Weakly Non-Negative Supermartingales for Omega-Regular Verification

2026-07-20

AI Summary

This work addresses the limitation imposed by strong global non-negativity, which hinders the automated synthesis of simple certificates for probabilistic programs, while naive relaxation of this condition compromises verification soundness. The paper introduces lazy Streett supermartingales and their lexicographic extensions, leveraging weak non-negativity within polynomial templates to enable sound verification of almost-sure satisfaction of ω-regular properties under a broad class of sampling distributions with bounded support. It is the first to generalize the weak non-negativity approach from termination analysis to general ω-regular specifications and provides a one-dimensional compositional characterization of lexicographic certificates. Experimental evaluation on 170 polynomial probabilistic program benchmarks demonstrates a 20.0–23.5 percentage point improvement in verification success rate over strong non-negativity baselines.

Abstract

Martingale-based methods are central to probabilistic program verification, but strong global non-negativity requirements can exclude simple certificates from tractable template classes. Relaxing this requirement enlarges the search space for automated synthesis, but naive relaxations are unsound in the probabilistic setting. We introduce lazy Streett supermartingales and their lexicographic extension, showing that weak non-negativity can nevertheless be used soundly to certify almost-sure satisfaction of $ω$-regular properties with polynomial templates under a broad class of sampling distributions, including all bounded-support distributions. This extends prior weakly non-negative methods from termination to general $ω$-regular verification. We further give a compositional account of lexicographic certificates in terms of one-dimensional ones. Experiments on 170 polynomial probabilistic-program benchmarks show increases of 20.0-23.5 percentage points in verification success over the strongly non-negative baseline.

Problem

• martingales
• probabilistic verification
• omega-regular properties
• non-negativity
• polynomial templates

Innovation

• weakly non-negative supermartingales
• omega-regular verification
• lazy Streett supermartingales
• lexicographic certificates
• probabilistic program verification