Behaviors of Processes and Synchronized Systems of Processes

Abstract

In these notes we consider processes as given by their sets of behaviors, including the infinite behaviors. The set HR∞ (p) of a process p is what we call an infinitary language, i.e. a subset of the set A∞ = A* ∪ Aω of all finite words (A*) and infinite words (Aω) on an alphabet of actions A. In a first part we study infinitary languages and then recognizability by transition systems with a special emphasis on infinitary rational languages which are defined as the family of those languages which are recognizable by finite transition systems. The definition of recognizability of a finite word by a transition system is the standard one which goes back to S. Kleene: the transition system S recognizes the word f ∈ A* iff there exists a computation sequence of S reading f which starts in an initial configuration and terminates in a final configuration (both sets of initial and final configurations are given as parts of the definition of S).