Abstract
Categorical compositional models of natural language exploit grammatical structure to calculate the meaning of sentences from the meanings of individual words. This approach outperforms conventional techniques for some standard NLP tasks. More recently, similar compositional techniques have been applied to conceptual space models of cognition.
Compact closed categories, particularly the category of finite dimensional vector spaces, have been the most common setting for categorical compositional models. When addressing a new problem domain, such as conceptual space models of meaning, a key problem is finding a compact closed category that captures the features of interest.
We propose categories of generalized relations as source of new, practical models for cognition and NLP. We demonstrate using detailed examples that phenomena such as fuzziness, metrics, convexity, semantic ambiguity and meaning that varies with context can all be described by relational models. Crucially, by exploiting a technical framework described in previous work of the authors, we also show how we can combine multiple features into a single model, providing a flexible family of new categories for categorical compositional modelling.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The slightly unusual formulation of identities is to avoid definition by cases. This means they can be interpreted in the internal language of an arbitrary topos.
- 2.
In fact, in order for composition to be associative, it is necessary to work with equivalence classes of spans. It is sufficient to consider representatives, and we do so to avoid distracting technicalities.
References
Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425. IEEE (2004)
Baez, J.C., Erbele, J.: Categories in control. Theory Appl. Categ. 30(24), 836–881 (2015)
Baez, J.C., Fong, B.: A compositional framework for passive linear networks. arXiv preprint arXiv:1504.05625 (2015)
Baez, J.C., Fong, B., Pollard, B.S.: A compositional framework for Markov processes. J. Math. Phys. 57(3), 033301 (2016)
Bankova, D., Coecke, B., Lewis, M., Marsden, D.: Graded entailment for compositional distributional semantics. arXiv preprint arXiv:1601.04908 (2015)
Bankova, D.: Comparing meaning in language and cognition - p-hypononymy, concept combination, asymmetric similarity. Master’s thesis, University of Oxford (2015)
Barr, M.: Exact categories. Exact Categories and Categories of Sheaves. LNM, vol. 236, pp. 1–120. Springer, Heidelberg (1971). doi:10.1007/BFb0058580
Barsalou, L.W.: Ideals, central tendency, and frequency of instantiation as determinants of graded structure in categories. J. Exp. Psychol. Learn. Mem. Cogn. 11(4), 629 (1985)
Bolt, J., Coecke, B., Genovese, F., Lewis, M., Marsden, D., Piedeleu, R.: Interacting conceptual spaces. In: Kartsaklis, D., Lewis, M., Rimell, L. (eds.) Proceedings of the 2016 Workshop on Semantic Spaces at the Intersection of NLP, Physics and Cognitive Science, SLPCS@QPL 2016, Glasgow, Scotland, 11 June 2016. EPTCS, vol. 221, pp. 11–19 (2016). http://dx.doi.org/10.4204/EPTCS.221.2
Bolt, J., Coecke, B., Genovese, F., Lewis, M., Marsden, D., Piedeleu, R.: Interacting conceptual spaces I: Grammatical composition of concepts. arXiv preprint arXiv:1703.08314 (2017)
Bonchi, F., Sobocinski, P., Zanasi, F.: Full abstraction for signal flow graphs. ACM SIGPLAN Not. 50(1), 515–526 (2015)
Borceux, F.: Handbook of Categorical Algebra: Volume 3, Categories of Sheaves. Cambridge University Press, Cambridge (1994)
Borceux, F.: Handbook of Categorical Algebra: Volume 2, Categories and Structures, vol. 2. Cambridge University Press, Cambridge (1994)
Coecke, B., Grefenstette, E., Sadrzadeh, M.: Lambek vs. Lambek: functorial vector space semantics and string diagrams for lambek calculus. Ann. Pure Appl. Logic 164(11), 1079–1100 (2013)
Coecke, B., Sadrzadeh, M., Clark, S.: Mathematical foundations for distributed compositional model of meaning. Lambek festschrift. Linguist. Anal. 36, 345–384 (2010)
Coecke, B., Kissinger, A.: Picturing Quantum Processes. A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press (2017, forthcoming)
Coecke, B., Paquette, E.O.: Categories for the practising physicist. In: Coecke, B. (ed.) New Structures for Physics, pp. 173–286. Springer, Heidelberg (2010)
Dale, R., Kehoe, C., Spivey, M.J.: Graded motor responses in the time course of categorizing atypical exemplars. Mem. Cogn. 35(1), 15–28 (2007)
Dostal, M., Sadrzadeh, M.: Many valued generalised quantifiers for natural language in the DisCoCat model. Technical report, Queen Mary University of London (2016)
Fong, B.: The algebra of open and interconnected systems. Ph.D. thesis, University of Oxford (2016)
Gärdenfors, P.: Conceptual Spaces: The Geometry of Thought. MIT Press, Cambridge (2004)
Gärdenfors, P.: The Geometry of Meaning: Semantics Based on Conceptual Spaces. MIT Press, Cambridge (2014)
Grefenstette, E., Sadrzadeh, M.: Experimental support for a categorical compositional distributional model of meaning. In: The 2014 Conference on Empirical Methods on Natural Language Processing, pp. 1394–1404 (2011). arXiv:1106.4058
Hampton, J.A.: Disjunction of natural concepts. Mem. Cogn. 16(6), 579–591 (1988)
Hampton, J.A.: Overextension of conjunctive concepts: evidence for a unitary model of concept typicality and class inclusion. J. Exp. Psychol. Learn. Mem. Cogn. 14(1), 12 (1988)
Hofmann, D., Seal, G.J., Tholen, W.: Monoidal Topology: A Categorical Approach to Order, Metric, and Topology, vol. 153. Cambridge University Press, Cambridge (2014)
Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium, vol. 1. Oxford University Press, Oxford (2002)
Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium, vol. 2. Oxford University Press, Oxford (2002)
Kartsaklis, D., Sadrzadeh, M.: Prior disambiguation of word tensors for constructing sentence vectors. In: The 2013 Conference on Empirical Methods on Natural Language Processing, pp. 1590–1601. ACL (2013)
Kissinger, A.: Finite matrices are complete for (dagger-)hypergraph categories. arXiv preprint arXiv:1406.5942 (2014)
Lambek, J.: Type grammar revisited. In: Lecomte, A., Lamarche, F., Perrier, G. (eds.) LACL 1997. LNCS, vol. 1582, pp. 1–27. Springer, Heidelberg (1999). doi:10.1007/3-540-48975-4_1
MacLane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer Science & Business Media, Heidelberg (2012)
Marsden, D., Genovese, F.: Custom hypergraph categories via generalized relations. In: CALCO 2017 (2017, to appear)
Marsden, D.: A graph theoretic perspective on CPM(Rel). In: Heunen, C., Selinger, P., Vicary, J. (eds.) Proceedings 12th International Workshop on Quantum Physics and Logic, QPL 2015, Oxford, UK, 15–17 July 2015. EPTCS, vol. 195, pp. 273–284 (2015). http://dx.doi.org/10.4204/EPTCS.195.20
Osherson, D.N., Smith, E.E.: Gradedness and conceptual combination. Cognition 12(3), 299–318 (1982)
Piedeleu, R., Kartsaklis, D., Coecke, B., Sadrzadeh, M.: Open system categorical quantum semantics in natural language processing. In: Moss, L.S., Sobocinski, P. (eds.) 6th Conference on Algebra and Coalgebra in Computer Science, CALCO 2015. LIPIcs, vol. 35, pp. 270–289. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)
Rosch, E., Mervis, C.B.: Family resemblances: studies in the internal structure of categories. Cogn. Psychol. 7(4), 573–605 (1975)
Sadrzadeh, M., Clark, S., Coecke, B.: The Frobenius anatomy of word meanings I: subject and object relative pronouns. J. Logic Comput. 23(6), ext044 (2013)
Sadrzadeh, M., Clark, S., Coecke, B.: The Frobenius anatomy of word meanings II: possessive relative pronouns. J. Logic Comput. 26(2), exu027 (2014)
Schütze, H.: Automatic word sense discrimination. Comput. Linguist. 24(1), 97–123 (1998)
Selinger, P.: Dagger compact closed categories and completely positive maps. Electron. Not. Theor. Comput. Sci. 170, 139–163 (2007)
Shepard, R.N., et al.: Toward a universal law of generalization for psychological science. Science 237(4820), 1317–1323 (1987)
Sobocinski, P.: Graphical linear algebra. Mathematical blog. https://graphicallinearalgebra.net/
Stubbe, I.: Categorical structures enriched in a quantaloid: categories and semicategories. Ph.D. thesis, Université Catholique de Louvain (2003)
Tversky, A.: Features of similarity. Psychol. Rev. 84(4), 327 (1977)
Acknowledgments
This work was funded by AFSOR grant “Algorithmic and Logical Aspects when Composing Meanings” and FQXi grant “Categorical Compositional Physics”.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer-Verlag GmbH Germany
About this paper
Cite this paper
Coecke, B., Genovese, F., Lewis, M., Marsden, D. (2017). Generalized Relations in Linguistics and Cognition. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-662-55386-2_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-55385-5
Online ISBN: 978-3-662-55386-2
eBook Packages: Computer ScienceComputer Science (R0)