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Computing Reachable Simulations on Transition Systems

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Reachability Problems (RP 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15050))

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Abstract

We study the problem of computing the reachable principals of the simulation preorder and the reachable blocks of simulation equivalence. Following a theoretical investigation of this problem, which highlights a sharp contrast with the already settled case of bisimulation, we design algorithms to solve this problem by leveraging the idea of interleaving reachability and simulation computation while possibly avoiding the computation of all the reachable states or the whole simulation preorder. In particular, we put forward a symbolic algorithm processing state partitions and, in turn, relations between their blocks, which is suited for processing infinite-state systems.

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Notes

  1. 1.

    The term principal comes from the well-known notion of principal ideal [13, Chapter I, sect. 3.4]. Detailed definitions are given in Sect. 3.

  2. 2.

    As shown in Sect. 4, an algorithm terminating on all infinite state systems cannot exist.

  3. 3.

    Observe that since \(R_{\textrm{sim}}\) is a preorder, we have that \(P_{\textrm{sim}}\in \textrm{Part}(\varSigma )\) coincides with the equivalence classes of the similarity equivalence \(R_{\textrm{sim}}\cap (R_{\textrm{sim}})^{-1}\).

  4. 4.

    For some systems, \((\textrm{rp}_1\)) could even be infinite, and \((\textrm{rp}_2\)) be a finite set.

  5. 5.

    We distinguish the states of \(\textsf{post}^*(I)\) from those in its subset \(\sigma \) by referring to the states in \(\sigma \) as provably reachable. We extend this notion to principals.

  6. 6.

    Note that line 11 might break transitivity of \(R\). In fact, \(R\) is not guaranteed to be a preorder during execution, and not even at termination.

  7. 7.

    Reachability analysis is mostly superfluous on explicitly represented systems as they usually do not encode unreachable states.

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Acknowledgements

Francesco Ranzato was partially funded by: the Italian MUR, under the PRIN 2022 PNRR project no. P2022HXNSC; Meta (formerly Facebook) Research, under a “Probability and Programming Research Award” and under a WhatsApp Research Award on “Privacy-aware Program Analysis”; by an Amazon Research Award for “AWS Automated Reasoning”. Nicolas Manini is supported by the grant PIPF-2022/COM-24370, funded by the Madrid Regional Government. This publication is part of the grant PID2022-138072OB-I00, funded by MCIN/AEI/10.13039/501100011033/FEDER, UE and part of the PRODIGY Project (TED2021-132464B-I00) funded by MCIN/AEI/10.13039/501100011033/and the European Union NextGenerationEU/PRTR.

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Authors and Affiliations

  1. IMDEA Software Institute, Pozuelo de Alarcón, Spain

    Pierre Ganty & Nicolas Manini

  2. Universidad Politécnica de Madrid, Madrid, Spain

    Nicolas Manini

  3. University of Padova, Padova, Italy

    Francesco Ranzato

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  1. Pierre Ganty
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Correspondence to Nicolas Manini .

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Editors and Affiliations

  1. TU Wien, Vienna, Austria

    Laura Kovács

  2. Paris Lodron University of Salzburg, Salzburg, Austria

    Ana Sokolova

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Ganty, P., Manini, N., Ranzato, F. (2024). Computing Reachable Simulations on Transition Systems. In: Kovács, L., Sokolova, A. (eds) Reachability Problems. RP 2024. Lecture Notes in Computer Science, vol 15050. Springer, Cham. https://doi.org/10.1007/978-3-031-72621-7_3

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