Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-12T21:48:37.325Z Has data issue: false hasContentIssue false
  • English
  • Français

A generalised quantifier theory of natural language in categorical compositional distributional semantics with bialgebras

Published online by Cambridge University Press:  10 April 2019

Jules Hedges  and
Mehrnoosh Sadrzadeh
Jules Hedges*
Affiliation:
Department of Computer Science, University of Oxford, Oxford, UK
Mehrnoosh Sadrzadeh
Affiliation:
School of Electronic Engineering and Computer Science, Queen Mary University of London, London, UK
*
*Corresponding author. Email: julian.hedges@cs.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Categorical compositional distributional semantics is a model of natural language; it combines the statistical vector space models of words with the compositional models of grammar. We formalise in this model the generalised quantifier theory of natural language, due to Barwise and Cooper. The underlying setting is a compact closed category with bialgebras. We start from a generative grammar formalisation and develop an abstract categorical compositional semantics for it, and then instantiate the abstract setting to sets and relations and to finite-dimensional vector spaces and linear maps. We prove the equivalence of the relational instantiation to the truth theoretic semantics of generalised quantifiers. The vector space instantiation formalises the statistical usages of words and enables us to, for the first time, reason about quantified phrases and sentences compositionally in distributional semantics.

Keywords

Compositional Distributional SemanticsCompact Closed CategoriesFrobenius and BialgebrasVector Space SemanticsGeneralised QuantifiersConservativityNatural Language Semantics
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 6 , June 2019 , pp. 783 - 809
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ajdukiewicz, K. (1935). Die syntaktische konnexitat. Studia Philosophica 1 1–27.Google Scholar
Bar-Hillel, Y. (1953). A quasi-arithmetical notation for syntactic description. Language 29 47–58.CrossRefGoogle Scholar
Bar-Hillel, Y., Gaifman, C. and Shamir, E. (1960). On categorial and phrase-structure grammars. Bulletin of the Research Council of Israel 9F, 1–16.Google Scholar
Barwise, J. and Cooper, R. (1981). Generalized quantifiers and natural language. Linguistics and Philosophy 4 159–219.CrossRefGoogle Scholar
Bonchi, F., Sobocinski, P. and Zanasi, F. (2014). Interacting bialgebras are Frobenius. In: Muscholl, A. (ed.) Proceedings of FoSSaCS 2014, vol. 8412, Grenoble, France Springer, 351–365.Google Scholar
Bullinaria, J. A. and Levy, J. P. (2007). Extracting semantic representations from word co-occurrence statistics: A computational study. Behavior Research Methods 39 510–526.CrossRefGoogle ScholarPubMed
Buszkowski, W. (1988). Generative power of categorial grammars. In: Oehrle, R., Bach, E., and Wheeler, D. (eds.) Categorial Grammars and Natural Language Structures, Studies in Linguistics and Philosophy, vol. 32, Springer Netherlands, 69–94.CrossRefGoogle Scholar
Buszkowski, W. (2001). Lambek grammars based on pregroups. In: Logical Aspects of Computational Linguistics, Lecture Notes in Computer Science, vol. 2099, Springer Berlin Heidelberg, 95–109.CrossRefGoogle Scholar
Chomsky, N. (1956). Three models for the description of language. IRE Transactions on Information Theory 2 113–124.CrossRefGoogle Scholar
Clark, S., Coecke, B. and Sadrzadeh, M. (2008). A compositional distributional model of meaning. In: Bruza, P., Lawless, W., Coecke, B. (eds.) Proceedings of the Second Symposium on Quantum Interaction (QI), Oxford University, College Publications, 133–140.Google Scholar
Clark, S., Coecke, B. and Sadrzadeh, M. (2013). The Frobenius anatomy of relative pronouns. In: Kornai, A., Kuhlmann, M. (eds.), 13th Meeting on Mathematics of Language (MoL)., Sofia, Bulgaria, ACL, 41–51.Google Scholar
Clark, S. and Pulman, S. (2007). Combining symbolic and distributional models of meaning. In: Bruza, P., Lawless, W., van Rijsbergen, C. J. (eds.) Proceedings of the AAAI Spring Symposium on Quantum Interaction, Technical Report SS-07-08, Stanford University, AAAI Press, 52–55.Google Scholar
Coecke, B., Grefenstette, E. and Sadrzadeh, M. (2013). Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus. Annals of Pure and Applied Logic 164(11) 1079–1100, special issue on Seventh Workshop on Games for Logic and Programming Languages (GaLoP VII).CrossRefGoogle Scholar
Coecke, B., Sadrzadeh, M. and Clark, S. (2010). Mathematical foundations for distributed compositional model of meaning. Lambek Festschrift. Linguistic Analysis 36 345–384.Google Scholar
Firth, J. (1957). A synopsis of linguistic theory 1930–1955. In: Palmer, F. R. (ed.) Studies in Linguistic Analysis Longmans, 168–205.Google Scholar
Frege, G. (1948). On sense and reference. The Philosophical Review 57 209–230.CrossRefGoogle Scholar
Geffet, M. and Dagan, I. (2005). The distributional inclusion hypotheses and lexical entailment. In: Proceedings of the 43rd Annual Meeting on Association for Computational Linguistics, ACL ‘05, Association for Computational Linguistics, 107–114.Google Scholar
Grefenstette, E., Dinu, G., Zhang, Y., Sadrzadeh, M. and Baroni, M. (2013). Multi-step regression learning for compositional distributional semantics. In: 10th International Conference on Computational Semantics (IWCS). Postdam.Google Scholar
Grefenstette, E. and Sadrzadeh, M. (2011). Experimental support for a categorical compositional distributional model of meaning. In: Proceedings of Conference on Empirical Methods in Natural Language Processing (EMNLP), Computational Linguistics 41. MIT Press 1394–1404.Google Scholar
Grefenstette, E. and Sadrzadeh, M. (2015). Concrete models and empirical evaluations for the categorical compositional distributional model of meaning. Computational Linguistics 41 71–118.CrossRefGoogle Scholar
Harris, Z. (1954). Distributional structure. Word 10, 146–162, Routledge.CrossRefGoogle Scholar
Kartsaklis, D. (2015). Compositional Distributional Semantics with Compact Closed Categories and Frobenius Algebras. PhD thesis, Department of Computer Science, University of Oxford.Google Scholar
Kartsaklis, D., Kalchbrenner, N. and Sadrzadeh, M. (2014). Resolving lexical ambiguity in tensor regression models of meaning. In: Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics, Volume 2: Short Papers, June 22–27, 2014, ACL 2014, Baltimore, MD, USA, 212–217.Google Scholar
Kartsaklis, D. and Sadrzadeh, M. (2013). Prior disambiguation of word tensors for constructing sentence vectors. In: Proceedings of Conference on Empirical Methods in Natural Language Processing (EMNLP) Association for Computational Linguistics, 1590–1601.Google Scholar
Kartsaklis, D., Sadrzadeh, M. and Pulman, S. (2012). A unified sentence space for categorical distributional-compositional semantics: Theory and experiments. In: Proceedings of 24th International Conference on Computational Linguistics (COLING 2012): Posters, Mumbai, India, 549–558.Google Scholar
Kartsaklis, D., Sadrzadeh, M., Pulman, S. and Coecke, B. (2013). Reasoning about meaning in natural language with compact closed categories and Frobenius algebras. In: Chubb, A., Eskandarian, J. and Harizanov, V. (eds.) Logic and Algebraic Structures in Quantum Computing and Information, Cambridge University Press. 199–222.Google Scholar
Kelly, G. and Laplaza, M. (1980). Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213. http://www.sciencedirect.com/science/article/pii/0022404980901012CrossRefGoogle Scholar
Kock, A. (1972). Strong functors and monoidal monads. Archiv der Mathematik 23 113–120.CrossRefGoogle Scholar
Lambek, J. (1958). The mathematics of sentence structure. American Mathematics Monthly 65 154–170.CrossRefGoogle Scholar
Lambek, J. (1997). Type grammars revisited. In: Proceedings of LACL 97, Lecture Notes in Artificial Intelligence, vol. 1582, Springer Verlag. 1–27.Google Scholar
Lambek, J. (2008). From Word to Sentence: A Computational Algebraic Approach to Grammar. Polimetrica.Google Scholar
Lambek, J. (2010). Compact monoidal categories from linguistics to physics. In: Coecke, B. (ed.) New Structures for Physics, Lecture Notes in Physics, Springer, 451–469.Google Scholar
Landauer, T. and Dumais, S. (1997). A solution to Plato’s problem: The latent semantic analysis theory of acquisition, induction, and representation of knowledge. Psychological Review 104 211–240.CrossRefGoogle Scholar
Lapesa, G. and Evert, S. (2014). A large scale evaluation of distributional semantic models: Parameters, interactions and model selection. Transactions of the Association for Computational Linguistics 2 531–545.CrossRefGoogle Scholar
Lin, D. (1998). Automatic retrieval and clustering of similar words. In: Proceedings of the 17th international conference on Computational linguistics, vol. 2, Association for Computational Linguistics, 768–774.Google Scholar
Lund, K. and Burgess, C. (1996). Producing high-dimensional semantic spaces from lexical co-occurrence. Behavior Research Methods Instruments and Computers 28 (2) 203–208.CrossRefGoogle Scholar
McCurdy, M. (2012). Graphical methods for Tannaka duality of weak bialgebras and weak Hopf algebras. Theory and Applications of Categories 26 (9) 233–280.Google Scholar
Milajevs, D., Kartsaklis, D., Sadrzadeh, M. and Purver, M. (2014). Evaluating neural word representations in tensor-based compositional settings. In: Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), Association for Computational Linguistics, 708–719.CrossRefGoogle Scholar
Mitchell, J. and Lapata, M. (2010). Composition in distributional models of semantics. Cognitive Science 34 1388–1439.CrossRefGoogle ScholarPubMed
Montague, R. (1970). English as a formal language. In: Visentini, B. (ed.) Linguaggi nella Società e nella Tecnica, Edizioni di Comunità, 189–224.Google Scholar
Polajnar, T., Fagarasan, L. and Clark, S. (2014). Reducing dimensions of tensors in type-driven distributional semantics. In: Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), 1036–1046. Association for Computational Linguistics.CrossRefGoogle Scholar
Preller, A. (2013). From logical to distributional models. In: Proceedings of the 10th International Workshop on Quantum Physics and Logic, QPL 2013, Castelldefels (Barcelona), Spain, July 17–19, 113–131.Google Scholar
Preller, A. (2014). Natural language semantics in biproduct dagger categories. Journal of Applied Logic 12 (1) 88–108. https://doi.org/10.1016/j.jal.2013.08.001CrossRefGoogle Scholar
Preller, A. and Lambek, J. (2007). Free compact 2-categories. Mathematical Structures in Computer Science 17 309–340.CrossRefGoogle Scholar
Preller, A. and Sadrzadeh, M. (2010). Bell states and negative sentences in the distributed model of meaning. In: Coecke, B., Panangaden, P., Selinger, P. (eds.) Proceedings of the 6th QPL Workshop on Quantum Physics and Logic, Electronic Notes in Theoretical Computer Science, University of Oxford. 141–153.Google Scholar
Preller, A. and Sadrzadeh, M. (2011). Semantic vector models and functional models for pregroup grammars. Journal of Logic Language and Information 20 419–443.CrossRefGoogle Scholar
Rubenstein, H. and Goodenough, J. (1965). Contextual correlates of synonymy. Communications of the ACM 8 (10) 627–633.CrossRefGoogle Scholar
Rypacek, O. and Sadrzadeh, M. (2014). A low-level treatment of generalised quantifiers in categorical compositional distributional semantics. In: Joint Proceedings of the Second International Workshop on Natural Language and Computer Science (NLCS14) and First International Workshop on Natural Language Services for Reasoners (NLSR 2014), TR 2014/02, Center for Informatics and Systems of the University of Coimbra, 165–177.Google Scholar
Sadrzadeh, M., Clark, S. and Coecke, B. (2013). Frobenius anatomy of word meanings i: Subject and object relative pronouns. Journal of Logic and Computation 23 1293–1317.CrossRefGoogle Scholar
Sadrzadeh, M., Clark, S. and Coecke, B. (2014). Frobenius anatomy of word meanings 2: Possessive relative pronouns. Journal of Logic and Computation 26 785–815.CrossRefGoogle Scholar
Salton, G., Wong, A. and Yang, C. S. (1975). A vector space model for automatic indexing. Communications of the ACM 18 613–620.CrossRefGoogle Scholar
Schuetze, H. (1998). Automatic word sense discrimination. Computational Linguistics 24 (1) 97–123.Google Scholar
Turney, P. D. (2006). Similarity of semantic relations. Computational Linguistics 32 (3) 379–416.CrossRefGoogle Scholar
van Benthem, J. (1987). Categorial grammar and lambda calculus. In: Skordev, Dimiter G. (ed.) Mathematical Logic and Its Applications, Springer, 39–60.CrossRefGoogle Scholar
Weeds, J., Weir, D. and McCarthy, D. (2004). Characterising measures of lexical distributional similarity. In: Proceedings of the 20th International Conference on Computational Linguistics, COLING ‘04, Association for Computational Linguistics. 1015.Google Scholar