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Speeding up the Constraint-Based Method in Difference Logic

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Theory and Applications of Satisfiability Testing – SAT 2016 (SAT 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9710))

Abstract

Over the years the constraint-based method has been successfully applied to a wide range of problems in program analysis, from invariant generation to termination and non-termination proving. Quite often the semantics of the program under study as well as the properties to be generated belong to difference logic, i.e., the fragment of linear arithmetic where atoms are inequalities of the form \(u - v \le k\). However, so far constraint-based techniques have not exploited this fact: in general, Farkas’ Lemma is used to produce the constraints over template unknowns, which leads to non-linear SMT problems. Based on classical results of graph theory, in this paper we propose new encodings for generating these constraints when program semantics and templates belong to difference logic. Thanks to this approach, instead of a heavyweight non-linear arithmetic solver, a much cheaper SMT solver for difference logic or linear integer arithmetic can be employed for solving the resulting constraints. We present encouraging experimental results that show the high impact of the proposed techniques on the performance of the VeryMax verification system.

Partially supported by Spanish MINECO under grant TIN2015-69175-C4-3-R.

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Notes

  1. 1.

    Note that equalities can be considered as conjunctions of inequalities.

  2. 2.

    For \(\varphi \in \mathcal {F}(\mathcal{X})\), the formula \(\varphi ' \!\in \! \mathcal {F}(\mathcal{X}')\) is the version of \(\varphi \) using primed variables.

  3. 3.

    Here a simplified procedure for proving an assertion is described in order to highlight the key contribution of this work, that is, how to circumvent non-linearities.

  4. 4.

    When the generated invariants consisting of a single inequality do not prove the assertion, as indicated in Section 2 the procedure can be iterated by strengthening the transitions, thereby allowing the synthesis of invariant conjunctions of inequalities.

  5. 5.

    This yields a non-linear formula if templates and \(\mathcal{S}\) include general linear inequalities.

  6. 6.

    Executables, benchmarks and detailed tables with the results of all the experiments in this paper can be found at www.cs.upc.edu/~erodri/sat16.tgz.

  7. 7.

    This is the time limit used in VeryMax for this type of queries in previous works [20].

  8. 8.

    Note that this does not mean that they are unsafe.

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Authors and Affiliations

  1. SpazioDati, Trento, Italy

    Lorenzo Candeago

  2. Universitat Politècnica de Catalunya, Barcelona, Spain

    Daniel Larraz, Albert Oliveras, Enric Rodríguez-Carbonell & Albert Rubio

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  1. Lorenzo Candeago
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  2. Daniel Larraz
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  3. Albert Oliveras
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  4. Enric Rodríguez-Carbonell
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Correspondence to Enric Rodríguez-Carbonell .

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Editors and Affiliations

  1. Aix-Marseille Université, Marseille, France

    Nadia Creignou

  2. Université d'Artois, Lens, France

    Daniel Le Berre

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Candeago, L., Larraz, D., Oliveras, A., Rodríguez-Carbonell, E., Rubio, A. (2016). Speeding up the Constraint-Based Method in Difference Logic. In: Creignou, N., Le Berre, D. (eds) Theory and Applications of Satisfiability Testing – SAT 2016. SAT 2016. Lecture Notes in Computer Science(), vol 9710. Springer, Cham. https://doi.org/10.1007/978-3-319-40970-2_18

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