10 research outputs found
Compositional Distributional Semantics with Compact Closed Categories and Frobenius Algebras
This thesis contributes to ongoing research related to the categorical
compositional model for natural language of Coecke, Sadrzadeh and Clark in
three ways: Firstly, I propose a concrete instantiation of the abstract
framework based on Frobenius algebras (joint work with Sadrzadeh). The theory
improves shortcomings of previous proposals, extends the coverage of the
language, and is supported by experimental work that improves existing results.
The proposed framework describes a new class of compositional models that find
intuitive interpretations for a number of linguistic phenomena. Secondly, I
propose and evaluate in practice a new compositional methodology which
explicitly deals with the different levels of lexical ambiguity (joint work
with Pulman). A concrete algorithm is presented, based on the separation of
vector disambiguation from composition in an explicit prior step. Extensive
experimental work shows that the proposed methodology indeed results in more
accurate composite representations for the framework of Coecke et al. in
particular and every other class of compositional models in general. As a last
contribution, I formalize the explicit treatment of lexical ambiguity in the
context of the categorical framework by resorting to categorical quantum
mechanics (joint work with Coecke). In the proposed extension, the concept of a
distributional vector is replaced with that of a density matrix, which
compactly represents a probability distribution over the potential different
meanings of the specific word. Composition takes the form of quantum
measurements, leading to interesting analogies between quantum physics and
linguistics.Comment: Ph.D. Dissertation, University of Oxfor
A first-order logic for string diagrams
Equational reasoning with string diagrams provides an intuitive means of
proving equations between morphisms in a symmetric monoidal category. This can
be extended to proofs of infinite families of equations using a simple
graphical syntax called !-box notation. While this does greatly increase the
proving power of string diagrams, previous attempts to go beyond equational
reasoning have been largely ad hoc, owing to the lack of a suitable logical
framework for diagrammatic proofs involving !-boxes. In this paper, we extend
equational reasoning with !-boxes to a fully-fledged first order logic called
with conjunction, implication, and universal quantification over !-boxes. This
logic, called !L, is then rich enough to properly formalise an induction
principle for !-boxes. We then build a standard model for !L and give an
example proof of a theorem for non-commutative bialgebras using !L, which is
unobtainable by equational reasoning alone.Comment: 15 pages + appendi
A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras
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 Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras
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
Compositional distributional semantics with compact closed categories and Frobenius algebras
The provision of compositionality in distributional models of meaning, where a word is represented as a vector of co-occurrence counts with every other word in the vocabulary, offers a solution to the fact that no text corpus, regardless of its size, is capable of providing reliable co-occurrence statistics for anything but very short text constituents. The purpose of a compositional distributional model is to provide a function that composes the vectors for the words within a sentence, in order to create a vectorial representation that re ects its meaning. Using the abstract mathematical framework of category theory, Coecke, Sadrzadeh and Clark showed that this function can directly depend on the grammatical structure of the sentence, providing an elegant mathematical counterpart of the formal semantics view. The framework is general and compositional but stays abstract to a large extent. This thesis contributes to ongoing research related to the above categorical model in three ways: Firstly, I propose a concrete instantiation of the abstract framework based on Frobenius algebras (joint work with Sadrzadeh). The theory improves shortcomings of previous proposals, extends the coverage of the language, and is supported by experimental work that improves existing results. The proposed framework describes a new class of compositional models thatfind intuitive interpretations for a number of linguistic phenomena. Secondly, I propose and evaluate in practice a new compositional methodology which explicitly deals with the different levels of lexical ambiguity (joint work with Pulman). A concrete algorithm is presented, based on the separation of vector disambiguation from composition in an explicit prior step. Extensive experimental work shows that the proposed methodology indeed results in more accurate composite representations for the framework of Coecke et al. in particular and every other class of compositional models in general. As a last contribution, I formalize the explicit treatment of lexical ambiguity in the context of the categorical framework by resorting to categorical quantum mechanics (joint work with Coecke). In the proposed extension, the concept of a distributional vector is replaced with that of a density matrix, which compactly represents a probability distribution over the potential different meanings of the specific word. Composition takes the form of quantum measurements, leading to interesting analogies between quantum physics and linguistics.This thesis is not currently available in OR