Resourceful Traces for Commuting Processes

2025-07-24

AI Summary

This paper addresses the lack of resource-sensitive, action-commuting algebraic structure for effectful categories. We introduce a novel semantic framework based on Mazurkiewicz traces, modeling actions not as atomic operations but as resource transformations constrained by input/output types. This yields the first commuting tensor product on the free effectful category, rigorously capturing concurrent composition where actions commute and resources are shared. Our approach establishes, for the first time, a categorical correspondence between typed trace representations and generalized Freyd categories, enabling graphical reasoning. It provides a new algebraic foundation for compositional semantics of systems with computational effects.

Technology Category

• Knowledge Representation and Reasoning: Action, Change, and Causality
• Reasoning under Uncertainty: Causality
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We show that, when the actions of a Mazurkiewicz trace are considered not merely as atomic (i.e., mere names) but transformations from a specified type of inputs to a specified type of outputs, we obtain a novel notion of presentation for effectful categories (also known as generalised Freyd categories), a well-known algebraic structure in the semantics of side-effecting computation. Like the usual representation of traces as graphs, our notion of presentation gives rise to a graphical calculus for effectful categories. We use our presentations to give a construction of the commuting tensor product of free effectful categories, capturing the combination of systems in which the actions of each must commute with one another, while still permitting exchange of resources

Problem

• Develop a novel presentation for effectful categories
• Create a graphical calculus for effectful categories
• Construct commuting tensor product for free effectful categories

Innovation

• Transform actions into input-output transformations
• Graphical calculus for effectful categories
• Construct commuting tensor product for free categories