Prop ositional Pro of Complexity Past Present and Future Paul Beame and Toniann Pitassi 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 circuit complexity 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 all 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 up 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 PROOF COMPLEXITY 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 Frege system 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 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