Like most models used in model-checking, timed automata are an idealized mathematical model used for representing systems with strong timing requirements. In such mathematical models, properties can be violated, due to unlikely (sequences of) events. We propose two new semantics for the satisfaction of LTL formulas, one based on probabilities, and the other one based on topology, to rule out these sequences. We prove that the two semantics are equivalent and lead to a PSPACE-Complete model-checking problem for LTL over finite executions.

We investigate the relation between the behavior of non-deterministic systems under fairness constraints, and the behavior of probabilistic systems. To this end, first a framework based on computable stopping strategies is developed that provides a common foundation for describing both fair and probabilistic behavior. On the basis of stopping strategies it is then shown that fair behavior corresponds in a precise sense to random behavior in the sense of Martin-Löf’s definition of randomness. non-deterministic systems. Under this perspective the question is investigated what probabilistic properties are needed in such an implementation to guarantee (with probability one) certain required fairness properties in the behavior of the probabilistic system. Generalizing earlier concepts of ɛ-bounded transition probabilities, we introduce the notion of divergent probabilistic systems, which enables an exact characterization of the fairness properties of a probabilistic implementation. Looking beyond pure fairness properties, we also investigate what other qualitative system properties are guaranteed by probabilistic implementations of fair non-deterministic behavior. This leads to a completeness result which generalizes a well-known theorem by Pnueli and Zuck.