An Axiomatization of the Token Game Based on Petri Algebras

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 nets whose flow relations 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 numbers. 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 ...

• Type: Journal
• Year: 2007
• Where: FUIN
• Authors: Eric Badouel, Jules Chenou, Goulven Guillou
• Keywords: Commutative Monoid, FUIN 2007, Lexicographic Petri Nets, Petri Nets