Axiomatizing the algebra of net computations and processes

Descriptions of concurrent behaviors in terms of partial orderings (callednonsequential processes or simplyprocesses in Petri net theory) have been recognized as superior when information about distribution in space, about causal dependency or about fairness must be provided. However, at least in the general case of Place/Transition (P/T) nets, the proposed models lack a suitable, general notion ofsequential composition.In this paper, a new algebraic axiomatization is proposed, where, given a netN, a term algebraP[N] with two operations of parallel and sequential composition is defined. The congruence classes generated by a few simple axioms are proved isomorphic to a slight refinement of classical processes.Actually,P[N] is a symmetric strict monoidal category1, parallel composition is the monoidal operation on morphisms and sequential composition is morphism composition. BesidesP[N], we introduce a categorys[N] containing the classical occurrence and step sequences. The term algebras ofP[N] and ofs[N] are in general incomparable, thus we introduce two more categoriesK[N] and ℐ[N] providing an upper and a lower bound, respectively. A simple axiom expressing the functoriality of parallel composition mapsK[N] toP[N] ands[N] to ℐ[N], while commutativity of parallel composition mapsK[N] tos[N] andP[N] to ℐ[N] (see Fig. 4).Morphisms ofK[N] constitute a new notion of concrete net computation, while the strictly symmetric strict monoidal category ℐ[N] was introduced previously by two of the authors as a new algebraic foundation for P/T nets [22]. In the context of the present paper, the morphisms of ℐ[N] are proved isomorphic to the processes defined in terms of the “swap” transformation by Best and Devillers [5]. Thus the diamond of the four categories gives a full account in algebraic terms of the relations between interleaving and partial ordering observations of P/T net computations.