Density Matrices with Metric for Derivational Ambiguity

TL;DR

This paper introduces a novel approach to modeling derivational ambiguity in natural language using density matrices with a metric, integrating Lambek categorial grammar and linear lambda calculus.

Contribution

It replaces the pregroup grammar with a Lambek categorial grammar, introduces a metric for density matrices, and models derivational ambiguity directionally within the semantic framework.

Findings

Defines a symmetric bilinear form called a 'metric' for density matrices.

Models derivational ambiguity in syntax and semantics.

Provides a unified treatment of lexical and derivational ambiguity.

Abstract

Recent work on vector-based compositional natural language semantics has proposed the use of density matrices to model lexical ambiguity and (graded) entailment (e.g. Piedeleu et al 2015, Bankova et al 2019, Sadrzadeh et al 2018). Ambiguous word meanings, in this work, are represented as mixed states, and the compositional interpretation of phrases out of their constituent parts takes the form of a strongly monoidal functor sending the derivational morphisms of a pregroup syntax to linear maps in FdHilb. Our aims in this paper are threefold. Firstly, we replace the pregroup front end by a Lambek categorial grammar with directional implications expressing a word's selectional requirements. By the Curry-Howard correspondence, the derivations of the grammar's type logic are associated with terms of the (ordered) linear lambda calculus; these terms can be read as programs for compositional meaning assembly with density matrices as the target semantic spaces. Secondly, we extend on the existing literature and introduce a symmetric, nondegenerate bilinear form called a "metric" that defines a canonical isomorphism between a vector space and its dual, allowing us to keep a distinction between left and right implication. Thirdly, we use this metric to define density matrix spaces in a directional form, modeling the ubiquitous derivational ambiguity of natural language syntax, and show how this alows an integrated treatment of lexical and derivational forms of ambiguity controlled at the level of the interpretation.