A generalised quantifier theory of natural language in categorical compositional distributional semantics with bialgebras

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.

A proof-theoretic approach to scope ambiguity in compositional vector space models

“…It has been argued before that modelling quantification in vector space models forces one to use non-linear maps [7]. However, this issue has been partially resolved by [8] when one admits a powerset structure to the basis vectors of the model. The then obtained bialgebra operations are linear in the algebraic sense, but non-linear in terms of typing information.…”

“…That this kind of operation would jeopardize a Lambek style grammar formalism is immediate as the bialgebra operations would correspond to contraction and expansion, respectively. Our argument will proceed by claiming that a continuation-passing-style translation that allows for lexical insertion of non-linear λ-terms can instead be interpreted by means of the bialgebra operations of [8].…”

“…We discuss this issue in more detail in the rest of the paper. Some of the above issues have been addressed in recent work by [20,8,19], giving accounts of subject/object relativisation, generalised quantifiers, and quantifier scope. In [20], the meaning of pronoun relative clauses is explained by using Frobenius algebras in the lexicon, and assigning different pregroup grammar types to the subject relative pronoun "who" and the object relative pronoun "whom".…”

“…Quantifier scope ambiguity as opposed to pronoun relativisation is more pressing as the scope ambiguity exists in English, and does not come from the lexicon, but rather different ways of reading the same surface form. The account of [8] explores the use of bialgebras to represent quantifiers, using context free grammars as the syntactic engine; its follow up discusses scope ambiguity but assumes the ambiguity to be given before detailing the direct scope and inverse scope readings of phrases of the shape "Every man loves some woman". In order to explain how the ambiguity comes about, we need to detail the syntactic process, and integrate it with a vector based semantics.…”

A Type-Driven Vector Semantics for Ellipsis with Anaphora Using Lambek Calculus with Limited Contraction

“…In each of these, the Frobenius algebras allow one to use element wise multiplication of arbitrary tensors, corresponding to the usual intersective interpretation one finds in formal semantics [8]. A treatment of quantification was also given using the bialgebraic nature of vector spaces over powersets of elements [14,42]. An explanation of the derivational processes resulting in these compositional meanings requires more elaborate grammatical mechanisms: Wijnholds [46] repeats the exercise to give a compositional distributional model for a symmetric extension of the Lambek calculus.…”