On Interpolation in Automated Theorem Proving

Abstract

Given two inconsistent formulae, a (reverse) interpolant is a formula implied by one, inconsistent with the other, and only containing symbols they share. Interpolation finds application in program analysis, verification, and synthesis, for example, towards invariant generation. An interpolation system takes a refutation of the inconsistent formulae and extracts an interpolant by building it inductively from partial interpolants. Known interpolation systems for ground proofs use colors to track symbols. We show by examples that the color-based approach cannot handle non-ground refutations by resolution and paramodulation/superposition. We present a two-stage approach that works by tracking literals, computes a provisional interpolant, which may contain non-shared symbols, and applies lifting to replace non-shared constants by quantified variables. We obtain an interpolation system for non-ground refutations, and we prove that it is complete, if the only non-shared symbols in provisional interpolants are constants.

Classifying and Solving Horn Clauses for Verification

“…, ¬p n } being the set of all possible literals, and (l, l ) ∈ E if and only if there is a clause ¬l ∨ l ∨ C ∈ C (that means, a clause containing the literal l , and the literal l with reversed sign). 7 The graph (V, E) is acyclic. To see this, suppose there is a cycle l 1 , l 2 , .…”

“…The refinement problem saw a significant progress when Craig interpolants extracted from unsatisfiability proofs were used as relevant predicates [20]. While interpolation has enjoyed a significant progress for various logical constraints [7][8][9]24], there have been substantial proposals for more general forms of interpolation [1,19,24].…”

“…(1) r1(X, Res) ← true (2) r2(X', Res) ← r1(X, Res) ∧ X' ≥ 0 (3) r3(X, Res') ← r2(X, Res) ∧ rf(X, Res') (5) false ← r3(X, Res) ∧ Res X + 1 (6) r5(N, Rec, Tmp) ← true (7) …”

First-Order Interpolation and Interpolating Proof Systems

“…The result of extracting relational reverse interpolants from superposition refutations was also proved in [12] and [3]. Our proof is considerably shorter and simpler than either of these proofs.…”

“…The notion of relational interpolant can already be found in [12] and was used in a number of publications, including [1] and [3]. In this section we will use an arbitrary first-order theory T , whenever possible.…”