The categorical contours of the Chomsky-Schützenberger representation theorem

2023-12-29

arXiv.org

AI Summary

This paper unifies and generalizes context-free grammars (CFGs) and nondeterministic finite automata (NFAs) within a categorical framework. To achieve this, it introduces the “outline category” as the left adjoint to the concatenation functor and employs operadic structures, fibrations, and colored tree languages to model CFGs and NFAs as functors over appropriate base categories. The main contributions are threefold: (1) a fundamental generalization of the Chomsky–Schützenberger theorem, applicable to tree-outline word languages and arbitrary base categories; (2) a conceptual, categorical proof of closure of context-free languages (CFLs) under intersection with regular languages; and (3) a categorical characterization of CFLs as functorial images of intersections between colored outline languages and regular languages, thereby establishing closure properties of generalized context-free and regular languages.

Technology Category

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

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: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We develop fibrational perspectives on context-free grammars and on nondeterministic finite-state automata over categories and operads. A generalized CFG is a functor from a free colored operad (aka multicategory) generated by a pointed finite species into an arbitrary base operad: this encompasses classical CFGs by taking the base to be a certain operad constructed from a free monoid, as an instance of a more general construction of an emph{operad of spliced arrows} $mathcal{W},mathcal{C}$ for any category $mathcal{C}$. A generalized NFA is a functor from an arbitrary bipointed category or pointed operad satisfying the unique lifting of factorizations and finite fiber properties: this encompasses classical word automata and tree automata without $epsilon$-transitions, but also automata over non-free categories and operads. We show that generalized context-free and regular languages satisfy suitable generalizations of many of the usual closure properties, and in particular we give a simple conceptual proof that context-free languages are closed under intersection with regular languages. Finally, we observe that the splicing functor $mathcal{W} : Cat o Oper$ admits a left adjoint $mathcal{C}: Oper o Cat$, which we call the emph{contour category} construction since the arrows of $mathcal{C},mathcal{O}$ have a geometric interpretation as oriented contours of operations of $mathcal{O}$. A direct consequence of the contour / splicing adjunction is that every pointed finite species induces a universal CFG generating a language of emph{tree contour words.} This leads us to a generalization of the Chomsky-Sch""utzenberger Representation Theorem, establishing that a subset of a homset $L subseteq mathcal{C}(A,B)$ is a CFL of arrows if and only if it is a functorial image of the intersection of a $mathcal{C}$-chromatic tree contour language with a regular language.

Problem

• Generalizes context-free grammars and automata over categories and operads.
• Proves closure properties for generalized context-free and regular languages.
• Extends Chomsky-Schützenberger theorem to functorial images of tree contour languages.

Innovation

• Generalized CFGs as functors from free colored operads
• Generalized NFAs as functors from bipointed categories
• Contour category construction with geometric interpretation