Sheaf theoretic semantics for vector space models of natural language
Contextuality is a fundamental feature of quantum physical theories and one that distinguishes it from classical mechanics. In a recent paper by Abramsky and Brandenburger, the categorical notion of sheaves has been used to formalize contextuality. This has resulted in generalizing and extending contextuality to other theories which share some structural properties with quantum mechanics. A consequence of this type of modeling is a succinct logical axiomatization of properties such as non-local correlations and as a result of classical no go theorems such as Bell and Kochen-Soecker. Like quantum mechanics, natural language has contextual features; these have been the subject of much study in distributional models of meaning, originated in the work of Firth and later advanced by Schutze. These models are based on vector spaces over the semiring of positive reals with an inner product operation. The vectors represent meanings of words, based on the contexts in which they often appear, and the inner product measures degrees of word synonymy. Despite their success in modeling word meaning, vector spaces suffer from two major shortcomings: firstly they do not immediately scale up to sentences, and secondly, they cannot, at least not in an intuitive way, provide semantics for logical words such as `and', `or', `not'. Recent work in our group has developed a compositional distributional model of meaning in natural language, which lifts vector space meaning to phrases and sentences. This has already led to some very promising experimental results. However, this approach does not deal so well with the logical words.
The goal of this project is to use sheaf theoretic models to provide both a contextual and logical semantics for natural language. We believe that sheaves provide a generalization of the logical Montague semantics of natural language which did very well in modeling logical connectives, but did not account for contextuality. The project will also aim to combine these ideas with those of the distributional approach, leading to an approach which combines the advantages of Montague-style and vector-space semantics.
The interested student should have taken the category theory and computational linguistics courses, or be familiar with the contents of these.