# Higher-Order DisCoCat (Peirce-Lambek-Montague semantics)

arXiv: https://arxiv.org/abs/2311.17813 · HTML: https://mlsanity.com/paper/2311.17813/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 33 references that are in the map.
- First posted on arXiv 29 Nov 2023.

## Builds on (most cited first, 9 of 9)

- [Foundations for Near-Term Quantum Natural Language Processing](https://mlsanity.com/paper/2012.03755/) (2020)
- [The Mathematics of Text Structure](https://mlsanity.com/paper/1904.03478/) (2019)
- [A CCG-Based Version of the DisCoCat Framework](https://mlsanity.com/paper/2105.07720/) (2021)
- [Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus](https://mlsanity.com/paper/1302.0393/) (2013)
- [Functorial Question Answering](https://mlsanity.com/paper/1905.07408/) (2019)
- [Categorical Tools for Natural Language Processing](https://mlsanity.com/paper/2212.06636/) (2022)
- [Functorial Language Games for Question Answering](https://mlsanity.com/paper/2005.09439/) (2020)
- [Category Theory for Quantum Natural Language Processing](https://mlsanity.com/paper/2212.06615/) (2022)
- [Coend Optics for Quantum Combs](https://mlsanity.com/paper/2205.09027/) (2022)

## Built on by (most cited first, 2 of 2)

- [Towards Compositional Interpretability for XAI](https://mlsanity.com/paper/2406.17583/) (2024)
- [A Diagrammatic Calculus for a Functional Model of Natural Language Semantics](https://mlsanity.com/paper/2507.00782/) (2025)

## Abstract (arXiv)

We propose a new definition of higher-order DisCoCat (categorical compositional distributional) models where the meaning of a word is not a diagram, but a diagram-valued higher-order function. Our models can be seen as a variant of Montague semantics based on a lambda calculus where the primitives act on string diagrams rather than logical formulae. As a special case, we show how to translate from the Lambek calculus into Peirce's system beta for first-order logic. This allows us to give a purely diagrammatic treatment of higher-order and non-linear processes in natural language semantics: adverbs, prepositions, negation and quantifiers. The definition presented in this article comes with a proof-of-concept implementation in DisCoPy, the Python library for string diagrams.