Articles
Petri nets with generalized algebra: a comparison

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Abstract

In the last decade we can see substantial effort to develop an abstract and uniform constructions for Petri nets. Most of such abstractions are based on algebraic characterizations of Petri nets. They work mostly over commutative monoids and their various subclasses, namely cancellative commutative monoids or cones of Abelian groups. In the paper we study relationships between Petri nets with generalized underlying algebra. More precisely, we study Petri nets over commutative monoids, cancellative commutative monoids, cones of Abelian groups, and fully ordered cones of Abelian groups. As the main result, we show that classes of reachability graphs of Petri nets over cancellative commutative monoids and cones of Abelian groups coincide (up to isomorphism). In other words, partial order on used cancellative commutative monoid plays no role in expressive power of Petri nets. However, as shows the fact that the class of reachability graphs of nets over fully ordered cones is a proper subclass of the class of reachability graphs of nets over cancellative commutative monoids, the total order on used monoids plays an important role in expressive power of Petri nets.

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This work is a part of the joint research project “DFG-Forschergruppe Petrinetz-Technologie” between H. Weber (Coordinator), H. Ehrig (both from Technical University Berlin) and W. Reisig (Humboldt University Berlin) supported by the German Research Council (DFG). Part of this work was done during the author's stay at Institute of Control Theory and Robotics, Slovak Academy of Sciences, and BRICS (Basic Research in Computer Science, Centre of the Danish National Research Foundation) Department of Computer Science, University of Aarhus.