Low- and High-Level Petri Nets with Individual Tokens

In this article, we present a new variant of Petri nets with markings called Petri nets with individual tokens, together with rule-based transformation following the double pushout approach. The most important change to former Petri net transformation approaches is that the marking of a net is no longer a collective set of tokens, but each each has an own identity leading to the concept of Petri nets with individual tokens. This allows us to formulate rules that can change the marking of a net arbitrarily without necessarily manipulating the structure. As a first main result that depends on nets with individual markings we show the equivalence of transition firing steps and the application of firing-simulating rules. We define categories of low-level and of algebraic high-level nets with individual tokens, called PTI nets and AHLI nets, respectively and relate them with each other and their collective counterparts by functors. To be able to use the properties and analysis results of \MCALM-adhesive HLR systems (formerly know as weak adhesive high-level replacement systems) we show in further main results that both categories of PTI nets and AHLI nets are \MCALM-adhesive categories. By showing how to construct initial pushouts we also give necessary and sufficient conditions for the applicability of transformation rules in these categories, known as gluing condition in the literature.