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Categorical Vector Space Semantics for Lambek Calculus with a Relevant Modality

Lachlan McPheat, Mehrnoosh Sadrzadeh, Hadi Wazni, Gijs Wijnholds

TL;DR

This work develops a categorical–vector-space semantics for the Lambek Calculus with a relevant modality $!L^*$ to model controlled contraction and permutation, enabling DisCoCat-style compositional semantics of parasitic-gap phenomena. It defines a $ extbf{!L^*}$-category $oldsymbol{ ext{C}( ext{!L}^*)}$ with a coalgebra modality, and instantiates it in $ extbf{FdVect}$ via a quantisation functor, offering three concrete $!$-functors. The authors introduce diagrammatic clasp tooling and provide linguistic derivations for parasitic-gap examples, complemented by an experimental validation using extended disambiguation data and multiple neural embeddings; Full copying and Cofree-inspired copying perform best among linear models, with BERT providing the strongest overall baseline. The results suggest that the proposed categorical, vector-space approach can rival traditional additive embeddings in handling grammatically guided copying phenomena and motivates further work on coherence, differential-category connections, and bounded modalities for scalable parsing.

Abstract

We develop a categorical compositional distributional semantics for Lambek Calculus with a Relevant Modality !L*, which has a limited edition of the contraction and permutation rules. The categorical part of the semantics is a monoidal biclosed category with a coalgebra modality, very similar to the structure of a Differential Category. We instantiate this category to finite dimensional vector spaces and linear maps via "quantisation" functors and work with three concrete interpretations of the coalgebra modality. We apply the model to construct categorical and concrete semantic interpretations for the motivating example of !L*: the derivation of a phrase with a parasitic gap. The effectiveness of the concrete interpretations are evaluated via a disambiguation task, on an extension of a sentence disambiguation dataset to parasitic gap phrases, using BERT, Word2Vec, and FastText vectors and Relational tensors.

Categorical Vector Space Semantics for Lambek Calculus with a Relevant Modality

TL;DR

This work develops a categorical–vector-space semantics for the Lambek Calculus with a relevant modality to model controlled contraction and permutation, enabling DisCoCat-style compositional semantics of parasitic-gap phenomena. It defines a -category with a coalgebra modality, and instantiates it in via a quantisation functor, offering three concrete -functors. The authors introduce diagrammatic clasp tooling and provide linguistic derivations for parasitic-gap examples, complemented by an experimental validation using extended disambiguation data and multiple neural embeddings; Full copying and Cofree-inspired copying perform best among linear models, with BERT providing the strongest overall baseline. The results suggest that the proposed categorical, vector-space approach can rival traditional additive embeddings in handling grammatically guided copying phenomena and motivates further work on coherence, differential-category connections, and bounded modalities for scalable parsing.

Abstract

We develop a categorical compositional distributional semantics for Lambek Calculus with a Relevant Modality !L*, which has a limited edition of the contraction and permutation rules. The categorical part of the semantics is a monoidal biclosed category with a coalgebra modality, very similar to the structure of a Differential Category. We instantiate this category to finite dimensional vector spaces and linear maps via "quantisation" functors and work with three concrete interpretations of the coalgebra modality. We apply the model to construct categorical and concrete semantic interpretations for the motivating example of !L*: the derivation of a phrase with a parasitic gap. The effectiveness of the concrete interpretations are evaluated via a disambiguation task, on an extension of a sentence disambiguation dataset to parasitic gap phrases, using BERT, Word2Vec, and FastText vectors and Relational tensors.

Paper Structure

This paper contains 19 sections, 1 theorem, 47 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. !L*: Lambek Calculus with a Relevant Modality
  4. Categorical Semantics for !L*
  5. Vector Space Semantics for C(!L*)
  6. Concrete Constructions
  7. ! as the Dual of an Algebra
  8. ! as the Identity Functor
  9. Clasp Diagrams
  10. Linguistic Examples
  11. "John signed the papers"
  12. "The papers that John signed"
  13. "The papers that John signed without reading"
  14. Experimental Validation
  15. The Parasitic Gap Phrase Disambiguation Task
  16. ...and 4 more sections

Key Result

Proposition 1

Functorial models of $\mathbf{!L^*}$ are in bijection with categorical models of $\mathbf{!L^*}$. That is, given a $\mathbf{!L^*}$-category $\mathcal{D}$ and an interpretation of $(\!| \, |\!): \mathbf{!L^*} \to \mathcal{D}$ there is a unique functor $\mathcal{C}(\mathbf{!L^*})\to \mathcal{D}$ facto

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Proof 1: Sketch
  • Definition 5
  • Definition 6
  • Definition 7