Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

§1. Introduction. The classical propositional calculus has an undeserved reputation among logicians as being essentially trivial. I hope to convince the reader that it presents some of the most challenging and intriguing problems in modern logic. Although the problem of the complexity of propositional proofs is very natural, it has been investigated systematically only since the late 1960s. Interest in the problem arose from two fields connected with computers, automated theorem proving and computational complexity theory. The earliest paper in the (...) subject is a ground-breaking article by Tseitin [62], the published version of a talk given in 1966 at a Leningrad seminar. In the three decades since that talk, substantial progress has been made in determining the relative complexity of proof systems, and in proving strong lower bounds for some restricted proof systems. However, major problems remain to challenge researchers. The present paper provides a survey of the field, and of some of the techniques that have proved successful in deriving lower bounds on the complexity of proofs. A major area only touched upon here is the proof theory of bounded arithmetic and its relation to the complexity of propositional proofs. The reader is referred to the book by Buss [10] for background in bounded arithmetic. The forthcoming book by Krajíček [40] also gives a good introduction to bounded arithmetic, as well as covering most of the basic results in complexity of propositional proofs. (shrink)

Partial consistency statements can be expressed as polynomial-size propositional formulas. Frege proof systems have polynomial-size partial self-consistency proofs. Frege proof systems have polynomial-size proofs of partial consistency of extended Frege proof systems if and only if Frege proof systems polynomially simulate extended Frege proof systems. We give a new proof of Reckhow's theorem that any two Frege proof systems p-simulate each other. The proofs depend on polynomial size propositional formulas defining the truth of propositional formulas. These are already known to (...) exist since the Boolean formula value problem is in alternating logarithmic time; this paper presents a proof of this fact based on a construction which is somewhat simpler than the prior proofs of Buss and of Buss-Cook- Gupta-Ramachandran. (shrink)

Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

We consider tautologies formed form a pseudo-random number generator, defined in Krajicek [11] and in Alekhnovich et al. [2]. We explain a strategy of proving their hardness for Extended Frege systems via a conjecture about bounded arithmetic formulated in Krajicek [11]. Further we give a purely finitary statement, in the form of a hardness condition imposed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at non-specialists, of the relation between prepositional proof complexity and (...) bounded arithmetic. (shrink)

We study from the proof complexity perspective the proof search problem : •Is there an optimal way to search for propositional proofs?We note that, as a consequence of Levin’s universal search, for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal proof search algorithm exists without restricting proof systems iff a p-optimal proof system exists.To characterize precisely the time proof search algorithms need for individual formulas (...) we introduce a new proof complexity measure based on algorithmic information concepts. In particular, to a proof system P we attach information-efficiency function $i_P$ assigning to a tautology a natural number, and we show that: • $i_P$ characterizes time any P-proof search algorithm has to use on $\tau $,•for a fixed P there is such an information-optimal algorithm,•a proof system is information-efficiency optimal iff it is p-optimal,•for non-automatizable systems P there are formulas $\tau $ with short proofs but having large information measure $i_P$.We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for $i_P$ where no super-polynomial size lower bounds are known. We also point out connections of the new measure with some topics in proof complexity other than proof search. (shrink)

Assume that the problem P0 is not solvable in polynomial time. Let T be a first-order theory containing a sufficiently rich part of true arithmetic. We characterize T∪{ConT} as the minimal extension of T proving for some algorithm that it decides P0 as fast as any algorithm B with the property that T proves that B decides P0. Here, ConT claims the consistency of T. As a byproduct, we obtain a version of Gödelʼs Second Incompleteness Theorem. Moreover, we characterize problems (...) with an optimal algorithm in terms of arithmetical theories. (shrink)

In proof theory one distinguishes sequent proofs with cut and cut-free sequent proofs, while for proof complexity one distinguishes Frege systems and extended Frege systems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cut-free systems with extension, which is neither possible with Frege systems, nor with the sequent calculus. We show that the propositional pigeonhole principle admits polynomial-size proofs in a cut-free system with extension. We also define (...) cut-free systems with substitution and show that the cut-free system with extension p-simulates the cut-free system with substitution. (shrink)

Let g be a map defined as the Nisan–Wigderson generator but based on an NP ∩ coNP -function f. Any string b outside the range of g determines a propositional tautology τb expressing this fact. Razborov [27] has conjectured that if f is hard on average for P/poly then these tautologies have no polynomial size proofs in the Extended Frege system EF. We consider a more general Statement that the tautologies have no polynomial size proofs in any propositional proof system. (...) This is equivalent to the statement that the complement of the range of g contains no infinite NP set. We prove that Statement is consistent with Cook' s theory PV and, in fact, with the true universal theory T PV in the language of PV. If PV in this consistency statement could be extended to "a bit" stronger theory then Razborov's conjecture would follow, and if TPV could be added too then Statement would follow. We discuss this problem in some detail, pointing out a certain form of reflection principle for propositional logic, and we introduce a related feasible disjunction property of proof systems. (shrink)

A fast consistency prover is a consistent polytime axiomatized theory that has short proofs of the finite consistency statements of any other polytime axiomatized theory. Krajíček and Pudlák have proved that the existence of an optimal propositional proof system is equivalent to the existence of a fast consistency prover. It is an easy observation that NP = coNP implies the existence of a fast consistency prover. The reverse implication is an open question. In this paper we define the notion of (...) an unlikely fast consistency prover and prove that its existence is equivalent to NP = coNP. Next it is proved that fast consistency provers do not exist if one considers RE axiomatized theories rather than theories with an axiom set that is recognizable in polynomial time. (shrink)

The purpose of this paper is to survey the correspondence between bounded arithmetic and propositional proof systems. In addition, it also contains some new results which have appeared as an extended abstract in the proceedings of the conference TAMC 2008 [11].Bounded arithmetic is closely related to propositional proof systems; this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Krajíček and (...) Pudlák [46]. Instead of focusing on the relation between particular proof systems and theories, we favour a general axiomatic approach to this correspondence. In the course of the development we particularly highlight the role played by logical closure properties of propositional proof systems, thereby obtaining a characterization of extensions of EF in terms of a simple combination of these closure properties (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

A notion of feasible function of finite type based on the typed lambda calculus is introduced which generalizes the familiar type 1 polynomial-time functions. An intuitionistic theory IPVω is presented for reasoning about these functions. Interpretations for IPVω are developed both in the style of Kreisel's modified realizability and Gödel's Dialectica interpretation. Applications include alternative proofs for Buss's results concerning the classical first-order system S12 and its intuitionistic counterpart IS12 as well as proofs of some of Buss's conjectures concerning IS12, (...) and a proof that IS12 cannot prove that extended Frege systems are not polynomially bounded. (shrink)

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(m d+1) symbols. We show that renaming (...) Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. (shrink)

We consider tautologies formed from a pseudo-random number generator, defined in Krajíček [11] and in Alekhnovich et al. [2]. We explain a strategy of proving their hardness for Extended Frege systems via a conjecture about bounded arithmetic formulated in Krajíček [11]. Further we give a purely finitary statement, in the form of a hardness condition imposed on a function, equivalent to the conjecture.

Leonid Levin showed that every algorithm computing a function has an optimal inverter. Recently, we applied his result in various contexts: existence of optimal acceptors, existence of hard sequences for algorithms and proof systems, proofs of Gödel’s incompleteness theorems, analysis of the complexity of the clique problem assuming the nonuniform Exponential Time Hypothesis. We present all these applications here. Even though a simple diagonalization yields Levin’s result, we believe that it is worthwhile to be aware of the explicit result. The (...) purpose of this survey is to convince the reader of our view. (shrink)