Axiomatizing the algebra of net computations and processes

Abstract.

Descriptions of concurrent behaviors in terms of partial orderings (called nonsequential processes or simply processes 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 of sequential composition. In this paper, a new algebraic axiomatization is proposed, where, given a net $N$, a term algebra ${\cal P}[N]$ ith 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, ${\cal P}[N]$ is a symmetric strict monoidal category [See the Appendix for a precise definition of this notion. However, the basic idea is simple. There is a binary operation, defined both on the objects and on the morphisms, that is functorial and satisfies the axioms of a monoid up to a natural isomorphism. If the monoid peration is commutative (again, up to a natural isomorphism), the monoidal category is called symmetric. For example, the cartesian product of sets is a symmetric monoidal operation. If the natural isomorphisms are identities, then we get strict versions of the notion.], parallel composition is the monoidal operation on morphisms and sequential composition is morphism composition. Besides ${\cal P}[N]$, we introduce a category ${\cal S}[N]$ containing the classical occurrence and step sequences. The term algebras of ${\cal P}[N]$ and of ${\cal S}[N]$ are in general incomparable, thus we introduce two more categories ${\cal K}[N]$ and ${\cal T}[N]$ providing an upper and a lower bound, respectively. A simple axiom expressing the functoriality of parallel composition maps ${\cal K} [N]$ to ${\cal P}[N]$ and ${\cal S}[N]$ to ${\cal T}[N]$, while commutativity of parallel composition maps ${\cal K} [N]$ to ${\cal S}[N]$ and ${\cal P}[N]$ to ${\cal T}[N]$ (see Fig. 4). Morphisms of ${\cal K} [N]$ constitute a new notion of concrete net computation, while the strictly symmetric strict monoidal category ${\cal T}[N]$ 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 ${\cal T} [N]$ 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.