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International Joint Conference on Automated Reasoning

IJCAR 2016: Automated Reasoning pp 499–515Cite as

Model Checking Parameterised Multi-token Systems via the Composition Method

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9706))

Abstract

We study the model checking problem of parameterised systems with an arbitrary number of processes, on arbitrary network-graphs, communicating using multiple multi-valued tokens, and specifications from indexed-branching temporal logic. We prove a composition theorem, in the spirit of Feferman-Vaught [21] and Shelah [31], and a finiteness theorem, and use these to decide the model checking problem. Our results assume two constraints on the process templates, one of which is the standard fairness assumption introduced in the cornerstone paper of Emerson and Namjoshi [18]. We prove that lifting any of these constraints results in undecidability. The importance of our work is three-fold: (i) it demonstrates that the composition method can be fruitfully applied to model checking complex parameterised systems; (ii) it identifies the most powerful model, to date, of parameterised systems for which model checking indexed branching-time specifications is decidable; (iii) it tightly marks the borders of decidability of this model.

Benjamin Aminof is supported by the Vienna Science and Technology Fund (WWTF) through grant ICT12-059. Sasha Rubin is a Marie Curie fellow of the Istituto Nazionale di Alta Matematica.

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Notes

  1. 1.

    The direction-labels on the edges (also called a local orientation) represent network port numbers [14, 27]. All of our results also hold for the case that each edge has a single direction-label that combines send and receive directions, e.g., “clockwise”.

  2. 2.

    Indeed, there are infinitely many \(\textsf {CTL}^*_{1}\backslash \textsf {X}\) formulas that are pairwise logically-inequivalent. E.g., every finite word over \(\{0,1\}\) can be represented as an LTS, which itself can be axiomatised by a \(\textsf {CTL}^*_{1}\backslash \textsf {X}\) formula that uses the \(\mathbin {\mathsf {U}}\) operator.

  3. 3.

    The existence of a cutoff is independent of whether \(\mathcal {G}\) is computable. However, deciding whether a given number is a cutoff may not be easy. Consider for example the limited setting of [2]: there exists a computable \(\mathcal {G}\) and a fixed \({\mathbf {P}}\) such that it is impossible, given \(k,d \in \mathbb {N}\) (even fixing \(d = 1\)), to compute a cutoff [2]. Nonetheless, by [3], in the same setting (and we believe that also in our broader setting) one can compute a cutoff for many natural parameterized topologies \(\mathcal {G}\).

  4. 4.

    Fortunately, we only have to mimic such transitions that cross blocks in \(\rho ^t\).

  5. 5.

    The full version of this lemma contains two more conclusions.

  6. 6.

    Here, the empty set \(\emptyset \) is a letter in \(2^{[k]}\), not to be confused with the empty string \(\epsilon \).

  7. 7.

    Communication in [22] is by rendezvous, powerful enough to express token-passing.

  8. 8.

    Moreover, our work inherits from [2, 13] the non-uniformity of the decision problem. We leave for future work the problem of calculating explicit cutoffs for concrete classes of network-graphs, as was done in [3].

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Authors and Affiliations

  1. Technische Universität Wien, Vienna, Austria

    Benjamin Aminof

  2. Università di Napoli “Federico II”, Naples, Italy

    Sasha Rubin

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  1. Benjamin Aminof
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  1. Aix-Marseille University, Marseille, France

    Nicola Olivetti

  2. SRI International, Menlo Park, California, USA

    Ashish Tiwari

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Aminof, B., Rubin, S. (2016). Model Checking Parameterised Multi-token Systems via the Composition Method. In: Olivetti, N., Tiwari, A. (eds) Automated Reasoning. IJCAR 2016. Lecture Notes in Computer Science(), vol 9706. Springer, Cham. https://doi.org/10.1007/978-3-319-40229-1_34

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  • DOI: https://doi.org/10.1007/978-3-319-40229-1_34

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