Decisiveness of Stochastic Systems and its Application to Hybrid Models (Full Version)

TL;DR

This paper advances the concept of decisiveness in stochastic systems, providing a general criterion for its proof, and applies it to stochastic hybrid systems to enable decidability of reachability problems in certain classes.

Contribution

It introduces a broad, natural criterion for decisiveness in general stochastic transition systems and applies it to stochastic hybrid systems to achieve decidability results.

Findings

Provided a general criterion for decisiveness of STSs.

Proved decisiveness for a large class of stochastic hybrid systems.

Established decidability of reachability in certain stochastic hybrid systems.

Abstract

In [ABM07], Abdulla et al. introduced the concept of decisiveness, an interesting tool for lifting good properties of finite Markov chains to denumerable ones. Later, this concept was extended to more general stochastic transition systems (STSs), allowing the design of various verification algorithms for large classes of (infinite) STSs. We further improve the understanding and utility of decisiveness in two ways. First, we provide a general criterion for proving decisiveness of general STSs. This criterion, which is very natural but whose proof is rather technical, (strictly) generalizes all known criteria from the literature. Second, we focus on stochastic hybrid systems (SHSs), a stochastic extension of hybrid systems. We establish the decisiveness of a large class of SHSs and, under a few classical hypotheses from mathematical logic, we show how to decide reachability problems in this class, even though they are undecidable for general SHSs. This provides a decidable stochastic extension of o-minimal hybrid systems. [ABM07] Parosh A. Abdulla, Noomene Ben Henda, and Richard Mayr. 2007. Decisive Markov Chains. Log. Methods Comput. Sci. 3, 4 (2007).