Mathematical Foundations for a Compositional Distributional Model of Meaning

We propose a mathematical framework for a unification of the distributional of meaning in terms of vector space models, and a compositional theory grammatical types, for which we rely on the algebra of Pregroups, by Lambek. This mathematical framework enables us to compute the of a well-typed sentence from the meanings of its constituents., the type reductions of Pregroups are `lifted' to morphisms in a, a procedure that transforms meanings of constituents into a meaning the (well-typed) whole. Importantly, meanings of whole sentences live in a space, independent of the grammatical structure of the sentence. Hence inner-product can be used to compare meanings of arbitrary sentences, as it for comparing the meanings of words in the distributional model. The structure we employ admits a purely diagrammatic calculus which how the information flows between the words in a sentence in order to up the meaning of the whole sentence. A variation of our `categorical' which involves constraining the scalars of the vector spaces to the of Booleans results in a Montague-style Boolean-valued semantics.