On lexicographic proof rules for probabilistic termination

• Main Authors: CHATTERJEE, Krishnendu, GOHARSHADY, Ehsan Kafshdar, NOVOTNÝ, Petr, ZÁREVUCKÝ, Jiří, ZIKELIC, Dorde
• Format: text
• Language: English
• Published: Institutional Knowledge at Singapore Management University 2025
• Subjects: Probabilistic programs; termination; martingales; Software Engineering
• Online Access: https://ink.library.smu.edu.sg/sis_research/9069; https://ink.library.smu.edu.sg/context/sis_research/article/10072/viewcontent/3585391.pdf
• Institution: Singapore Management University

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 a LexRSM not existing even for simple terminating programs. Our contributions are twofold. First, we introduce a generalization of LexRSMs that 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.