A New Proof Rule for Almost-Sure Termination

TL;DR

This paper introduces a novel proof rule for establishing almost-sure termination of probabilistic programs, accommodating both probabilistic and demonic choices with state-dependent probabilities, advancing the verification of complex probabilistic systems.

Contribution

It presents a new parametric variant of super-martingale-based proof rules for almost-sure termination, applicable directly to source code with probabilistic and demonic choices.

Findings

Proves soundness of the new proof rule.

Shows applicability beyond existing rules.

Connects to classical Markov chain results.

Abstract

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.