Trace-based process algebras for real-time probabilistic systems

We study the problem of defining trace-based models for systems with real-time and stochastic behaviour. Trace semantics is a simple and intuitive model that describes the behaviour of a system in terms of the possible actions it can perform, and it can be used for the specification of safety properties. We present two process algebras, one featuring real-time and another stochastic behaviour, and investigate their trace semantics. In the case of real-time systems, we propose a real-time extension to the process algebra CSP. Inspired by timed automata, a successful formalism for the specification and verification of real-time systems, we handle real time by means of clocks, i.e. real-valued variables that increase at the same rate as time. We give a discrete trace semantics to our language and define the resulting refinement and equivalence relations. One advantage of our proposal is that it is possible to automatically verify refinement relations between processes. We demonstrate how this can be achieved and also describe the syntactic restrictions needed and the limitations of this approach. We take a different approach in the case of stochastic systems: in this case we focus on the operational model, and investigate ways to define a trace semantics. We study the interaction between non-deterministic and probabilistic behaviour in systems with continuous state spaces, arbitrary probability distributions and uncountable branching. We identify the class of schedulers that ensures measurability properties on executions, and show that such measurability properties are preserved by parallel composition. Under these restrictions, we define trace semantics and show that a more general notion of bisimulation is needed when dealing with continuous state spaces in order to preserve linear-time semantics. Finally, we show how this model can be used to give semantics to a stochastic process algebra.