Diagonal Compositionality of Partial Petri Nets

Abstract

A categorical semantic domain is constructed for Petri nets which satisfies the diagonal compositionality requirement, i.e., Petri nets are equipped with a hierarchical specification mechanism (vertical compositionality) that distributes through net combinators (horizontal compositionality). Therefore, a net may be hierarchically specified before or after the joint behavior of component parts (nets) in order to obtain the same resulting system. The abstraction mechanism proposed is based on graph transformations using the so-called single pushout approach. A finitely complete and cocomplete category of partial marked Petri nets and partial morphisms is introduced and it is claimed that, with respect to partial morphisms, “Petri nets are semi-groups” . Classes of transformations stand for hierarchical specification mechanism, named reification, where part of a net (usually a transition) is replaced by another (possible complex) net. The composition of reifications (i.e., composition of pushouts) is defined, leading to a category of nets and reifications which is also complete and cocomplete. Since the reification operation composes, the vertical compositionality requirement of Petri nets is achieved. Then, it is proven that the reification also satisfies the horizontal compositionality requirement, i.e., the reification of nets distributes through the parallel composition. A specification grammar and the induced subcategory of nets and reifications stand for a system specification and all possible levels of system abstractions and their relationship, respectively. Also, a generalization of the top-down approach for concurrent systems is introduced.