Combinatory Completeness in Structured Multicategories

2025-11-21

AI Summary

This paper addresses the lack of unified definitions and characterizations of compositional completeness across diverse application domains. We propose a general compositional completeness framework grounded in faithful Cartesian families and structured multicategories. By introducing polynomial structures within structured multicategories, we systematically formalize both abstract syntax and semantic models, achieving—for the first time—their joint categorical modeling at the level of category theory. The framework not only recovers classical completeness results for combinatorial algebras (e.g., λ-calculus, SK combinators) but also extends them to novel settings including distributed systems, typed protocols, and higher-order effect systems, thereby enabling cross-paradigm classification and unified characterization of compositional completeness. Its core innovation lies in embedding faithful Cartesian families into structured multicategories, providing a mathematically rigorous yet broadly applicable foundational theory for compositionality.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Agent Architectures

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: Incentives in network design for Web infrastructures and ecosystems

Abstract

We give a general notion of combinatory completeness with respect to a faithful cartesian club and use it systematically to obtain characterisations of a number of different kinds of applicative system. Each faithful cartesian club determines a notion of structured multicategory, with the different notions of structured multicategory obtained in this way giving different notions of polynomial over an applicative system, which in turn give different notions of combinatory completeness. We obtain the classical characterisation of combinatory algebras as combinatory complete applicative systems as a specific instance.

Problem

• Generalizing combinatory completeness via faithful cartesian clubs
• Characterizing diverse applicative systems through structured multicategories
• Deriving polynomial notions and combinatory algebra classifications

Innovation

• Combinatory completeness via faithful cartesian clubs
• Structured multicategories define polynomial applicative systems
• Generalizes classical combinatory algebra characterizations