Interpolation theorems, lower Bounds for proof systems, and independence results for bounded arithmetic

Journal of Symbolic Logic 62 (2):457-486 (1997)
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Abstract

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 (α) (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 in the bounded arithmetic theory S 2 2 (α). In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle

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Citations of this work

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References found in this work

Proof Theory.Gaisi Takeuti - 1990 - Studia Logica 49 (1):160-161.
On the Scheme of Induction for Bounded Arithmetic Formulas.A. J. Wilkie & J. B. Paris - 1987 - Annals of Pure and Applied Logic 35 (C):261-302.
The Foundations of Mathematics.Charles Parsons & Evert W. Beth - 1961 - Philosophical Review 70 (4):553.

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