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Compositional Verification of Concurrent Systems by Combining Bisimulations

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Formal Methods – The Next 30 Years (FM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11800))

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Abstract

One approach to verify a property expressed as a modal \(\mu \)-calculus formula on a system with several concurrent processes is to build the underlying state space compositionally (i.e., by minimizing and recomposing the state spaces of individual processes, keeping visible only the relevant actions occurring in the formula), and check the formula on the resulting state space. It was shown previously that, when checking the formulas of the \(L_{\mu }^{ dsbr }\) fragment of \(\mu \)-calculus (consisting of weak modalities only), individual processes can be minimized modulo divergence-preserving branching (divbranching) bisimulation. In this paper, we refine this approach to handle formulas containing both strong and weak modalities, so as to enable a combined use of strong or divbranching bisimulation minimization on concurrent processes depending whether they contain or not the actions occurring in the strong modalities of the formula. We extend \(L_{\mu }^{ dsbr }\) with strong modalities and show that the combined minimization approach preserves the truth value of formulas of the extended fragment. We implemented this approach on top of the CADP verification toolbox and demonstrated how it improves the capabilities of compositional verification on realistic examples of concurrent systems.

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Notes

  1. 1.

    For instance, the composition of P and Q where action a of P synchronizes with either b or c of Q, can be written as \(\rho (P)\,|[b, c]|\,Q\), where \(\rho \) maps a onto \(\{b, c\}\).

  2. 2.

    For instance, \(P_1\,|[a]|\,(P_2\,|[]|\,P_3)\) is equivalent to \(\rho _0((\rho _1(P_1)\,|[a_1]|\,\rho _2(P_2))\,|[a_2]|\,\rho _3(P_3))\) —observe the different associativity— where \(\rho _1\) maps a onto \(\{a_1, a_2\}\), \(\rho _2\) renames a into \(a_1\), \(\rho _3\) renames a into \(a_2\), and \(\rho _0\) renames \(a_1\) and \(a_2\) into a.

  3. 3.

    \(\mu \)ACTL\(\setminus \)X denotes ACTL\(\setminus \)X plus fixed points. The authors of [24] claim that \(L_{\mu }^{ dsbr }\) is as expressive as \(\mu \)ACTL\(\setminus \)X, but they omit that the \(\langle \varphi _1?.\alpha _{\tau } \rangle \,@\) weak infinite looping modality cannot be expressed in \(\mu \)ACTL\(\setminus \)X.

  4. 4.

    Theorem 2 generalizes easily to more general compositions \(P\,||\,Q\) (with admissible action mappings) if Q does not contain any action that maps onto a strong action.

  5. 5.

    Otherwise, our approach coincides with the mono-bisimulation approach of [24]. In all the examples addressed in this section, there is always a unique minimal set \(A_s\), whose identification is made easy using Lemma 1.

  6. 6.

    Specification available at ftp://ftp.inrialpes.fr/pub/vasy/demos/demo_05.

  7. 7.

    http://rers-challenge.org.

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Authors and Affiliations

  1. Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP (Institute of Engineering Univ. Grenoble Alpes), LIG, 38000, Grenoble, France

    Frédéric Lang & Radu Mateescu

  2. ISTI-CNR, Pisa, Italy

    Franco Mazzanti

Authors
  1. Frédéric Lang
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  2. Radu Mateescu
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Correspondence to Frédéric Lang .

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Editors and Affiliations

  1. Consiglio Nazionale delle Ricerche, Pisa, Italy

    Maurice H. ter Beek

  2. Macquarie University, Sydney, NSW, Australia

    Annabelle McIver

  3. University of Minho, Braga, Portugal

    José N. Oliveira

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Lang, F., Mateescu, R., Mazzanti, F. (2019). Compositional Verification of Concurrent Systems by Combining Bisimulations. In: ter Beek, M., McIver, A., Oliveira, J. (eds) Formal Methods – The Next 30 Years. FM 2019. Lecture Notes in Computer Science(), vol 11800. Springer, Cham. https://doi.org/10.1007/978-3-030-30942-8_13

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