Skip to main content
SpringerLink
Log in
Book cover

Categories and Types in Logic, Language, and Physics pp 108–135Cite as

Learning Lambek Grammars from Proof Frames

  • Chapter

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8222))

Abstract

In addition to their limpid interface with semantics, categorial grammars enjoy another important property: learnability. This was first noticed by Buszkowski and Penn and further studied by Kanazawa, for Bar-Hillel categorial grammars.

What about Lambek categorial grammars? In a previous paper we showed that product free Lambek grammars are learnable from structured sentences, the structures being incomplete natural deductions. Although these grammars were shown to be unlearnable from strings by Foret ad Le Nir, in the present paper, we show that Lambek grammars, possibly with product, are learnable from proof frames i.e. incomplete proof nets.

After a short reminder on grammatical inference à la Gold, we provide an algorithm that learns Lambek grammars with product from proof frames and we prove its convergence. We do so for 1-valued ”(also known as rigid) Lambek grammars with product, since standard techniques can extend our result to k-valued grammars. Because of the correspondence between cut-free proof nets and normal natural deductions, our initial result on product free Lambek grammars can be recovered.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amblard, M.: Calculs de représentations sémantiques et syntaxe générative: les grammaires minimalistes catégorielles. PhD thesis, Université Sciences et Technologies - Bordeaux I (September 2007)

    Google Scholar 

  2. Amblard, M., Lecomte, A., Retoré, C.: Categorial minimalist grammars: From generative grammar to logical form. Linguistic Analysis 36(1-4), 273–306 (2010)

    Google Scholar 

  3. Amblard, M., Retoré, C.: Natural deduction and normalisation for partially commutative linear logic and lambek calculus with product. In: Cooper, S.B., Kent, T.F., Löwe, B., Sorbi, A. (eds.) Computation and Logic in the Real World (Computing in Europe 2007). Quaderni del Dipartimento di Scienze Matematiche e Informatiche Roberto Magari, vol. ID487, pp. 28–35. Università degli Studi di Siena (September 2007)

    Google Scholar 

  4. Angluin, D.: Finding patterns common to a set of strings. Journal of Computer and Sytem Science 21(1), 46–62 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  5. Angluin, D.: Inductive inference of formal languages from positive data. Information and Control 45, 117–135 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bar-Hillel, Y.: A quasi arithmetical notation for syntactic description. Language 29, 47–58 (1953)

    Article  MATH  Google Scholar 

  7. Bar-Hillel, Y., Gaifman, C., Shamir, E.: On categorial and phrase-structure grammars. Bulletin of the Research Council of Israel F(9), 1–16 (1963)

    Google Scholar 

  8. Berwick, R.C., Pietroski, P., Yankama, B., Chomsky, N.: Poverty of the stimulus revisited. Cognitive Science 35(5), 1207–1242 (2011)

    Article  Google Scholar 

  9. Bonato, R.: Uno studio sull’apprendibilità delle grammatiche di Lambek rigide — A study on learnability for rigid Lambek grammars. Tesi di Laurea & Mémoire de D.E.A, Università di Verona & Université Rennes 1 (2000)

    Google Scholar 

  10. Bonato, R., Retoré, C.: Learning rigid Lambek grammars and minimalist grammars from structured sentences. In: Popelìnskỳ, L., Nepil, M. (eds.) Proceedings of the third workshop on Learning Language in Logic, LLL 2001. FI MU Report series, vol. FI-MU-RS-2001-08, pp. 23–34. Faculty of Informatics – Masaryk University, Strabourg (September 2001)

    Google Scholar 

  11. Buszkowski, W.: Discovery procedures for categorial grammars. In: van Benthem, J., Klein, E. (eds.) Categories, Polymorphism and Unification. Universiteit van Amsterdam (1987)

    Google Scholar 

  12. Buszkowski, W., Penn, G.: Categorial grammars determined from linguistic data by unification. Studia Logica 49, 431–454 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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), http://hal.archives-ouvertes.fr/hal-00823554

    Google Scholar 

  14. Foret, A., Le Nir, Y.: Lambek rigid grammars are not learnable from strings. In: COLING 2002, 19th International Conference on Computational Linguistics, Taipei, Taiwan, vol. 1, pp. 274–279 (August 2002)

    Google Scholar 

  15. Fulop, S.: The Logic and Learning of Language. Trafford on Demand Pub. (2004)

    Google Scholar 

  16. Gleitman, L., Liberman, M. (eds.): An invitation to cognitive sciences, vol. 1. Language. MIT Press (1995)

    Google Scholar 

  17. Gold, E.M.: Language identification in the limit. Information and control 10, 447–474 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guerrini, S.: A linear algorithm for mll proof net correctness and sequentialization. Theoretical Computer Science 412(20), 1958–1978 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Johnson, K.: Gold’s theorem and cognitive science. Philosophy of Science 71, 571–592 (2004)

    Article  MathSciNet  Google Scholar 

  20. Joshi, A., Vijay-Shanker, K., Weir, D.: The convergence of mildly context-sensitive grammar formalisms. In: Sells, P., Schieber, S., Wasow, T. (eds.) Fundational Issues in Natural Language Processing. MIT Press (1991)

    Google Scholar 

  21. Kanazawa, M.: Learnable classes of categorial grammars. PhD thesis, Universiteit van Amsterdam (1994)

    Google Scholar 

  22. Kanazawa, M.: Learnable classes of categorial grammars. Studies in Logic, Language and Information. FoLLI & CSLI distributed by Cambridge University Press (1998)

    Google Scholar 

  23. Lambek, J.: The mathematics of sentence structure. American Mathematical Monthly, 154–170 (1958)

    Google Scholar 

  24. Melliès, P.A.: A topological correctness criterion for multiplicative non commutative logic. In: Ehrhard, T., Girard, J.Y., Ruet, P., Scott, P. (eds.) Linear Logic in Computer Science. London Mathematical Society Lecture Notes, vol. 316, pp. 283–321. Cambridge University press (2004)

    Google Scholar 

  25. Moot, R.: Semi-automated extraction of a wide-coverage type-logical grammar for French. In: Proceedings of Traitement Automatique des Langues Naturelles (TALN), Montreal (2010)

    Google Scholar 

  26. Moot, R., Retoré, C.: The logic of categorial grammars: A deductive account of natural language syntax and semantics. LNCS, vol. 6850. Springer, Heidelberg (2012), http://www.springer.com/computer/theoretical+computer+science/book/978-3-642-31554-1

  27. Morrill, G.: Categorial Grammar: Logical Syntax, Semantics, and Processing. OUP, Oxford (2011)

    Google Scholar 

  28. Morrill, G.: Incremental processing and acceptability. Computational Linguistics 26(3), 319–338 (2000); preliminary version: UPC Report de Recerca LSI-98-46-R (1998)

    Google Scholar 

  29. Nicolas, J.: Grammatical inference as unification. Rapport de Recherche RR-3632. INRIA (1999)

    Google Scholar 

  30. Pentus, M.: Lambek grammars are context-free. In: Logic in Computer Science. IEEE Computer Society Press (1993)

    Google Scholar 

  31. Piattelli-Palmarini, M. (ed.): Théories du langage, théories de l’apprentissage — le débat Chomsky Piaget. Editions du Seuil. Number 138 in Points (1975)

    Google Scholar 

  32. Pinker, S.: Language acquisition. In: [16], ch. 6, pp. 135–182

    Google Scholar 

  33. Pinker, S.: Why the child holded the baby rabbits. In: [16], ch. 5, pp. 107–133

    Google Scholar 

  34. Pullum, G.K., Scholz, B.C.: Empirical assessment of stimulus poverty arguments. The Linguistic Review 19, 9–50 (2002)

    Google Scholar 

  35. Reali, F., Christiansen, M.H.: Uncovering the richness of the stimulus: Structure dependence and indirect statistical evidence. Cognitive Science 29(6), 1007–1028 (2005)

    Article  Google Scholar 

  36. Retoré, C.C.: Le système F en logique linéaire. Mémoire de D.E.A. (dir.: J.-y. girard), Université Paris 7 (1987)

    Google Scholar 

  37. Sandillon-Rezer, N.-F., Moot, R.: Using tree transducers for grammatical inference. In: Pogodalla, S., Prost, J.-P. (eds.) LACL 2011. LNCS (LNAI), vol. 6736, pp. 235–250. Springer, Heidelberg (2011)

    Google Scholar 

  38. Stabler, E.: Derivational minimalism. In: Retoré, C. (ed.) LACL 1996. LNCS (LNAI), vol. 1328, pp. 68–95. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  39. Tellier, I.: How to split recursive automata. In: Clark, A., Coste, F., Miclet, L. (eds.) ICGI 2008. LNCS (LNAI), vol. 5278, pp. 200–212. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  40. Tiede, H.J.: Deductive Systems and Grammars: Proofs as Grammatical Structures. PhD thesis, Illinois Wesleyan University (1999), http://www.iwu.edu/htiede/

  41. Zucker, J.: The correspondence between cut-elimination and normalisation i, ii. Annals of Mathematical Logic 7, 1–156 (1974)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Questel SAS, Sophia Antipolis, France

    Roberto Bonato

  2. IRIT, Toulouse, France

    Christian Retoré

  3. Univ. Bordeaux, France

    Christian Retoré

Authors
  1. Roberto Bonato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Retoré
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Philosophy, Education and Quantitative Economics, Gabriele D’Annunzio University, Chieti-Pescara, Via dei Vestini 31, 60100, Chieti, Italy

    Claudia Casadio

  2. Department of Computer Science, Oxford University, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Bob Coecke

  3. Utrecht Institute of Linguistics OTS, Utrecht University, Trans 10,, 3512 JK, Utrecht, The Netherlands

    Michael Moortgat

  4. Department of Mathematics, University of Ottawa, 585 King Edward, K1N 6N5, Ottawa, ON, Canada

    Philip Scott

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bonato, R., Retoré, C. (2014). Learning Lambek Grammars from Proof Frames. In: Casadio, C., Coecke, B., Moortgat, M., Scott, P. (eds) Categories and Types in Logic, Language, and Physics. Lecture Notes in Computer Science, vol 8222. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54789-8_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-54789-8_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-54788-1

  • Online ISBN: 978-3-642-54789-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation