\task)h(n)m(um)m(b)s(er")f(input)e([CF91)q(].)85 b(A)45 b(surprising)448 1824 y(recen)m(t)d(pap)s(er)e(of)h(Hertrampf)f([Her97) r(])g(uses)h(ordered)f(access)i(in)m(v)m(olving)d(m)m(ultiple)448 1942 y(queries,)50 b(com)m(bined)c(with)f(quan)m(ti\014er-based)g(and)h (mo)s(dulo-based)f(computation,)448 2061 y(to)j(completely)e(c)m (haracterize)j(the)e(languages)g(accepted)h(b)m(y)f(certain)f(b)s (ottlenec)m(k)448 2180 y(mac)m(hine)30 b(classes|classes)h(that)g(had)f (long)f(eluded)g(crisp)g(c)m(haracterization.)448 2441 y Ft(Advice)76 b(Classes,)85 b(Self-Sp)s(ecifying)76 b(Mac)m(hines,)86 b(and)75 b(Completeness)448 2559 y(T)m(yp)s(es)448 2742 y FA(A)64 b(n)m(um)m(b)s(er)f(of)h(other)g(seemingly)f(di\013eren) m(t)g(notions)g(are)h(related)g(to)h(query)448 2860 y(order.)46 b(Hemaspaandra,)33 b(Hemp)s(el,)f(and)f(W)-8 b(ec)m(hsung)33 b([HHW97)r(])g(ha)m(v)m(e)g(studied)e(self-)448 2979 y(sp)s(ecifying)39 b(mac)m(hines|nondeterministic)e(mac)m(hines)k(that) g(dynamically)d(sp)s(ecify)448 3097 y(the)47 b(path)g(sets)g(on)g(whic) m(h)e(they)i(will)d(accept.)92 b(They)46 b(completely)g(c)m (haracterize)448 3216 y(the)k(t)m(w)m(o)g(most)f(natural)f(suc)m(h)h (classes)g(in)f(terms)h(of)g(query-order)f(classes)h(with)448 3334 y(a)54 b(\\p)s(ositiv)m(e)f(\014nal)f(query")i(restriction.)109 b(They)53 b(sho)m(w)g(that)i(the)e(classes)h(ha)m(v)m(e)448 3453 y(equiv)-5 b(alen)m(t)35 b(c)m(haracterizations)i(as)f(the)g(#P)o (-closures)g(of)f(P)h(and)f(NP)-8 b(,)36 b(resp)s(ectiv)m(ely)-8 b(,)448 3572 y(and)23 b(they)f(establish)g(a)h(query)f(order)g(result)g (mixing)f(function)g(and)i(language)g(classes:)448 3690 y(P)510 3657 y Fu(#P[1])724 3690 y FA(=)29 b(P)886 3657 y Fu(#P)o(:NP)1174 3690 y Fl(\()-15 b(\))57 b FA(P)1460 3657 y Fu(#P[1])1674 3690 y FA(=)29 b(P)1836 3657 y Fu(#P)o(:NP[)p Fk(O)r Fu(\(1\)])2286 3690 y FA(\(where)k(\\)p Fl(C)h FA(:)29 b Fl(D)s FA(")j(access)i(means)448 3809 y(one)23 b(query)g(eac)m(h)h(except)f(when)f Fl(O)s FA(\(1\))i(queries)e(are)h (explicitly)d(stated)k(for)e(the)h(queried)448 3927 y(class\).)59 b(They)36 b(also)g(sho)m(w)g(that)h(the)f(classes)h(ha)m(v)m(e)g(c)m (haracterizations)h(in)d(terms)h(of)448 4046 y(the)31 b(\\input-sp)s(eci\014c)d(advice")j(notation)f(of)h(K\177)-45 b(obler)29 b(and)h(Thierauf)e([KT94)q(].)589 4167 y(Agra)m(w)m(al,)f (Beigel,)f(and)d(Thierauf)g([ABT96)q(],)j(indep)s(enden)m(tly)21 b(of)j([HHW95)r(],)i(also)448 4286 y(study)h(input-sp)s (eci\014c-advice)f(classes.)40 b(As)28 b(noted)g(b)m(y)g(Hemaspaandra)g (et)g(al.)g(in)f(the)448 4404 y(journal)c(v)m(ersion)g(of)h([HHW95)r (],)i(this)d(can)h(b)s(e)g(seen)g(as)g(equiv)-5 b(alen)m(t)24 b(to)g(studying)f(query)448 4523 y(order)j(with)f(a)h(\\p)s(ositiv)m(e) g(\014nal)e(query")i(restriction|i.e.,)g(the)h(mac)m(hines)e(m)m(ust)h (\\do")448 4641 y(exactly)34 b(what)f(the)h(resp)s(onse)e(to)i(their)e (second)h(query)g(is.)48 b(A)33 b(detailed)f(and)h(careful)448 4760 y(discussion)g(of)i(the)h(relationship)d(b)s(et)m(w)m(een)i(the)h (t)m(w)m(o)h(pap)s(ers)d(can)h(b)s(e)g(found)f(in)g(the)448 4878 y(journal)29 b(v)m(ersion)h(of)h([HHW95)r(].)589 5000 y(Finally)-8 b(,)50 b(a)d(long)f(line)f(of)i(researc)m(h)h(has)e (ask)m(ed)i(whether)61 b Fl(\024)2813 4967 y Fu(p)2813 5022 y(m)2891 5000 y FA(-completeness)448 5118 y(and)h Fl(\024)728 5074 y Fu(p)728 5145 y(T)798 5118 y FA(-completeness)48 b(stand)g(or)f(fall)f(together)j(for)f(classes)f(that)h(p)s(oten)m (tially)448 5237 y(lac)m(k)54 b(complete)g(sets.)109 b(Gurevic)m(h)52 b([Gur83)q(])h(and)g(Am)m(b)s(os-Spies)e([Am)m(b86)r (])i(ha)m(v)m(e)448 5355 y(sho)m(wn)43 b(that,)j(for)d(all)f(classes)h Fl(C)k FA(closed)c(do)m(wn)m(w)m(ards)g(under)e(T)-8 b(uring)41 b(reductions,)448 5474 y(it)c(holds)e(robustly)g(that:)54 b Fl(C)41 b FA(has)52 b Fl(\024)1742 5441 y Fu(p)1742 5496 y(m)1820 5474 y FA(-complete)37 b(sets)g(if)f(and)g(only)g(if)f Fl(C)42 b FA(has)52 b Fl(\024)3337 5430 y Fu(p)3337 5501 y(T)3407 5474 y FA(-)448 5592 y(complete)36 b(sets.)56 b(Nonetheless,)36 b(b)m(y)g(studying)d(a)j(strong)f(nondeterministic)e (closure)p eop %%Page: 13 13 13 12 bop 448 399 a FA(of)32 b(NP)f(that,)h(it)f(turns)f(out,)i (exactly)g(equals)f(the)g(query-order)g(class)g(P)2972 366 y Fu(NP)p Fk(\\)p Fu(coNP:NP)3412 399 y FA(,)448 517 y(Hemaspaandra)d(et)f(al.)g(ha)m(v)m(e)i(recen)m(tly)e(sho)m(wn)g (that)g(on)g(some)h(reducibilit)m(y)c(closures)448 636 y(of)40 b(NP)-8 b(,)55 b Fl(\024)834 603 y Fu(p)834 658 y(m)912 636 y FA(-completeness)40 b(and)54 b Fl(\024)1770 591 y Fu(p)1770 663 y(T)1840 636 y FA(-completeness)40 b(do)g(not)g(robustly)e(stand)h(or)448 754 y(fall)29 b(together)j([HHH97c)r(].)448 1053 y FC(References)448 1260 y FA([ABT96])47 b(M.)g(Agra)m(w)m(al,)k(R.)46 b(Beigel,)k(and)c (T.)g(Thierauf.)85 b(Mo)s(dulo)45 b(information)827 1373 y(from)60 b(nonadaptiv)m(e)h(queries)f(to)h(NP.)131 b(T)-8 b(ec)m(hnical)61 b(Rep)s(ort)f(96-001,)827 1486 y(Electronic)51 b(Collo)s(quium)d(on)k(Computational)f(Complexit)m(y)-8 b(,)56 b(Jan)m(uary)827 1599 y(1996.)135 b(App)s(ears)61 b(as)h(\\Pinp)s(oin)m(ting)d(Computation)h(with)g(Mo)s(dular)827 1711 y(Queries)20 b(in)f(the)i(Bo)s(olean)g(Hierarc)m(h)m(y")h(in)d (the)i(16th)h(FSTTCS)d(conference.)448 1908 y([Am)m(b86])47 b(K.)d(Am)m(b)s(os-Spies.)81 b(A)44 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b(June)d(1997.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF