The Equational Theory of Pomsets

Pomsets have been introduced as a mode2 of concurrency. Since a pomset is a string in which the total order has been relaxed to be a partial order, in this paper we view them as a generalization cf Strings, and investigate their algebraic properties. In particular, we investigate the axiomatic properties of pornsets, sets of pomsets and ideals of pornsets, under such operations as concatenation, parallel composition, union and their associated closure operations. We find that the equational theory of sets, pomsets under concatenation, parallel composition and union is finitely axiomatizable, whereas the theory of languages under the analogous operations is not. A similar result is obtained for ideals of pornsets, which incorporate the notion of subsumption which is also known as auaentation. Finally, we show that the addition of any closure operation (parallel or serial) Ieads to nonfinite axiomatizability of the resulting equational theory. Recently, many researchers have turned to partial orders in order to aid in understanding the semantics of parallel programs. Consider the work of Lamport [ l&19], and Winskei [30,31] as just two examples of this trend. The advocates of net theory have long held this point of view [3,9]. The particular formalism which is inspired by partial orders that we concern ourselves with is known as the partially ordered multiset or pomset. These were first suggested as a solution to what has become known as the Brock-Ackerman anomaly [4] to the dataffow networks of Kahn and MacQueen [16]. Pratt has written many papers showing how pomsets can be used to represent parallel processes of a very general nature and how they can be composed [23,24]. This work was cgrried out independently from that of Grabowski [9], who introduced the notion of a partial string and a partial language, which coincide exactiy with the notions of pomset and process in this paper. Grabowski was concerned with characterizing the notion of firability of a partial string by a Petri net. His paper explores the relationship between partial strings an well as discussing the notion of Y-product, which is closely reidted t hope that the reader will indulge et " and " process " where Grabows " partial language " .