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(ab)r(out)g(ho)n(w)g(certain)f(computational)456 2178 y(resources)25 b(are)i(consumed)g(while)h(solving)e(the)i(problem.)555 2278 y(Upp)r(er)d(b)r(ounds)f(are)f(t)n(ypically)g(pro)n(v)n(ed)g(b)n (y)g(exhibiting)h(an)g(algorithm)f(and)h(lo)n(w)n(er)e(b)r(ounds)456 2378 y(are)k(pro)n(v)n(ed)g(b)n(y)i(reasoning)d(ab)r(out)j(what)f(an)n (y)g(algorithm)g(m)n(ust)g(do)h(to)f(solv)n(e)f(the)i(problem.)555 2477 y(The)36 b(fact)g(that)g(complexit)n(y)f(b)r(ounds)h(seem)f(to)g (b)r(e)i(consisten)n(t)e(across)e(man)n(y)i(di\013eren)n(t)456 2577 y(mo)r(dels)i(of)g(computation)g(is)g(a)g(bit)h(astounding)e(and)h (suggests)f(that)i(there)f(are)f(ph)n(ysical)456 2677 y(constrain)n(ts)26 b(on)h(information.)456 2834 y(1.2.)40 b Fr(Some)23 b(history)i(of)g(computation.)40 b Fp(In)22 b(1936,)f(Alan)g(T)-7 b(uring)22 b(wrote)e(a)i(pap)r(er)f(en)n(titled) 456 2934 y Fn(On)28 b(Computable)i(Numb)l(ers)25 b Fp(describing)h(a)g (h)n(yp)r(othetical)g(device)h(whic)n(h)f(b)r(ecame)g(kno)n(wn)g(as)456 3033 y(the)g(T)-7 b(uring)25 b(mac)n(hine.)36 b(This)26 b(mac)n(hine)g(op)r(erated)f(b)n(y)h(reading)e(and)i(writing)g(sym)n(b) r(ols)f(on)h(an)456 3133 y(in\014nite)g(pap)r(er)f(tap)r(e.)37 b(Its)25 b(in)n(ternal)g(state)h(w)n(as)f(a)g(\014nite)h(state)g(mac)n (hine.)35 b(This)26 b(design)f(w)n(as)g(a)456 3233 y(crucial)h(step)i (in)g(building)g(the)g(\014rst)f(programmable)e(computers.)555 3332 y(In)h(1945,)e(mathematician)h(John)f(v)n(on)h(Neumann)g(sho)n(w)n (ed)f(that)i(a)f(computer)g(could)g(ha)n(v)n(e)456 3432 y(a)33 b(simple)h(\014xed)g(structure)f(common)g(to)h(an)n(y)f (program.)53 b(The)34 b(program)e(w)n(as)h(stored)g(as)g(a)456 3532 y(sequence)27 b(of)g(instructions)h(on)f(the)h('tap)r(e'.)37 b(This)28 b(b)r(ecame)g(kno)n(wn)f(as)g(the)h('stored)f(program)456 3631 y(mo)r(del'.)67 b(Another)37 b(imp)r(ortan)n(t)h(feature)f(of)h(v) n(on)f(Neumann's)g(arc)n(hitecture)g(w)n(as)f(random)456 3731 y(access)31 b(memory)h(whic)n(h)h(allo)n(w)n(ed)e(constan)n(t)h (time)i(access)d(to)i(an)n(y)f(cell)h(on)f(the)i(tap)r(e.)52 b(Tw)n(o)456 3830 y(y)n(ear)26 b(later,)h(this)h(design)f(had)g(b)r (een)h(ph)n(ysically)f(realized.)555 3930 y(In)33 b(the)g(1930s,)e (Alonzo)h(Ch)n(urc)n(h)g(and)g(Stephen)i(Kleene)e(form)n(ulated)f(the)i (Lam)n(b)r(da)f(cal-)456 4030 y(culus)39 b(initially)h(as)f(a)h (foundation)f(for)h(mathematics)f(with)h(an)g(e\013ectiv)n(e)f('pro)r (cedure')g(or)456 4129 y('algorithm'.)g(This)29 b(calculus)f(dealt)h (with)g(the)g(manipulation)g(of)f(v)-5 b(ariables,)28 b(functional)h(ab-)456 4229 y(straction)19 b(and)h(function)h (application.)34 b(This)20 b(language)f(b)r(ecame)h(the)g(foundation)g (of)h(mo)r(dern)456 4329 y(programming)j(languages)h(and)i(w)n(as)f (sho)n(wn)g(to)h(b)r(e)g(equiv)-5 b(alen)n(t)26 b(to)h(T)-7 b(uring)27 b(mac)n(hines.)36 b(This)456 4428 y(\014nal)27 b(p)r(oin)n(t)h(is)f(kno)n(wn)g(as)g(the)h(Ch)n(urc)n(h-T)-7 b(uring)26 b(thesis.)456 4586 y(1.3.)40 b Fr(Classi\014cation)i(of)g (complexit)m(y)-8 b(.)40 b Fp(The)d(initial)g(fora)n(ys)e(in)n(to)h (computational)g(com-)456 4686 y(plexit)n(y)27 b(w)n(ere)g(concerned)f (with)i(T)-7 b(uring)28 b(decidabilit)n(y)-7 b(.)456 4837 y Fr(De\014nition)45 b(1.1.)i Fn(Given)42 b(a)g(\014nite)e(alphab) l(et)j Fp(\006)e Fn(and)h(language)g Fm(L)i(\032)f Fp(\006)2904 4807 y Fl(\003)2943 4837 y Fn(,)h(we)e(say)f(that)456 4937 y(a)f(pr)l(oblem)h(is)g Fr(T)-8 b(uring-decidable)39 b Fn(if)i(ther)l(e)f(exists)g(a)g(T)-6 b(uring)40 b(machine)i Fk(T)12 b Fn(,)42 b(which)f(c)l(an)456 5037 y(determine)30 b(if)h(any)f(wor)l(d)g Fk(w)c Fm(2)d Fp(\006)1508 5006 y Fl(\003)1576 5037 y Fn(is)30 b(in)g Fm(L)g Fn(in)g(\014nite)f(time.)p 456 5122 499 4 v 555 5216 a Fj(Date)5 b Fq(:)23 b(Decem)n(b)r(er)g (5th,)h(2001.)1933 5315 y Fi(1)p eop %%Page: 2 2
2 1 bop 456 251 a Fi(2)295 b(ANDREW)22 b(LADD)h(PHYSICAL)f(COMPUT)-5 b(A)g(TION)22 b(GR)n(OUP)45 b(RICE)23 b(UNIVERSITY)555 450 y Fp(The)j(di\016cult)n(y)g(is)g(that)g(a)f(program)f(whic)n(h)i (nev)n(er)e(halts)i(can)f(easily)g(b)r(e)h(form)n(ulated.)36 b(F)-7 b(or)456 550 y(example)27 b(in)h Fh(C)p Fp(,)456 667 y Fh(void)42 b(never_halts\(voi)o(d\))456 766 y({)543 866 y(while\(1\))e({)j(do_nothing\(\);)38 b(})456 966 y(})555 1083 y Fp(will)33 b(nev)n(er)e(halt.)51 b(It)32 b(turns)g(out)g(that)h(man)n(y)e(computational)h(problems)f(are)g(suc)n (h)h(that)456 1182 y(no)j(decidable)h(program)e(exists)i(to)g(solv)n(e) f(them.)63 b(An)37 b(example)f(of)g(this)g(is)g(the)h(infamous)456 1282 y Fr(Halting)27 b(Problem)p Fp(.)34 b(This)24 b(fact)h(is)f(a)g (sp)r(ecial)g(case)f(of)h(G\177)-42 b(odel's)24 b(Incompleteness)g (Theorem.)555 1382 y(In)29 b(computer)f(science,)h(a)f(commonly)g(held) g(dogma)g(is)g(that)h(T)-7 b(uring)28 b(computabilit)n(y)g(is)h(a)456 1481 y(ph)n(ysical)d(phenomenon.)37 b(De\014nitiv)n(e)28 b(evidence)f(to)h(the)g(con)n(trary)d(has)i(not)h(b)r(een)g(pro)r (duced.)456 1631 y(1.4.)40 b Fr(Asymptotic)34 b(time)e(and)j(space.)42 b Fp(A)30 b(more)f(re\014ned)h(notion)g(of)g(complexit)n(y)f(can)h(b)r (e)456 1730 y(obtained)41 b(b)n(y)g(measuring)f(the)i(consumption)f(of) g(computational)g(resources)e(during)i(the)456 1830 y(solution)20 b(of)h(a)g(problem.)34 b(By)21 b(studying)g(ho)n(w)g(the)g(amoun)n(t)g (of)g(resources)e(v)-5 b(aries)21 b(as)f(a)h(function)456 1930 y(of)j(the)g(input)h(size,)f(w)n(e)g(can)g(obtain)f(v)-5 b(aluable)24 b(information)f(ab)r(out)h(the)h(e\016ciency)f(of)g(a)f (giv)n(en)456 2029 y(program)i(or)i(the)h(di\016cult)n(y)g(of)f(a)h (giv)n(en)e(problem.)555 2129 y(The)g(t)n(w)n(o)f(most)g(common)g(quan) n(tities)h(measured)f(are)f(time)i(and)g(space,)f(i.e.)37 b(the)26 b(n)n(um)n(b)r(er)456 2228 y(of)d(steps)h(the)h(T)-7 b(uring)23 b(mac)n(hine)h(tak)n(es)f(during)g(the)h(solution)g(of)g(a)f (problem)h(and)f(the)i(n)n(um)n(b)r(er)456 2328 y(of)i(memory)g(cells)g (used)h(during)f(this)h(computation.)555 2428 y(The)d(asymptotic)f(gro) n(wth)g(in)h(these)f(t)n(w)n(o)g(quan)n(tities)h(is)f(used)h(to)g(sep)r (erate)f(problems)g(in)n(to)456 2527 y(v)-5 b(arious)26 b(complexit)n(y)h(classes.)456 2677 y(1.5.)40 b Fr(What)45 b(is)f Fk(P)12 b Fr(?)41 b Fp(The)e(complexit)n(y)f(class)g Fk(P)50 b Fp(is)39 b(set)g(of)f(all)g(languages)f(whic)n(h)i(can)f(b)r (e)456 2776 y(decided)27 b(b)n(y)h(a)f(T)-7 b(uring)27 b(mac)n(hine)g(in)h(p)r(olynomial)f(time.)37 b(More)27 b(formally)-7 b(,)456 2911 y Fr(De\014nition)38 b(1.2.)44 b Fn(A)36 b(language)h Fm(L)d(2)h Fk(P)47 b Fn(when)37 b(ther)l(e)f(exists)f(a)h(T)-6 b(uring)36 b(machine)h Fk(T)47 b Fn(such)456 3011 y(that)29 b(for)i(every)g Fk(w)25 b Fm(2)f Fp(\006)1201 2981 y Fl(\003)1269 3011 y Fn(with)30 b Fm(j)p Fk(w)r Fm(j)24 b Fp(=)e Fk(n)p Fn(,)30 b(a)g(p)l(olynomial)i Fk(g)s Fp(\()p Fk(n)p Fp(\))e Fn(c)l(an)g(b)l(e)g(found)g(such)g(that)1717 3193 y Fp(lim)1688 3243 y Fg(n)p Fl(!1)1886 3137 y Fk(f)9 b Fp(\()p Fk(n)p Fp(\))p 1886 3174 164 4 v 1890 3250 a Fk(g)s Fp(\()p Fk(n)p Fp(\))2083 3193 y(=)22 b(1)555 3376 y Fn(for)31 b Fk(f)9 b Fp(\()p Fk(n)p Fp(\))p Fn(,)30 b(the)g(numb)l(er)f(of)h (steps)g(by)g Fk(T)41 b Fn(as)30 b(it)g(de)l(cides)h Fk(w)26 b Fm(2)d(L)p Fn(.)555 3511 y Fk(P)40 b Fp(is)27 b(frequen)n(tly)g(view)n(ed)h(as)f(the)h(class)e(of)i(tractable)f (problems.)456 3661 y(1.6.)40 b Fr(What)33 b(is)f Fk(N)9 b(P)j Fr(?)41 b Fp(The)29 b(complexit)n(y)f(class)f Fk(N)9 b(P)40 b Fp(is)28 b(set)h(of)f(all)g(languages)e(with)j(p)r(olyno-)456 3760 y(mial)e(time)h(v)n(eri\014ers.)35 b(A)28 b(formal)f(description)g (is)456 3895 y Fr(De\014nition)32 b(1.3.)41 b Fn(A)31
b(language)g Fm(L)26 b(2)g Fk(N)9 b(P)42 b Fn(when)32 b(ther)l(e)f(exists)g(a)g(T)-6 b(uring)31 b(machine)h Fk(T)42 b Fn(such)456 3995 y(that)665 4129 y Fm(L)23 b Fp(=)g Fm(f)p Fk(w)d Fm(j)f Fk(T)41 b Fn(de)l(cides)31 b Fm(h)p Fk(w)r(;)14 b(c)p Fm(i)31 b Fn(for)f(some)h(string)e Fk(c)h Fn(in)g(time)f(p)l(olynomial)j(in)e Fm(j)p Fk(w)r Fm(j)h(g)456 4264 y Fn(The)f(string)g Fk(c)g Fn(c)l(an)f(b)l(e)h(viewe) l(d)h(as)f(a)g(pr)l(o)l(of)h(or)g(c)l(erti\014c)l(ate)e(of)i Fk(w)h Fn(b)l(eing)e(in)g(or)g(out)g(of)g Fm(L)p Fn(.)555 4399 y Fp(Another)18 b(c)n(haracterization)e(of)i Fk(N)9 b(P)31 b Fp(is)18 b(via)g(p)r(olynomial)f(time)i(deciding)f (non-deterministic)456 4498 y(T)-7 b(uring)27 b(mac)n(hine.)36 b(Man)n(y)27 b(imp)r(ortan)n(t)g(problems)g(lie)h(in)g(the)g(class)e (as)h(w)n(e)g(shall)h(see.)456 4648 y(1.7.)40 b Fr(So)e(y)m(ou)h(w)m (an)m(t)g(to)g(b)s(e)e(a)i(millionaire...)d Fp(It)e(easy)e(to)h(see)g (that)g Fk(P)45 b Fm(\032)31 b Fk(N)9 b(P)j Fp(.)54 b(A)33 b(k)n(ey)456 4747 y(problem)27 b(in)h(theoretical)e(computer)i(science) f(in)h(the)g(question)f(of)1772 4882 y Fk(P)35 b Fp(=)23 b Fk(N)9 b(P)j Fp(?)456 5016 y(It)31 b(is)g(curren)n(tly)f(b)r(eliev)n (ed)h(that)h Fk(P)41 b Fm(6)p Fp(=)28 b Fk(N)9 b(P)43 b Fp(and)31 b(there)g(are)f(man)n(y)h(strong)f(argumen)n(ts)g(wh)n(y)-7 b(.)456 5116 y(The)28 b(resolution)g(of)g(this)h(problem)f(is)g(on)h (the)f(top)h(ten)g(list)g(of)f(mathematical)g(problems)g(for)456 5216 y(the)g(21st)e(cen)n(tury)-7 b(,)27 b(eac)n(h)g(one)g(with)i(a)e (million)h(dollar)e(prize)h(attac)n(hed)g(to)h(it.)p eop %%Page: 3 3 3 2 bop 459 251 a Fi(SUR)-7 b(VEY)29 b(ON)g(THE)g(COMPLEXITY)g(OF)g (CER)-5 b(T)g(AIN)29 b(APPR)n(O)n(A)n(CHS)g(TO)g(STR)n(UCTURE)h (PREDICTION)-29 b(3)456 450 y Fp(1.8.)40 b Fr(What)29 b(is)e(reduction?)42 b Fp(A)25 b(k)n(ey)f(to)r(ol)g(classifying)f (complexit)n(y)h(of)h(v)-5 b(arious)23 b(complexit)n(y)456 550 y(results)33 b(in)h(the)g(idea)g(of)g(reduction.)55 b(In)n(tuitiv)n(ely)-7 b(,)36 b(giv)n(en)d(t)n(w)n(o)g(problems)g Fk(A)h Fp(and)g Fk(B)t Fp(,)i(if)e(the)456 649 y(problem)25 b Fk(A)i Fp(can)f(b)r(e)g(translated)f(e\016cien)n(tly)i(in)n(to)f(an)f (instance)h(of)h Fk(B)j Fp(and)c(solv)n(ed)f(as)h Fk(B)k Fp(then)456 749 y(w)n(e)d(sa)n(y)f(that)i Fk(A)g Fp(reduces)f(to)g Fk(B)t Fp(.)555 849 y(Since)h(w)n(e)g(are)f(concerned)f(with)j(p)r (olynomial)e(time)h(complexit)n(y)g(problems,)f(w)n(e)g(are)g(con-)456 948 y(cerned)35 b(with)g(p)r(olynomial)g(time)h(reduction,)h(i.e.)61 b(the)35 b(translation)g(step)g(can)g(b)r(e)h(done)f(in)456 1048 y(p)r(olynomial)27 b(time.)661 1170 y Fm(\017)41 b Fp(If)c Fk(A)f Fp(can)g(b)r(e)h(reduced)f(to)g(b)r(e)g Fk(B)k Fp(and)c Fk(B)41 b Fp(can)36 b(b)r(e)g(reduced)g(to)g Fk(A)p Fp(,)j(w)n(e)c(sa)n(y)g(these)744 1269 y(problems)27 b(are)g(p)r(olynomial)g(time)h(equiv)-5 b(alen)n(t.)661 1369 y Fm(\017)41 b Fp(W)-7 b(e)28 b(sa)n(y)f(that)h Fk(B)j Fp(is)d Fk(N)9 b(P)j Fp(-hard)26 b(if)i(ev)n(ery)f(problem)g Fk(A)c Fm(2)g Fk(N)9 b(P)40 b Fp(reduces)27 b(to)g Fk(B)t Fp(.)661 1469 y Fm(\017)41 b Fp(If)28 b Fk(B)k Fp(is)c Fk(N)9 b(P)j Fp(-hard)26 b(and)h Fk(B)g Fm(2)d Fk(N)9 b(P)j Fp(,)27 b(then)h Fk(B)k Fp(is)c Fk(N)9 b(P)j Fp(-complete.)1250 1665 y(2.)41 b Fo(HP)30 b(Model)h(of)h(Pr)n(otein)f(F)n(olding)555 1814 y Fp(In)i(this)h(approac)n(h)d(to)i(protein)f(folding,)j(the)e (20-letter)f(alphab)r(et)h(of)g(amino)f(acids)h(in)g(a)456 1914 y(string)21 b(is)g(reduced)g(to)h(2,)g Fk(H)29 b Fp(and)21 b Fk(P)12 b Fp(.)35 b(These)21 b(refer)g(to)g(h)n(ydrophobic) g(and)g(p)r(olar)g(\(h)n(ydrophilic\))456 2014 y(amino)31
b(acids)g(resp)r(ectiv)n(ely)-7 b(.)49 b(Hydrophobic)31 b(forces)g(are)f(b)r(eliev)n(ed)i(to)g(b)r(e)g(the)g(predominan)n(t)456 2113 y(force)26 b(in)h(the)h(folding)f(of)g(globular)f(proteins.)36 b(The)27 b(energy)f(function)i(for)f(adjacen)n(t)f(...