First-Order Interpolation and Interpolating Proof Systems

16 pagesPublished: May 4, 2017

Abstract

It is known that one can extract Craig interpolants from so-called local
proofs. An interpolant extracted from such a proof is a boolean
combination of formulas occurring in the proof. However, standard complete
proof systems, such as superposition, for theories having the interpolation
property are not necessarily complete for local proofs: there are formulas
having non-local proofs but no local proof.

In this paper we investigate interpolant extraction from non-local refutations
(proofs of contradiction) in the superposition calculus and prove a number
of general results about interpolant extraction and complexity of extracted
interpolants. In particular, we prove that the number of quantifier alternations
in first-order interpolants of formulas without quantifier alternations is
unbounded. This result has far-reaching consequences for using local proofs
as a foundation for interpolating proof systems: any such proof system should
deal with formulas of arbitrary quantifier complexity.

To search for alternatives for interpolating proof systems, we consider several
variations on interpolation and local proofs. Namely, we give an algorithm for
building interpolants from resolution refutations in logic without equality and
discuss additional constraints when this approach can be also used for logic
with equality. We finally propose a new direction related to interpolation via
local proofs in first-order theories.

Keyphrases: interpolation, Resolution Calculus, superposition

In: Thomas Eiter and David Sands (editors). LPAR-21. 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 46, pages 49--64