Henk T. C. Stoof

Adriana D. Correia, Henk T. C. Stoof, Michael Moortgat

Extended versions of the Lambek Calculus currently used in computational linguistics rely on unary modalities to allow for the controlled application of structural rules affecting word order and phrase structure. These controlled structural operations give rise to derivational ambiguities that are missed by the original Lambek Calculus or its pregroup simplification. Proposals for compositional interpretation of extended Lambek Calculus in the compact closed category of FVect and linear maps have been made, but in these proposals the syntax-semantics mapping ignores the control modalities, effectively restricting their role to the syntax. Our aim is to turn the modalities into first-class citizens of the vectorial interpretation. Building on the directional density matrix semantics, we extend the interpretation of the type system with an extra spin density matrix space. The interpretation of proofs then results in ambiguous derivations being tensored with orthogonal spin states. Our method introduces a way of simultaneously representing co-existing interpretations of ambiguous utterances, and provides a uniform framework for the integration of lexical and derivational ambiguity.

Adriana D. Correia, Michael Moortgat, Henk T. C. Stoof

Recent work on vector-based compositional natural language semantics has proposed the use of density matrices to model lexical ambiguity and (graded) entailment (e.g. Piedeleu et al 2015, Bankova et al 2019, Sadrzadeh et al 2018). Ambiguous word meanings, in this work, are represented as mixed states, and the compositional interpretation of phrases out of their constituent parts takes the form of a strongly monoidal functor sending the derivational morphisms of a pregroup syntax to linear maps in FdHilb. Our aims in this paper are threefold. Firstly, we replace the pregroup front end by a Lambek categorial grammar with directional implications expressing a word's selectional requirements. By the Curry-Howard correspondence, the derivations of the grammar's type logic are associated with terms of the (ordered) linear lambda calculus; these terms can be read as programs for compositional meaning assembly with density matrices as the target semantic spaces. Secondly, we extend on the existing literature and introduce a symmetric, nondegenerate bilinear form called a "metric" that defines a canonical isomorphism between a vector space and its dual, allowing us to keep a distinction between left and right implication. Thirdly, we use this metric to define density matrix spaces in a directional form, modeling the ubiquitous derivational ambiguity of natural language syntax, and show how this alows an integrated treatment of lexical and derivational forms of ambiguity controlled at the level of the interpretation.