Abstract

Abstract: This document gives an algebraic and two polygraphic translations of Petri nets, all three providing an easier way to describe reductions and to identify some of them. The first one sees places as generators of a commutative monoid and transitions as rewriting rules on it: this setting is totally equivalent to Petri nets, but lacks any graphical intuition. The second one considers places as 1-dimensional cells and transitions as 2-dimensional ones: this translation recovers a graphical meaning but raises many difficulties since it uses explicit permutations. Finally, the third translation sees places as degenerated 2-dimensional cells and transitions as 3-dimensional ones: this is a setting equivalent to Petri nets, equipped with a graphical interpretation. Outline In this document, we study Petri nets in order to give two possible polygraphic presentations for them. This work follows Albert Burroni’s intuitions: many computer science and proof theory objects have natural translations into polygraphs. These are topology-flavoured objects consisting of collections of directed cells of various dimensions, equipped with a rich algebraic structure. In section 1, we recall some basic facts about Petri nets, describe their representations and associate them reduction graphs, equipped with a relation that identifies paths that intuitively represent the same

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