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

arXiv: https://arxiv.org/abs/1302.0393 · HTML: https://mlsanity.com/paper/1302.0393/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2013.

## Built on by (most cited first, 13 of 13)

- [A CCG-Based Version of the DisCoCat Framework](https://mlsanity.com/paper/2105.07720/) (2021)
- [A Quantum Natural Language Processing Approach to Musical Intelligence](https://mlsanity.com/paper/2111.06741/) (2021)
- [Grover's Algorithm for Question Answering](https://mlsanity.com/paper/2106.05299/) (2021)
- [A Quantum Natural Language Processing Approach to Pronoun Resolution](https://mlsanity.com/paper/2208.05393/) (2022)
- [Higher-Order DisCoCat (Peirce-Lambek-Montague semantics)](https://mlsanity.com/paper/2311.17813/) (2023)
- [Functorial Language Models](https://mlsanity.com/paper/2103.14411/) (2021)
- [Lambek pregroups are Frobenius spiders in preorders](https://mlsanity.com/paper/2105.03038/) (2021)
- [Towards Transparency in Coreference Resolution: A Quantum-Inspired Approach](https://mlsanity.com/paper/2312.00688/) (2023)
- [A vector logic for intensional formal semantics](https://mlsanity.com/paper/2602.02940/) (2026)
- [Pregroup Grammars, their Syntax and Semantics](https://mlsanity.com/paper/2109.11237/) (2021)
- [DisCoCat for Donkey Sentences](https://mlsanity.com/paper/2308.04519/) (2023)
- [Quantum Methods for Managing Ambiguity in Natural Language Processing](https://mlsanity.com/paper/2504.00040/) (2025)
- [Functorial Neural Architectures from Higher Inductive Types](https://mlsanity.com/paper/2603.16123/) (2026)

## Abstract (arXiv)

The Distributional Compositional Categorical (DisCoCat) model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, by integrating qualitative compositional type-logic and quantitative modelling of meaning 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, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambek's previous type-logic, the Lambek calculus, which has been extensively used to formalise and reason about various linguistic phenomena. The apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings.