In: EATCS Bulletin, No. 61, Formal Specification Column, pages 52-58. February 1997.
Abstract: Petri nets have been first introduced by C. A. Petri in his famous PhD thesis and have been studied from a theoretical and practical point of view in numerous articles and books. In the first 20 years mainly low-level nets including condition-event systems, elementary net systems and place-transition nets have been considered. The first type of high-level nets, called predicate-transition nets, was developed by Genrich and Lautenbach, to support the modelling of concurrent and distributed systems. In the last decade several other high-level nets were introduced and studied for similar purposes including coloured Petri nets by Jensen, Petri nets with individual tokens by Reisig, and combinations of nets with algebraic specifications by Krmer and Schmidt as well as by Vautherin and Reisig, leading to several variants of algebraic nets, which are also called algebraic high-level nets. From an algebraic point of view methods of linear algebra are used to calculate invariants for all kinds of Petri nets. Moreover, rings and modules are considered by Reisig, monoids, algebras and morphisms of Petri nets leading to Petri net categories by Winskel, Meseguer and Montanari. This categorical view of Petri nets has been used to study compositionality problems, transformations of nets in the sense of graph transformations and high-level-replacement systems, and to present a unified approach for different types of Petri nets. In this overview we show how to come from the conventional representation of Petri nets in terms of sets and multisets to algebraic versions of low-level and high-level nets. These types of nets are currently studied in the literature using different kinds of terminology. We only present the basic definitions from an algebraic point of view and the reader is kindly requested to consult the corresponding papers in our references concerning examples, theory and applications.