Lexicographic Ranking Supermartingales with Lazy Lower Bounds

Abstract

AbstractLexicographic Ranking SuperMartingale (LexRSM) is a probabilistic extension of Lexicographic Ranking Function (LexRF), which is a widely accepted technique for verifying program termination. In this paper, we are the first to propose sound probabilistic extensions of LexRF with a weaker non-negativity condition, called single-component (SC) non-negativity. It is known that such an extension, if it exists, will be nontrivial due to the intricacies of the probabilistic circumstances.Toward the goal, we first devise the notion of fixability, which offers a systematic approach for analyzing the soundness of possibly negative LexRSM. This notion yields a desired extension of LexRF that is sound for general stochastic processes. We next propose another extension, called Lazy LexRSM, toward the application to automated verification; it is sound over probabilistic programs with linear arithmetics, while its subclass is amenable to automated synthesis via linear programming. We finally propose a LexRSM synthesis algorithm for this subclass, and perform experiments.