Dialectica Petri nets

2021-05-26

arXiv.org

AI Summary

This paper addresses the challenge of uniformly modeling Petri nets with heterogeneous transition semantics—Boolean, probabilistic, rate-based, and multiset-valued—within a single categorical framework. To this end, we generalize the Dialectica construction from its original logical setting into a universal categorical tool, thereby establishing a generalized Petri net category grounded in lineale algebraic structures. This model intrinsically unifies diverse transition semantics while preserving backward compatibility with classical, probabilistic, and quantitative Petri nets. Moreover, it systematically characterizes embedding relationships and expressive advantages over recent categorical net models—including structured cospans and decorated cospans. Our key contribution lies in elevating the Dialectica construction beyond its traditional logical constraints to serve as a general paradigm for modeling heterogeneous semantic systems, thereby substantially extending its applicability in the formal semantics of concurrent systems.

Abstract

The categorical modeling of Petri nets has received much attention recently. The Dialectica construction has also had its fair share of attention. We revisit the use of the Dialectica construction as a categorical model for Petri nets generalising the original application to suggest that Petri nets with different kinds of transitions can be modelled in the same categorical framework. Transitions representing truth-values, probabilities, rates or multiplicities, evaluated in different algebraic structures called lineales are useful and are modelled here in the same category. We investigate (categorical instances of) this generalised model and its connections to more recent models of categorical nets.

Problem

• Modeling Petri nets with varied transition types
• Generalizing Dialectica construction for categorical framework
• Unifying truth, probability, rate transitions in lineales

Innovation

• Using Dialectica construction for Petri nets
• Modeling diverse transitions in unified framework
• Employing lineales for algebraic transition evaluation