Prop ositional Pro of Complexity Past Present

and Future

Paul Beame and

Abstract Pro of complexity the study of the lengths of pro ofs in

prop ositional logic is an area of study that is fundamentally connected

b oth to ma jor op en questions of computational complexity theory and

to practical prop erties of automated theorem provers In the last decade

there have b een a number of signicant advances in pro of complexity

lower b ounds Moreover new connections b etween pro of complexity

and have b een uncovered and the interplay b etween

these two areas has b ecome quite rich In addition attempts to extend

existing lower b ounds to ever stronger systems of pro of have spurred the

introduction of new and interesting pro of systems adding b oth to the

practical asp ects of pro of complexity as well as to a rich theory This

note attempts to survey these developments and to lay out some of the

op en problems in the area

Introduction

One of the most basic questions of logic is the following Given a uni

versally true statement tautology what is the length of the shortest pro of

of the statement in some standard axiomatic pro of system The prop osi

tional logic version of this question is particularly imp ortant in computer

science for b oth theorem proving and complexity theory Imp ortant related

algorithmic questions are Is there an ecient algorithm that will pro duce

a pro of of any tautology Is there an ecient algorithm to pro duce the

shortest pro of of any tautology Such questions of theorem proving and

complexity inspired Co oks seminal pap er on NPcompleteness notably enti

tled The complexity of theoremproving pro cedures and were contem

plated even earlier by Go del in his now wellknown letter to von Neumann

see

The ab ove questions have fundamental implications for complexity the

ory As formalized by Co ok and Reckhow there exists a prop ositional

pro of system giving rise to short p olynomialsize pro ofs of tautologies

if and only if NP equals coNP Co ok and Reckhow were the rst to prop ose

a program of research aimed at attacking the NP versus coNP problem by

systematically studying and proving strong lower b ounds for standard pro of

BEAME AND PITASSI

systems of increasing complexity This program has several imp ortant side

eects

First standard pro of systems are interesting in their own right Almost

all theoremproving systems implement a deterministic or randomized pro

cedure that is based on a standard prop ositional pro of system and thus

upp er and lower b ounds on these systems shed light on the inherent com

plexity of any theoremproving system on which it is based The most

striking example is Resolution on which almost all prop ositional theorem

provers and even rstorder theorem provers are based

Secondly and of equal or greater imp ortance lower b ounds on standard

pro of systems additionally prove that a certain class of algorithms for the

satisability problem will fail to run in p olynomialtime

This program has led to many b eautiful results as well as to new con

nections with circuit complexity within the last twenty years In this article

we will try to highlight some of the main discoveries with emphasis on the

interplay b etween logic circuit complexity theory and combinatorics that

has arisen We omit all pro ofs see for a quite readable survey that

includes detailed pro ofs of many of the earlier results

In section we dene various pro of systems that will b e discussed

throughout this article In section we review some of the main lower

b ounds that have b een proven for standard pro of systems emphasizing the

combinatorial techniques and connections to circuit complexity that have

b een shown Finally in section we list some of the main op en questions

and promising directions in the area

Prop ositional pro of systems

What exactly is a prop ositional pro of Co ok and Reckhow were p ossibly

the rst to make this and related questions precise They saw that it is

useful to separate the idea of providing a pro of from that of b eing ecient

Since there are only nitely many truth assignments to check why not allow

the statement itself as a pro of What extra value is there in a lled out

truth table or a derivation using some axiominference scheme The key

observation is that a pro of is easy to check unlike the statement itself Of

course we also need to know the format in which the pro of will b e presented

in order to make this check That is in order to identify some character

string as a pro of we must see it as an instance of some general format for

presenting pro ofs Therefore a propositional proof system S is dened to b e

a p olynomialtime computable predicate S such that for all F

F TAUT p S F p

That is we identify a pro of system with a p olynomial time pro cedure that

checks the correctness of pro ofs Prop erty ensures that the system S is

Co ok and Reckhows denition is formally dierent although essentially equivalent

to this one They dene a pro of system as a p olynomialtime computable onto function



f TAUT which can b e thought of as mapping each string viewed as p otential pro of

PROPOSITIONAL PAST PRESENT AND FUTURE

logically b oth sound and complete The complexity comp of a prop ositional

S

pro of system S is then dened to b e the smallest b ounding function b N

N on the lengths of the pro ofs in S as a function of the tautologies b eing

proved ie for all F

F TAUT pjpj bjF j S F p

Ecient pro of systems corresp ond to those of p olynomial complexity these

are called pbounded

Given these denitions many natural questions arise How ecient are

existing pro of systems How can one compare the relative eciencies of

pro of systems Can one classify pro of systems using reduction as we do

languages Is there a pro of system of optimal complexity up to a p olyno

mial

The key to ol for comparing pro of systems is psimulation A pro of sys

tem T psimulates a pro of system S i there is a p olynomialtime computable

function f mapping pro ofs in S into pro ofs in T that is for all F TAUT

S F p T F f p We use nonstandard notation and write S T in

p

O

this case Clearly it implies that comp comp One says that T

T

S

weakly psimulates S i we have this latter condition but we do not know if

such a reducing function f exists One says that S and T are pequivalent

i each psimulates the other Obviously two pequivalent pro of systems

either are b oth pb ounded or neither is

Frege and extendedFrege pro ofs Co ok and Reckhow did more

than merely formalize the intuitive general notions of the eciency of prop o

sitional pro ofs They also identied two ma jor classes of pequivalent pro of

systems which they called Frege and extendedFrege systems in honor of

Gottlob Frege who made some of the rst attempts to formalize mathe

matics based on logic and set theory and whose work is now b est

known as the unfortunate victim of Russells famous paradox concerning the

set of all sets that are not members of themselves

A F is dened in terms of a nite implicationally complete

enough to derive every true statement set A of axioms and inference rules

F

A A

k

The general form of an inference rule is written as where A A

k

B

and B are prop ositional formulas the rule is an axiom if k A formula

H follows from formulas G G using this inference rule if there is a

k

consistent set of substitutions of formulas for the variables app earing in

for i k and H B the rule such that G A

i

i

For a Frege system F a typical set of axioms A might include the

F

or identity as well as the cut rule axiom of the excluded middle

AA AA

AAB AC C B

or modus ponens A pro of of a tautology F in F consists

AB B

of a nite sequence F F of formulas called lines such that F F

r

onto the tautology it proves The analogous function f in our case would map F p to

F if S F p were true and would map it to a trivial tautology x x otherwise In the

converse direction one would dene S F p to b e true i f p F

BEAME AND PITASSI

and each F either is an an instance of an axiom in A or follows from some

j F

previous lines F F for i i j using some inference rule of A

i i F

k

k

An equivalent way of using a Frege system works backwards from F to

derive a contradiction such as p p The size of a Frege pro of is typically

dened to b e the total number of symbols o ccurring in the pro of The pro of

can also b e treelike or daglike in the treelike case each intermediate

formula can b e at used at most once in subsequent derivations in the more

general daglike case an intermediate formula can b e used unboundedly

many times Kra jcek has shown that for Frege systems there is not

much loss in eciency in going from a daglike pro of to a treelike pro of

Various Frege systems which have also b een called Hilb ert systems or

Hilb ertstyle deduction systems app ear frequently in logic textb o oks How

ever it is dicult to nd two logic textb o oks that dene precisely the same

such system Co ok and Reckhow showed that these distinctions do not mat

ter namely all Frege systems are pequivalent Furthermore they showed

that Frege systems were also pequivalent to another class of pro of systems

app earing frequently in logic textb o oks called sequent calculus or Gentzen

systems These systems manipulate pairs of sequences or sets of formulas

written as where and are sequences of formulas with the in

tended interpretation b eing that the conjunction of the formulas in implies

the disjunction of the formulas in Therefore to prove a formula F in the

sequent calculus one proves the corresp onding sequent F

In any sequent calculus system there is an underlying basis set of connec

tives B B can consist of unbounded fanin or b ounded fanin connectives

and the only requirement is that B b e a complete basis Typically B is the

standard basis consisting of and The only initial sequent is A A

for any formula A dened over B

Additionally there are three types of rules for deriving new sequents from

previous ones i structural rules ii logical rules and iii the cut rule

Structural rules do not actually manipulate the underlying formulas of the

sequent but instead they allow one to op erate on the sequences of formulas

as sets rather than sequences A typical structural rule is contraction which

allows us to derive A from A A The logical rules allow

us to build larger formulas from previous ones according to the truthtable

denition of each of the connectives in B More precisely for each connective

in B there are two logical rules one for introducing the connective on the

left and one for introducing the connective on the right For example if

is in B then the left rule would allow us to derive A B from

A B and the right rule would allow us to derive A B

from A and B

Of particular imp ortance for sequent calculi is the cut rule From A

and A derive Gentzen showed that the cut rule is

unnecessary but it may have a huge impact on pro of length If the cut

rule is removed then one obtains a much weaker system than Frege systems

This system is known as analytic tableaux or cutfree LK While analytic

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

tableaux are often more ecient than truth tables somewhat surprisingly

they cannot even psimulate truth tables b ecause their worstcase complexity

n

is n rather than O n for truth tables see

The other ma jor class of pro of systems identied by Co ok and Reck

how includes systems that p ermit one to extend Frege pro ofs by introducing

new prop ositional variables to stand for arbitrary formulas app earing in the

pro of All such systems which are called extendedFrege are pequivalent

to each other These systems app ear to b e much more succinct than Frege

pro ofs and can conveniently express many mathemtical arguments quite nat

urally It has b een shown by Dowd unpublished and also by Kra jcek and

Pudlak that extendedFrege pro ofs are also pequivalent to substitution

Frege sF pro ofs In sF one is allowed to use each line of a Frege pro of

immediately as if it were an axiom that is new lines follow from existing

ones by substituting arbitrary formulas for their prop ositional variables

CNF refutations and Resolution Using the construction of

the standard reduction from SAT to SAT one can take an arbitrary prop o

sitional formula F and convert it to a CNF or CNF formula in such a way

that it has only p olynomially larger size and is unsatisable i the original

formula was a tautology To do this one adds new variables x to stand

A

for each of its subformulas A and clauses to sp ecify that the value at each

connective is computed correctly as well as one clause of the form x In

F

this way one can consider any sound and complete system that pro duces

refutations for CNF formulas as a general prop ositional pro of system

In the s several such refutation systems were developed The most

p owerful of these systems is Resolution which in its prop ositional form

is a very sp ecialized form of a Frege pro of system that can only manipulate

clauses and has only one inference rule the resolution rule

A x B x

A B

called resolution on variable x and is a sp ecial form of cut The contradictory

formula to b e derived is simply an empty clause which can b e seen as the

result of resolving on clauses p and p

Resolution was predated by two systems known as DavisPutnam pro ce

dures which are still the most widely used in prop ositional theorem proving

The general idea of these pro cedures is to convert a problem on n variables

to problems on n variables by eliminating all references to some variable

The former which we call DP do es this by applying all p ossible uses

of the resolution rule on a given variable to eliminate it The latter

which we call DLL and is the form used to day branches based on the p os

sible truth assignments to a given variable although at rst this do es not

lo ok like Resolution it is an easy argument to show that this second form is

equivalent to the sp ecial class of treelike Resolution pro ofs As a pro of sys

tem Resolution is strictly stronger than DP which is strictly stronger

BEAME AND PITASSI

than DLL The reasons for DLLs p opularity are related to its pro of

search prop erties which we discuss b elow

A more general but still restricted form of Resolution is called regular

Resolution which was introduced and analyzed by Tseitin A regular

Resolution refutation is a Resolution refutation whose underlying directed

acyclic graph has the prop erty that along each path from the ro ot empty

clause to a leaf initial clause each variable is resolved up on at most once

It is not to o hard to see that any minimal treelike Resolution refutation

is regular also the DP algorithm trivially pro duces a regular Resolution

pro of For a p erio d of years after Tseitins analysis although there were

improvements in the b ounds derived for regular resolution understand

ing general resolution seemed out of reach

Circuitcomplexitybased Pro of Systems One of the most

p owerful insights that has developed in the study of prop ositional pro of com

plexity is that there is a parallel b etween circuitbased complexity classes

and prop ositional pro of systems This insight was rst made by Co ok

where he established a close connection b etween p olynomialsize extended

Frege pro ofs and pro ofs using p olynomialtime reasoning In more famil

iar terms he showed that extended Frege pro ofs are the nonuniform analog

in the same way that of p olynomialtime pro ofs systems such as PV or S

p olynomialsize circuits are the nonuniform analog of the

P The same intuition applied to other pro of systems was subsequently

used to obtain other imp ortant results by Ajtai and Buss More

generally the parallel b etween circuit classes and pro of systems has greatly

broadened the range of pro of systems that are typically considered and has

led to new techniques for analyzing pro of systems and circuit classes We

rst briey outline the general form of this corresp ondence and then we

examine and rene some of the pro of systems ab ove in this light

Typically circuitbased complexity classes are dened by giving a struc

tural characterization of a class of circuits and then placing some b ound on

the size of the circuits involved For many circuitbased complexity classes

C this size b ound is p olynomial For any such class C we can consider a

Fregestyle pro of system whose lines are circuits with the same structural

characterization as the circuits dening C but which do not necessarily sat

isfy the size b ound The set of circuits of this type must b e closed under

substitution into any formula app earing in an axiom or inference rule of

the system Although the notation is not precise in general we call such a

pro of system CFrege Since we can always assume that our goal formula

is in CNF there is no problem representing it in CFrege for virtually any

nontrivial C

For example the complexity class corresp onding to the set of all p olyno

mial size prop ositional formulas is NC so NC Frege would just b e another

name for Frege It is also easy to observe that the extension rule of extended

Frege pro ofs builds circuits in terms of the original prop ositional variables

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

in which the new variables give the values computed by subcircuits Thus

extendedFrege could also b e called Pp olyFrege Resolution is a Frege sys

tem that manipulates simple clauses or alternatively terms if one views it

dually as a pro of system rather than a refutation system but there isnt a

convenient name for this complexity class of depth formulas Note that

with this intuition it is clear that extendedResolution the natural gener

alization of Resolution that p ermits the introduction of new prop ositional

variables is pequivalent to extendedFrege since it clearly can generate any

circuit in Pp oly with a p olynomial number of extensions

A very natural new pro of system arising from this corresp ondence is a

generalization of Resolution to arbitrary constantdepth unbounded fanin

formulascircuits for constant depth there is no dierence b etween p olynomial

size formulas and circuits This new system AC Frege also known as

constant or b oundeddepth Frege rst arose from the study of b ounded

rstorder arithmetic the word b ounded in that case derives from p oly

nomial complexity b ounds rather than from the depth AC Frege pro ofs

derive from translations of pro ofs in certain systems of b ounded rstorder

arithmetic which are restrictionsextensions of Peano arithmetic

that mo del feasible inference Although these motivations are imp ortant

in the study of constructive logic space considerations do not p ermit us

to go into detail ab out them we refer the interested reader to where

many of these connections are describ ed in detail Typically lower b ounds

on the size of AC Frege pro ofs can show that related rstorder tautologies

are unprovable in a given system of b ounded arithmetic These transla

tions are analogous to those of Furst Saxe and Sipser and Sipser

which convert oracle computations in the p olynomial hierarchy to constant

depth unbounded fanin circuits In addition to AC Frege pro of systems

for AC pFrege and TC Frege and their sub classes have also b een studied

extensively

The corresp ondence b etween circuit classes and pro of systems has not

only b een fruitful in developing ideas for new pro of systems It has also

b een the avenue for applying circuit lower b ound techniques to prop ositional

pro ofs Some of the ma jor progress of the last decade building on the original

insight due to Ajtai has b een in achieving lower b ounds for AC

Frege pro of systems and their extensions In general the intuition for this

approach is that any tautology that needs to use in its pro of some concept

that is not representable in complexity class C will not b e eciently provable

in CFrege

The State of the Art in Pro of Complexity

We give a quick tour of the state of the art in prop ositional pro of com

plexity As is always inevitable in a short survey such as ours space consid

erations do not p ermit us to do justice to the full range of results available

Although we will not always emphasize the connections very strongly many

BEAME AND PITASSI

of these results have b een inspired or derived from metho ds in circuit com

plexity and conversely lower b ounds for particular pro of systems imply lower

b ounds for restricted families of algorithms for solving SAT We exp ect this

crossfertilization to continue

In keeping with the general program of proving that pro of systems are

not ecient much work has b een devoted to proving lower b ounds on the

sizes of pro ofs of sp ecic tautologies One can broadly characterize several

classes of formulas for which these lower b ounds have b een shown The

rst two classes consist of the prop ositional translations of combinatorial or

counting principles The rst of these involves highly symmetric counting

principles A canonical example here is the translation of the pigeonhole

principle which prohibits functions from m to n for m n as an unsat

m

isable CNF formula PHP with mn atoms p denoting whether or not

ij

n

n

i is mapp ed to j Related principles include the counting principles C ount

p

for n mo d p which express the prop erty that n cannot b e p erfectly

m

partitioned into sets of size p and onto PHP which only prohibits bijec

n

tions rather than all functions The second class consists of much less

symmetric counting principles such as the o ddcharged graph principles for

b oundeddegree graphs one for each graph which express the prop erty that

the sum of the degrees in any of its subgraphs is even these are of partic

ular interest when the underlying graph is a b oundeddegree expander

The third class of formulas are mintermmaxterm formulas asso ciated with

any monotone function which express the fact that any minterm and any

maxterm of such a function must overlap these are of particular interest

when the function is known to require exp onentialsize monotone circuits

A related family of formulas is obtained by taking a language that is in

NP coNP and writing the formula expressing the fact that any instance

x cannot have b oth a yes witness and a no witness Finally there are

k CNF formulas randomly chosen from an appropriate distribution

We summarize known b ounds for these formulas by considering the var

ious pro of systems one by one Four basic metho ds can b e identied for

proving these lower b ounds the b ottleneck counting metho d the

metho d of restrictions the interpolation metho d algebraic metho ds

The rst three metho ds are quite sp ecic whereas the last metho d is the

youngest probably the most p owerful and already has many facets

Some of the state of our knowledge of pro of complexity lower b ounds

can b e summarized by the following chain

Resolution AC Frege AC pFrege TC Frege

p p p

The separations match all the ma jor circuit complexity separations known

with the notable exception of the last one However the prosp ect for pro of

complexity seems b etter than that for circuit complexity The results of

indicate that to make further progress in circuit lower b ounds will likely re

quire very new nonconstructive techniques However such barriers do not

currently exist in pro of complexity that is proving sup erp olynomial lower

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

b ounds for Frege systems might b e no harder than what we currently know

how to do The techniques used to obtain some of the most recent results

in pro of complexity use insights from areas of apparently un

related to those applied to show circuit complexity b ounds

Resolution Resolution is the most wellstudied mo del Exp o

nential lower b ounds are now known for all of the ma jor classes of formulas

listed ab ove The rst sup erp olynomial lower b ound for Resolution was ob

tained by Tseitin in the s for the o ddcharged graph tautologies in the

sp ecial case of regular Resolution Interestingly obtaining an improve

ment of this b ound to an exp onential one by Galil was a driving force

b ehind some of the early work in the development of the theory of expander

graphs

There was a year gap b efore the rst sup erp olynomial lower b ound

for pro ofs in general Resolution was obtained by Haken who showed ex

p onential lower b ounds for the pigeonhole principle Subsequently exp onen

tial b ounds have also b een shown for the o ddcharged graph formulas

random k CNF formulas with various clausevariable ratios

and mintermmaxterm formulas The pro ofs of all of the strongest forms

of these b ounds for Resolution other than those for the mintermmaxterm

formulas involve a technique known as bottleneck counting due to Haken

In this metho d one views the pro of as a directed acyclic graph of clauses

and views the truth assignments as owing from the ro ot of the directed

acyclic graph to a leaf where an assignment ows through a clause C if and

only if i it ows through the parent clause of C and ii the assignment

falsies C Each assignment can b e seen to ow through a unique path in any

Resolution refutation The idea is to show that for the formula in question

there must exist a large set of truth assignments with the prop erty that

each must pass through a large clause Since a large clause cannot falsify

to o many assignments this implies that there must exist many large clauses

and hence the pro of must b e large

An essential lemma in any b ottleneck counting argument is to show

that any Resolution refutation of F must involve a large clause Recently

it has b een shown using ideas from that for a suitable choice of

parameters this lemma is also sucient namely any Resolution refutation

of small size can b e converted into a refutation with only small clauses

Another metho d used to obtain exp onential Resolution lower b ounds

used for example for the mintermmaxterm formulas is the metho d of in

terp olation which will b e discussed in section

In addition to lower b ounds for general resolution there is also practi

cal interest in understanding the b ehavior of the sp ecial cases of DP and

DLL algorithms See Random k CNF formulas have b een of

particular interest in this regard and there is a variety of results giving more

precise b ounds on their prop erties b oth as pro of systems and as satisability

algorithms at various clausevariable ratios

BEAME AND PITASSI

AC Frege systems and their extensions While Hakens b ound

for resolution was a ma jor breakthrough it is the pap er by Ajtai giving

sup erp olynomial lower b ounds for pro ofs of the pigeonhole principle in AC

Frege systems that has formed the basis of much of the research in pro of

complexity over the last decade As mentioned ab ove this gave the rst

connection b etween the techniques of circuit complexity and those of pro of

complexity Ajtais result has b een improved to exp onential lower b ounds

and these apply to all the symmetric counting principles in the rst class

ab ove eg Despite this success no lower b ounds are known for

AC Frege pro ofs of formulas in the other classes ab ove although there are

certain other tautologies for which we know a sup erp olynomial separation

as a function of the depth

The restriction method by which the lower b ounds ab ove were shown

comes from Ajtais pap er The essence of this idea is to apply restrictions

to try to simplify each of the formulas in the pro of yet leave the input tau

tology still highly nontrivial Thus the basic metho d is very similar to the

randomrestriction metho d used to show that AC cannot compute parity

However it is necessarily more complex Since the circuits app earing in a

sound pro of always compute the constant function the usual simplica

tion induced by restrictions applied to circuits must b e replaced by one that

includes a form of approximation as well Using this metho d one shows

that if the pro of is to o short then there exists a restriction such that after

applying the restriction to the short pro of what results is a very trivial pro of

of a formula of the same basic form but on a reduced number of variables

Then a contradiction can b e reached by showing that such a trivial pro of

cannot exist

Once it is known that the pigeonhole principle is not provable in AC

n

Frege one can immediately obtain a stronger system by adding PHP as

n

an axiom schema for arbitrary n ie one is p ermitted to derive lines in the

n

pro of by substituting arbitrary formulas for the variables of some PHP

n

n

do es not have Ajtai showed that even in this stronger system C ount

p olynomialsize pro ofs There is now some quite interesting structure known

Frege systems in ab out the relative pro of strength of these augmented AC

which some axiom scheme is added to the basic system

n

was added as an axiom schema In the system ab ove in which PHP

n

n

principles in fact are now known to require exp onen all of the C ount

p

n

axiom schema any tial size pro ofs conversely given any C ount

p

log n

np

n

b oundeddepth Frege pro ofs of PHP or onto PHP requires ex

n

n

n n

p onential size but onto PHP is trivial Thus PHP is exp onen

n n

n

tially stronger than onto PHP Quite precise conditions are now known

n

4 used the language of forcing to describ e this approximation the cleanest way of

expressing this approximation is in terms of socalled k evaluations which are describ ed

in 13 , 92, 11 and are a mo dication of the denitions in 63

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

n

under which exp onential separations exist b etween the various C ount prin

p

ciples in this context

The pro ofs for these results b egin with the same restriction metho d

strategy describ ed ab ove However with the additional axiom schema there

is the further requirement that each line in the pro of where the axiom

schema is applied b e simplied just as the rest of the pro of is The need to

prove this latter requirement motivated the introduction of Nullstellensatz

pro ofs see b elow and the exp onential separations are all derived from

lower b ounds on the degrees of such pro ofs

The metho d involving Nullstellensatz degree lower b ounds is not the

only one that has b een used for obtaining such separations In proving the

n

rst sup erp olynomial separations b etween the various C ount Ajtai

p

used certain prop erties of concisely represented symmetric systems of linear

equations which he proved using structural results in the theory of repre

sentations of the symmetric group over GF p to show that the axiom

schemas are appropriately simplied This technique has recently b een em

ployed to give sup erp olynomial lower b ounds for very p owerful algebraic

pro of systems see sections and

Cutting Planes and Interpolation The metho d of using cut

ting planes for inference in the study of p olytop es in integer programming

was rst describ ed by Gomory mo died and shown to b e complete by

Chvatal and rst analyzed for its eciency as a pro of system in

It is one of two classes of pro of systems developed by representing unsatis

ably problems as integer or programming problems the other b eing a

collection of systems due to Lovasz and Schrijver which are describ ed

in detail in the op en problems section

Cutting Planes pro ofs manipulate integer linear inequalities One can

add inequalities or multiply them by p ositive constants but the truly p ow

erful rule is the rounded division rule

ca x ca x ca x b a x a x a x dbce

k k k k

A refutation of a set of integer linear inequalities is a sequence of inequali

ties where each inequality is either one of the original ones or follows from

previous inequalities by applying one of the ab ove rules and where the nal

inequality is To refute a CNF formula one rst converts each clause

into an equivalent integer linear inequality Cutting planes pro ofs can sim

ulate Resolution eciently and easily prove all of the symmetric counting

tautologies mentioned ab ove

Exp onential lower b ounds have b een shown for cutting planes pro ofs of

the mintermmaxterm formulas using a metho d called interpolation In this

metho d one b egins with an unsatisable formula of the form F Ax z

B y z where we view x and y each as a vector of private variables and

z as a vector of shared variables After any assignment to the common

z variables is made in the remaining formula Ax B y it must b e

the case that either A is unsatisable or B is unsatisable The asso ciated

BEAME AND PITASSI

interpolation problem for F takes as input an assignment to the common

variables and outputs A only when Ax is unsatisable and outputs B

only when B y is unsatisable Of course sometimes b oth A and B may

b e acceptable answers This problem is called the interpolation problem

since it is equivalent to Craig Interpolation In the typical formulation G is

a tautological formula in the form A x z B y z In our formulation

F is G A is A and B is B

For arbitrary F it may b e very dicult to solve the interpolation prob

lem asso ciated with F if L is a decision problem in NP coNP then if

we dene Ax z to b e a formula stating that x is a yes witness for the

instance z and let B y z state that y is a no witness for z then the ex

istence of a p olynomial time algorithm for the interpolation problem would

have surprising consequences for complexity theory See However it

might still b e that whenever F of the ab ove form has a short refutation

in some pro of system S the interpolation problem asso ciated with F has a

p olynomialtime solution This p ossibility was rst suggested by Kra jcek

If this situation exists for pro of system S then we say that S has the fea

sible interpolation property There is also a monotone version of feasible

interpolation Namely F Ax z B y z is monotone if z o ccurs only

p ositively in A and in this case the interpolation problem is monotone S

has the monotone feasible interpolation prop erty if whenever F is mono

tone and has a short S pro of then the asso ciated interpolation problem has

p olynomialsize uniform monotone circuits

Razb orov and indep endently Bonet Pitassi and Raz were the

rst to use the ab ove idea to obtain exp onential lower b ounds for certain

pro of systems In a formula formalizing that SAT do es not have

p olynomialsize circuits is constructed with the prop erty that the asso ciated

interpolant problem has no p olynomialtime circuits under cryptographic

assumptions constructs a monotone formula with the prop erty that

the asso ciated interpolant problem has no monotone p olynomialtime cir

cuits under no complexity assumptions On the other hand they show

that smallweight Cutting Planes has monotone feasible interpolation thus

implying exp onential lower b ounds Pudlak signicantly extended the

ab ove ideas by showing that unrestricted Cutting Planes also has a form of

monotone feasible interpolation This combined with new exp onential lower

b ounds for monotone real circuits gives unconditional lower b ounds

for Cutting Planes

Feasible interpolation can thus give very go o d lower b ounds for many

pro of systems sometimes only under cryptographic assumptions In addi

tion to the unconditional lower b ounds mentioned ab ove conditional lower

b ounds have b een shown for all of the following systems Resolution Cutting

Planes Nullstellensatz Polynomial Calculus as well as any pro of

system where the underlying formulas in the pro of have small probabilistic

communication complexity Unfortunately there are strong negative

results showing that the interpolation metho d cannot b e applied to give

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

lower b ounds for the following pro of systems again under various crypto

graphic assumptions Extended Frege Frege TC Frege and even

AC Frege

A disadvantage of the interpolation metho d for obtaining lower b ounds

is that it applies only to formulas of a very sp ecial form Thus for example

nothing is known ab out the length of the shortest Cutting Planes pro ofs for

either the o ddcharged graph formulas or for random formulas

Algebraic pro of systems The Nullstellensatz and Polyno

mial Calculus pro of systems are based on a sp ecial case of Hilb erts

famous Nullstellensatz which relates the question of the nonexistence of si

multaneous zeros of a family of multivariable p olynomials in certain elds to

the question of the existence of co ecient p olynomials witnessing that is in

the ideal generated by these p olynomials In fact they use a generalization

of this sp ecial case to rings as well as elds

To use this relation one rst expresses an unsatisable Bo olean formula

as a system of constantdegree p olynomial equations in some p olynomial

ring For the prop ositional versions of the symmetric counting principles

m

these translations are quite natural for example PHP when translated

n

P

0

has p olynomials x for each i m as well as x x for

ij ij

i j

j n

each i i m and j n More generally there are natural lowdegree

translations of arbitrary CNF formulas that are similar to those used in

probabilistically checkable pro ofs PCP To use these mechanisms

to detect solutions only we add the equations x x for each

variable x Hilb erts Nullstellensatz implies that such a system fQx g

do es not have a solution if and only if there exists a family of multivariate

P

p olynomials P such that P xQ x It is easy to see that the

i i

i

x x equations guarantee that degree at most n is sucient

The complexity in the Nullstellensatz pro of system is simply the size of

the dense representation of the co ecient p olynomials and thus of the form

O d

n where d is the largest degree required In the Polynomial Calculus

pro of system one do es not need to explicitly write down these p olynomials

all at once but rather one can give a derivation that demonstrates their

existence involving p olynomials of low degree along the way The size of

each of these p olynomials is also based on this dense representation

A variety of Nullstellensatz lower b ounds are known for the symmetric

counting principles but no Nullstellensatz lower b ounds are known for the

other formulas ab ove For example the o ddcharged graph tautologies have

easy pro ofs over Z

One drawback of the Nullstellensatz system although not Polynomial

Calculus is that a simple chain of inference of length n using mo dus p onens

requires nonconstant degree log n It is even p ossible that certain

principles may b e proved using degree using Polynomial Calculus but

p

require degree n Nullstellensatz pro ofs

BEAME AND PITASSI

n

Although it is trivial to prove C ount in constant degree in Z degree

r

r

m n

lower b ounds of n have b een shown in Z for PHP C ount for

r

n s

nr

most s r and onto PHP So far all of the Nullstellensatz

n

b ounds mentioned were shown using the notion of a dual design

This is typically a combinatorial construction that guarantees that one can

derive a solution to a set of dual equations that express the co ecient of

P

the constant term in P Q as a linear combination of higher degree

i i

i

P

co ecients in the indeterminate co ecients of the P Since P Q

i i i

i

is supp osed to represent the p olynomial this would b e imp ossible

introduced a nice technique used in that makes such designs easier to

construct

m

Using a more general class of designs the degree b ounds for PHP were

n

were improved to degree n by Razb orov Remarkably Razb orovs

result also applies to the Polynomial Calculus pro of system and was not

only the rst nontrivial lower b ound shown for that system but also is one

of strongest known for the Nullstellensatz system It involves an explicit

m

computation of the Gro ebner basis of the ideal generated by the PHP

n

equations Razb orov also extends this lower b ound to obtain stronger nearly

linear degree lower b ounds in Polynomial Calculus for related tautologies

as a function of the number of variables

Recently Kra jicek has shown nonconstant degree lower b ounds for

n

C ount in the Polynomial Calculus pro of system over Z by extending the

r

s

results of Ajtai regarding symmetric linear equations and the structure of

representations of the symmetric group mentioned earlier

AC r Frege systems We have already seen how one can extend

AC Frege pro ofs by adding axioms for counting mo dulo r A far more gen

eral and in some sense more natural way to add the p ower of mo dular

counting to a pro of system is to include it fundamentally in the structure of

the ob jects ab out which one reasons that is to introduce mo dular count

ing connectives into the lines of the pro ofs themselves and new inference

rules for manipulating these formulas AC r Frege is precisely such a sys

tem Even the Polynomial Calculus pro of system mo dulo r may b e viewed

as a subsystem of an AC r Frege pro of system in which all the mo dular

connectives are at the top

At present there are no lower b ounds known for AC r Frege even when

r is a prime One program for obtaining such b ounds was laid out in

where it is shown how to convert an AC pFrege pro of of F to a Polynomial

Calculus pro of of a system that involves the p olynomials for F plus certain

low degree extension p olynomials which mimic the low degree approxima

tions used by for AC p lower b ounds

Frege systems and TC Frege systems Just as AC r Frege

pro ofs include counting mo dulo r as a rstclass concept so TC Frege pro ofs

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

include counting up to a threshold as a rstclass concept Lo osely sp eak

ing TC Frege pro ofs are pro ofs where the underlying class of formulas are

smallweight constantdepth threshold formulas Thus for example Cut

ting Planes pro ofs can b e viewed as a sp ecial case of restricted TC Frege

pro ofs

We only have partial results on TC Frege pro ofs Maciel and Pitassi

have shown pro oftheoretic analogues of circuit complexity constructions

to relate TC and AC pFrege pro ofs In particular they show how one can

convert p olynomialsize AC Frege pro ofs into restricted quasip olynomial

size depth TC Frege pro ofs

The p otential list of candidate hard problems for TC Frege is quite

short in part b ecause there are ecient TC Frege pro ofs for so many for

mulas for which lower b ounds are known in other systems and b ecause the

basic techniques for dealing with these formulas fundamentally break down

There are p olynomialsize TC Frege pro ofs of all of the symmetric count

ing principles as well as the o ddcharged graph principles And

as mentioned ab ove the interpolation metho d cannot apply to TC Frege

assuming that factoring Blum integers is hard Since TC Frege pro ofs

are sp ecial cases of Frege pro ofs the same problems apply to Frege pro ofs

as well Some candidates for hard tautologies for these systems have b een

suggested in

Optimal pro of systems Research in pro of complexity was orig

inally motivated in part as a way of proving NP coNP by proving su

p erp olynomial lower b ounds for increasingly p owerful pro of systems An

imp ortant question is whether such a chain of results will ever actually lead

us to a pro of of NP coNP In other words is there an optimal pro of

system This question is quite imp ortant and is still op en However some

partial results have b een obtained relating this existence to the

equivalence of certain complexity classes

Pro of Search While lengths of pro ofs are imp ortant it is also

imp ortant to b e able to nd pro ofs quickly Clearly if we know that a

pro of system S is not p olynomiallyb ounded then no ecient deterministic

pro cedure can exist that will pro duce short pro ofs of all tautologies But is

it p ossible to nd short pro ofs of all tautologies that have short pro ofs To

this end denes a pro of system S to b e automatizable if there exists a

deterministic algorithm that takes as input a tautology F and outputs an

S pro of of F in time p olynomial in the size of the shortest S pro of of F

Automatizability is very imp ortant for automated theorem proving and

is very similar to an older concept k provability The k provability problem

for a pro of system S is as follows The input is a formula F and a number

k and the input should b e accepted if and only if F has an S pro of of size at

most k Clearly a pro of system S is not automatizable if the k provability

problem cannot b e approximated to within any p olynomial factor

BEAME AND PITASSI

Which pro of systems are automatizable and for which pro of systems is

k provability hard It has b een shown that the k provability

problem is NP hard for essentially every standard prop ositional pro of sys

tem and furthermore using the PCP theorem that the k provability prob

lem cannot b e approximated to within any constant factor unless P NP

The ab ove hardness results show that nding go o d estimates of the pro of

length is hard even for very simple pro of systems such as Resolution But

it may still b e that most pro of systems are automatizable However under

stronger assumptions one can show that many pro of systems are not au

tomatizable These results are shown by exploiting a connection b etween

interpolation and automatizability In particular it can b e shown that if a

pro of system S do es not have feasible interpolation then this implies that S

is not automatizable Thus feasible interpolation gives us a formal tradeo

b etween the complexitystrength of S and the ability to nd short pro ofs

quickly Using this connection it has b een shown that AC Frege pro ofs as

well as any pro of system that can psimulate AC Frege is not automatiz

able under cryptographic assumptions See

Are there any pro of systems that are automatizable Both the Nullstel

lensatz and Polynomial Calculus pro of systems as well as DLL are actually

search pro cedures as well as nondeterministic algorithms But unlike DLL

the Nullstellensatz and Polynomial Calculus algorithms are guaranteed to

nd short pro ofs if they exist that is they are automatizable Any

pro of of degree d in n variables may b e found using linear algebra in time

O d

n Nullstellensatz pro ofs may b e exp onentially smaller than b ounded

depth Frege pro ofs but they also may b e exp onentially larger than Resolu

tion pro ofs Polynomial Calculus pro ofs are at worst quasip olynomially

larger than the b est pro ofs under any DLL algorithm and from this one can

derive a metho d of searching for short DLL pro ofs that is guaranteed to

succeed In this algorithm is converted to a direct search pro ce

dure for such pro ofs It app ears fruitful to investigate Polynomial Calculus

pro ofs as a theoremproving to ol to see if they can b e rened to comp ete

with DLL algorithms Preliminary results of this form app ear in

Op en Problems

Find hard tautologies for TC Frege and Frege No examples of

tautologies are known for which Frege pro ofs even require a sup erlinear

number of distinct subformulas One dicult problem that is faced when

trying to prove lower b ounds for Frege or Extended Frege systems is that

there is a surprising lack of hard candidate tautologies Most of the lower

b ounds proven thus far have b een for various counting principles all of

which have p olynomialsize TC Frege or Frege pro ofs Even the minterm

maxterm formulas can b e viewed as an application of a counting principle

Some candidate hard examples have b een suggested in including

random k CNF formulas Another family of examples that is of particular

interest for complexity theory are at least as hard to prove as many of

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

the classes of formulas discussed in this pap er but are only b elieved to b e

tautological are particular formulas given by Razb orov stating that NP

is not contained in Pp oly

There is an even larger gap in tautologies that seem to separate extended

Frege from Frege systems In fact we know of no convincing combinatorial

tautologies that might have p olynomialsize extended Frege pro ofs but re

quire exp onentialsize Frege pro ofs In several tautologies based on

linear algebra are suggested to give a quasip olynomial separation b etween

extended Frege and Frege systems A very simple such example suggested

by Co ok and Racko is the prop ositional form of the Bo olean matrix pro d

uct implication AB I BA I

How hard are random k CNF formulas The only lower b ounds

known for unsatisability pro ofs of random formulas are for forms of Resolu

tion What ab out other pro of systems such as b oundeddepth Frege pro ofs

cutting planes pro ofs Nullstellensatz or p olynomial calculus pro ofs The

absence of random unsatisable formulas in the list of lower b ounds for sys

tems other than Resolution is quite noteworthy esp ecially given the lack

of go o d upp er b ounds for pro ofs of these formulas in any system even

extendedFrege

Many NPcomplete graph problems are easy on the average for the natu

ral random graph probability distributions Random k CNF formulas under

the analogous probability distributions seem surprisingly hard in the region

of probabilities for which the formulas are likely unsatisable Is

k UNSAT hard on the average in this sense

For example the b est upp er b ound for any search algorithm for unsat

O n m

isability pro ofs of random mclause nvariable CNF formulas is

with probability o in n and this is tight for a class of DLL al

gorithms For Resolution the lower b ound for this problem is nearly

n m

with probability O in n Can these b e improved

Superpolynomial lower bounds for AC r Frege Studying AC r

Frege is a natural next step in proving lower b ounds for pro of systems

in particular when r is a prime We already mentioned the program for

obtaining such lower b ounds in For this system we do have a natural

n

candidate for a hard tautology namely C ount for prime p r Such

p

a lower b ound would further the circuitpro of system corresp ondence by

extending pro of complexity lower b ounds to the natural analogue of the

Razb orovSmolensky circuit lower b ounds

Polynomial Calculus in a theorem prover Better Resolution

proof search Designing ecient theorem provers for the prop ositional

calculus is an imp ortant practical question To date DLL algorithms are

the champion theorem provers although they are theoretically quite weak as

pro of systems A recent challenger seems promising A variant of the Gro eb

ner basis algorithm has b een used to nd Polynomial Calculus pro ofs

and build a fairly ecient theorem prover Can this b e tuned to comp ete

with DLL algorithms

BEAME AND PITASSI

Clegg et al also give simulations of DLL and Resolution by the

Polynomial Calculus One metho d of improving the comp etitiveness of Poly

nomial Calculus as a theorem prover would b e to improve these simulations

This is also related to the question of improving the more direct metho ds for

pro of search for Resolution and DLL that were inspired by these

Polynomial Calculus simulations

New proof systems from NPcomplete problems An app ealing di

rection in pro of complexity is the p ossibility of using natural domains that

contain NPhard problems to seek out new and interesting pro of systems

which reason ab out ob jects from radically dierent domains from Bo olean

formulas Cutting planes come from integer programming Nullstellensatz

and Polynomial Calculus systems come from systems of p olynomial equa

tions Pitassi and Urquhart considered the Ha jos calculus for non

colorability which they found surprisingly to b e equivalent to extended

Frege pro ofs It is likely that there is more to b e mined in this search

Weak pigeonhole principle and the limits of Resolution It is

m

known that for n m n log n PHP requires sup erp olynomialsize

n

Resolution pro ofs Originally it was conjectured that for any m n

m

any Resolution refutation of PHP would require size exp onential in n

n

However this conjecture was shown to b e false for large enough m

Moreover it app ears that the standard metho d the b ottleneck counting

technique cannot b e applied to obtain lower b ounds for m n An inter

m

esting op en problem is to prove lower b ounds for PHP for m n This

n

would likely give rise to a new lower b ound metho d for Resolution Ad

ditionally the complexity of the weak pigeonhole principle in various pro of

systems is interesting in its own right it is known that one can prove the

existence of innitely many primes in systems as weak as p olynomialsize

b oundeddepth Frege assuming the weakpigeonhole principle as an axiom

schema More generally the weak pigeonhole principle can b e used to carry

out most combinatorial counting arguments and is closely connected to

approximate counting

Partial results for the weak pigeonhole principle have recently b een ob

tained There is a very tight connection b etween regular Resolution

refutations and readonce branching programs which generalizes the equiv

alence b etween tree resolution and DLL mentioned earlier Let F b e an

unsatisable formula in conjunctive normal form The search problem as

so ciated with F takes as input a truth assignment to the underlying vari

ables of F and outputs a clause in F that is set to false by Kra jcek

has shown that for any F the minimal size Resolution refutation of F is

essentially equivalent to the minimal size readonce branching program to

solve the related search problem This idea was exploited in to obtain

some restricted lower b ounds for Resolution pro ofs of weak versions of the

pigeonhole principle

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

Another even older principle for which no sup erp olynomial Resolution

lower b ounds are known is the dominotiling principle More details ab out

this old problem can b e found in

LovaszSchrijver proof systems A variety of inference systems for

programming are describ ed by Lovasz and Schrijver in a pap er that

is primarily concerned with their implications for linear programming Like

cutting planes these pro of systems represent statements using systems of

linear inequalities but unlike cutting planes they replace rounding by an

ability to take linear combinations of derived degree terms in intermediate

steps obtained by multiplying certain inequalities and including the equa

tions x x provided all degree terms cancel There are several versions

dep ending on whether one can add the squares of arbitrary linear terms

to the appropriate side of the inequalities or multiply inequalities by

x or x or more generally multiply by any linear term previously

shown to b e p ositive Lovasz and Schrijver prove a number of prop erties of

their systems that allow one to precisely determine the depth of the pro ofs

involved but they do not consider the issue of pro of size directly

It turns out that despite the absence of the rounded division of cutting

planes one can easily simulate Resolution and prove the pigeonhole principle

in p olynomial size in their weakest system To see an example of the system

in action consider that if a b a c b c holds

for a b c then at most one of a b c is and so

a b c This is obtained by computing a a b a a ab

ab a a c a a ac ac and a b c

a b c ab ac Adding these inequalities leads to the desired result

However other problems are less clear In particular Lovasz private

communication suggested the problem of deriving the maximum indep en

dent set size of graphs that can b e expressed as the line graphs of o dd cliques

One can easily show that this problem is completely equivalent to the parity

principle and requires linear depth in the number of vertices but this is

far from either an algorithm or a lower b ound This question is intimately

tied to the question of whether these systems can simulate cutting planes

eciently Assuming that they cannot it may b e even more interesting to

consider what the combination of these new systems with cutting planes is

capable of

Oddcharged graphs hard for AC Frege Urquhart has suggested

that a study of Tseitins o ddcharged graph tautologies for appropriate

graphs in b oundeddepth Frege systems might also lead to lower b ounds

for random formulas since it app ears very dicult to apply Ajtais program

to them No results are know for these even on depth Frege systems

More general lower bound technique for Cutting Planes The

interpolation metho d has b een successfully applied to obtain unconditional

lower b ounds for Cutting Planes pro ofs However we are quite far from un

derstanding more generally what types of tautologies are hard for Cutting

BEAME AND PITASSI

Planes In particular are random formulas hard Other families of tautolo

gies p ossibly hard for Cutting Planes are the o ddcharged graph tautologies

n

Exponential bounds for C ount ontoPHP in Polynomial Cal

p

culus The lower b ounds of are barely sup erp olynomial and it seems

unlikely that the metho ds used can pro duce much larger lower b ounds The

lower b ounds of Razb orov involve a direct computation of the low degree

m

elements of the Gro ebner basis of the PHP p olynomials This computa

n

m

tion relies heavily on the indep endence of the degree one PHP p olynomi

n

als All the other counting principles ab ove do not have this prop erty and

it do es not app ear that the same metho d can b e applied to them Is there a

more general metho dology that do es not require direct computation of the

basis

Probabilistically checkable algebraic proofs The Nullstellensatz

and Polynomial Calculus pro of systems use the dense representation of mul

tivariate p olynomials What happ ens if one mo dies these pro of systems to

manipulate p olynomials as straightline programs that compute them rather

than writing them out explicitly One diculty is that testing the equality of

p olynomials represented this way is not known to b e eciently computable

deterministically However there are ecient probabilistic algorithms to do

this check Such a pro of system would lie outside our pro of system

denition since the verication predicate S would only b e probabilistically

checkable See for more details

Unsatisfiability threshold for random k SAT Although this is not

a pro of complexity question per se it is of interest in understanding the

pro of complexity of random k CNF formulas There have b een a number of

pap ers analyzing the satisability prop erties of these formulas as a function

of their clausevariable ratios Recently it has b een shown that there is a

sharp threshold b ehavior for such formulas but it is not known precisely

where such a threshold lies or even if it approaches some xed limit In

k k

general it is known that it lies b etween k and ln and for

k it is known to lie b etween and and is conjectured

to b e around A related question is whether or not the satisability

problem is easy right up to the threshold

Natural Proofs in Proof Complexity In circuit complexity Razb orov

and Rudich suggest that sub ject to some plausible cryptographic con

jectures current techniques will b e inadequate for obtaining sup erp olynomial

lower b ounds for TC circuits To this p oint pro of complexity has made

steady progress at matching the sup erp olynomial lower b ounds currently

known in the circuit world alb eit using dierent techniques and the ma

jor remaining analogous result a lower b ound for AC pFrege pro ofs also

may b e within reach Unlike the circuit world however there is no ana

logue of Shannons counting argument for size lower b ounds for random

functions and there do es not seem any inherent reason for TC Frege to b e

b eyond current techniques While it is true that one can show the failure of

TC Frege interpolation also dep ending on cryptographic conjectures this

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

applies to AC Frege as well for which we do have lower b ounds Is there

any analogue of natural pro ofs in pro of complexity Razb orov has

lo oked at the quite dierent question of examining particular pro of systems

and showing that any ecient pro ofs of lower b ounds for circuits using such

systems automatically naturalize

M Ajtai The complexity of the pigeonhole principle Combinatorica

M Ajtai The indep endence of the mo dulo p counting principles In Proceedings of

the TwentySixth Annual ACM Symposium on Theory of Computing pages

Montreal Quebec Canada May

M Ajtai Symmetric systems of linear equations mo dulo p Technical

Rep ort TR Electronic Collo quium in Computation Complexity

httpwwwecccunitrierd eec

Miklos Ajtai The complexity of the pigeonhole principle In th Annual Symposium

on Foundations of Computer Science pages White Plains NY Octob er

IEEE

Miklos Ajtai Parity and the pigeonhole principle In Samuel R Buss and P J Scott

editors Feasible Mathematics pages A Mathematical Sciences Institute Work

shop Ithaca NY Birkhauser

M Alekhnovich S Buss S Moran and T Pitassi Minimum Prop ositional pro of

length is NPhard to linearly approximate Manuscript

E Allender and U Hertrampf Depth reduction for circuits of unbounded fanin

Information and Computation August

S Arora C Lund Ra jeev Motwani M Sudan and Mario Szegedy Pro of verication

and hardness of approximation problems In Proceedings rd Annual Symposium on

Foundations of Computer Science Pittsburgh PA Octob er IEEE

P Beame R Karp T Pitassi and M Saks On the complexity of unsatisability

of random k CNF formulas In Proceedings of the th Annual ACM Symposium on

Theory of Computing Dallas TX May

P Beame and T Pitassi Simplied and improved resolution lower b ounds In th

Annual Symposium on Foundations of Computer Science pages Burlington

VT Octob er IEEE

P Beame and S Riis More on the relative strength of counting principles In P Beame

and S Buss editors Proof Complexity and Feasible Arithmetics DIMACS American

Mathematical So ciety To app ear

Paul W Beame Stephen A Co ok Je Edmonds Russell Impagliazzo and Toni

ann Pitassi The relative complexity of NP search problems In Proceedings of the

TwentySeventh Annual ACM Symposium on Theory of Computing pages

Las Vegas NV May

Paul W Beame Russell Impagliazzo Jan Kra jcek Toniann Pitassi and Pavel

Pudlak Lower b ounds on Hilb erts Nullstellensatz and prop ositional pro ofs Proc

London Math Soc

Paul W Beame Russell Impagliazzo Jan Kra jcek Toniann Pitassi Pavel Pudlak

and Alan Woo ds Exp onential lower b ounds for the pigeonhole principle In Proceed

ings of the TwentyFourth Annual ACM Symposium on Theory of Computing pages

Victoria BC Canada May

Paul W Beame and Toniann Pitassi An exp onential separation b etween the matching

principle and the pigeonhole principle In th Annual IEEE Symposium on Logic in

Computer Science pages Montreal Queb ec June

S BenDavid and A Gringauze On the existence of prop ositional pro of systems and

oraclerelativized prop ositional logic Technical Rep ort TR Electronic Collo

quium in Computation Complexity httpwwwecccunitrierdee ccc

BEAME AND PITASSI

E Ben Sasson and A Wigderson Private communication

M Bonet S Buss and T Pitassi Are there hard examples for Frege systems In

P Clote and J Remmel editors Feasible Mathematics II pages Birkhauser

M Bonet C Domingo and R Gavalda No feasible interpolation or automatization

for AC Frege pro of systems Manuscript

M Bonet T Pitassi and R Raz Lower b ounds for cutting planes pro ofs with small

co ecients Journal of Symbolic Logic September

M Bonet T Pitassi and R Raz No feasible interpolation for TC frege pro ofs In

th Annual Symposium on Foundations of Computer Science IEEE Octob er

S Buss On Go dels theorems on lengths of pro ofs I I Lower b ounds for recognizing

k symbol provability In P Clote and J Remmel editors Feasible mathematics II

pages BirkhauserBoston

S Buss R Impagliazzo J Kra jcek P Pudlak A A Razb orov and J Sgall Pro of

complexity in algebraic systems and b ounded depth Frege systems with mo dular

counting Computation Complexity

S Buss and T Pitassi Resolution and the weak pigeonhole principle In Proceedings

of Computer Science Logic Lecture Notes in Computer Science SpringerVerlag

S R Buss Lower b ounds on Nullstellensatz pro ofs via designs In P W Beame

and S R Buss editors Proof Complexity and Feasible Arithmetics DIMACS pages

American Math So c

S R Buss and T Pitassi Go o d degree b ounds on Nullstellensatz refutations of

the induction principle In Proceedings of the Eleventh Annual Conference on Com

putational Complexity formerly Structure in Complexity Theory pages

Philadelphia PA May IEEE

S R Buss and G Turan Resolution pro ofs of generalized pigeonhole principles

Theoretical Computer Science December

Samuel R Buss Bibliop olis Nap oli Volume of Studies

in Pro of Theory

Samuel R Buss Polynomial size pro ofs of the pigeonhole principle Journal of Sym

bolic Logic

V Chvatal Edmonds p olytop es and a hierarchy of combinatorial problems Discrete

Mathematics

V Chvatal and B Reed Mick gets some the o dds are on his side In nd Annual

Symposium on Foundations of Computer Science pages Pittsburgh PA

Octob er IEEE

V Chvatal and Endre Szemeredi Many hard examples for resolution Journal of the

ACM

M Clegg J Edmonds and R Impagliazzo Using the Grobner basis algorithm to

nd pro ofs of unsatisability In Proceedings of the TwentyEighth Annual ACM Sym

posium on Theory of Computing pages Philadelphia PA May

S Co ok and D Mitchell Finding hard instances of the satisability problem A

survey In DIMACS Series in Theoretical Computer Science

Stephen A Co ok The complexity of theorem proving pro cedures In Conference

Record of Third Annual ACM Symposium on Theory of Computing pages

Shaker Heights OH May

Stephen A Co ok Feasibly constructive pro ofs and the prop ositional calculus In

Proceedings of the Seventh Annual ACM Symposium on Theory of Computing pages

Albuquerque NM May

Stephen A Co ok and Rob ert A Reckhow Time b ounded random access machines

Journal of Computer and System Sciences

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

W Co ok C R Coullard and G Turan On the complexity of cutting plane pro ofs

Discrete Applied Mathematics

M Davis G Logemann and D Loveland A machine program for theorem proving

Communications of the ACM

M Davis and H Putnam A computing pro cedure for quantication theory Commu

nications of the ACM

U Feige S Goldwasser L Lovasz S Safra and M Szegedy Approximating clique

is almost NP complete In Proceedings nd Annual Symposium on Foundations of

Computer Science pages San Juan Puerto Rico Octob er IEEE

G Frege Begrisschrift eine der arithmetischen nachgebildete Formelsprache des

reinben Denkens Neb ert Halle

G Frege Grundgesetze der Arithmetik Neb ert Halle

E Friedgut Necessary and sucient conditions for sharp thresholds of graph prop

erties and the k sat problem Preprint May

A Frieze and S Suen Analysis of two simple heuristics on a random instance of k

SAT Journal of Algorithms

Xudong Fu On the complexity of proof systems PhD thesis

M Furst J B Saxe and M Sipser Parity circuits and the p olynomialtime hier

archy Mathematical Systems Theory April

O Gabb er and Z Galil Explicit constructions of linear size sup erconcentrators In

th Annual Symposium on Foundations of Computer Science pages New

York NY IEEE

Z Galil On the complexity of regular resolution and the DavisPutnam pro cedure

Theoretical Computer Science

A Go erdt Regular resolution versus unrestricted resolution SIAM Journal on Com

puting

RE Gomory Outline of an algorithm for integer solutions to linear programs Bul

letin of the American Mathematical Society

A Haken The intractability of resolution Theoretical Computer Science

A Haken and S A Co ok An exp onential lower b ound for the size of monotone real

circuits Preprint

K Iwama Complexity of nding short resolution pro ofs In I Privara and P Ruzicka

editors Lecture Notes in Computer Science pages SpringerVerlag

G D James The Representation Theory of the Symmetric Groups Number in

Lecture Notes in Mathematics SpringerVerlag

S Kirkpatrick and B Selman Critical b ehavior in the satisability of random for

mulas Science May

L M Kirousis E Kranakis and D Krizanc Approximating the unsatisability

threshold of random formulas In Proceedings of the Fourth Annual European Sym

posium on Algorithms pages Barcelona Spain September

J Kra jcek On the degree of ideal membership pro ofs from uniform families of p oly

nomials over nite elds Manuscript

J Kra jicek Lower b ounds to the size of constantdepth prop ositional pro ofs Journal

of Symbolic Logic March

J Kra jcek Bounded Arithmetic Propositional Logic and Complexity Theory Cam

bridge University Press

J Kra jicek and P Pudlak Prop ositional pro of systems the consistency of rst order

theories and the complexity of computations J Symbolic Logic

BEAME AND PITASSI

K Kra jicek and P Pudlak Some consequences of cryptographic conjectures for S

and EF In D Leivant editor Logic and Computational Complexity pages

SpringerVerlag

J Kra jcek P Pudlak and A Woo ds Exp onential lower b ounds to the size of

b ounded depth Frege pro ofs of the pigeonhole principle Random Structures and Al

gorithms

L Lovasz and A Schrijver Cones of matrices and setfunctions and optimization

SIAM J Optimization

A Maciel and T Pitassi Nonautomatizability of b oundeddepth Frege pro ofs Man

uscript

k

A Maciel and T Pitassi On AC C p frege pro ofs In Proceedings of the Twenty

Ninth Annual ACM Symposium on Theory of Computing pages May

J Mener and J Toran Optimal pro of systems for prop ositional logic and complete

sets In th Annual Symposium on Theoretical Aspects of Computer Science Lecture

Notes in Computer Science Paris France February SpringerVerlag

D Mitchell Hard problems for csp algorithms In Proceedings from AAAI

D Mundici A lower b ound for the complexity of Craigs interpolants in sentential

logic Archiv fur Mathematische Logik und Grund lagenforschung

J Paris and A Wilkie Counting problems in b ounded arithmetic In Methods in

Mathematical Logic Proceedings of the th Latin American Symposium on Mathe

matical Logic volume of Lecture notes in Mathematics pages

Berlin SpringerVerlag

JB Paris A J Wilkie and A R Woo ds Provability of the pigeonhole principle

and the existence of innitely many primes Journal of Symbolic Logic

T Pitassi Algebraic prop ositional pro of systems In DIMACS Series in Discrete

Mathematics volume pages American Math So c

T Pitassi and A Urquhart The complexity of the Ha jos calculus In nd Annual

Symposium on Foundations of Computer Science pages Pittsburgh PA

Octob er IEEE

Toniann Pitassi Paul W Beame and Russell Impagliazzo Exp onential lower b ounds

for the pigeonhole principle Computational Complexity

P Pudlak Lower b ounds for resolution and cutting plane pro ofs and monotone com

putations Journal of Symbolic Logic September

P Pudlak and J Sgall Algebraic mo dels of computation and interpolation for alge

braic pro of systems In P W Beame and S R Buss editors Proof Complexity and

Feasible Arithmetics DIMACS pages American Math So c

A A Razb orov Lower b ounds for the p olynomial calculus November

A A Razb orov Lower b ounds for the size of circuits with b ounded depth with basis

f g Mat Zametki

A A Razb orov Bounded arithmetic and lower b ounds in Bo olean complexity In

P Clote and J Remmel editors Feasible Mathematics II pages Birkhauser

A A Razb orov Unprovability of lower b ounds on the circuit size in certain fragments

of b ounded arithmetic Izvestiiya of the RAN

A A Razb orov and S Rudich Natural pro ofs Journal of Computer and System

Sciences August

A A Razb orov A Wigderson and A C Yao Readonce branching programs rect

angular pro ofs of the pigeonhole principle and the transversal calculus In Proceedings

of the Twenty Ninth Annual ACM Symposium on Theory of Computing pages

El Paso TX May

Sren Riis Independence in Bounded Arithmetic PhD thesis Oxford University

PROPOSITIONAL PROOF COMPLEXITY PAST PRESENT AND FUTURE

J A Robinson A machine oriented logic based on the resolution principle Journal

of the ACM

J T Schwartz Probabilistic algorithms for verication of p olynomial identities Jour

nal of the ACM pages

M Sipser The history and status of the P versus NP question In Proceedings of the

TwentyFourth Annual ACM Symposium on Theory of Computing pages

Victoria BC Canada May

Michael Sipser A complexity theoretic approach to randomness In Proceedings of

the Fifteenth Annual ACM Symposium on Theory of Computing Boston MA April

Roman Smolensky Algebraic metho ds in the theory of lower b ounds for Bo olean

circuit complexity In Proceedings of the Nineteenth Annual ACM Symposium on

Theory of Computing pages New York NY May

G S Tseitin On the complexity of derivation in the prop ositional calculus In A O

Slisenko editor Studies in Constructive Mathematics and Part

II

A Urquhart Manuscript

A Urquhart Hard examples for resolution Journal of the ACM

A Urquhart The complexity of prop ositional pro ofs Bul letin of Symbolic Logic

December

Computer Science and Engineering University of Washington Seattle

WA

Email address beamecswashingtonedu

Computer Science Tucson AZ

Email address tonicsarizonaedu