Functorial Semantics for Petri Nets under the Individual Token Philosophy

Although the algebraic semantics of place/transition Petri nets under the collective token philosophy has been fully explained in terms of (strictly) symmetric (strict) monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory because it lacks universality and also functoriality. We introduce the notion of pre-net to recover these aspects, obtaining a fully satisfactory categorical treatment centered on the notion of adjunction. This allows us to present a purely logical description of net behaviours under the individual token philosophy in terms of theories and theory morphisms in partial membership equational logic, yielding a complete match with the theory developed by the authors for the collective token view of net

On the Category of Petri Net Computations

We introduce the notion of strongly concatenable process as a refinement of concatenable processes [DMM89] which can be expressed axiomatically via a functor $Q[-]$ from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net $N$, the strongly concatenable processes of $N$ are isomorphic to the arrows of $Q[-]$. In addition, we identify a coreflection right adjoint to $Q[-]$ and characterize its replete image, thus yielding an axiomatization of the category of net computations

Functorial Analysis of Algebraic Higher-Order Net Systems with Applications to Mobile Ad-Hoc Networks

Algebraic higher-order (AHO) net systems are Petri nets with place/ transition systems, i.e. place/transition nets with initial markings, and rules as tokens. In several applications, however, there is the need for explicit data modeling. The main idea of this paper is to introduce AHO net systems with high-level net systems and corresponding rules as tokens. We relate them to AHO net systems with low-level net systems as tokens and analyze the firing and transformation properties of the corresponding net class transformation defined as functors between the corresponding categories of AHO net systems. All concepts and results are explained with an example in the application area of mobile ad-hoc networks. From an abstract point of view, mobile ad-hoc networks consist of mobile nodes which communicate with each other independent of a stable infrastructure, while the topology of the network constantly changes depending on the current position of the nodes and their availability. To ensure satisfactory team cooperation in workflows of mobile ad-hoc networks we use the modeling technique of AHO net systems

On the Model of Computation of Place/Transition Petri Nets

In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical "token game", one can model the behaviour of Petri nets via non-sequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. More precisely, we introduce the new notion of decorated processes of Petri nets and we show that they induce on nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net N can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification