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On Interpolation in Automated Theorem Proving

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Abstract

Given two inconsistent formulæ, 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 formulæ 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.

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Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Verona, Strada Le Grazie 15, 37134, Verona, Italy

    Maria Paola Bonacina

  2. Department of Computer Science, Chalmers University of Technology, Göteborg, Sweden

    Moa Johansson

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  1. Maria Paola Bonacina
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  2. Moa Johansson
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Correspondence to Maria Paola Bonacina.

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Research supported in part by grant no. 2007-9E5KM8 of the Ministero dell’Istruzione Università e Ricerca, Italy, and by COST Action IC0901 Rich-model Toolkit of the European Union.

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Bonacina, M.P., Johansson, M. On Interpolation in Automated Theorem Proving. J Autom Reasoning 54, 69–97 (2015). https://doi.org/10.1007/s10817-014-9314-0

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