TheHistory andStatus of thePversus NP Que stion

Michael Sipser

DepartmentofMathematics

Mas sachus ettsInstituteofTechnology

Cambr idge MA

As long as a branch of science oers an abundance of problems

so long it i s alive a lack of problems fore shadows extinction or

thecessation of indep endentdevelopment

David Hilbert f rom a lecture delivere d b efore the

Inter national Congre ss of Mathematicians atPar i s in

When

Juris Har tmanis

somewhere in a large space of p oss ibilitie s Computers Signicance

may greatly sp ee d up thesearch butthe extremely large

TheP versus NP que stion grew outofdevelopmentsin

space s that really do o ccur in cas e s of intere st would

mathematical logic and electronic technology dur ingthe

still require geologic timetosearchexhaustively even on

middle part of thetwentieth century It i s now con

thefastest machines imaginable The only way to solve

sidere d tobeoneofthe most imp ortant problems in

such cas e s practically would b e tondametho d which

contemp orary mathematics andtheoretical computer sci

avoided searchingbybrute force Roughly sp eaking the

ence The literature on thi s sub ject i s divers e butnot

Pversus NP que stion asks whether in general sucha

huge it i s p oss ible for thestudentto acquaint hims elf

method exists

with all of the imp ortant relevantworkinasemester or

tracte d cons iderable attention Thi s que stion has at

hi s articleIhaveattempted to orga two In wr itingt

Itsintuitivestatementissimpleand acce ss ible to non

nize andde scr ib e thi s literature including an o ccas ional

sp eciali sts even thos e outside of science By embracing

opinion aboutthe most f ruitful directions butnotech

i ssue s in the foundations of mathematics as well as in the

nical details

practice of computing it gains somethingincharacter

In thersthalf of thi s centuryworkonthepower of

beyondthatofamere puzzle butrather withappar

formal systems le d tothe formalization of thenotion of

ently deep er s ignicance and broader cons equence s

algor ithm andthe realization that certain problems are

ChurchandTur ing in their workonHilb erts

algor ithmically unsolvable At aroundthi s time fore

entschiedungsproblemhaveshown thattestingwhether

runners of the programmable computingmachinewere

an ass ertion has a pro of i s algor ithmically unsolvable

b eginningtoappear As mathematicians contemplated

TheP versus NP que stion can b e viewe d as a more ni

the practical capabilitie s and limitations of suchdevice s

tary vers ion of the entschiedungsproblem Suppose we

computational complexitytheory emerge d f rom thethe

wi sh totest whether an ass ertion has a short pro of Of

ory of algor ithmic unsolvability

cours e thi s can b e solved by a brute force algor ithm but

Early on a particular typ e of computational task b e

canitbedone eciently

came evident where one i s s eeking an ob ject which lie s

The entschiedungsproblem does have practical im

Supp orted by NSF Grant CCR and Air Force Con

portance in addition to its philosophical s ignicance

tract AFOSR

Mathematical pro of i s a co dication of more general hu

man reasoning An automatic theorem prover would

havewideapplication within computer science if it op

erate d eciently enough Even though thi s i s hop ele ss

in general there may b e imp ortant sp ecial cas e s which Neumann wroteaboutthe i ssue in a remarkably mo d

are solvable It would b e nice if Churchs or Tur ings er n way formulatingitinterms of thetime require d on

pro ofs gaveussome information aboutwhere the eas ier aTur ingmachinetotest whether a formula in thepred

cas e s might lie Unfortunatelythe ir argumentsreston icate calculus has a pro of of length nHealsoasked

s elfreference a contr ived phenomenon whichnever howmuchwe can improveuponbrute force in general

app ears sp ontaneously Thi s do e s not tell us whatmakes when solvingcombinator ial problems It i s not known

the problem hardinintere stingcases Conce ivablya whether von Neumann whowas dying of cancer atthe

pro of that P i s not equal toNPwould b e more informa time thoughtaboutGodels letter or re sp onded toit

tive Von Neumann do e s appear tohavehad someawareness

of theunde s irability of brute force s earch A couple of

years earlier hehad wr itten a pap er giving an algor ithm

avoidingbrute force s earchforthe ass ignment problem

Hi story

Vo

Edmonds Edgavetherstlucid accountofthe

Amongthemany equivalent formulations of thePversus

i ssue of brute force s earchappearinginwester n litera

NP que stion the followingone i s particularly s imple to

ture Themain contr ibution of hi s pap er was a p oly

de scr ib e

nomial time algor ithm for themaximummatching prob

Is there an algorithm which when presentedwithan

lem Thi s algor ithm its elf still stands as a b eautiful ex

undirectedgraph determines whether it contains a com

ample of howbrute force can sometimes be avoided if

plete subgraph on half its nodes and whose running

oneisclever enough Edmonds feltthenee d to explain

time is boundedabove by a polynomial in the number

whyhiswas a re sultat all when there i s an obvious

of nodes

nite algor ithm for solvingthe problem In so doinghe

di scuss e d the problem of brute force s earch in general

By theterm algorithm wemean a nite s et of instruc

as an algor ithmicmethod

tions on anyofanumber of relate d mo dels of computa

In the s re s earchers in theSoviet Union were

tion eg themultitap e Tur ingmachine Theterm P

aware of the i ssue of brute force s earch calle d perebor

for Polynomial time refers tothe class of language s lan

in Russ ian Yablonski Yaadescribed the i ssue for

guage s decidable on a Tur ingmachineintimebounded

general problems and fo cus e d on the sp ecic problem

aboveby a p olynomial It i s in var iantover reason

of constructingthesmalle st circuit for a given function

able computational mo dels NP for Nondetermini stic

Cur iouslyinalater pap er Yabhe obscure d the i ssue

Polynomial time refers totheanalogous class for non

by claimingto provethat p erebor is unavoidable for thi s

determini stic Tur ingmachines NP cons i stsofthos e

problem That pap er whos e main theorem makes no

language s where memb ership i s veriable in p olynomial

reference tothe notion of algor ithm certainly did not

time Thequestion of whether P i s equal toNPisequiv

do as claimed at least in the s ens e wewould mean it

alenttowhether an NPcomplete problem suchasthe

today Trakhtenbrot sheds some lightonthi s murky

clique problem described above can b e solve d in p oly

situationinhissurvey of early Russ ian work on p erebor

nomial time The books byHop croft and Ullman HU

Tr

and Garey and Johnson GJ contain excellentintro

ductions tothi s subject

PandNP

Several early pap ers notably thos e of Rabin Ra

BruteForce Se arch

Hartmani s andStear ns HS andBlum Bl pro

posed anddeveloped th enotionofmeasur ingthecom

Theidea of us ing brute force s earchto solve certain

plexity of a problem in terms of thenumber of steps

problems i s certainly very old In pr inciple manynat

require d tosolveitwith an algor ithm Cobham Co

urally o ccurr ing problems may b e solve d in thi s way

Edmonds Ed andRabin Ra sugge sted the class

though if the s earch space i s large thi s b ecomes obvi

P as a reasonable approximationtothe class of reali s

ously impractical People realize d thi s dicultyinpar

tically solvable problems

ticular cas e s and o ccas ionally were able tondalter na

Theimportance of the class NP stems f rom thelarge

tivemetho ds Oneofthe early accompli shmentsofthe

numb er of computational problems containe d within

theory of algor ithms was to recognize brute force s earch

Co ok Coand Levin Le rst dened thi s class and

asaphenomenon indep endentofany sp ecic problem

proved theexistence of complete problems Somewhat

Anumb er of re s earchers working s eparately hit up on

t hi s idea

A translation of Godels letter along with quotations f rom a

number of histor ical papers are include d in an appendix Godel in a recently di scovere d letter tovon

earlier Edmonds Edb capture d the e ss ence of the of language s eciently computable in the pre s ence of

notion of NP withhisgoodcharacter ization whereby that oracle Baker Gill andSolovay BGSdemon

an ass i stant can convince hi s sup ervi sor of the solution strate d an oracle relativeto whichPisequaltoNPand

to a problem Since the solution may b e e ither p os itive another relativetowhichthey dier Astepbystep

or negativeinthe cas e of a problem involving language simulation of onemachineby another suchasthatused

memb ership Edmondss notionisgenerally asso ciated inaconventional diagonalization still succee ds in the

withthe class NP coNP Karp Kademonstrated pre s ence of an oracle as thesimulator nee ds only query

the surprisingly wide extent of NPcompleteness when the oracle whenever thesimulated machine do e s Any

heshowed thatmany f amiliar problems havethi s prop argumentwhich relie s only on stepbystep s imulation

erty tocollapseorseparatePandNPwould thus also apply

in the pre s ence of any oracle As thi s would contradict

the BGS theorem stepbystep s imulation will not b e

Explicit Conjecture

sucienttosettle the que stion

Co ok rst preci s ely formulated thePversus NP conjec

Theaboveisaheuristic argumentasthenotion of

ture in hi s pap er More or le ss clos e approxima

stepbystep s imulation i s not preci s ely dened In

tions toitappeared somewhat earlier in thepapersof

recent pap ers Lund Fortnow Karlo and Ni san and

Edmonds Levin Yablonski andinGodels letter

Shamir LFKN Sh intro duce d a new typ e of s imu

lation to prove equivalence for the complexity class e s IP

andPSPACE Thi s s imulation i s not of thestepbystep

Status

sort butisrather indirect andindee d do e s not relativize

to all oracle s as can b e s een f rom the pap er of Fortnow

Diagonalization and Relativization

an d Sips er FS Conce ivablyanindirect s imulation

suchasthi s could b e combined withthe diagonalization

Rabin Raand Hartmani s andStear ns HS proved

method to s eparate certain complexity class e s

the exi stence of decidable problems with arbitrar ily high

computational complexityThey us e d a class ical diag

onalization giving an algor ithm which byde s ign ac

Indep endence

cepts a language dier ing f rom anylanguage of low

Followingthecelebrated workofGodel andCohen we

complexity Thi s technique allows us to conclude for

knowthatcertain mathematical que stions suchasthe

example thatthere exi st problems requir ing exp onen

Continuum Hyp othe s i s cannot b e answere d within ac

tial time Problems constructe d in thi s way are articial

cepted formal systems of mathematical reasoning There

havingnomeaning or s ignicance outsideoftheir role

has b een sp eculation thatthePv ersus NP que stion may

in thi s theorem Meyer andStockmeyer MSused

be unre solvable in a s imilar s ens e Thi s would mean that

thi s theorem todemonstratenatural problems provably

if there were no p olynomial time algor ithm for the clique

outside ofPbyshowingthatthey are complete for high

problem wewould never b e able toprovethenonexi s

complexity class e s

tence On theother handifsuch an algor ithm were to

A priori it might s eem p oss ible to us e diagonaliza

exi st wewould not b e able toproveits p erformance

tion to provethat some problem in NP require s exp o

Iwillmention here someoftheliterature on indep en

nential time Onewould give a nondetermini stic p oly

dence though I stateattheoutset myopinion thatour

nomial timeTur ingmachinewhich o ver the cours e of

currentinabilitytoresolvethi s que stion der ives from

its inputs has an opp ortunitytorun eachofthedeter

her than anylackofpower insucient brain p ower rat

mini stic p olynomial timeTur ingmachines and arrange

in our formal systems

to accept a dier inglanguage Thi s s imple argument

In a s ens e theworkonrelativization already sugge sts

f ails b ecaus e thenondetermini stic machine runningin

k

a sort of limited indep endence for thePversus NP que s

some xe d n time i s not able tosimulatea determin

tion One mightbeable toformulate an axiomatic sys

istic machinewhos e timebound i s a larger p olynomial

tem corre sp ondingto pure recurs ivefunction theoryin

Poss ibly there i s a more clever nondetermini stic s imu

whichthe re sults of Baker Gill andSolovay pre sumably

lation of thedetermini stic machinewhichwould allow

would showthatthequestion i s unre solvable However

the nondetermini stic machineto carry outthe diagonal

thi s would b e very f ar f rom e stabli shingindep endence

ization pro ce ss Themetho d of relativization described

with re sp ect tostrong systems suchasnu mber theory

b elow sugge ststhatthi s will not b e straightforward

or s et theory

In a relativize d computation themachineisprovided

with a s et calle d an oracleandthe capabilitytodeter In the direction of stronger indep endence a num

minememb ership in thi s s et without cost For eachor b er of pap ers haveattempted to oer a b eginning

acle there i s asso ciated the relativize d complexity class notably HH giving an oracle under whichtheP

versus NP que stionisindep endentofsettheory and formulas are unsatisable The axioms are the claus e s

Li DLSa Bu CU BH e stabli shingin of theformula In the inference rule if clauses x

dep endence and relate d re sultswithin weak f ragments andx are already pre s ent then thenew claus e

of number theory or relate d systems While I do not may b e added Unsatisability i s proved when

profe ss exp ertise in thi s area I doubt whether the re theempty claus e i s der ived The complexityofthe pro of

sults obtained so far in thi s vein do more than rai s e the is thenumb er of claus e s added Anyunsati sable for

p oss ibilityofmeaningful indep endence mula has a re solution pro of Haken solving a que stion

thathad b een op en for manyyears proved thatthere

are formulas requir ing exp onential s ize pro ofs Ha

Expre s s ibility in Logic

Hi s work followed thatofTseitin Tswhohad solved

regular re solution a further re str icte d form s ee also

Another p otential application of mathematical logic

Ga Extende d re solution a generalize d form re

arises becaus e it provide s a dierentway of dening

mains open see Co Kozen Koconsidere d other

f amiliar complexity classes Fagin Fashowed that

pro of systems andestabli shed lower b ounds for them

NP equals the class of language s repre s enting struc

Pro of systems are related tothe class NP If all unsat

ture s denable by s econdorder s entence s He and

isable formulas had p olynomial length re solution pro ofs

Jones and Selman JSshowed thatthe class NEXP

then it would eas ily followthat NPcoNPHakens re

TIME equals the class of language s repre s entingthecol

sultprove s a sp ecial cas e of theNPcoNP conjecture

lections of cardinalitie s of univers e s of such structure s

showingthatoneparticular nondetermini stic algor ithm

Much earlier Scholz Schad cons idere d such collec

i s sup erp olynomial

tions and asked whether they could b e character ize d

s ee also Ass er As Immerman Imhas extended

the corre sp ondence withnitemod el theory toPNL

Circuit Complexity

andother complexity class e s us ing rstorder calculus

ador ne d withvar ious op erators Hope thatthi s reco d

Shannon Sh proposed the s ize of Bo olean circuitsas

inginto logic may help tosettle que stions in complexity

ameasure of the complexityoffunctions Savage Sa

theory has b een partly b or n outby Immermans di scov

demonstrate d a clos e relationship b etween circuit s ize

ery Imthat NLcoNL inspire d bycertain obs er

andtimeona Tur ingmachine Circuits are an app ealing

vations concer ning rstorder denability followingan

mo del of computation for provinglower b ounds Their

earlier weaker re sult of LJK Szelep csenyi Sz

relatively s imple denition renders them more amenable

indep endently obtaine d NLcoNL ataroundthesame

to combinator ial analys i s and allows for natural var ia

time

tions and re str ictions Thi s has b een key toachieving

anumb er of imp ortantresults Many re s earchers con

sider circuit complexitytobethe most viable direction

Re str icte d Systems

for re solvingthePversus NP que stion We will sur

vey recentworkinthi s area withaneyetowards thi s

glower b ounds on complexity The dicultyinprovin

goal ThesurveypaperofBoppanaand Sips er BS

stems f rom therichness of the class of algor ithms One

contains details of muchofthi s work de scr ib e d in a

approachthat allows for partial progre ss i s to re str ict the

readable f ashion Dunne DuandWegener We

class of algor ithms cons idere d There i s a substantial lit

haverecently wr itten b o oks on Bo olean complexity

erature on b ounds in var ious re str icted models These

A circuit or straight line programover a bas i s typ fall intotwocategor ie s whichmay b e calle d natural

ically and or and not i s a s equence of instructions models and handicappedmodelsInthe former a mo del

each pro ducingafunction byapplyinganoperation f rom i s s electe d which allows only for operations sp ecially tai

thebasisto previously obtained functions Initiallywe

lore d tothe problem athand Thi s includes sorting

start outwiththefunctions nat urally asso ciated with

mo dels p olynomial evaluation mo dels var ious mo dels

or iented tow ards data structure s andothers In thelat eachofthe inputvar iable s If eachfunction is used at

ter we s eek tomaintain the generalityofthemodel but most once then the circuit i s calle d a formula

weaken it suciently so thatanintere stinglower b ound

By Savages theorem any problem in P has a p oly

i s obtainable Bothapproache s b ear a connection to s ep

nomial s ize f amily of circuits Thus toshowthata

arating complexity classes We will treathandicapped

problem i s outsideofPitwould suce toshowthatits

mo dels shortly in thesection on circuit complexity Here

circuit complexity i s sup erp olynomial A s imple count

we cons ider re str icte d pro of systems as a form of natural

ing argumentshows that most Bo olean functions on n

mo del

var iable s require exp onential circuit s ize Sh Mu

The best lower b oundwehavetodate for a problem in A proof system i s a collection of axioms andrulesof

NP i s n on due toNBlumBl It do e s not inference Resolution i s a pro of system for showingthat

s eem likely thatthetechnique s us e d to get thi s re sult tomerge A circuit may thus b e s implie d inductively

will extendto s ignicantly b etter b ounds while pre s ervingthefactthatitcompute s a par ityfunc

tion Ajtais argument i s also an induction He showed

In Va Valiant sugge ststhat a direct attempt to

thatanysetdenable byalowdepth circuit i s well ap

provelower b ounds on the s ize of Bo olean circuitsmay

proximated bya setofavery sp ecial form a di sjoint

not b e the r ightapproach There are natural algebraic

union of large cylinders A cylinder is a maximal col

generalizations of certain NPcomplete problems such

lectionofstrings which all agree on some collection of

as the Hamiltonian cycle problem to an arbitrary r ing

var iable s Clearly the par ityfunction cannot b e approxi

Wemay also generalize circuitsover thebasisand xor

mate d in thi s wayHismain combinator ial lemmashows

for such problems to arbitrary r ings Circuitswhich

thatthe propertyofbeingwell approximable i s clos e d

solvethe generalize d problem form a proper sub class of

under p olynomial union andinters ection

he circuits whichmerely workcorrectly in the Bo olean t

Yao Yacombine d probabili stic re str iction with

cas e As suchitmay be easier toprove stronglower

bounds in thi s re str icted case Valiantpointsoutthat atyp e of approximation togiveanexponential lower

our f ailure todothi s even then argue s p o orly for our bound for constantdepthparity circuits Hastad Ha

chance s in the Bo olean cas e He sugge sts directing our s implie d and strengthened thi s b oundby improvingthe

energie s toward solvingthe algebraic cas e rst core of the FSS argumentusingatypeofanalys i s of con

ditional probabilitie s developed byYao

There are mer itstothi s line or reasoning Oneof

the imp e dimentsinthelower b ounds area i s a shortage Sips er Sishowed thatthere are functions com

putable bydepth d linear s ize circuitsand which require

of problems of intermediate diculty whichlend ins ight

sup erp olynomial s ize for depth d byatechnique very

intotheharder problems The algebraic generaliza tions

may b e imp ortantsteps towards the Bo olean goal On similar tothatusedbyFSSYao claime d without pro of

theother hand I do not b elieve as Valiant argue s that an exp onential b ound for thi s depth hierarchyUsing

the algebraic cas e i s prerequi s ite True thealgebraic hi s s implie d method Hastad pre s ente d a pro of of thi s

re sult

cas e i s more re str ictive andhence eas ier Thesame

may b e said for uniform mo dels of computation ver

Razb orov Raintro duce d a metho d of obt aining

sus nonuniform mo dels Nonuniform mo dels suchas

lower b ounds on the s ize of limited depth circuitsover

circuits are more p owerful than uniform mo dels such

a larger bas i s He proved thatthemajorityfunction re

as Tur ingmachine s b ecaus e the algor ithm us e d may

quire s exp onential s ize for circuitshaving parity gates

change arbitrar ily dep endinguponthelengthofthein

as well as and orand not gates He showed thatany

put Accordingly it may b e more diculttoprovelower

set denable withasmall shallow circuit over this basis

bounds on the s e more p owerful mo dels Neverthele ss

is well approximated by a p olynomial of lowdegree over

theharder problem may allowoneto s ee more eas

thetwo elementeldand argue s thatthemajorityfunc

ily theheart of thematter unobscure d byunnece ssary

tion cannot b e well approximate d in thi s wayBarring

feature s

ton Baa extended thi s to circuits with mod gate s for

q

anyprime q instead of only parity ie mod gates

Smolensky Smfurther s implie d and improved thi s

Bounde d Depth Circuits

showingthat computingthe mod function require s ex

p

ponential s ize when computed byshallow circuitshaving

Thenext few s ections treathandicapped var iantsofthe

and orand mod gates for p and q powers of dierent

q

circuit mo del By placingvar ious re str ictions on struc

pr imes

ture such as limitingthedepth or re str ictingthe bas i s it

has b een p oss ible toobtain s everal stronglower b ounds

The rst re sultofthi s kindisdue toFurst Saxe and

Monotone Circuit Size

Sips er FSS andindep endently Ajtai Aj who

proved that certain s imple functions suchasthepar A monotone circuit i s onehaving and and or gates but

ityfunction require sup erp olynomial circuit s ize if the no negations A function i s monotone if x y implie s

depthisheld xe d f x f y under the usual Bo olean order ing It i s

easy toseethatmonotone circuitscompute exactly the

These two pro ofs of thi s theorem us e dierent though

class of monotonefunctions

related methods Furst Saxe and Sips er intro duce d the

notion of probabilistic restriction a randomly s elected The rst stronglower b oundconcerningthi s mo del

ass ignmentto someofthevar iable s They showed that is due toRazborov Raa whoproved thatthe

for any circuit of the form and of small ors the circuit clique function has sup erp olynomial monotone complex

induce d byarandom re str iction may likely b e putin ity Shortly thereafter Andreev An us ing s imilar

the form or of small ands Thi s interchange allows ad metho ds prove d an exp onential lower b ound further

jacent levels of orstofollowoneanother andtherefore tightened by Alon andBoppana AB

Razb orovs theorems on monotoneandbounded depth Karchmer andWigderson KWgavea very nice

circuits as well as the aforementioned proof of Ajtai character izationoflanguage s denable withsuchcir

rely up on a technique whichhas cometo b e calle d the cuits whichmay b e us eful in obtaininglower b ounds

approximation method Oneofthe dicultie s in at on depth Let A be suchalanguage In the commu

temptingtoanalyze the computation of a circuit re nication game for Athere are two players onehaving

str icte d or general for the purp os e s of provinglower astringinA andtheother having a str ingofthesame

bounds i s thatthesubfunctions computed bythe cir lengthnotinATheplayers communicatewitheach

cuit may b e complicated andhard to grasp In theap other tonda position where their two str ings dier

proximation method weshowthatthesubfunctions are The minimumnumber of bitsthatthey require todo

in a certain s ens e near functions whicharemuch s im thi s over all str ings of length n is the complexityofthe

pler Take for example Razb orovs lower b oundforthe game Karchmer and Wigderson showthatthi s exactly

clique function Cons ider any monotone circuit com equals the minimum circuit depthnece ssary for thi s lan

putingthi s function The plan i s to adjust theresult guage

of eachofthe and and or gates in suchaway that

Amonotonevar iantofthi s gamehas playe d an im

each adjustmentalters the re sultonlyslightly while

portant role in thediscovery of lower b ounds on mono

the adjusted output i s f ar f rom the clique function

tone circuit depth If onedemands in addition thatthe

Thi s means thatthere must havebeenmanyadjustments

foundpositionisforthestringinAthen thecom

andhence manygates

plexityofthegameistheminimum monotone circuit

If general circuitscomputing monotonefunctions

depth Thi s character ization of monotonedepthcom

could b e converted into monotone circuitswithonlya

plexityledthem to a log n log log nlower b ou nd

p olynomial increas e in s ize then strong monotonelower

on the monotonedepth for the stconnectivityfunction

bounds would yield s eparations of complexity class e s in

Thi s was subs equently improved to log nbyBop

general complexitytheory Razb orov Rabshowed

pana BS whoavoided thecommunication gameby

thatthi s i s not true when heusedtheapproximation

arguing directly on the monotone circuit indep endently

toshowthatthethatthe problem of testing foraper

obtained byHastad unpubli she d Raz and Wigderson

fect matching require s sup erp olynomial s ize monotone

RWusedthe communication gameto obtain a lin

circuits As thi s problem i s in P Ed weknowthat

ear depthlower b ound for monotone circuits computing

it has p olynomial s ize nonmonotone circuits Us inga

hingfunction Their proof uses a lower b ound thematc

s imilar method Tardos Tashowed thatthere i s a

on the probabili stic communication complexityofthe

monotone problem in P which require s exp onential s ize

s et di sjointne ss problem due toKalyanasundaram and

monotone circuits

Schnitger KSsimplie d byRazborov Raa Be

There i s a class of functions where the monotoneand

caus e of thi s it s eems that it will b e dicultto phras e the

nonmonotone complexitie s are p olynomially related

RazWigderson argument withoutapp ealingtothecom

The s e are the slice functions intro duce d byBerkowitz

munication game Razb orov RabandGoldmann and

Be A function f is a slice function if for some k

Hastad GHobtaine d additional lower b ounds us ing

thevalue of f xiswhen thenumberofsinx is

the communication game

fewer than k andthev alue i s when thenumber of s

in x i s more than k Heshowed thatany general cir

hatmanyof Gr igni and Sips er GSpointoutt

cuit computing a slice function may b e converted into

theresults in monotone complexity can b e viewed in

a monotonefunction by adding only a p olynomial num

terms of monotoneanalogs mPmNPetc of thefa

k k

ber of gate s Hence if onewere able toprove a strong

miliar complexity class e s PNPAC NC andoth

lower b oundonthemonotone circuit complexityofa

ers Thus we already knowthatmPmNP Raa

slice function then thi s would showthatPNP

mP fP monog Rab andmNC mNL KW

Weuse mono tode s ignatethe class of all monotonelan

guage s Gr igni andSipserdemonstratethat mNLm

Monotone Circuit Depth

coNL us ingthetechnique of Karchmer and Wigder

son Thi s shows theinherently nonmonotonen ature of Circuitsover thestandard bas i s re str icted to f anin two

epth form an imp ortantsub class of the theImmermanSzelep cs enysimulation In a later pa with O log nd

p olynomial s ize circuits Such circuits are equivalent p er GS they strengthen the KW re sultby s eparat

in p ower to p olynomial s ize formulas Provingthata ingmNC f rom mL There are a numberofopen ques

language i s not denable with a circuit of thi s kindwould tions in monotone complexity for example whether Bar

showthat it i s not in the class NC Thi s would b e ringtons b eautiful s imulation of logdepth circuitsby

weaker than showingitisoutsideofPbutwe are still p olynomials ize width branching programs i s inher

unable todothi s for any language in NP ently nonmonotone s ee GS

Active Research Directions cuits computingthe par ityfunction on n var iable s i s the

theorem thatthere are no countables ize depth dcir

In thi s s ection we will examine a few directions which

cuits computing a par ityfunction on manyvar iable s

app ear to b e plaus ible means of s eparating complexity

A Bo olean function on many Bo olean var iables is a

class e s These are the approximation methodwhich

parity function if theoutputchange s whenever any s in

has b een us e d succe ssfully in analyzing re str icte d circuit

gle inputchange s The pro of of the innitary theorem

mo dels the function composition methodwhichmay b e

did prece deandmotivatethe pro of in thenitecase

us eful to provefunctions require large circuit depth and

Thetwo pro ofs are very s imilar in structure A s ec

an innite analog whichhas le d tolower b ounds in the

ond related application of thi s analogy app ears in the

bounded depthcircuitmodel

constantdepth hierarchytheorem Si

There i s a p oss ibilitythattheapproximationmethod

Thi s sameapproach sugge sts an inniteanalog tothe

whichhas b een us e d so succe ssfully toobtain lo wer

class NP theanalytic s ets also calle d sets The

bounds for monotoneandbounded depth circuits may

class ical theorem due toLeb e sgue Lestatingthat

apply in the general cas e as well In Ra Razb orov

thethere i s an analytic s et whos e complementisnot

cons iders thi s que stion andgives both a positiveand

analytic may thus b e taken as an inniteanalog tothe

negative answer He shows that for oneformalization of

statementthatNPcoNPLeb e sgues pro of i s a diag

themethod it is too weak to giveintere stingbounds

onalization anditdoesnotseemlikely thatithas a

On theother hand a generalization of themethod is in

nitecounterpart In SiIgive a dierent pro of of

pr inciple strong enough though there are much greater

Leb e sgues theorem Thi s new pro of do e s not re st up on

technical dicultie s when applying it Roughly sp eak

thenotion of universal set essential tothe diagonaliza

ing t hese twovers ions of theapproximation metho d dif

tion in Leb e sgues pro of It oers more information of

fer in theway the class of approximatingfunctions are

acombinator ial nature thatmay b e us eful in the nite

chos en In theweaker vers ion the class i s s electe d in ad

cas e

vance andapplie s to all circuits In the stronger vers ion

the class dep ends up on the circuit

Karchmer Raz andWigderson KRW proposed a

Acknowle dgments

direction for investigatingtheNC versus P que stion

Let B denotethe s et of all Bo olean functions on n

n

Iwishtothank Maur icio Karchmer and Alexander

g B wedenethe var iable s Given f B and

m n

Razb orovwho read an earlier draft of thi s pap er and

nm

comp os ition f g f g f g by

oere d s everal sugge stions and corrections

I am grateful to Jur i s Hartmani s for providingmewith

f g X X f g X g X

n n

acopyofGodels letter andto Sor in Istrail andArthur

S Wens inger for its translation It app ears here with

m

where X f g for i n It i s clear that

i

te for Advance d Study the p ermi ss ion of the Institu

df g df dg KRW conjecture thatthese

Finallyitiswonderful tothank my wife Ina for

two quantitie s are clos e to eachother and argue that if

puttingupwiththewritingofthi s pap er as our daugh

so NC P Edmonds Impagliazzo RudichandSgall

ter Rachel was b e ingbornandforother ass i stance at

EIRSand slightly later Hastad and Wigderson

cr itical moments

HWinasomewhat dierentway proved a weak

form of thi s conjecture

In Si I sugge st a way touseideas f rom de scr ip

Appendix Hi stor ical Quotes

tive s et theory Moschovaki s Mohas an excellent

text on thi s sub ject to gain ins ightintocertain prob

lems in circuit complexitytheoryThisstems f rom a

Shannon di scuss ing circuit design metho ds Sh

proposed analogy b etween p olynomialityand countabil

The problem of synthesizing nons er ie sparallel cir

ity A justication for thi s analogy i s thatu ncountabil

cuits i s excee dingly dicult It i s even more dicultto

ityand sup erp olynomiality often have a common or igin

showthat a circuit foundinthi s way i s the most eco

thepower s et op eration In thi s way certain problems in

nomical oneto realize a given function The diculty

circuit complexityhave innitary analogs often eas ier

spr ings f rom th e large number of essentially dierent

to solve When thi s o ccurs it may lead toanapproach

networks availableandmoreparticularly f rom the lack

in the nitary cas e

of a s imple mathematical idiom for repre s entingthese

Asucce ssful application of thi s strategy o ccurre d in

circuits

provingthelower b oundofFurst Saxe andmys elf

mentione d earlier FSS The inniteanalog tothe von Neumann pre s enting an algor ithm for solving

theorem thatthere are no p olynomials ize depth d cir the ass ignment problem Vo

It tur ns outthatthi s number of steps require d by giving a p olynomial time algor ithm for themaximum

the algor ithm is a moderatepower of n ie cons ider matching problem Ed

ably smaller than theobvious e stimate nmentioned

An explanation i s due on theuseofthewords e

earlier

cient algor ithm First whatI present i s a conceptual

description of an algor ithm and not a particular formal

Yablonski di scuss ingalter natives toperebor in de

hm or co de ize d algor it

s igning circuitsYaa

For practical purp os e s computational details are vital

At presentthere i s an extens ive eld of problems in

However my purp os e i s only toshowasattractively as I

cyb er netics where theexistence of certain ob jectsor

can thatthere i s an ecientalgorithm Accordingtothe

f actsmay be establi she d quite tr ivially and within the

dictionaryecient means adequateinop eration or

limitsofthe class ical denition of algor ithms completely

p erformance Thi s i s roughly themeaningIwantin

eectivelyyet a solution i s in practice often imp oss ible

thesensethatitisconceivable for maximummatching

b ecaus e of itscumb ersomenature Such for example

tohave no ecient algor ithm Perhaps a b etter word i s

are someofthe problems involving information co ding

go o d

andtheanalys i s andsynthesis of circuits It i s here

I am claiming as a mathematical re sult theexistence

thatthenece ss ityofmakingthe class ical denition an

of a go od algor ithm for ndinga maximum cardinality

algor ithm more preci s e naturally ar i s e s It i s tobeex

matching in a graph

p ected thatthi s will to a greater extentthan at pre s ent

There i s an obvious nite algor ithm butthatalgo

takeinto accountthe p eculiar itie s of certain class e s of

rithm increas e s in diculty exp onentially withthe s ize

problems andmay p oss iblyleadtosuchdevelopment s

of the graph It i s bynomeans obvious whether or not

in the concept of algor ithm that dierenttypes of al

there exi sts an algor ithm whos e diculty increas e s only

gor ithms will not b e comparable Atthemomentitis

algebraically withthesizeofthe graph

to o early tomake pre dictions of howthe notion of an

Themathematical s ignicance of thi s pap er re sts

algor ithm may b e mo die d for wehave as yet very

largely on the assumption thatthetwo prece ding s en

little idea of howthevar ious class e s of problems should

tence s havemathematical meaning I am not prepare d

b e sp ecie d In thepresentarticle weattempt toex

t o s et up themachinery nece ssary togivethem formal

plore the algor ithmic dicultie s ar i s inginthe solution

meaningnoristhe pre s entcontext appropr iate for doing

of cyb er netic problems which while admitting of a tr iv

thi s butIshould liketo explain theidea a little further

ial solution on the bas i s of the class ical denition of

informallyItmay b e that s ince one i s customar ily con

an algor ithm are not practically solvable b ecaus e of the

cer ned withexistence convergence nitene ss andso

mass ivenature of tha t solution

forth one i s not inclined totake s er iously the que stion

of the exi stence of a betterthannite algor ithm

Cobham studyingwhy certain problems suchamul

The relative cost in timeorwhatever of thevar ious

tiplication are computationally more dicultthan oth

applications of a particular algor ithm i s a f airly clear

ers suchasaddition Co

notion at least as a natural phenomenon Pre sumably

Thus the pro ce ss of adding m and n can b e carr ie d

thenotion can b e formalize d Here algor ithm i s us e d

outinanumber of steps b ounded by a linear p olynomial

in the str ict s ens e tomean theidealization of somephys

in l mand l n Similarlywecanmultiply m and n in

ical machinery whichgives a deniteoutput cons i sting

anumber of steps b ounded by a quadratic p olynomial in

of cost plus thede s ire d re sult for eachmember of a

volved in l mand l n So too thenumberofsteps in

sp ecie d domain of inputs theindividual problems

the extraction of square ro ots calculation of quotients

The problemdomain of applicability for an algor ithm

etc can b e b ounded by p olynomials in the lengths of

often sugge sts for its elf p oss ible measure s of s ize for the

thenumb ers involved andthi s s eems to b e a property

individual problems for maximummatching for ex

of s imple function in general Thi s sugge ststhatwe

ample thenumberofedgesorthenumber of vertice s in

cons ider the class which I will call L of all functions

the graph Once a measure of problems ize i s chos en

havingthi s prop erty

F N tobethe least upper boundonthe wecandene

A

For s everal reasons theclassL seems a natural oneto

cost of applyingalgorithm A to problems of s ize N

cons ider For onething if we formalize theabovedef

When themeasure of problems ize i s reasonable and

inition relativetovar ious general class e s of computin g

when the s ize s assumevalue s arbitrar ily large an

machines we s eem always toendupwiththesamewell

asymptotic e stimateof F N let us call it the order

A

dene d class of functions Thus wecangivea mathemat

of diculty of algorithm Aistheoretically imp ortant

ical character ization of L havingsome condence thatit

It cannot b e r igge d bymakingthe algor ithm articially

character ize s correctly our informally dene d class

dicultforsmaller s ize s It i s onecriter ion showinghow

Edmonds a s ection marke d Digre ss ion in a pap er good the algor ithm i s not merely in compar i son with

other given algor ithms for the same class of problems with a minimum partition and whichthe sup ervi sor can

but also on thewhole how go o d in compar i son withit then us e with eas e tover ify withmathematical cer

self There are of cours e other equally valuable cr iter ia taintythatthe partition i s indee d minimum Having

And in practice thi s one i s rough one reason b e ingthat a good character ization do e s not mean nece ssar ily that

the s ize of a problem whichwould ever b e cons idere d i s there i s a go o d algor ithm The assistantmay haveto

bounded kill hims elf withworktondthe information andthe

partition

It i s plaus ible toassumethatany algor ithm i s equiva

lent b othinthe problems towhichitapplie s andinthe

Rabin in a survey pap er on automatatheory Ra

costsofitsapplications to a normal algor ithm which

decomp os e s intoelemental steps of certain pre scr ib e d

Weshall cons ider an algor ithm tobepractical if for

typessothatthe costsofthesteps of all normal algo

k

automatawith n states it requires at most cn k is

tiswemay us e something rithms are comparable Tha

a xe d integer and c a xe d constant computational

likeChurchs the s i s in logic Then it i s p oss ible to ask

steps Thi s stipulation i s admittedlybothvague and

Does there or do e s there not exi st an algor ithm of given

arbitrary We do not in f act cannot denewhatis

order of diculty for a given class of problems

meantbyacomputational step thus have no preci s e

One can ndmany class e s of problems b e s ides maxi

andgeneral measure for the complexity of algor ithms

mummatchingandits generalizations whichhave algo

Furthermore there i s no comp elling reason to class ify

rithms of exp onential order but s eemingly nonebetter

k

algor ithms requir ing cn steps as practical

An example known to organic chemi stsisthatofde

Several p ointsmay b e rai s e d in defens e of theabove

cidingwhether two given graphs are i somorphic For

stipulation In every given algor ithm the notion of a

practical purp os e s the dierence b etween algebraic and

computational step i s quiteobvious Hence there i s not

exp onential order i s often more crucial than the dier

thatmuchvagueness aboutthemeasure of complexityof

ence b etw een niteand nonnite

exi sting algor ithms Another s ignicant pragmatic f act

It would b e unfortunateforany r igid cr iter ion toin

is that all exi stingalgorithms e ither require up toabout

hibit the practical development of algor ithms whichare

n

n steps or els e require or wors e steps Thus drawing

either not known or known not to conform nicely tothe

k

theline of practicalitybetween algor ithms requir ing n

cr iter ion Manyofthe best algorithmic ideas known

steps and algor ithms for whichnosuchboundexists

today would suer bysuchtheoretical p e dantry In

seems to b e reasonable

f act an outstandingop en que stion i s e ss entially how

go o d i s a particular algor ithm for linear programming

Co ok cons ider ingthe complexityoftheorem proving

the s implex method And on theother hand manyim

Co The class P i s denoted by L

p ortant algor ithmic ideas in electr ical switchingtheory

Furthermore thetheorems sugge st thattautologies is

are obviously not go o d in our s ens e

a good candidate for an intere sting s et not in L andI

However if only to motivatethe s earch for go o d prac

feel thatitisworthspending cons iderable eort trying

tical algor ithms it i s imp ortantto realize that itismath

toprovethi s conjecture Suchaproofwould b e a major

ematically s ens ible even to que stion the ir exi stence For

breakthrough in complexitytheory

onethingthetask can then b e described in terms of

concrete conjecture s

Levin f rom Trakhtenbrots survey on p ereb or Le

Fortunatelyinthe cas e of maximummatchingthe re

Tr

sults are positive Butpossiblythi s f avourable p os ition

Thi s i s thesituation with so calle d exhaustive s earch

is very s eldom the cas e Perhaps thetwoness of edges

problems including the minimization of Bo olean func

makes the algebraic order for ma tchingrather sp ecial for

tions the s earch for pro ofs of nitelength thedeter

matching in compar i son withthe order of dicultyfor

mination of the i somorphi sm of graphs etc All of

more general combinator ial extremum problems

the s e problems are solved by tr ivial algor ithms entailing

the s equential scanning of all p oss ibilitie s Theop erat

Edmonds di scuss ingthematroid partition problem

ingtimeofthealgorithm i s however exp onential and

Edb

mathematicians nurture the conviction that it i s imp os

Weseeka goodcharacter ization of the minimumnum

sible tond s impler algor ithms

ber of indep endentsetsintowhichthe columns of a ma

G tr ix of can b e partitioned As thecriter ion of go o d odel in a letter tovon Neumann Translated by

for thecharacter ization weapply the pr inciple of theab Arthur S Wens inger with Sor in Istrails guidance The

solute sup ervi sor The good character ization will de or iginal letter i s in the Manuscr ipt Divi s ion of the Li

scr ib e certain information aboutthematr ix whichthe brary of Congre ss Washington DC The translator had

sup ervi sor can require hi s ass i stanttosearchout along acce ss to Jur i s Hartmani s article Ha Di scuss ions

with Hartley Rogers and Philip Scowcroft provided con for theunsolvabilityoftheEntscheidungsproblem

tr ibutions tothe translation n Kn or Kn means of cours e s imply that

thenumber of steps vi savi s dem blossen Probieren can

b e re duce d f rom N tologN or log N Such strong

Dear Mr von Neumann

re ductions do indee d o ccur however in the cas e of

other nite problems eg in the cas e of calculatinga

Tomygreat regret I haveheard aboutyour illness

quadratic re s idue bymeans of rep eated application of

Thenews reached me most unexp ectedly Morgenster n

the law of recipro cityItwould b e intere stingtoknow

had to b e sure told meinthesummer about some inr

for example whatthesituation i s in the cas e of deter

mityyou had b een suer ing butatthetimehethought

miningwhether a numberisaprimenumb er andinthe

thatnoma jor s ignicance was tobeattached to it I

cas e of nitecombinator ial problems how strongly in

hear thatyou have b een undergoing radical treatment

general thenumberofsteps vi savi s the bloss en Pro

in the past s everal months and I am happy thatthi s

bieren can b e re duce d

has achieved thedesired resultsandthatthings are now

goingbetter for you I hope andwishthatyour con

Iwonder if you haveheard thatPosts problem as

dition continue s to improveandthatthelatest me dical

towhether there are degree s of unsolvability amongthe

advance s if p oss ible can lead to a completerecovery

problems y y x where i s recurs ive has b een

solve d in the p os itivebya very youngman named

Since I haveheard thatyou are feeling stronger now

Richard Fr ie db erg The solution i s very elegant Un

Itakethe lib erty of wr itingtoyou aboutamathemat

fortunatelyFriedberg does not wantto pursue graduate

ical problem about whichyour views would intere st me

workinmathematics butinme dicine evidently under

greatly It i s evidentthatone can eas ily construct a Tur

the inuence of hi s f ather

ingmachine which for eachformula F of the pre dicate

calculus and for every natural number n will allowone

Whatisyour opinion bythewayoftheattemptsre

todecideifF has a pro of of length n lengthnumber

cently in vogue again to bas e Analys i s on ramie d

of symb ols Let F nbethenumberofsteps thatthe

typ etheory You are probably aware that in connec

machine require s for thatandletn max F n

F

tion withthi s Paul Lorenzen has progre ss e d as f ar as

The que stion i s how f ast do e s ngrow for an opti

Leb e sgues Measure TheoryBut I b elievethatinim

mal machine One can showthat n Kn If there

portant asp ectsofAnalys i s nonpre dicativemethods of

actually were a machine with n Kn or even only

pro of cannot b e eliminated

with Kn thi s would have cons equence s of the great

est magnitude Thatistosayitwould clearly indicate

Iwould b e very happytohear f rom you p ersonally

that despitetheunsolvabilityoftheEntscheidungsprob

and pleas e let meknowifIcandoanythingat all for

thematician in thecase lem themental eort of thema

you

of ye sorno que stions could b e completely Godels fo ot

note apart f rom thepostulation of axioms replace d byma

With b e st greetings andwishes also toyour wife

chines Onewould indee d haveto s imply s elect an n

Your very devoted

so large that if themachine yields no re sult there

Kurt Godel

would then also b e no reason tothink further aboutthe

problem Now it s eems tome however tobetotally

PS I congratulateyou heartily on the

within the realm of p oss ibilitythat n grows slowly

Kn is the only e stimate For it s eems that n

lit thesimplytesting or trying out exhaustive s earch

or brute force are the currently e stablished terms

that can b e der ive d f rom a generalization of the pro of

The exp onenthere is p ossibly an n



Verr ingerungen to hear

 

Re stsymbol lit I should liketoallowmys elf





severely radically howmuch is not quiteadequateasa de s engeren Funktionenkalkuls lit of thenarrower function

translation calculus





It is worthremarkingthatthis work was donebyFriedberg It is not completely clear if theword is nur onlyornun

in his undergraduate s enior honors thesis now however only s eems to t the context better

 

Godel uses the English word Analysis here not the German Tragweite sugge sts great breadth of range or impact lit

term Analytik carrying p ower far into many elds or areas whereitwould have



eor ie its eect verzwe igteTypenth

 

an ihre Frau Gemahlin is an unusually p oliteandold or large enough



fashione d formula quite in accord withthe formal toneofthis Godel says so langsam wachst which doe s not mean grows

letter to a man clearly highly e steemed byGodel so slowly but more somethinglike accordingly grows slowly



The balance of the p ostscr ipt is missing f rom thephoto where accordinglymeans consider ing thetype of growthof

copyofthedocumentavailable to us nIh ave been discussing

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Pa M S Paterson An intro duction toBoolean

KW M Karchmer and A Wigderson Mono

function complexity Asterisque

tone circuits for connectivity require sup er

logar ithmic depth Proceedings of the th

Ra M O Rabin Degree of dicutly of comput

Annual ACM Symposium on Theory of Com

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sivesets Tech Rep No Hebrew Univer

Ka R M Karp Re ducibility among combinato

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r ial problems Complexity of Computer Com

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Ra M O Rabin Mathematical theory of au

tomata Proceedings of th ACM Symposium

Ko D Kozen Lower b ounds for natural pro of

in Applied Mathematics

systems Proceedings of the th Annual Sym

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RW R Raz and A Wigderson Monotonecir

cuitsform atching require linear depth Pro

ceedings of the nd Annual ACM Symposium

LJK K J Lange B Jenner B Kirsig The

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on Theory of Computing

A logar ithmic hierarchy collaps e s A

Proceedings of the th International Confer

Raa A A Razborov Alower b oundonthe

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monotonenetwork complexityofthe logical

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p ermanent Matematicheskie Zametki

Le H Lebesgue Sur le s fonctions repres en in Russ ian Engli sh translation in

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table s analytiquement Journal de Math Mathematical Notes of the Academy of Sci

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Rab A A Razborov Alower b oundonthe Si M Sipser Atop ological view of some prob

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p ermanent Matematicheskie Zametki maticaSocietatis Janos Bolyai

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Mathematical Notes of the Academy of Sci

Sm R Smolensky Algebraic methods in the

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theory of lower b ounds for Bo olean circuit

complexity Proceedings of th Annual ACM

Ra A A Razborov Lower b ounds on the

Symposium on Theory of Computing

s ize of b ounded depthnetworks overacom

plete basis with logical addition Matematich

eskie Zametki in Russ ian En

Sz R Szelepcsenyi Themetho d of force d enu

gli sh translation in Mathematical Notes of the

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Ra A A Razborov On themetho d of approxi

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Symposium on Theory of Computing

Tr B A Trakhtenbrot A survey of Russ ian

approaches toPerebor bruteforce s earch al

Raa A A Razborov On the di str ibutional com

gor ithms Annals of the History of Computing

plexityofdisjointne ss Proceeding of the In

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ternational ConferenceonAutomata Lan

Ts G S Tseitin On the complexityofder iva

guages and Computation toapp ear in

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Theoretical Computer Science

Constructive Mathematics and Mathemati

cal Logic Part Consultants Bureau New

Rab A A Razborov Applications of matr ix

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methods for thetheory of lower b ounds in

computational complexity Combinatorica

Va L G Valiant Why i s Bo olean complex

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workshop on Boolean Function Complexity

Sa J E Savage Computational workandtime

Durham e dited byMSPaterson Cam

on nitemachines Journal of the A CM

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V o J von Neumann A certain zerosumtwo

Sa V Y Sazanov A logical approachto

p erson game equivalenttotheoptimal ass ign

the problem PNP Mathematical Founda

ment problem Contributions to the Theory of

tions of Computer Science Lecturenotesin

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Computer Science

We I Wegener The Complexity of Boolean

Sc H Scholz Problem Journal of Symbolic

Functions WileyTeubner

Logic

Yaa S Yablonski The algor ithmic dicultie s of

Sh A Shamir IPPSPACE Proceedings of

synthe s izingminimal switching circuits Prob

th Annual ACM Symposium on Theory of

lemy Kibernetiki Moscow Fizmatgiz

Computing toapp ear in the

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netics Pergamon Pre ss

Sh C E Shannon The synthesis of two

Yab S Yablonski On the imp oss ibility of elimi

terminal switching circuits Bel l Systems

natingbrute force s earch in solving some prob

Technical Journal

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Si M Sipser On p olynomial vs exp onential

growth unpubli she d

ao Separatingthe p olynomialtime Ya A C Y

hierarchyby oracle s Proceedings of th An

Si M Sipser Borel s etsand circuit complexity

nual IEEE Symposium on Foundations of

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