## Journal of Symbolic Logic

### Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic

Jan Krajicek

#### 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 $S^2_2(\alpha)$ (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 $S^2_2(\alpha)$. 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.

#### Article information

Source
J. Symbolic Logic, Volume 62, Issue 2 (1997), 457-486.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745237

Mathematical Reviews number (MathSciNet)
MR1464108

Zentralblatt MATH identifier
0891.03029

JSTOR

#### Citation

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