What Will Be Eventually True of Polynomial Hybrid Automata?

Abstract

Hybrid automata have been introduced in both control engineering and computer science as a formal model for the dynamics of hybrid discrete-continuous systems. While computability issues concerning safety properties have been extensively studied, liveness properties have remained largely uninvestigated. In this article, we investigate decidability of state recurrence and of progress properties.

First, we show that state recurrence and progress are in general undecidable for polynomial hybrid automata. Then, we demonstrate that they are closely related for hybrid automata subject to a simple model of noise, even though these automata are infinite-state systems. Based on this, we augment a semi-decision procedure for recurrence with a semidecision method for length-boundedness of paths in such a way that we obtain an automatic verification method for progress properties of linear and polynomial hybrid automata that may only fail on pathological, practically uninteresting cases. These cases are such that satisfaction of the desired progress property crucially depends on the complete absence of noise, a situation unlikely to occur in real hybrid systems.

This article reflects work that has been partially funded by the Deutsche Forschungsgemeinschaft under contract DFG Da 206/5-2.