A New Definition of Morphism on Petri Nets

Petri nets are a fundamental model of concurrent processes and have a wide range of applications. They can be viewed as generalisation of transition systems in which concurrency is not simulated by non-deterministic interleaving. They were invented by C. A. Petri in the 60's. (A reference work is [Br].) It can be argued that the main effort and success of Petri Net Theory has been in developing techniques for showing properties of arbitrary Petri nets, e.g. Kurt Lautenbach has used techniques of linear algebra to discover invariants (properties which hold at all reachable markings). These techniques can be used to prove properties of concurrent programs. First represent the program as one big net and then prove properties about that. The problem is that big nets get out of hand, and more easily out of mind. For this reason chiefly, Hartmann Genrich, Kurt Lauteubach and Kurt Jensen invented predicate transition nets and coloured nets [GL, J] and accompanying techniques to find their invariants. Although they certainly do give a more compact way to model programs and systems they are necessarily more complicated, are more like programs, and need a semantics to relate them to structures which are more simple and universal. We address another problem, that of constructions on Petri nets and how to prove properties of a compound process by proving properties of its components. The constructions follow from a new notion of morphism on Petri nets—it is not the same as Petri's original notion. The morphisms respect the token game unlike Petri's original. The category of nets with the new morphisms has a product which is closely related to various parallel compositions which have been defined on labelled Petri nets for synchronising processes (see e.g. the compositions on nets defined in [LS,...] and section 3). It has a coproduct which is a generalised form of the "sum" operation as used for example in [M]. One can use Petri nets to give semantics to programming languages. But, what is the semantics of nets? In themselves nets are complicated objects whose behaviour is rather intricate. When do Petri nets have the same behaviour? Attempting to answer these questions leads naturally to occurrence nets first introduced in [NPW1, 2]. Occurrence nets form' a subcategory which bears a pleasant relation to the larger category of nets; the inclusion functor has a right adjoint which is an operation taking a net to its unfolding …