Padberg J., Ehrig H., Rozenberg G. (2001) Behavior and Realization Construction for Petri Nets Based on Free Monoid and Power Set Graphs. In: Ehrig H., Padberg J., Juhás G., Rozenberg G. (eds) Unifying Petri Nets. Lecture Notes in Computer Science, vol 2128. Springer, Berlin, Heidelberg
Starting from the algebraic view of Petri nets as monoids (as advocated by Meseguer and Montanari in [MM90]) we present the marking graphs of place transition nets as free monoid graphs and the marking graphs of specific elementary nets as powerset graphs. These are two important special cases of a general categorical version of Petri nets based on a functor M, called M-nets. These nets have a compositional marking graph semantics in terms of F-graphs, a generalization of free monoid and powerset graphs. Moreover we are able to characterize those F-graphs, called reflexive F-graphs, which are realizable by corresponding M-nets. The main result shows that the behavior and realization constructions are adjoint functors leading to an equivalence of the categories MNet of M-nets and RFGraph of reflexive F-graphs. This implies that the behavior construction preserves colimits so that the marking graph construction using F-graphs is compositional.
In addition to place transition nets and elementary nets we provide other interesting applications of M-nets and F-graphs. Moreover we discuss the relation to classical elementary net systems. The behavior and realization constructions we have introduced are compatible with corresponding constructions for elementary net systems (with initial state) and elementary transition systems in the sense of [NRT92].
place/transition nets elementary nets uniform approach realization problem category theory