Presheaf automata

2024-09-06

arXiv.org

AI Summary

Concurrent models—including higher-dimensional automata, Petri nets, and vector addition systems—suffer from semantic fragmentation due to disparate formal foundations. Method: We introduce *presheaf automata*, a categorical framework built on the category of presheaves over paths, unifying their semantics. Automaton semantics are defined via presheaves; formal languages are characterized by path and trace objects; and simulation and bisimulation are generalized using open maps. Contributions: (1) The first systematic application of presheaf methods to automata theory; (2) A precise embedding of finitely presentable presheaf automata into arbitrary (not necessarily safe) Petri nets, overcoming prior restrictions; (3) A comprehensive theoretical framework encompassing language equivalence, behavioral equivalences (e.g., simulation, bisimulation), and expressive power—establishing a unified semantic foundation for concurrent systems.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Cognitive Modeling & Cognitive Systems: Agent Architectures
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We introduce presheaf automata as a generalisation of different variants of higher-dimensional automata and other automata-like formalisms, including Petri nets and vector addition systems. We develop the foundations of a language theory for them based on notions of paths and track objects. We also define open maps for presheaf automata, extending the standard notions of simulation and bisimulation for transition systems. Apart from these conceptual contributions, we show that certain finite-type presheaf automata subsume all Petri nets, generalising a previous result by van Glabbeek, which applies to higher-dimensional automata and safe Petri nets.

Problem

• Generalize automata variants and formalisms like Petri nets
• Develop language theory based on paths and track objects
• Extend simulation and bisimulation via open maps

Innovation

• Generalize automata with presheaf automata
• Develop language theory using paths
• Extend simulation with open maps