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Michael Moortgat

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3 papers

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FLAP Journal 2020 Journal Article

A Frobenius Algebraic Analysis for Parasitic Gaps.

Michael Moortgat
Mehrnoosh Sadrzadeh
Gijs Wijnholds

The interpretation of parasitic gaps is an ostensible case of non-linearity in natural language composition. Existing categorial analyses, both in the typelogical and in the combinatory traditions, rely on explicit forms of syntactic copying. We identify two types of parasitic gapping where the duplication of semantic content can be confined to the lexicon. Parasitic gaps in adjuncts are analysed as forms of generalized coordination with a polymorphic type schema for the head of the adjunct phrase. For parasitic gaps affecting arguments of the same predicate, the polymorphism is associated with the lexical item that introduces the primary gap. Our analysis is formulated in terms of Lambek calculus extended with structural control modalities. A compositional translation relates syntactic types and derivations to the interpreting compact closed category of finite dimensional vector spaces and linear maps with Frobenius algebras over it. When interpreted over the necessary semantic spaces, the Frobenius algebras provide the tools to model the proposed instances of lexical polymorphism.

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FLAP Journal 2020 Journal Article

Density Matrices with Metric for 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. 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 allows an integrated treatment of lexical and derivational forms of ambiguity controlled at the level of the interpretation.

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FLAP Journal 2020 Journal Article

Vector Spaces as Kripke Frames.

Giuseppe Greco
Fei Liang
Michael Moortgat
Alessandra Palmigiano
Apostolos Tzimoulis

In recent years, the compositional distributional approach in computational linguistics has opened the way for an integration of the lexical aspects of meaning into Lambek’s type-logical grammar program. This approach is based on the observation that a sound semantics for the associative, commutative and unital Lambek calculus can be based on vector spaces by interpreting fusion as the tensor product of vector spaces. In this paper, we build on this observation and extend it to a ‘vector space semantics’ for the general Lambek calculus, based on algebras over a field K (or K-algebras), i.e. vector spaces endowed with a bilinear binary product. Such structures are well known in algebraic geometry and algebraic topology, since Lie algebras and Hopf algebras are important instances of K-algebras. Applying results and insights from duality and representation theory for the algebraic semantics of nonclassical logics, we regard K-algebras as ‘Kripke frames’ the complex algebras of which are complete residuated lattices. This perspective makes it possible to establish a systematic connection between vector space semantics and the standard Routley-Meyer semantics of (modal) substructural logics.

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