Coherent Diagrammatic Reasoning in Compositional Distributional Semantics

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10388)


The framework of Categorical Compositional Distributional models of meaning [3], inspired by category theory, allows one to compute the meaning of natural language phrases, given basic meaning entities assigned to words. Composing word meanings is the result of a functorial passage from syntax to semantics. To keep one from drowning in technical details, diagrammatic reasoning is used to represent the information flow of sentences that exists independently of the concrete instantiation of the model. Not only does this serve the purpose of clarification, it moreover offers computational benefits as complex diagrams can be transformed into simpler ones, which under coherence can simplify computation on the semantic side. Until now, diagrams for compact closed categories and monoidal closed categories have been used (see [2, 3]). These correspond to the use of pregroup grammar [12] and the Lambek calculus [9] for syntactic structure, respectively. Unfortunately, the diagrammatic language of Baez and Stay [1] has not been proven coherent. In this paper, we develop a graphical language for the (categorical formulation of) the nonassociative Lambek calculus [10]. This has the benefit of modularity where extension of the system are easily incorporated in the graphical language. Moreover, we show the language is coherent with monoidal closed categories without associativity, in the style of Selinger’s survey paper [17].


Diagrammatic reasoning Coherence theorem Proof nets Compositional distributional semantics 



The author is greatly indebted for many fruitful discussions with Michael Moortgat during the writing of the MSc thesis on which this paper is largely based. Also, a thanks goes out to Mehrnoosh Sadrzadeh for discussions culminating in the existence of this paper. A thanks as well to John Baez and Peter Selinger for giving some advice a long time ago on the topic of diagrammatic reasoning. Finally, the author would like to thank the two anonymous referees of this paper. The author was supported by a Queen Mary Principal’s Research Studentship during the writing of this paper.


  1. 1.
    Baez, J., Stay, M.: Physics, topology, logic and computation: a rosetta stone. In: Coecke, B. (ed.) New Structures for Physics, pp. 95–172. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-12821-9_2 CrossRefGoogle Scholar
  2. 2.
    Coecke, B., Grefenstette, E., Sadrzadeh, M.: Lambek vs. Lambek: functorial vector space semantics and string diagrams for lambek calculus. Ann. Pure Appl. log. 164(11), 1079–1100 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Coecke, B., Sadrzadeh, M., Clark, S.: Mathematical foundations for a compositional distributional model of meaning. arXiv preprint arXiv:1003.4394 (2010)
  4. 4.
    Freyd, P., Yetter, D.N.: Coherence theorems via knot theory. J. Pure Appl. Algebr. 78(1), 49–76 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Girard, J.Y.: Linear logic. Theor. Comput. Sci. 50(1), 1–101 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Joyal, A., Street, R.: The geometry of tensor calculus, I. Adv. Math. 88(1), 55–112 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. J. Pure Appl. Algebr. 19, 193–213 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lambek, J.: The mathematics of sentence structure. Am. Math. Mon. 65(3), 154–170 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lambek, J.: On the calculus of syntactic types. Struct. Lang. Math. Asp. 166, C178 (1961)Google Scholar
  11. 11.
    Lambek, J.: Deductive systems and categories. Theory Comput. Syst. 2(4), 287–318 (1968)zbMATHGoogle Scholar
  12. 12.
    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 CrossRefGoogle Scholar
  13. 13.
    Lambek, J., Scott, P.J.: Introduction to Higher-Order Categorical Logic, vol. 7. Cambridge University Press, Cambridge (1988)zbMATHGoogle Scholar
  14. 14.
    Moortgat, M.: Multimodal linguistic inference. J. Log. Lang. Inf. 5(3–4), 349–385 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moot, R.: Proof Nets for Linguistic Analysis. Ph.D. thesis, Utrecht University (2002)Google Scholar
  16. 16.
    Penrose, R.: Applications of negative dimensional tensors. In: Combinatorial Mathematics and its Applications, vol. 1, pp. 221–244 (1971)Google Scholar
  17. 17.
    Selinger, P.: A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics, pp. 289–355. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Wijnholds, G.J.: Categorical foundations for extended compositional distributional models of meaning. MSc. thesis, University of Amsterdam (2014)Google Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Queen Mary University of LondonLondonUK

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