Advertisement

Semantic Unification

A Sheaf Theoretic Approach to Natural Language
  • Samson Abramsky
  • Mehrnoosh Sadrzadeh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8222)

Abstract

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.

Keywords

Local Section Relation Symbol Sheaf Theory Contravariant Functor Discourse Referent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aone, C., Bennet, S.W.: Applying machine learning to anaphora resolution. In: IJCAI-WS 1995. LNCS, vol. 1040, pp. 302–314. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  2. 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
  3. 3.
    Abramsky, S., Brandenburger, A.: The sheaf-theoretic structure of non-locality and contextuality. New Journal of Physics 13(11), 113036 (2011)CrossRefGoogle Scholar
  4. 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. 5.
    Abramsky, S., Hardy, L.: Logical Bell Inequalities. Physical Review A 85, 062114 (2012)Google Scholar
  6. 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. 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. 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. 9.
    Geach, P.T.: Reference and Generality, An examination of some medieval and modern theories, vol. 88. Cornell University Press (1962)Google Scholar
  10. 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
  11. 11.
    Groenendijk, J., Stokhof, M.: Dynamic Predicate Logic. Linguistics and Philisophy 14, 39–100 (1991)CrossRefMATHGoogle Scholar
  12. 12.
    Harris, Z.S.: Mathematical structures of language, Interscience Tracts in Pure and Applied Mathematics, vol. 21. University of Michigan (1968)Google Scholar
  13. 13.
    Jaynes, E.T.: Information theory and statistical mechanics. Physical Review 106(4), 620 (1957)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kamp, H., van Genabith, J., Reyle, U.: Discourse Representation Theory. In: Handbook of Philosophical Logic, vol. 15, pp. 125–394 (2011)Google Scholar
  15. 15.
    Lambek, J.: Type Grammars as Pregroups. Grammars 4, 21–39 (2001)CrossRefMATHGoogle Scholar
  16. 16.
    Lane, S.M., Moerdijk, I.: Sheaves in geometry and logic: A first introduction to topos theory. Springer (1992)Google Scholar
  17. 17.
    Mitkov, R.: Anaphora Resolution. Longman (2002)Google Scholar
  18. 18.
    Dowty, D.R., Wall, R.E., Peters, S.: Introduction to Montague Semantics. D. Reidel Publishing Company, Dodrecht (1981)Google Scholar
  19. 19.
    Visser, A.: The Donkey and the Monoid: Dynamic Semantics with Control Elements. Journal of Logic, Language and Information Archive 11, 107–131 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Samson Abramsky
    • 1
  • Mehrnoosh Sadrzadeh
    • 2
  1. 1.Department of Computer ScienceUniversity of OxfordUK
  2. 2.School of Electronic Engineering and Computer ScienceQueen Mary University of LondonUK

Personalised recommendations