Abstract
The categorical compositional approach to meaning has been successfully applied in natural language processing, outperforming other models in mainstream empirical language processing tasks. We show how this approach can be generalized to conceptual space models of cognition. In order to do this, first we introduce the category of convex relations as a new setting for categorical compositional semantics, emphasizing the convex structure important to conceptual space applications. We then show how to construct conceptual spaces for various types such as nouns, adjectives and verbs. Finally we show by means of examples how concepts can be systematically combined to establish the meanings of composite phrases from the meanings of their constituent parts. This provides the mathematical underpinnings of a new compositional approach to cognition.
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Notes
- 1.
This paper is a significantly extended version of the workshop paper Bolt et al. (2016).
- 2.
Authors in alphabetical order.
- 3.
It could be argued that these are not transitive verbs, but intransitive verbs plus preposition. However, we can parse the combination as a transitive verb, since a preposition has type s r sn l and therefore the combination reduces to type of a transitive verb:
$$\displaystyle \begin{aligned} (n^r s)(s^r s n^l) \leq n^r s n^l \end{aligned}$$
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Acknowledgements
This work was partially funded by AFSOR grant “Algorithmic and Logical Aspects when Composing Meanings”, the FQXi grant “Categorical Compositional Physics”, and EPSRC PhD scholarships.
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Bolt, J., Coecke, B., Genovese, F., Lewis, M., Marsden, D., Piedeleu, R. (2019). Interacting Conceptual Spaces I: Grammatical Composition of Concepts. In: Kaipainen, M., Zenker, F., Hautamäki, A., Gärdenfors, P. (eds) Conceptual Spaces: Elaborations and Applications. Synthese Library, vol 405. Springer, Cham. https://doi.org/10.1007/978-3-030-12800-5_9
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