Lexicographic Ranking Supermartingales with Lazy Lower Bounds

Lexicographic 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.