A Simple Categorical Calculus of Interacting Processes

2026-03-18

AI Summary

This work addresses the formal modeling of interactive concurrent processes by proposing a multi-categorical parameterized term rewriting system. For the first time, process interaction is characterized as a confluent and terminating rewrite relation, and it is shown that the resulting term structures naturally form a virtual double category. The central contribution lies in the construction of a denotational semantic functor from this computational syntax to the free corner construction in free monoidal categories, thereby providing a rigorous semantic foundation and mathematical guarantee for interactive behavior.

Technology Category

• Cognitive Modeling & Cognitive Systems: Social Cognition And Interaction
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

We present a calculus that models a simple sort of process interaction. Our calculus consists of a collection of terms together with a rewrite relation, parameterised by an arbitrary multicategory whose morphisms we understand as non-interactive processes. We show that our calculus is confluent and terminating, and that terms modulo the induced convertibility relation form a virtual double category. We relate our calculus to the free cornering of a monoidal category, which is a double-categorical model of process interaction that is similar in spirit to the calculus presented herein. Precisely, we construct a functor from the virtual double category given by our calculus into the underlying virtual double category of the free cornering of the free monoidal category on the multicategory of non-interacting processes. If we think of the terms of our calculus as programs and the rewriting system as an operational semantics for these programs, this functor gives a sound denotational semantics for our calculus in terms of the free cornering.

Problem

• process interaction
• categorical semantics
• virtual double category
• multicategory
• operational semantics

Innovation

• virtual double category
• process interaction
• multicategory
• confluent rewriting
• free cornering