# A New Proof Rule for Almost-Sure Termination

arXiv: https://arxiv.org/abs/1711.03588 · HTML: https://mlsanity.com/paper/1711.03588/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Nov 2017.

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

- [Probabilistic unifying relations for modelling epistemic and aleatoric uncertainty: semantics and automated reasoning with theorem proving](https://mlsanity.com/paper/2303.09692/) (2023)
- [Supermartingale Certificates for Parametric MDPs](https://mlsanity.com/paper/2609.12715/) (2026)

## Abstract (arXiv)

An important question for a probabilistic program is whether the probability mass of all its diverging runs is zero, that is that it terminates "almost surely". Proving that can be hard, and this paper presents a new method for doing so; it is expressed in a program logic, and so applies directly to source code. The programs may contain both probabilistic- and demonic choice, and the probabilistic choices may depend on the current state. As do other researchers, we use variant functions (a.k.a. "super-martingales") that are real-valued and probabilistically might decrease on each loop iteration; but our key innovation is that the amount as well as the probability of the decrease are parametric. We prove the soundness of the new rule, indicate where its applicability goes beyond existing rules, and explain its connection to classical results on denumerable (non-demonic) Markov chains.