Comparing models of parallel computation by homomorphisms

Abstract

Since the early 1960's many different models of parallel computation have been developed, some being based on finite state n1achine approaches, others on program schemata models, with many having underlying directed graph representations.[1,2,7,8,10,14,16,17] On the surface these models often appear quite different from one another. Also, the particular types of parallelism problems studied, and the results obtained, seemed often to bear no relation to one another. Thus, the area of modelling parallel computation seemed rather fragmented. In attempts to help unify the area, papers have been written comparing the various models and showing relationships between them. [2.9-13,15] Thus, results obtained in terms of one model often could be translated into results in another model. It is fairly well known by now, for example, that vector addition systems are in some sense equivalent to Petri nets,[6,13,16] and that some of the other graphical models for parallel computation are somehow less powerful than Petri nets or vector addition systems. This paper presents a comparative study of models of parallel computation using a rather general notion of a homomorphism between computation systems. This notion provides a precise relationship of one computation system being simulated by another computation system. In comparison to previous studies, this approach provides a more precise and deeper understanding of how the properties of the various models are interrelated. Many of the previously discovered relationships follow as corollaries of results obtained here. In particular, the concept of an isomorphism between two models turns out to be a particular type of homomorphism which is shown to be both length preserving and bijective. Results in this paper that may be of particular interest include a long list of properties that are preserved under homomorphism, that vector replacement systems can be simulated by vector addition systems, and that generalized Petri nets can be simulated by Petri nets. Here we outline our approach without giving all the precise definitions, and we omit all proofs. A more complete account is given in [9].