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International Symposium on Formal Methods

FM 2021: Formal Methods pp 619–639Cite as

On Lexicographic Proof Rules for Probabilistic Termination

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 13047)

Abstract

We consider the almost-sure (a.s.) termination problem for probabilistic programs, which are a stochastic extension of classical imperative programs. Lexicographic ranking functions provide a sound and practical approach for termination of non-probabilistic programs, and their extension to probabilistic programs is achieved via lexicographic ranking supermartingales (LexRSMs). However, LexRSMs introduced in the previous work have a limitation that impedes their automation: all of their components have to be non-negative in all reachable states. This might result in LexRSM not existing even for simple terminating programs. Our contributions are twofold: First, we introduce a generalization of LexRSMs which allows for some components to be negative. This standard feature of non-probabilistic termination proofs was hitherto not known to be sound in the probabilistic setting, as the soundness proof requires a careful analysis of the underlying stochastic process. Second, we present polynomial-time algorithms using our generalized LexRSMs for proving a.s. termination in broad classes of linear-arithmetic programs.

Keywords

  • Probabilistic programs
  • Termination
  • Martingales

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Acknowledgements

This research was partially supported by the ERC CoG 863818 (ForM-SMArt), the Czech Science Foundation grant No. GJ19-15134Y, and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665385.

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  1. IST Austria, Klosterneuburg, Austria

    Krishnendu Chatterjee & Đorđe Žikelić

  2. Ferdowsi University of Mashhad, Mashhad, Iran

    Ehsan Kafshdar Goharshady

  3. Masaryk University, Brno, Czech Republic

    Petr Novotný & Jiří Zárevúcky

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  1. University of Twente, Enschede, The Netherlands

    Prof. Dr. Marieke Huisman

  2. NASA Ames Research Center, Moffett Field, CA, USA

    Corina Păsăreanu

  3. Institute of Software, Chinese Academy of Sciences, Beijing, China

    Naijun Zhan

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Chatterjee, K., Goharshady, E.K., Novotný, P., Zárevúcky, J., Žikelić, Đ. (2021). On Lexicographic Proof Rules for Probabilistic Termination. In: Huisman, M., Păsăreanu, C., Zhan, N. (eds) Formal Methods. FM 2021. Lecture Notes in Computer Science(), vol 13047. Springer, Cham. https://doi.org/10.1007/978-3-030-90870-6_33

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