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

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.

Bio‐based thermoplastic polyurethane and polyamide 11 bioalloys with excellent shape memory behavior

“…Modification of polyamides by melt blending has been well documented in the literature. [9][10][11][12][13][14][15][16] Previous researches [10][11][12][13][14][15][16] used thermoplastic polyurethane (TPU) to improve the material's mechanical performance. Polyamides are expected to be more compatible with TPU and can possibly modify the final properties of blends because of the formation of intermolecular hydrogen bonds between amide groups of polyamide and urethane groups of TPU.…”

“…Further addition of TPU, however, resulted in larger droplet size and lower impact strength. 12,13 Genovese and Shanks 14 used simulation method to discuss the specific interactions between PA6 and TPU. Some researchers 15,16 have attempted to use functionalized polyethyleneoctene elastomer (POE) to improve PAs impact strength.…”

Petri nets based on Lawvere theories

“…Lending nets are also equipped with a partial labeling of the places and transitions so they can be composed and are required to have no transitions which can be fired spontaneously. In 2018, Genovese and Herold introduced integer nets ( Genovese and Herold, 2018). Let ABGRP be the Lawvere theory of abelian groups.…”

“…There is a correspondence between lending Petri nets and propositional contract logic; a form of logic useful for ensuring that complex networks of contracts are honored (Bartoletti et al, 2015). Genovese and Herold constructed a categorical semantics for integer nets (Genovese and Herold, 2018). In Section 6, we will construct a variation of this semantics which uses the general framework developed in this paper.…”

“…The semantics of these showcase the individual token philosophy, which distinguishes between identical tokens and keeps track of causality within sequences of processes. In 2018, Herold and Genovese introduced integer nets, a type of Petri net which has a free abelian group of places (Genovese and Herold, 2018) and are similar to lending nets introduced in Bartoletti et al (2015). These are useful for modeling credit in propositional contract logic (Bartoletti and Zunino, 2010).…”