A maximal entropy stochastic process for a timed automaton

Several ways of assigning probabilities to runs of timed automata (TA) have been proposed recently. When only the TA is given, a relevant question is to design a probability distribution which represents in the best possible way the runs of the TA. We give an answer to this question using a maximal entropy approach. We introduce our variant of a stochastic model, the stochastic process over runs, which permits to simulate random runs of any given length with a linear number of atomic operations. We adapt the notion of Shannon (continuous) entropy to such processes. Our main contribution is an explicit formula defining a process Y * which maximizes the entropy. This ensures that, among the stochastic process over runs of a given TA, Y * is the one that permits to sample runs of the TA in the most uniform way possible. Hence, our method could be used in a statistical model checking framework, providing a non-trivial yet natural way to generate runs in a quasi-uniform manner (described in the article). The formula defining Y * is an adaptation of the so-called Shannon-Parry measure to the timed automata setting. We also show that Y * enjoys well known properties in ergodic and information theory, namely, Y * is ergodic and satisfies an asymptotic equipartition property.