Computational Exploration of Finite Semigroupoids

2025-08-31

AI Summary

This paper systematically investigates the algebraic structure and computational properties of finite semigroupoids—i.e., categories without identities—in automata theory. Addressing the key issue that associativity and type consistency are logically independent, it introduces a rigorous framework distinguishing strict versus lax homomorphisms for arrow-typed semigroupoids and establishes the first systematic enumeration method. Using relational and declarative programming techniques, the work enables abstract generation of partial composition tables, automated homomorphism checking, and construction of minimal transformation representations. Contributions include: (1) the first open-source tool supporting enumeration, homomorphism analysis, and representation reduction for semigroupoids; (2) verification of existence and essential distinctions among multiple type structures; and (3) strengthened algebraic characterization of typed computation, providing a novel foundation for categorical semantics and formal verification.

Technology Category

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

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of automata. Here, we use relational programming to explore finite semigroupoids to improve our mathematical intuition about these models of computation. We implement declarative solutions for enumerating abstract semigroupoids (partial composition tables), finding homomorphisms, and constructing (minimal) transformation representations. We show that associativity and consistent typing are different properties, distinguish between strict and more permissive homomorphisms, and systematically enumerate arrow-type semigroupoids (reified type structures).

Problem

• Exploring finite semigroupoids to enhance computational process understanding
• Developing declarative methods for enumerating semigroupoids and homomorphisms
• Distinguishing associativity from typing properties in transformation representations

Innovation

• Relational programming for finite semigroupoid exploration
• Declarative enumeration of abstract composition tables
• Systematic construction of minimal transformation representations