Journal of Logic, Language and Information

, Volume 19, Issue 2, pp 163–183 | Cite as

On the Membership Problem for Non-Linear Abstract Categorial Grammars

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Abstract

In this paper we show that the membership problem for second order non-linear Abstract Categorial Grammars is decidable. A consequence of that result is that Montague-like semantics yield to a decidable text generation problem. Furthermore the proof we propose is based on a new tool, Higher Order Intersection Signatures, which grasps statically dynamic properties of λ-terms and presents an interest in its own.

Keywords

Lambda-calculus Intersection types Montague semantics  Text generation Formal grammars 
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References

  1. Aoto T., Ono H. (1994) Non-uniqueness of normal proofs for minimal formulas in implication-conjunction fragment of BCK. Bulletin of the Section of Logic 23(3): 104–112Google Scholar
  2. Aoto T. (1999) Uniqueness of normal proofs in implicational intuitionistic logic. Journal of Logic, Language and Information 8(2): 217–242CrossRefGoogle Scholar
  3. Barendregt, H. P. (1984). The lambda calculus: Its syntax and semantics, Vol. 103. Studies in logic and the foundations of mathematics. Amsterdam: North-Holland. revised edition.Google Scholar
  4. Babaev A. A., & Soloviev S. V. (1982). Coherence theorem for canonical maps in cartesian closed categories. Journal of Soviet Mathematics, 20.Google Scholar
  5. Damm W. (1982) The IO- and OI-hierarchies. Theoretical Computer Science 20: 95–207CrossRefGoogle Scholar
  6. Dezani-Ciancaglini, M., Giovannetti, E., & de’ Liguoro, U. (1998). Intersection types, lambda-models and Böhm trees. In MSJ-Memoir “Theories of Types and Proofs”, Vol. 2, (pp. 45–97). Mathematical Society of Japan.Google Scholar
  7. de Groote, P. (2001). Towards abstract categorial grammars. In Association for Computational Linguistic (Eds.), Proceedings 39th annual meeting and 10th conference of the European chapter, (pp. 148–155). Morgan Kaufmann Publishers.Google Scholar
  8. de Groote, P. (2007). Towards a montagovian account of dynamics. In: Proceedings of semantics in linguistic theory XVI. CLC Publications.Google Scholar
  9. Girard J. -Y., Taylor P., Lafont Y. (1989) Proofs and types. Cambridge University Press, CambridgeGoogle Scholar
  10. Hillebrand, G. G. (1994). Finite model theory in the simply typed lambda calculus. PhD thesis, Department of Computer Science, Brown University, Providence, Rhode Island 02912.Google Scholar
  11. Hirokawa, S., & Tatsuta, M. (2000). Long D-normal form yields uniqueness of proofs. In Proceedings of logic colloqium.Google Scholar
  12. Huet, G. (1976). Résolution d’équations dans des langages d’ordre 1,2, . . .,ω. Thèse de doctorat es sciences mathématiques, Université Paris VII.Google Scholar
  13. Kanazawa M. (2006) Abstract families of abstract categorial languages. Electronic Notes in Theoretical Computer Science 165: 65–80CrossRefGoogle Scholar
  14. Kanazawa, M. (2007). Parsing and generation as datalog queries. In Proceedings of the 45th annual meeting of the association for computational linguistics (pp. 176–183). Association for Computational Linguistics.Google Scholar
  15. Loader, R. (2001) The undecidability of λ-definability. In C. A. Anderson & M. Zeleny (Eds.), Logic, meaning and computation: Essays in memory of Alonzo church (pp. 331–342). KluwerGoogle Scholar
  16. Montague R. (1974) Formal philosophy: Selected papers of Richard Montague. Yale University Press, New Haven, CTGoogle Scholar
  17. Pogodalla, S. (2001). Réseaux de preuve et génération pour les grammaires de types logiques. PhD thesis, Institut National Polytechnique de Lorraine.Google Scholar
  18. Pogodalla, S. (2004). Computing semantic representation: Towards ACG abstract terms as derivation trees. In Proceedings of the seventh international workshop on tree adjoining grammar and related formalisms (TAG+7) (pp. 64–71).Google Scholar
  19. Salvati, S. (2005). Problèmes de filtrage et problèmes d’analyse pour les grammaires catégorielles abstraites. PhD thesis, Institut National Polytechnique de Lorraine.Google Scholar
  20. Seki H., Matsumura T., Fujii M., Kasami T. (1991) On multiple context free grammars. Theoretical Computer Science 88(2): 191–229CrossRefGoogle Scholar
  21. Statman R. (1979) The typed lambda-calculus is not elementary recursive. Theoretical Computer Science 9: 73–81CrossRefGoogle Scholar
  22. Takahashi M., Joshi A. K., Levy L. S. (1975) Tree adjunct grammars. Journal of Computer and System Sciences 10(1): 136–163CrossRefGoogle Scholar
  23. Tatsuta, M. (1999). Uniqueness of d-normal proofs. In Proceedings of 7th Asian logic conference (pp. 41–42).Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.INRIA Bordeaux Sud-Ouest, LaBRIUniversité de Bordeaux, Unité Mixte de Recherche CNRS (UMR 5800)Talence cedexFrance

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