Adequate path expressions

Abstract

The object of this paper is to present some results in the formal study of path expressions. Path expressions, or GR-paths, are syntactic objects, that is, terminals strings of a grammar. GR-paths themselves may also be regarded as grammars; each GR-path determines a trace language, in the sense of [M_az 77]. In this sense, GR-paths may be thought of as a defining systems, in that each is associated with a set of possible histories of concurrent behaviour, represented formally by traces. In this exposition, traces are represented as n-tuples of strings and it is briefly shown how such n-tuples determine labelled, partially ordered sets, a popular way of modelling concurrent histories.

A general problem in the study of path expressions is that of being able to deduce dynamic properties of paths from their static properties, that is to deduce properties of a set of traces from combinatorial properties of a string, the GR-path which generates this set. In this paper we show that the formalism associated with path expressions supports rigorous mathematical analysis of paths with respect to a dynamic property called adequacy, a notion related to freedom from deadlock and to the internal consistency of a specification. Several theorems are proved about this property, including a generalisation of the well-known liveness theorem for 1-safe marked graphs (3.3) and a number of results on adequacy-preserving transformations, which may, in some cases, be used to reduce large programs to smaller and more manageable ones with the same adequacy properties. These results are illustrated with examples.

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