We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie.

The Haj'os Calculus is a simple, nondeterministic procedure which generates the class of non-3-colorable graphs. Mansfield and Welsch [MW] posed the question of whether there exist graphs which require exponential-sized Haj'os constructions. Unless NP 6= coNP , there must exist graphs which require exponential-sized constructions, but to date, little progress has been made on this question, despite considerable effort. In this paper, we prove that the Haj'os Calculus generates polynomial-sized constructions for all non-3-colorable graphs if and only if Extended Frege systems are polynomially bounded. Extended Frege systems are a very powerful family of proof systems for proving tautologies, and proving superpolynomial lower bounds for these systems is a long-standing, important problem in logic and complexity theory. We also establish a relationship between a complete subsystem of the Haj'os Calculus, and boundeddepth Frege systems; this enables us to prove exponential lower bounds on ...

This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.

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. Depth d Frege proofs of m lines can be transformed into depth d proofs of O(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.

The Haj'os Calculus is a simple, nondeterministic procedure which generates precisely the class of non-3-colorable graphs. In this note, we prove exponential lower bounds on the size of tree-like Haj'os constructions. 1 Introduction The Haj'os calculus is a simple, nondeterministic procedure for generating the class of graphs that are not k-colorable [Haj]. Mansfield and Welsh [MW] have posed the problem of determining the complexity of this procedure; in particular, it is an open problem whether or not there exists a polynomial-size Haj'os construction for every non3 -colorable graph. Because graph 3-colorability is NP-complete, if there were polynomial-size Haj'os constructions of all non-3-colorable graphs, then NP = coNP , so we expect that the Haj'os calculus is not polynomially bounded. However, there has been very little progress toward a proof of this conjecture, despite considerable effort. Pitassi and Urquhart [PU] have recently shown that the Haj'os calculus is polynomiall...

We show that tree-like (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cut-free PK. Generally, we exhibit a family of sequents that have polynomial size cut-free proofs but requires superpolynomial tree-like proofs even when the cut rule is allowed on a class of cut-formulas that satisfies some plausible hardness assumption. This gives (in some cases, conditional) negative answers to several questions from a recent work of Maciel and Pitassi (LICS 2006). Our technique is inspired by the technique from Maciel and Pitassi. While the sequents used in earlier work are derived from the Pigeonhole principle, here we generalize Statman’s sequents. This gives the desired separation, and at the same time provides stronger results in some cases. 1

. In this paper we introduce Gentzen-style quantied propositional proof systems L i for the theories R i 2 . We formalize the systems L i within the bounded arithmetic theory R 1 2 and we show that for i 1, R i 2 can prove the validity of a sequent derived by an L i -proof. This statement is formally called i-RFN(L i ). We show if R i 2 ` 8xA(x) where A 2 b i , then for each integer n there is a translation of the formula A into quantied propositional logic such that R i 2 proves there is an L i -proof of this translated formula. Using the proofs of these two facts we show that L i is in some sense the strongest system for which R i 2 can prove i-RFN and we show for i j 2 that the 8 b j -consequences of R i 2 are nitely axiomatized. 1. Introduction Propositional proof systems and bounded arithmetic are closely connected. Cook [10] introduced the equational arithmetic theory PV of polynomial time computable functions and showed PV co...

It is known that constant-depth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider tree-like Sequent Calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning small-depth Boolean circuits, we prove exponential lower bounds for such proofs. We prove these lower bounds directly from the computational hardness assumption. We start with a lower bound for cut-free proofs and “lift ” it so it applies to proofs with constant-depth cuts. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known unconditional lower bound in the case where modular connectives are not used. We establish a conditional exponential separation between the power of constant-depth proofs that use different modular connectives. We show that these tree-like proofs with constant-depth cuts cannot polynomially simulate similar dag-like proofs, even when the dag-like proofs are cut-free. We present a new proof of the non-finite axiomatizability of the theory of bounded arithmetic I∆0(R). Finally, under a plausible hardness assumption concerning the polynomial-time hierarchy, we show that the hierarchy G ∗ i of quantified propositional proof systems does not collapse.