Reasoning about Algebraic Generalisation of Petri Nets

In this paper we study properties of and (as we hope) a uniform frame for Petri net models, which enables us to generalise algebra as well as enabling rule of Petri nets. Our approach of such a frame is based on using partial groupoids in Petri nets. Properties of Petri nets constructed in this manner are investigated through related labelled transition systems. In particular, we investigate the relationships between properties of partial groupoids used in Petri nets and properties of transition systems crucial for the existence of the state equation and linear algebraic techniques. We show that partial groupoids embeddable into Abelian groups play an important role in preserving these properties.

[1]  C. Petri.  Kommunikation mit Automaten , 1962 .

[2]  A. Clifford,et al.  The algebraic theory of semigroups , 1964 .

[3]  Richard M. Karp,et al.  Parallel Program Schemata , 1969, J. Comput. Syst. Sci..

[4]  Robert M. Keller,et al.  Formal verification of parallel programs , 1976, CACM.

[5]  James Lyle Peterson,et al.  Petri net theory and the modeling of systems , 1981 .

[6]  Jirí Adámek,et al.  Theory of Mathematical Structures , 1983 .

[7]  Grzegorz Rozenberg.  Behaviour of Elementary Net Systems , 1986 .

[8]  P. S. Thiagarajan.  Elementary Net Systems , 1986 .

[9]  José Meseguer,et al.  Petri Nets Are Monoids , 1990, Inf. Comput..

[10]  Grzegorz Rozenberg,et al.  Diamond properties of elementary net systems , 1991, Fundam. Informaticae.

[11]  Grzegorz Rozenberg,et al.  High-level Petri Nets: Theory And Application , 1991 .

[12]  Mogens Nielsen,et al.  Models for Concurrency , 1992 .

[13]  Maciej Koutny,et al.  Invariant Semantics of Nets with Inhibitor Arcs , 1991, CONCUR.

[14]  Grzegorz Rozenberg,et al.  Elementary Transition Systems , 1990, Theor. Comput. Sci..

[15]  Madhavan Mukund.  Petri Nets and Step Transition Systems , 1992, Int. J. Found. Comput. Sci..

[16]  Kurt Jensen,et al.  Coloured Petri Nets: Basic Concepts, Analysis Methods and Practical Use. Vol. 2, Analysis Methods , 1992 .

[17]  Xudong He,et al.  A Formal Definition of Hierarchical Predicate Transition Nets , 1996, Application and Theory of Petri Nets.

[18]  Alessandro Giua,et al.  A Survey of Petri Net Methods for Controlled Discrete Event Systems , 1997, Discret. Event Dyn. Syst..

[19]  Kurt Jensen,et al.  Coloured Petri Nets , 1997, Monographs in Theoretical Computer Science An EATCS Series.

[20]  Eric Badouel.  Representations of Reversible Automata and State Graphs of Vector Addition Systems , 1998 .

[21]  Gabriel Juhás.  The essence of Petri nets and transition systems through Abelian groups , 1998, Electron. Notes Theor. Comput. Sci..