# On Lexicographic Proof Rules for Probabilistic Termination

arXiv: https://arxiv.org/abs/2108.02188 · HTML: https://mlsanity.com/paper/2108.02188/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2021.

## Built on by (most cited first, 2 of 2)

- [Learning Provably Stabilizing Neural Controllers for Discrete-Time Stochastic Systems](https://mlsanity.com/paper/2210.05304/) (2022)
- [Supermartingale Certificates for Parametric MDPs](https://mlsanity.com/paper/2609.12715/) (2026)

## Abstract (arXiv)

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.