Advertisement

A Logic for Categorial Grammars: Lambek’s Syntactic Calculus

  • Richard Moot
  • Christian Retoré
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6850)

Summary

Our second chapter is a rather complete study of the Lambek calculus, which enables a completely logical treatment of categorial grammar.

We first present its syntax in full detail, both with sequent calculus and natural deduction, and explain the relationship between these two presentations. Then we turn our attention to the normal forms for such proofs. Normalization and its dual namely interpolation are not only pleasant mathematical properties; they also are key properties for the correspondence between Lambek grammars and more familiar phrase structure grammars; we give a detailed proof of the theorem of Pentus establishing the weak equivalence between context-free grammars and Lambek grammars.

In addition, we prove completeness for the Lambek calculus with respect to linguistically natural models: in these models categories are interpreted as subsets of a free monoid (eg. as strings of words or lexical items). Providing such a simple and natural interpretation provides another strong justification for the categorial approach.

Keywords

Natural Deduction Sequent Calculus Elimination Rule Primitive Type Introduction Rule 
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. Abramsky, S.: Computational interpretations of linear logic. Theoretical Computer Science 111, 3–57 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Autebert, J.M., Boasson, L., Sénizergues, G.: Langages de parenthèses, langages n.t.s. et homomorphismes inverses. RAIRO Informatique Théorique 18(4), 327–344 (1984)MathSciNetzbMATHGoogle Scholar
  3. van Benthem, J.: Categorial grammar. In: Essays in Logical Semantics, ch. 7, pp. 123–150. Reidel, Dordrecht (1986)CrossRefGoogle Scholar
  4. van Benthem, J.: Categorial grammars and lambda calculus. In: Skordev, D. (ed.) Mathematical Logic and its Applications. Plenum Press (1987)Google Scholar
  5. van Benthem, J.: Language in Action: Categories, Lambdas and Dynamic Logic. Sudies in logic and the foundation of mathematics, vol. 130. North-Holland, Amsterdam (1991)zbMATHGoogle Scholar
  6. Buszkowski, W.: Compatibility of a categorial grammar with an asssociated category system. Zeitschrift für Matematische Logik und Grundlagen der Mathematik 28, 229–238 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Buszkowski, W.: Mathematical linguistics and proof theory. In: van Benthem, J., ter Meulen, A. (eds.) Handbook of Logic and Language, ch. 12, pp. 683–736. North-Holland Elsevier, Amsterdam (1997)CrossRefGoogle Scholar
  8. Carpenter, B.: Lectures on Type-Logical Semantics. MIT Press, Cambridge (1996)Google Scholar
  9. Chomsky, N.: Syntactic structures. Janua linguarum. Mouton, The Hague (1957)Google Scholar
  10. Chomsky, N.: Formal properties of grammars. In: Handbook of Mathematical Psychology, vol. 2, pp. 323–418. Wiley, New-York (1963)Google Scholar
  11. Chomsky, N.: The minimalist program. MIT Press, Cambridge (1995)zbMATHGoogle Scholar
  12. Cohen, J.M.: The equivalence of two concepts of categorial grammars. Information and Control 10, 475–484 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Engelfriet, J.: Context-free graph grammars. In: Rosenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages 3: Beyond Words, pp. 125–213. Springer, New York (1997)CrossRefGoogle Scholar
  14. Engelfriet, J., Maneth, S.: Tree Languages Generated by Context-Free Graph Grammars. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) TAGT 1998. LNCS, vol. 1764, pp. 15–29. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. Gécseg, F., Steinby, M.: Tree languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, ch. 1. Springer, Berlin (1997)Google Scholar
  16. Gentzen, G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 176–210, 405–431 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Girard, J.Y.: Linear logic. Theoretical Computer Science 50(1), 1–102 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Girard, J.Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge Tracts in Theoretical Computer Science, vol. 7. Cambridge University Press (1988)Google Scholar
  19. de Groote, P., Retoré, C.: Semantic readings of proof nets. In: Kruijff, G.J., Morrill, G., Oehrle, D. (eds.) Formal Grammar, pp. 57–70. FoLLI, Prague (1996)Google Scholar
  20. Hendriks, H.: Studied flexibility: Categories and types in syntax and semantics. PhD thesis, University of Amsterdam, ILLC Dissertation Series (1993)Google Scholar
  21. Hepple, M.: Normal form theorem proving for the Lambek calculus. In: Proceedings of COLING 1990, Helsinki, pp. 173–178 (1990)Google Scholar
  22. Kanazawa, M., Salvati, S.: On the derivations of lambek grammars (2009) (unpublished manuscript)Google Scholar
  23. König, E.: Parsing as natural deduction. In: Proceedings of the Annual Meeting of the Association for Computational Linguistics, pp. 272–297 (1989)Google Scholar
  24. Lambek, J.: The mathematics of sentence structure. American Mathematical Monthly, 154–170 (1958)Google Scholar
  25. Moortgat, M.: Categorial Investigations: Logical and Linguistic Aspects of The Lambek Calculus. Foris, Dordrecht (1988)Google Scholar
  26. Nivat, M.: Congruences de thue et t-langages. Studia Scientiarum Mathematicarum Hungarica 6, 243–249 (1971)MathSciNetzbMATHGoogle Scholar
  27. Pentus, M.: Lambek calculus is L-complete. Tech. Rep. LP-93-14, Institute for Logic, Language and Computation, Universiteit van Amsterdam (1993a)Google Scholar
  28. Pentus, M.: Lambek grammars are context-free. In: Logic in Computer Science. IEEE Computer Society Press (1993b)Google Scholar
  29. Pentus, M.: Product-free Lambek calculus and context-free grammars. Journal of Symbolic Logic 62(2), 648–660 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Pentus, M.: Lambek calculus is NP-complete. Theoretical Computer Science 357(1), 186–201 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Retoré, C., Stabler, E.: Generative grammar in resource logics. Research on Language and Computation 2(1), 3–25 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Roorda, D.: Resource logic: proof theoretical investigations. PhD thesis, FWI, Universiteit van Amsterdam (1991)Google Scholar
  33. Thatcher, J.W.: Characterizing derivation trees of context free grammars through a generalization of finite automata theory. Journal of Computer and System Sciences 1, 317–322 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Tiede, H.-J.: Lambek Calculus Proofs and Tree Automata. In: Moortgat, M. (ed.) LACL 1998. LNCS (LNAI), vol. 2014, pp. 251–265. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  35. Zielonka, W.: Axiomatizability of the Adjukiewicz-Lambek calculus by means of cancellation schemes. Zeitschrift für Matematische Logik und Grundlagen der Mathematik 27(13-14), 215–224 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Zielonka, W.: A simple and general method of solving the finite axiomatizability problems for Lambek’s syntactic calculi. Studia Logica 48(1), 35–39 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Richard Moot
  • Christian Retoré

There are no affiliations available

Personalised recommendations