Petri Algebras

Eric Badouel 1 Jules Chenou 2 Goulven Guillou 3
1 S4 - System synthesis and supervision, scenarios
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
3 Lab-STICC_UBO_CACS_MOCS
Lab-STICC - Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance, UBO - Université de Brest
Abstract : The firing rule of Petri nets relies on a residuation operation for the commutative monoid of natural number. We identify a class of residuated commutative monoids, called Petri algebras, for which one can mimic the token game of Petri nets to define the behaviour of generalized Petri net whose flow relation and place contents are valued in such algebraic structures. The sum and its associated residuation capture respectively how resources within places are produced and consumed through the firing of a transition. We show that Petri algebras coincide with the positive cones of lattice-ordered commutative groups and constitute the subvariety of the (duals of) residuated lattices generated by the commutative monoid of natural number. We however exhibit a Petri algebra whose corresponding class of nets is strictly more expressive than the class of Petri nets. More precisely, we introduce a class of nets, termed lexicographic Petri nets, that are associated with the positive cones of the lexicographic powers of the additive group of real numbers. This class of nets is universal in the sense that any net associated with some Petri algebras canbe simulated by a lexicographic Petri net. All the classical decidable properties of Petri nets however (termination, covering, boundedness, structural boundedness, accessibility, deadlock, liveness ...) are undecidable on the class of lexicographic Petri nets. Finally we turn our attentionto bounded nets associated with Petri algebras and show that their dynamic can be reformulated in term of MV-algebras.
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Submitted on : Friday, May 19, 2006 - 9:06:48 PM
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  • HAL Id : inria-00070648, version 1

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Eric Badouel, Jules Chenou, Goulven Guillou. Petri Algebras. [Research Report] RR-5355, INRIA. 2004, pp.28. 〈inria-00070648〉

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