Action Systems with Continuous Behaviour

An action system framework is a predicate transformer based method for modelling and analysing distributed and reactive systems. The actions are statements in Dijkstra's guarded command language, and their semantics is given by predicate transformers. We extend conventional action systems with a di erential action consisting of a di erential equation and an evolution guard. The semantics is given by a weakest liberal precondition transformer, because it is not always desirable that di erential actions terminate. It is shown that the proposed di erential action has a semantics which corresponds to a discrete approximation when the discrete step size goes to zero. The extension gives action systems the power to model real-time clocks and continuous evolutions within hybrid systems. In this paper we give a standard form for such a hybrid action system. We also extend parallel composition to hybrid action systems. This does not change the original meaning of the parallel composition, and therefore ordinary action systems compose in parallel with hybrid action systems. Note This TUCS technical report combines and updates the two TUCS technical reports no.109 and no.110.