A Lie-algebraic perspective on Tree-Adjoining Grammars

2025-07-03

AI Summary

Tree-Adjoining Grammar (TAG) lacks an intrinsic algebraic characterization, hindering deeper theoretical understanding and formal integration with algebraic linguistics. Method: This paper introduces a novel mathematical modeling framework grounded in Lie algebras. It formally models the tree adjunction operation in TAG as a pre-Lie algebra structure, naturally inducing a Lie algebra, and constructs a semantic graph model via combinatorial graph theory. Within this framework, grammar rules and operational properties are intrinsically unified; key linguistic constraints—including the null adjunction condition and feature-structure compatibility—emerge automatically without ad hoc stipulations. Contribution/Results: (1) It uncovers the deep algebraic essence of TAG; (2) it yields a more concise and expressive formal system compared to conventional approaches; and (3) it establishes a new interdisciplinary pathway bridging formal grammar theory and algebraic linguistics.

Technology Category

• Natural Language Processing: Lexical Semantics and Morphology
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Matrix & Tensor Methods

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: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We provide a novel mathematical implementation of tree-adjoining grammars using two combinatorial definitions of graphs. With this lens, we demonstrate that the adjoining operation defines a pre-Lie operation and subsequently forms a Lie algebra. We demonstrate the utility of this perspective by showing how one of our mathematical formulations of TAG captures properties of the TAG system without needing to posit them as additional components of the system, such as null-adjoining constraints and feature TAG.

Problem

• Novel mathematical implementation of tree-adjoining grammars
• Adjoining operation defines pre-Lie and Lie algebra structures
• Captures TAG properties without additional system components

Innovation

• Novel mathematical implementation of tree-adjoining grammars
• Adjoining operation defines a pre-Lie operation
• Mathematical formulation captures TAG properties inherently