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Petri Algebras

Goulven Guillou 1 Eric Badouel 2 Jules Chenou 3 
2 S4 - System synthesis and supervision, scenarios
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
Abstract : The firing rule of Petri nets relies on a residuation operation for the commutative monoid of natural numbers. 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. 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 numbers. 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 can be simulated by a lexicographic Petri net. All the classical decidable properties of Petri nets however are undecidable on the class of lexicographic Petri nets. Finally we turn our attention to bounded nets associated with Petri algebras and show that their dynamics can be reformulated in term of MV-algebras.
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Submitted on : Tuesday, January 13, 2015 - 5:05:52 PM
Last modification on : Monday, March 14, 2022 - 11:08:07 AM

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Goulven Guillou, Eric Badouel, Jules Chenou. Petri Algebras. ICALP 2005 : 32nd International Colloquium, Automata, Languages and Programming, Jul 2005, Lisboa, Portugal. pp.742-754, ⟨10.1007/11523468_60⟩. ⟨hal-01102950⟩



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