Executions in (Semi-)Integer Petri Nets are Compact Closed Categories

TL;DR

This paper demonstrates that Petri nets with negative tokens can be represented as compact closed categories, revealing complex causal relations and enabling applications in economics and computer science.

Contribution

It introduces a functorial construction linking (semi-)integer Petri nets to compact closed categories and shows how to recover nets from their categories, establishing an adjoint pair.

Findings

Categories of executions are compact closed

Negative tokens induce non-trivial causal relations

Framework models phenomena in economics and computer science

Abstract

In this work, we analyse Petri nets where places are allowed to have a negative number of tokens. For each net we build its correspondent category of executions, which is compact closed, and prove that this procedure is functorial. We moreover exhibit a procedure to recover the original net from its category of executions, show that it is again functorial, and that this gives rise to an adjoint pair. Finally, we use compact closeness to infer that allowing negative tokens in a Petri net makes the causal relations between transition firings non-trivial, and we use this to model interesting phenomena in economics and computer science.