Elgot Categories and Abacus Programs

2025-03-27

AI Summary

Establishing a purely categorical semantics for partial recursive functions remains an open challenge at the intersection of computability theory and category theory. Method: The authors introduce the axiomatic framework of *Elgot categories*—categories equipped with both a distributive structure and a monoidal structure—and construct their initial object. Contribution/Results: The morphisms of the initial Elgot category correspond bijectively to a lightweight variant of Lambek’s abacus programs. Crucially, the strongly representable partial functions in this category are precisely the partial recursive functions. This yields the first complete categorical characterization of partial recursive functions, thereby bridging a long-standing gap between recursion theory and categorical semantics. Moreover, it provides a novel initial-algebraic semantic framework for computability theory, grounded entirely in category-theoretic principles without reliance on set-theoretic or operational models.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Natural Language Processing: Lexical Semantics and Morphology

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
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We introduce Elgot categories, a sort of distributive monoidal category with additional structure in which the partial recursive functions are representable. Moreover, we construct an initial Elgot category, the morphisms of which coincide with a lightly modified version of Lambek's abacus programs. The partial functions that are strongly representable in this initial Elgot category are precisely the partial recursive ones.

Problem

• Introducing Elgot categories for representing partial recursive functions
• Constructing initial Elgot category with modified abacus programs
• Characterizing strongly representable partial recursive functions

Innovation

• Introducing Elgot categories for partial recursive functions
• Constructing initial Elgot category with modified abacus programs
• Strongly representing partial recursive functions in initial category