A Model-Theoretic Characterization of Monadic Second Order Logic on Infinite Words

Ghilardi, Silvio; Gool, Samuel J. van

doi:10.1017/jsl.2016.70

Cited by 5 publications

(3 citation statements)

References 13 publications

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“…All in all, the present contribution is deeply rooted in the long-standing tradition of the application of model theory in computer science, as witnessed by notable approaches like the one in Ghilardi (2004), Baader et al (2006), Ghilardi et al (2008b), Ghilardi and van Gool (2017), Nicolini et al (2009aNicolini et al ( ,b, 2010, Sofronie-Stokkermans (2008, 2016, Ghilardi andGianola (2017, 2018). In particular, this paper applies these ideas in a genuinely novel 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 in the style of Ghilardi et al (2008a), Ghilardi and Ranise (2010a,b), Alberti et al (2014aAlberti et al ( ,b, 2017, Conchon et al (2012Conchon et al ( , 2015Conchon et al ( , 2018a, Delzanno (2018), Cimatti et al (2018), so as to make such techniques applicable to the timely, challenging settings of data-aware processes (Calvanese et al 2019d).…”

Section: Main Contributionsmentioning

confidence: 97%

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: Main Contributionsmentioning

confidence: 97%

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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“…Interestingly, model completeness has other well-known applications in computer science. It has been applied: (i) to reveal interesting connections between temporal logic and monadic second-order logic [29,30]; (ii) in automated reasoning to design complete algorithms for constraint satisfiability in combined theories over non-disjoint signatures [1,23,31,[49][50][51]; (iii) again in automated reasoning in relationship with interpolation and symbol elimination [59,60]; (iv) in modal logic and in software verification theories [24,25], to obtain combined interpolation results.…”

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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“…In algebraic logic some attention has been paid to the class of existentially closed structures in varieties coming from the algebraization of common propositional logics. In fact, there are relevant cases where such classes are elementary: this includes, besides the easy case of Boolean algebras, also Heyting algebras [GZ97,GZ02], diagonalizable algebras [Sha93,GZ02] and some universal classes related to temporal logics [GvG01], [GvG16]. However, very little is known about the related axiomatizations, with the remarkable exception of the case of the locally finite amalgamable varieties of Heyting algebras recently investigated in [DJ10] and of the simpler cases of posets and semilattices studied in [AB86].…”

Section: Introductionmentioning

confidence: 99%

Existentially Closed Brouwerian Semilattices

Carai¹,

Ghilardi²

2019

J. symb. log.

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The variety of Brouwerian semilattices is amalgamable and locally finite, hence by well-known results [Whe76], it has a model completion (whose models are the existentially closed structures). In this paper, we supply for such a model completion a finite and rather simple axiomatization.2010 Mathematics subject classification. Primary 03G25; Secondary 03C10, 06D20.A co-Brouwerian semilattice, CBS for short, is a structure obtained by reversing the order of a Brouwerian semilattice. We will work with CBSes instead of Brouwerian semilattices.Definition 2.1. A poset (P, ≤) is said to be a co-Brouwerian semilattice if it has a least element, which we denote with 0, and for every a, b ∈ P there exists the sup of {a, b}, which we call join of a and b and denote with a ∨ b, and the difference a − b satisfying for every c ∈ PClearly, there is also an alternative equational definition for co-Brouwerian semilattices (which we leave to the reader, because it is dual to the equational definition for Brouwerian semilattices given above).Moreover, we will call co-Heyting algebras the structures obtained reversing the order of Heyting algebras. Obviously any co-Heyting algebra is a CBS.Definition 2.2. Let A, B be co-Brouwerian semilattices. A map f : A → B is a morphism of co-Brouwerian semilattices if it preserves 0, the join and difference of any two elements of A.Notice that such a morphism f is an order preserving map because, for any a, b elements of a co-Brouwerian semilattice, we have a ≤ b iff a ∨ b = b.Definition 2.3. Let L be a CBS. We say that g ∈ L is join-irreducible iff for every n ≥ 0 and b 1 , . . . , b n ∈ L, we have that g ≤ b 1 ∨ . . . ∨ b n implies g ≤ b i for some i = 1, . . . , n.Notice that taking n = 0 we obtain that join-irreducibles are different from 0.Remark 2.4. Let L be a CBS and g ∈ L. Then the following conditions are equivalent:1. g is join-irreducible.2. g = 0 and for any b 1 , b 2 ∈ L we have that g3. For every n ≥ 0 and b 1 , . . . , b n ∈ L we have that g = b 1 ∨ . . . ∨ b n implies g = b i for some i = 1, . . . , n.4. g = 0 and for any b 1 , b 2 ∈ L we have that g = b 1 ∨ b 2 implies g = b 1 or g = b 2 .5. g = 0 and for any a ∈ L we have that g − a = 0 or g − a = g.Proof. The implications 1 ⇔ 2, 3 ⇔ 4 and 1 ⇒ 3 are straightforward. For the remaining ones see Lemma 2.1 in [Köh81].Definition 2.5. Let L be a CBS and a ∈ L.A join-irreducible component of a is a maximal element among the join-irreducibles of L that are smaller than or equal to a.

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A Model-Theoretic Characterization of Monadic Second Order Logic on Infinite Words

Cited by 5 publications

References 13 publications

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

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

Model Completeness, Uniform Interpolants and Superposition Calculus

Existentially Closed Brouwerian Semilattices

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