On Interpolation in Decision Procedures

Bonacina, Maria Paola; Johansson, Moa

doi:10.1007/978-3-642-22119-4_1

Cited by 10 publications

(12 citation statements)

References 28 publications

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“…Proofs by propositional resolution are trivially colorable, since no new literals are generated, and ground proofs in first-order logic are like propositional proofs in this respect. If equality enters the picture, already for ground proofs, equalities where one side is Acolored and the other B-colored are problematic: the assumption of a separating ordering for ground superposition prevents precisely such AB-mixed equalities, as explained in [11,12], where we give a systematic treatment of color-based interpolation systems for ground proofs. Proof transformation techniques to make ground proofs colored or colorable were studied in [18,19,38,41,52].…”

Section: State Of the Art And Problem Statementmentioning

confidence: 99%

On Interpolation in Automated Theorem Proving

Bonacina

Johansson

2014

J Autom Reasoning

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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.

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Section: State Of the Art And Problem Statementmentioning

confidence: 99%

On Interpolation in Automated Theorem Proving

Bonacina

Johansson

2014

J Autom Reasoning

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“…In Section 4.2, we have extensively discussed the closely related work of [32], where the authors illustrate a method to derive interpolants in a Nelson-Oppen combination procedure, provided that the component theories satisfy certain hypotheses. The work in [3], among other contributions, recasts the method of [32] in the context of the DP LL(T ) paradigm.…”

Section: Conclusion and Related Workmentioning

confidence: 99%

“…Let us call Σ A the signature Σ expanded with the free constants a ∪ c and Σ B the signature Σ expanded with the free constants b ∪ c (we put Σ C := Σ A ∩ Σ B = Σ ∪ {c}). As a first step, we build a maximal T -consistent set Γ of ground Σ A -formulae and a maximal T -consistent set ∆ of ground Σ B -formulae such that Θ 1 ⊆ Γ, Θ 2 ⊆ ∆, and Γ ∩ Σ C = ∆ ∩ Σ C 3.…”

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confidence: 99%

From Strong Amalgamability to Modularity of Quantifier-Free Interpolation

Bruttomesso

Ghilardi

Ranise

2012

Automated Reasoning

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The use of interpolants in verification is gaining more and more importance. Since theories used in applications are usually obtained as (disjoint) combinations of simpler theories, it is important to modularly re-use interpolation algorithms for the component theories. We show that a sufficient and necessary condition to do this for quantifierfree interpolation is that the component theories have the 'strong (sub-)amalgamation'property. Then, we provide an equivalent syntactic characterization, identify a sufficient condition, and design a combined quantifier-free interpolation algorithm capable of handling both convex and non-convex theories, that subsumes and extends most existing work on combined interpolation.

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“…Besides being an important logical property by itself, Craig interpolation has applications in definability [4], projective classes [24] and automated reasoning [6]. Moreover, Craig interpolation has been applied in modular specification [5], model checking of software applications [28,20,29] and in logics with interacting agents [21].…”

Section: Introductionmentioning

confidence: 99%