# Supermartingale Certificates for Quantitative Omega-regular Verification and Control

arXiv: https://arxiv.org/abs/2505.18833 · HTML: https://mlsanity.com/paper/2505.18833/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2025.

## Built on by (most cited first, 5 of 5)

- [Neural Proofs for Sound Verification and Control of Complex Systems](https://mlsanity.com/paper/2512.18389/) (2025)
- [Quantitative Verification of Omega-regular Properties in Probabilistic Programming](https://mlsanity.com/paper/2512.21596/) (2025)
- [Value Functions as Supermartingale Certificates](https://mlsanity.com/paper/2605.31524/) (2026)
- [Supermartingale Certificates for Parametric MDPs](https://mlsanity.com/paper/2609.12715/) (2026)
- [Runtime Monitoring of Supermartingale Certificates](https://mlsanity.com/paper/2610.19701/) (2026)

## Abstract (arXiv)

We present the first supermartingale certificate for quantitative ω-regular properties of discrete-time infinite-state stochastic systems. Our certificate is defined on the product of the stochastic system and a limit-deterministic B"uchi automaton that specifies the property of interest; hence we call it a limit-deterministic B"uchi supermartingale (LDBSM). Previously known supermartingale certificates applied only to quantitative reachability, safety, or reach-avoid properties, and to qualitative (i.e., probability 1) ω-regular properties. We also present fully automated algorithms for the template-based synthesis of LDBSMs, for the case when the stochastic system dynamics and the controller can be represented in terms of polynomial inequalities. Our experiments demonstrate the ability of our method to solve verification and control tasks for stochastic systems that were beyond the reach of previous supermartingale-based approaches.