This paper of Tseitin is a landmark as the first to give non-trivial lower bounds for propositional proofs; although it pre-dates the first papers on

Abstract not found

A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: 1. Feasible interpolation theorems for the following proof systems: (a) resolution. (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (ff). (c) linear equational calculus. (d) cutting planes. 2. New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]). (b) for the cutting planes proof system with coefficients written in unary ([4]). 3. An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds ...

Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic. $1. Introduction. The motivation for this paper comes primarily from two sources. First, Cook and Reckhow [2] and Statman [7] discussed connections between lengths of proofs in propositional logic and open questions in com-putational complexity such as whether NP = co-NP. Cook and Reckhow used the propositional pigeonhole principle as an example of a family of true formulae which

The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct reference to polynomials, time, or even computation. Complexity classes characterized in this way include polynomial time, the functional polytime hierarchy, the logspace decidable problems, and NC. After developing these "resource free" definitions, we apply them to redeveloping the feasible logical system of Cook and Urquhart, and show how this first-order system relates to the second-order system of Leivant. The connection is an interesting one since the systems were defined independently and have what appear to be very different rules for the principle of induction. Furthermore it is interesting to see, albeit in a very specific context, how to retract a second order statement, ("inducti...

We investigate the possibility to characterize (multi)functions that are \Sigma b i -definable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems defined over polynomial-time structures. We obtain the following results: 1. A reformulation of known characterizations of (multi)functions that are \Sigma b 1 - and \Sigma b 2 -definable in the theories S 1 2 and T 1 2 . 2. New characterizations of (multi)functions that are \Sigma b 2 - and \Sigma b 3 - definable in the theory T 2 2 . 3. A new non-conservation result: the theory T 2 2 (ff) is not 8\Sigma b 1 (ff)- conservative over the theory S 2 2 (ff). To prove that the theory T 2 2 (ff) is not 8\Sigma b 1 (ff)-conservative over the theory S 2 2 (ff), we present two examples of a \Sigma b 1 (ff)-principle separating the two theories: (a) the weak pigeonhole principle WPHP (a 2 ; f; g) formalizing that no function f is a bijection between a 2 and a with the inverse...

We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1st-order operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have the complexity classes between P and EXPTIME. Our parameterized fixpoint logics capture the complexity classes P, NP, PSPACE, and EXPTIME, but equality is achieved only over ordered structures. There is, however, an inherent mismatch between complexity and logic -- while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures. To overcome this mismatch, we develop a theory of relational complexity, which bridges tha gap between standard complexity and fixpoint logic. On one hand, we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complex...

Abstract. 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. Contents

We investigate the problem how to lift the non - 8\Sigma b 1 (ff) - conservativity of T 2 2 (ff) over S 2 2 (ff) to the expected non - 8\Sigma b i (ff) - conservativity of T i+1 2 (ff) over S i+1 2 (ff), for i ? 1. We give a non-trivial refinement of the "lifting method" developed in [4, 8], and we prove a sufficient condition on a 8\Sigma b 1 (f)-consequence of T2 (f) to yield the non-conservation result. Further we prove that Ramsey's theorem, a 8\Sigma b 1 (ff) - formula, is not provable in T 1 2 (ff), and that 8\Sigma b j (ff) - conservativity of T i+1 2 (ff) over T i 2 (ff) implies 8\Sigma b j (ff) - conservativity of the whole T2 (ff) over T i 2 (ff), for any j

Abstract not found