Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus

摘要

The Distributional Compositional Categorical (DisCoCat) model is amathematical framework that provides compositional semantics for meanings ofnatural language sentences. It consists of a computational procedure forconstructing meanings of sentences, given their grammatical structure in termsof compositional type-logic, and given the empirically derived meanings oftheir words. For the particular case that the meaning of words is modelledwithin a distributional vector space model, its experimental predictions,derived from real large scale data, have outperformed other empiricallyvalidated methods that could build vectors for a full sentence. This successcan be attributed to a conceptually motivated mathematical underpinning, byintegrating qualitative compositional type-logic and quantitative modelling ofmeaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambek's pregroup grammar.Pregroup types form a posetal compact closed category, which can be passed, ina functorial manner, on to the compact closed structure of vector spaces,linear maps and tensor product. The diagrammatic versions of the equationalreasoning in compact closed categories can be interpreted as the flow of wordmeanings within sentences. Pregroups simplify Lambek's previous type-logic, theLambek calculus, which has been extensively used to formalise and reason aboutvarious linguistic phenomena. The apparent reliance of the DisCoCat onpregroups has been seen as a shortcoming. This paper addresses this concern, bypointing out that one may as well realise a functorial passage from theoriginal type-logic of Lambek, a monoidal bi-closed category, to vector spaces,or to any other model of meaning organised within a monoidal bi-closedcategory. The corresponding string diagram calculus, due to Baez and Stay, nowdepicts the flow of word meanings.