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Solving Two-Player Games Under Progress Assumptions

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Verification, Model Checking, and Abstract Interpretation (VMCAI 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14499))

Abstract

This paper considers the problem of solving infinite two-player games over finite graphs under various classes of progress assumptions motivated by applications in cyber-physical system (CPS) design. Formally, we consider a game graph \(G\), a temporal specification \(\varPhi \) and a temporal assumption \(\psi \), where both \(\varPhi \) and \(\psi \) are given as linear temporal logic (LTL) formulas over the vertex set of \(G\). We call the tuple \((G,\varPhi ,\psi )\) an augmented game and interpret it in the classical way, i.e., winning the augmented game \((G,\varPhi ,\psi )\) is equivalent to winning the (standard) game \((G,\psi \Rightarrow \varPhi )\). Given a reachability or parity game \(\mathcal {G}= (G,\varPhi )\) and some progress assumption \(\psi \), this paper establishes whether solving the augmented game \(\mathfrak {G}= (G,\varPhi ,\psi )\) lies in the same complexity class as solving \(\mathcal {G}\). While the answer to this question is negative for arbitrary combinations of \(\varPhi \) and \(\psi \), a positive answer results in more efficient algorithms, in particular for large game graphs.

We therefore restrict our attention to particular classes of CPS-motivated progress assumptions and establish the worst-case time complexity of the resulting augmented games. Thereby, we pave the way towards a better understanding of assumption classes that can enable the development of efficient solution algorithms in augmented two-player games.

The randomization record is publicly available at www.aeaweb.org/journals/policies/random-author-order/search. S. P. Nayak, I. Saglam and A.-K. Schmuck are supported by the DFG projects 389792660 TRR 248-CPEC and SCHM 3541/1-1.

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Notes

  1. 1.

    This is because every \(\omega \)-regular specification can be reduced to a parity specification [3], which can be written as an LTL specification over V as given later.

  2. 2.

    The edges \(x_i \rightarrow x'_i\) mainly serve illustrative purposes, and the live outgoing edge of \(x'_i\) can actually be attributed directly to \(x_i\). Further, distributing live edges to separate vertices underscores the result’s validity for live edges with disjoint sources.

  3. 3.

    We note that all games can be converted to an equivalent alternating game by at most doubling the size of the vertex and edge sets of the game graph.

  4. 4.

    See [32] for a more in-depth version of this example.

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Author information

Authors and Affiliations

  1. Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany

    Anne-Kathrin Schmuck, Irmak Sağlam & Satya Prakash Nayak

  2. Department of Computer Science, University of Warwick, Coventry, UK

    K. S. Thejaswini

Authors
  1. Anne-Kathrin Schmuck
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  2. K. S. Thejaswini
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  3. Irmak Sağlam
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  4. Satya Prakash Nayak
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Correspondence to Irmak Sağlam .

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  1. CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

    Rayna Dimitrova

  2. Tel Aviv University, Tel Aviv, Israel

    Ori Lahav

  3. New York University, New York, NY, USA

    Sebastian Wolff

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Schmuck, AK., Thejaswini, K.S., Sağlam, I., Nayak, S.P. (2024). Solving Two-Player Games Under Progress Assumptions. In: Dimitrova, R., Lahav, O., Wolff, S. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2024. Lecture Notes in Computer Science, vol 14499. Springer, Cham. https://doi.org/10.1007/978-3-031-50524-9_10

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