Language is contextual and sheaf theory provides a high level mathematical framework to model contextuality. We show how sheaf theory can model the contextual nature of natural language and how gluing can be used to provide a global semantics for a discourse by putting together the local logical semantics of each sentence within the discourse. We introduce a presheaf structure corresponding to a basic form of Discourse Representation Structures. Within this setting, we formulate a notion of semantic unification — gluing meanings of parts of a discourse into a coherent whole — as a form of sheaf-theoretic gluing. We illustrate this idea with a number of examples where it can used to represent resolutions of anaphoric references. We also discuss multivalued gluing, described using a distributions functor, which can be used to represent situations where multiple gluings are possible, and where we may need to rank them using quantitative measures.
KeywordsLocal Section Relation Symbol Sheaf Theory Contravariant Functor Discourse Referent
Unable to display preview. Download preview PDF.
- 2.Abramsky, S.: Relational databases and Bells theorem. In: Tannen, V. (ed.) Festschrift for Peter Buneman (2013) (to appear); Available as CoRR, abs/1208.6416Google Scholar
- 4.Abramsky, S., Gottlob, G., Kolaitis, P.: Robust Constraint Satisfaction and Local Hidden Variables in Quantum Mechanics. To Appear in Proceedings of IJCAI 2013 (2013)Google Scholar
- 5.Abramsky, S., Hardy, L.: Logical Bell Inequalities. Physical Review A 85, 062114 (2012)Google Scholar
- 6.Coecke, B., Sadrzadeh, M., Clark, S.: Mathematical foundations for a compositional distributional model of meaning. Linguistic Analysis 36, 345–384 (2010)Google Scholar
- 7.Dagan, I., Itai, A.: Automatic processing of large corpora for the resolution of anaphora references. In: Proceedings of the 13th International Conference on Computational Linguistics (COLING 1990), Finland, vol. 3, pp. 330–332 (1990)Google Scholar
- 8.Firth, J.R.: A synopsis of linguistic theory 1930-1955. Studies in Linguistic Analysis, Special volume of the Philological Society. Blackwell, Oxford (1957)Google Scholar
- 9.Geach, P.T.: Reference and Generality, An examination of some medieval and modern theories, vol. 88. Cornell University Press (1962)Google Scholar
- 10.Grefenstette, E., Sadrzadeh, M.: Experimental Support for a Categorical Compositional Distributional Model of Meaning. In: Proceedings of the Conference on Empirical Methods in Natural Language Processing, EMNLP 2011 (2011)Google Scholar
- 12.Harris, Z.S.: Mathematical structures of language, Interscience Tracts in Pure and Applied Mathematics, vol. 21. University of Michigan (1968)Google Scholar
- 14.Kamp, H., van Genabith, J., Reyle, U.: Discourse Representation Theory. In: Handbook of Philosophical Logic, vol. 15, pp. 125–394 (2011)Google Scholar
- 16.Lane, S.M., Moerdijk, I.: Sheaves in geometry and logic: A first introduction to topos theory. Springer (1992)Google Scholar
- 17.Mitkov, R.: Anaphora Resolution. Longman (2002)Google Scholar
- 18.Dowty, D.R., Wall, R.E., Peters, S.: Introduction to Montague Semantics. D. Reidel Publishing Company, Dodrecht (1981)Google Scholar