Combined Covers and Beth Definability

Calvanese, Diego; Ghilardi, Silvio; Gianola, Alessandro; Montali, Marco; Rivkin, Andrey

doi:10.1007/978-3-030-51074-9_11

Cited by 12 publications

(16 citation statements)

References 26 publications

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“…This extended version contains additional material on complexity analysis and implementation. It contains also a proof about nonexistence of uniform interpolants (see [26,27,20,10,11,12] for the definition and more information on uniform interpolants).…”

Section: Introductionmentioning

confidence: 99%

Interpolation and Amalgamation for Arrays with MaxDiff

Ghilardi

Gianola

Kapur

2021

Lecture Notes in Computer Science

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In this paper, the theory of McCarthy’s extensional arrays enriched with a maxdiff operation (this operation returns the biggest index where two given arrays differ) is proposed. It is known from the literature that a diff operation is required for the theory of arrays in order to enjoy the Craig interpolation property at the quantifier-free level. However, the diff operation introduced in the literature is merely instrumental to this purpose and has only a purely formal meaning (it is obtained from the Skolemization of the extensionality axiom). Our maxdiff operation significantly increases the level of expressivity; however, obtaining interpolation results for the resulting theory becomes a surprisingly hard task. We obtain such results via a thorough semantic analysis of the models of the theory and of their amalgamation properties. The results are modular with respect to the index theory and it is shown how to convert them into concrete interpolation algorithms via a hierarchical approach.

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Section: Introductionmentioning

confidence: 99%

Interpolation and Amalgamation for Arrays with MaxDiff

Ghilardi

Gianola

Kapur

2021

Lecture Notes in Computer Science

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“…Symbol elimination of function and predicate variables should also be combined with cover computations. Combined cover algorithms (along the perspectives in [37]) could be crucial also in this setting: a first attempt to attack this problem, regarding the disjoint signatures combination, can be found in [16].…”

Section: Discussionmentioning

confidence: 99%

Model Completeness, Uniform Interpolants and Superposition Calculus

et al. 2021

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Uniform interpolants have been largely studied in non-classical propositional logics since the nineties; a successive research line within the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. This further research line is motivated by the fact that uniform interpolants offer an effective solution to tackle quantifier elimination and symbol elimination problems, which are central in model checking infinite state systems. This was first pointed out in ESOP 2008 by Gulwani and Musuvathi, and then by the authors of the present contribution in the context of recent applications to the verification of data-aware processes. In this paper, we show how covers are strictly related to model completions, a well-known topic in model theory. We also investigate the computation of covers within the Superposition Calculus, by adopting a constrained version of the calculus and by defining appropriate settings and reduction strategies. In addition, we show that computing covers is computationally tractable for the fragment of the language used when tackling the verification of data-aware processes. This observation is confirmed by analyzing the preliminary results obtained using the mcmt tool to verify relevant examples of data-aware processes. These examples can be found in the last version of the tool distribution.

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“…Proofs are omitted (sometimes in favor of intuitive justifications); however, they can all be found in the original papers. More precisely, proofs of the results from Section 3 are in [36], proofs of the results from Section 4 are in [37], and proofs of the results from Section 5 are in [31,32,[38][39][40].…”

Section: Structure Of the Papermentioning

confidence: 99%

“…This section presents results contained in [31,32,[38][39][40]64,65]. First, we analyze a strong form of of quantifier-free interpolation and its relationship with model-completeness [31,32]; we then show that, for convex theories, the same hypotheses allowing the transfer of the existence of ordinary interpolants also allow the transfer of the existence of these stronger interpolants [39,40].…”

Section: Uniform Interpolantsmentioning

confidence: 99%

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Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories

Ghilardi

Gianola

2022

Mathematics

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In this survey, we report our recent work concerning combination results for interpolation and uniform interpolation in the context of quantifier-free fragments of first-order theories. We stress model-theoretic and algebraic aspects connecting this topic with amalgamation, strong amalgamation, and model-completeness. We give sufficient (and, in relevant situations, also necessary) conditions for the transfer of the quantifier-free interpolation property to combined first-order theories; we also investigate the non-disjoint signature case under the assumption that the shared theory is universal Horn. For convex, strong-amalgamating, stably infinite theories over disjoint signatures, we also provide a modular transfer result for the existence of uniform interpolants. Model completions play a key role in the whole paper: They enter into transfer results in the non-disjoint signature case and also represent a semantic counterpart of uniform interpolants.

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Combined Covers and Beth Definability

Cited by 12 publications

References 26 publications

Interpolation and Amalgamation for Arrays with MaxDiff

Interpolation and Amalgamation for Arrays with MaxDiff

Model Completeness, Uniform Interpolants and Superposition Calculus

Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories

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