Craig Interpolation with Clausal First-Order Tableaux

TL;DR

This paper introduces a novel method for computing Craig-Lyndon interpolants from first-order tableau proofs, enabling efficient interpolation in theorem proving systems based on clausal tableaux.

Contribution

It presents the first interpolation technique for first-order proofs via closed tableaux, with a two-stage process including induction and interpolant lifting, justified by Herbrand's theorem.

Findings

Linear simulation of propositional interpolation methods

Correctness of interpolant lifting established

Applicability to goal-oriented and bottom-up tableau systems

Abstract

We develop foundations for computing Craig-Lyndon interpolants of two given formulas with first-order theorem provers that construct clausal tableaux. Provers that can be understood in this way include efficient machine-oriented systems based on calculi of two families: goal-oriented such as model elimination and the connection method, and bottom-up such as the hypertableau calculus. We present the first interpolation method for first-order proofs represented by closed tableaux that proceeds in two stages, similar to known interpolation methods for resolution proofs. The first stage is an induction on the tableau structure, which is sufficient to compute propositional interpolants. We show that this can linearly simulate different prominent propositional interpolation methods that operate by an induction on a resolution deduction tree. In the second stage, interpolant lifting, quantified variables that replace certain terms (constants and compound terms) by variables are introduced. We justify the correctness of interpolant lifting (for the case without built-in equality) abstractly on the basis of Herbrand's theorem and for a different characterization of the formulas to be lifted than in the literature. In addition, we discuss various subtle aspects that are relevant for the investigation and practical realization of first-order interpolation based on clausal tableaux.