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Journal of Logic, Language and Information

, Volume 5, Issue 2, pp 115–155 | Cite as

Identification in the limit of categorial grammars

  • Makoto Kanazawa
Article

Abstract

It is proved that for any k, the class of classical categorial grammars that assign at most k types to each symbol in the alphabet is learnable, in the Gold (1967) sense of identification in the limit from positive data. The proof crucially relies on the fact that the concept known as finite elasticity in the inductive inference literature is preserved under the inverse image of a finite-valued relation. The learning algorithm presented here incorporates Buszkowski and Penn's (1990) algorithm for determining categorial grammars from input consisting of functor-argument structures.

Key words

categorial grammar finite elasticity functor-argument struture identification in the limit inductive inference learnability 

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References

  1. Ajdukiewicz, K., 1935, “Die syntaktische Konnexität”, Studia Philosophica 1, 1–27.Google Scholar
  2. Angluin, D., 1980a, “Finding patterns common to a set of strings”, Journal of Computer and System Sciences 21, 46–62.Google Scholar
  3. Angluin, D., 1980b, “Inductive inference of formal languages from positive data”, Information and Control 45, 117–135.Google Scholar
  4. Angluin, D., 1982, “Inference of reversible languages”, Journal of the Association for Computing Machinery 29, 741–765.Google Scholar
  5. Angluin, D. and Smith, C. H., 1983, “Inductive inference: theory and methods”, Computing Surveys 15, 237–269.Google Scholar
  6. Barendregt, H. P., 1992, “Lambda calculi with types”, in Handbook of Logic in Computer Science, Volume 2, S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, eds., Oxford: Clarendon Press.Google Scholar
  7. Bar-Hillel, Y., 1953, “A quasi-arithmetical notation for syntactic description”, Language 29, 47–58. Reprinted in Bar-Hillel 1964.Google Scholar
  8. Bar-Hillel, Y., 1960, “Some linguistic obstacles to machine translation”, reprinted in Bar-Hillel 1964.Google Scholar
  9. Bar-Hillel, Y., 1964, Language and Information, Reading, MA: Addison-Wesley.Google Scholar
  10. Bar-Hillel, Y., Gaifman, H., and Shamir, E., 1960, “On categorial and phrase structure grammars”, The Bulletin of the Research Council of Israel, vol. 9F, 1–16. Reprinted in Bar-Hillel 1964.Google Scholar
  11. Buszkowski, W., 1987a, “Solvable problems for classical categorial grammars”, Bulletin of the Polish Academy of Sciences: Mathematics 35, 373–382.Google Scholar
  12. Buszkowski, W., 1987b, “Discovery procedures for categorial grammars”, in Categories, Polymorphism and Unification, E. Klein and J. van Benthem, eds., University of Amsterdam.Google Scholar
  13. Buszkowski, W., 1988, “Generative power of categorial grammars”, in Categorial Grammars and Natural Language Structures, R. T. Oehrle, E. Bach, and D. Wheeler, eds., Dordrecht: Reidel.Google Scholar
  14. Buszkowski, W. and Penn, G., 1990, “Categorial grammars determined from linguistic data by unification”, Studia Logica 49, 431–454.Google Scholar
  15. Chomsky, N., 1986, Knowledge of Language: Its Nature, Origin, and Use, New York: Praeger.Google Scholar
  16. Dowty, David. 1988. Type raising, functional composition, and non-constituent conjunction. In Richard T. Oehrle, Emmon Bach, and Deirdre Wheeler, eds., Categorial Grammars and Natural Language Structures, Dordrecht: Reidel.Google Scholar
  17. Gold, M. E., 1967, “Language identification in the limit”, Information and Control 10, 447–474.Google Scholar
  18. de Jongh, D. and Kanazawa, M., 1995, Learnability Theory, course material presented at the Seventh Summer School in Logic, Language and Information, University of Barcelona, August 1995.Google Scholar
  19. Kanazawa, M., 1994a, Learnable Classes of Categorial Grammars, Ph.D. dissertation, Stanford University. Available as ILLC Dissertation Series 1994–8, Institute for Logic, Language, and Computation, University of Amsterdam (illc@fwi.uva.nl).Google Scholar
  20. Kanazawa, M., 1994b, “A note on language classes with finite elasticity”, Reprot CS-R9471, CWI, Amsterdam.Google Scholar
  21. Kapur, S., 1991, Computational Learning of Languages, Ph.D. dissertation, Cornell University. Available as Technical Report 91-1234, Department of Computer Science, Cornell University.Google Scholar
  22. Kearns, M. J. and Vazirani, U. V., 1994, An Introduction to Computational Learning Theory, Cambridge, Mass.: MIT Press.Google Scholar
  23. Lambek, J., 1958, “The mathematics of sentence structure”, American Mathematical Monthly 65, 154–170. Reprinted in Categorial Grammars, W. Buszkowski, W. Marciszewski, and J. van Benthem, eds., Amsterdam: John Benjamins, 1988.Google Scholar
  24. Lambek, J., 1961, “On the calculus of syntactic types”, in Structure of Language and its Mathematical Aspects, R. Jakobson, ed., Providence, R.I.: American Mathematical Society.Google Scholar
  25. Lassez, J.-L., M. J. Maher, and K. Marriott. 1988. Unification revisited. In J. Minker, ed., Foundations of Deductive Databases and Logic Programming, pp. 587–625. Los Altos, Calif.: Morgan Kaufmann.Google Scholar
  26. Levy, Leon S. and Aravind K. Joshi. 1978. Skeletal structural descriptions. Information and Control 39, 192–211.Google Scholar
  27. Montague, R., 1973. “The proper treatment of quantification in ordinary English”, reprinted in Formal Philosophy: Selected Papers of Richard Montague, R. H. Thomason, ed., New Haven: Yale University Press, 1974.Google Scholar
  28. Moriyama, T. and Sato, M, 1993, “Properties of language classes with finite elasticity”, in Algorithmic Learning Theory, Proceedings, 1993, K. P. Jankte, S. Kobayashi, E. Tomita, and T. Yokomori, eds., Lecture Notes in Artificial Intelligence 744, Berlin: Springer.Google Scholar
  29. Motoki, T., Shinohara, T., and Wright, K., 1991, “The correct definition of finite elasticity: Corrigendum to identification of unions”, p. 375 in The Fourth Annual Workshop on Computational Learning Theory, San Mateo, CA: Morgan Kaufmann.Google Scholar
  30. Osherson, D., Stob, M., and Weinstein, S., 1986, Systems That Learn, Cambridge, MA: MIT Press.Google Scholar
  31. Osherson, D., Weinstein, S., de Jongh, D., and Martin, E., 1994, “Formal learning theory”, to appear in Handbook of Logic and Language, J. van Benthem and A. ter Meulen, eds.Google Scholar
  32. Sakakibara, Y., 1992, “Efficient learning of context-free grammars from positive structural examples”, Information and Computation 97, 23–60.Google Scholar
  33. Shinohara, T., 1990a, “Inductive inference from positive data is powerful”, pp. 97–110 in The 1990 Workshop on Computational Learning Theory, San Mateo, CA: Morgan Kaufmann.Google Scholar
  34. Shinohara, T., 1990b, “Inductive inference of monotonic formal systems from positive data”, pp. 339–351 in Algorithmic Learning Theory, S. Arikawa, S. Goto, S. Ohsuga, and T. Yokomori, eds., Tokyo: Ohmsha, and New York and Berlin: Springer.Google Scholar
  35. Steedman, Mark J. 1985. Dependency and coordination in the grammar of Dutch and English. Language 61, 523–568.Google Scholar
  36. Steedman, Mark J. 1987. Combinatory grammars and parasitic gaps. Natural Language and Linguistic Theory 5, 403–440.Google Scholar
  37. Steedman, Mark J. 1988. Combinators and grammars. In Richard T. Oehrle, Emmon Bach, and Deirdre Wheeler, eds., Categorial Grammars and Natural Language Structures. Dordrecht: Reidel.Google Scholar
  38. Wexler, K. and Culicover, P., 1980, Formal Principles of Language Acquisition, Cambridge, MA: MIT Press.Google Scholar
  39. Wright, K., 1989, “Identification of unions of languages drawn from an identifiable class”, pp. 328–333 in The 1989 Workshop on Computational Learning Theory, San Mateo, CA: Morgan Kaufmann.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Makoto Kanazawa
    • 1
  1. 1.Department of Cognitive and Information Sciences, Faculty of LettersChiba UniversityChiba-shiJapan

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