Shuffle on trajectories: Syntactic constraints

Abstract

We introduce and investigate new methods to define parallel composition of words and languages as well as of ω-words and ω-languages. The operation of parallel composition leads to new shuffle-like operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to define parallel composition of processes. It is also applicable to parallel computation. The operations are introduced using a uniform method based on the notion of trajectory. As a consequence, we obtain a very intuitive geometrical interpretation of the parallel composition operation. These operations lead in a natural way to a large class of semirings. The approach is amazingly flexible, diverse concepts from the theory of concurrency can be introduced and studied in this framework. For instance, we provide examples of applications to fairness property and to parallelization of non-context-free languages in terms of context-free and even regular languages. This paper concentrates on syntactic constraints. Semantic constraints will be dealt with in a forthcoming contribution.