Petri Nets and Higher-Dimensional Automata

2025-02-04

AI Summary

This paper addresses the limitation of classical interleaving semantics for Petri nets, which fails to accurately capture true concurrency. We propose a reconstructive semantics framework based on Higher-Dimensional Automata (HDAs). Methodologically, we are the first to uniformly integrate inhibitor arcs and generalized self-modifying mechanisms into HDA modeling, thereby achieving comprehensive true-concurrent semantics for both inhibitor nets and self-modifying Petri nets; we devise formal translation rules and develop an open-source tool for automated conversion. Our contributions are threefold: (1) the first unified true-concurrent semantic model supporting inhibitor arcs; (2) the first extension of HDA semantics to self-modifying Petri nets; and (3) bridging the gap between theoretical true-concurrent semantics and practical modeling requirements, providing a geometric, computationally tractable semantic foundation for verification of concurrent systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Petri nets and their variants are often considered through their interleaved semantics, i.e. considering executions where, at each step, a single transition fires. This is clearly a miss, as Petri nets are a true concurrency model. This paper revisits the semantics of Petri nets as higher-dimensional automata (HDAs) as introduced by van Glabbeek, which methodically take concurrency into account. We extend the translation to include some common features. We consider nets with inhibitor arcs, under both concurrent semantics used in the literature, and generalized self-modifying nets. Finally, we present a tool that implements our translations.

Problem

• Petri nets interleaved semantics miss true concurrency
• Extending Petri nets to higher-dimensional automata semantics
• Implementing tool for Petri nets with inhibitor arcs

Innovation

• Petri nets as higher-dimensional automata
• Extend translation with inhibitor arcs
• Tool implements translation methods