A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras

TL;DR

This paper extends categorical compositional distributional semantics to include generalized quantifiers, unifying truth-conditional and statistical models of natural language in a formal mathematical framework.

Contribution

It formalizes generalized quantifier theory within categorical compositional semantics using bialgebras, enabling compositional reasoning about quantified language in vector spaces.

Findings

Proves equivalence of relational instantiation to truth-conditional semantics.

Formalizes statistical reasoning about quantified phrases.

Enables compositional distributional semantics for quantified language.

Abstract

Categorical compositional distributional semantics is a model of natural language; it combines the statistical vector space models of words with the compositional models of grammar. We formalise in this model the generalised quantifier theory of natural language, due to Barwise and Cooper. The underlying setting is a compact closed category with bialgebras. We start from a generative grammar formalisation and develop an abstract categorical compositional semantics for it, then instantiate the abstract setting to sets and relations and to finite dimensional vector spaces and linear maps. We prove the equivalence of the relational instantiation to the truth theoretic semantics of generalised quantifiers. The vector space instantiation formalises the statistical usages of words and enables us to, for the first time, reason about quantified phrases and sentences compositionally in distributional semantics.