Craig Interpolation with Clausal First-Order Tableaux

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. Similar to known resolution-based interpolation methods our method proceeds in two stages. 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. Correctness of this second stage was apparently shown so far on the basis of resolution and paramodulation with an error concerning equality, on the basis of resolution with paramodulation and superposition for a special case, and on the basis of a natural deduction calculus without taking equality into special account. Here the correctness of interpolant lifting is justified abstractly on the basis of Herbrand's theorem and based on a different characterization of the formulas to be lifted than in the literature (without taking equality into special account). In addition, we discuss various subtle aspects that are relevant for the investigation and practical realization of first-order interpolation based on clausal tableaux. [23,24],1 for two first-order formulas F and G such that F entails G there exists a third first-order formula H that is entailed by F, entails G and is such that all predicate and function symbols and all free variables occurring in it occur in both F and G. Such a Craig interpolant H can be constructed from given formulas F and G with a calculus that allows to extract H from a proof that F entails G, or, equivalently, a proof