Model Completeness, Covers and Superposition

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

doi:10.1007/978-3-030-29436-6_9

Cited by 13 publications

(26 citation statements)

References 38 publications

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“…From the algorithmic point of view, we intend to develop more sophisticated techniques for the quantifier elimination module required in our backward reachability procedure. A first stepping stone in this direction, relying on a constrained version of the Superposition Calculus (SC) (Nieuwenhuis and Rubio 2001), can be found in Calvanese et al (2019c): indeed, thanks to our constrained version of the SC, suitably combined with congruence closure, we show that it is possible to obtain a quadratic bound for the complexity of the quantifier elimination procedure in the case of interest for our applications (Calvanese et al 2018a(Calvanese et al , 2019c.…”

Section: Discussionmentioning

confidence: 99%

“…First, contrary to what happens in linear arithmetics, the quantifier elimination needed to prove Proposition 3.2 has a much better behavior (from the complexity point of view) if obtained via a suitable version of the Knuth-Bendix procedure (Baader and Nipkow 1998) or of the Superposition Calculus (Nieuwenhuis and Rubio 2001). Since these aspects concerning quantifier elimination are rather delicate, we started studying them in a dedicated paper (Calvanese et al 2019c) and in its extended version (Calvanese et al 2018a) (our MCMT implementation, however, already partially takes into account such development), where, by using a constrained version of Superposition, we show that in the case of free unary functions and free relations the complexity has a quadratic bound even without assuming acyclicity.…”

Section: Proposition 32 T Has a Model Completion In Case It Is Axiomentioning

confidence: 99%

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SMT-based verification of data-aware processes: a model-theoretic approach

Calvanese

Ghilardi

Gianola

et al. 2020

Math. Struct. Comp. Sci.

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In recent times, satisfiability modulo theories (SMT) techniques gained increasing attention and obtained remarkable success in model-checking infinite-state systems. Still, we believe that whenever more expressivity is needed in order to specify the systems to be verified, more and more support is needed from mathematical logic and model theory. This is the case of the applications considered in this paper: we study verification over a general model of relational, data-aware processes, to assess (parameterized) safety properties irrespectively of the initial database (DB) instance. Toward this goal, we take inspiration from array-based systems and tackle safety algorithmically via backward reachability. To enable the adoption of this technique in our rich setting, we make use of the model-theoretic machinery of model completion, which surprisingly turns out to be an effective tool for verification of relational systems and represents the main original contribution of this paper. In this way, we pursue a twofold purpose. On the one hand, we isolate three notable classes for which backward reachability terminates, in turn witnessing decidability. Two of such classes relate our approach to conditions singled out in the literature, whereas the third one is genuinely novel. On the other hand, we are able to exploit SMT technology in implementations, building on the well-known MCMT (Model Checker Modulo Theories) model checker for array-based systems and extending it to make all our foundational results fully operational. All in all, the present contribution is deeply rooted in the long-standing tradition of the application of model theory in computer science. In particular, this paper applies these ideas in an original mathematical context and shows how these techniques can be used for the first time to empower algorithmic techniques for the verification of infinite-state systems based on arrays, so as to make such techniques applicable to the timely, challenging settings of data-aware processes.

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Section: Discussionmentioning

confidence: 99%

Section: Proposition 32 T Has a Model Completion In Case It Is Axiomentioning

confidence: 99%

SMT-based verification of data-aware processes: a model-theoretic approach

Calvanese

Ghilardi

Gianola

et al. 2020

Math. Struct. Comp. Sci.

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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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“…In these contributions, examples of cover computations were supplied and also some algorithms were sketched. The first formal proof about the existence of covers in EUF was however published only in [14] by the present authors; such a proof was equipped with powerful semantic tools (see the Cover-by-Extensions Lemma 3.1 below) obtained thanks to interesting connections with model completeness [56], and came with an algorithm for computing covers that is based on a constrained variant of the Superposition Calculus [54]. Both the model-theoretic tools and the algorithm are detailed in the present paper.…”

Section: Introductionmentioning

confidence: 99%

“…8 we give some details about our first implementation in our tool mcmt. This paper is the extended version of [14]: apart from containing more basic preliminary material, a thorough account of model-checking applications, full proofs and detailed examples, in Sects. 6 and 7 this paper covers additional new results on complexity analysis and extensions.…”

Section: Introductionmentioning

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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Model Completeness, Covers and Superposition

Cited by 13 publications

References 38 publications

SMT-based verification of data-aware processes: a model-theoretic approach

SMT-based verification of data-aware processes: a model-theoretic approach

Interpolation and Amalgamation for Arrays with MaxDiff

Model Completeness, Uniform Interpolants and Superposition Calculus

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