A Lie-algebraic perspective on Tree-Adjoining Grammars

TL;DR

This paper introduces a Lie-algebraic framework for Tree-Adjoining Grammars (TAG), revealing new mathematical structures that naturally encode properties of TAG without extra constraints.

Contribution

It provides a novel Lie-algebraic formulation of TAG using graph combinatorics, offering deeper mathematical insight into the system's properties.

Findings

Adjoining operation forms a pre-Lie algebra

The Lie algebra captures TAG properties inherently

Mathematical formulation simplifies understanding of TAG

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.