Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 62, Issue 2 (1997), 457-486.
Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic
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 $S^2_2(\alpha)$ (c) linear equational calculus (d) cutting planes. (2) New proofs of the exponential lower bounds (for new formulas) (a) for resolution () (b) for the cutting planes proof system with coefficients written in unary (). (3) An alternative proof of the independence result of  concerning the provability of circuit-size lower bounds in the bounded arithmetic theory $S^2_2(\alpha)$. 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.
J. Symbolic Logic, Volume 62, Issue 2 (1997), 457-486.
First available in Project Euclid: 6 July 2007
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Krajicek, Jan. Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic. J. Symbolic Logic 62 (1997), no. 2, 457--486. https://projecteuclid.org/euclid.jsl/1183745237