A Linear Specification Language for Petri Nets

This paper defines a category GNet with object set all Petri nets. A morphism in GNet from a net N to a net N’ gives a precise way of simulating every evolution of N by an evolution of N’. We exhibit a morphism from a simple message handler to one with error—correction, showing that the more refined message handler can simulate any behaviour ofits simple counterpart. The existence of such a morphism proves the correctness of the refinement. Earlier work [BroPO, BG9O, BG] defined a modular theory of elementary Petri nets based on de Paiva’s Dialectica categorical models of linear logic. We here modify her construction, defining categories MNC which model intuitionistic linear logic [GL87]. GNet arises naturally from MNSOt, inheriting the structure which models linear logic. This more general framework has several advantages over our previous one. The theory is simplified, we obtain precise results about morphisms as simulations, relating them to CCS, and we obtain a natural extension to marked nets. The linear connectives are modelled in GNet by net combinators. Being functonal, these combinators opt are such that, if each N~is refined by a net N~,then op~(Ns Nm) is refined by opt(N~ N~,,).We show that the operation of restriction also has this property, and thus (in the language of algebraic specification) our notion ofrefinement composes horizontally with respect to the linear connectives and restriction. Furthermore, our notion ofrefinement composes vertically because it corresponds to categorical morphisms. These properties of our notion of refinement are precisely those required to develop an algebra of nets in which complex nets can be built from smaller components, and refined in a modular and compositional way. We illustrate our approach with an extended example, analogous to Mllner’s Jobshop example.