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How Hard is It to Verify Flat Affine Counter Systems with the Finite Monoid Property?

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Automated Technology for Verification and Analysis (ATVA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9938))

Abstract

We study several decision problems for counter systems with guards defined by convex polyhedra and updates defined by affine transformations. In general, the reachability problem is undecidable for such systems. Decidability can be achieved by imposing two restrictions: (1) the control structure of the counter system is flat, meaning that nested loops are forbidden, and (2) the multiplicative monoid generated by the affine update matrices present in the system is finite. We provide complexity bounds for several decision problems of such systems, by proving that reachability and model checking for Past Linear Temporal Logic stands in the second level of the polynomial hierarchy \(\varSigma ^P_2\), while model checking for First Order Logic is PSPACE-complete.

R. Iosif—Supported by the French ANR project VECOLIB (ANR-14-CE28-0018).

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Notes

  1. 1.

    We ensure deadlock-freedom by adding a sink state \(\sigma \) to S, with a self-loop \(\sigma \xrightarrow [\scriptscriptstyle ]{\scriptscriptstyle \top } \sigma \), and a transition \(q\xrightarrow [\scriptscriptstyle ]{\scriptscriptstyle \top } \sigma \) from each state \(q\in Q\).

  2. 2.

    They are defined in the proof of Lemma 3.

  3. 3.

    We take here the classical semantics of \(\mathsf{PLTL}\) over infinite words.

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Author information

Authors and Affiliations

  1. Verimag, Université Grenoble Alpes, CNRS, Grenoble, France

    Radu Iosif

  2. IRIF, Université Paris Diderot, CNRS, Paris, France

    Arnaud Sangnier

Authors
  1. Radu Iosif
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  2. Arnaud Sangnier
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Correspondence to Arnaud Sangnier .

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Editors and Affiliations

  1. AIST , Osaka, Japan

    Cyrille Artho

  2. Inria Rennes , Rennes, France

    Axel Legay

  3. Bar Ilan University , Ramat Gan, Israel

    Doron Peled

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Iosif, R., Sangnier, A. (2016). How Hard is It to Verify Flat Affine Counter Systems with the Finite Monoid Property?. In: Artho, C., Legay, A., Peled, D. (eds) Automated Technology for Verification and Analysis. ATVA 2016. Lecture Notes in Computer Science(), vol 9938. Springer, Cham. https://doi.org/10.1007/978-3-319-46520-3_6

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  • DOI: https://doi.org/10.1007/978-3-319-46520-3_6

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  • Online ISBN: 978-3-319-46520-3

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