The IO and OI Hierarchies Revisited

  • Gregory M. Kobele
  • Sylvain Salvati
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7966)


We study languages of λ-terms generated by IO and OI unsafe grammars. These languages can be used to model meaning representations in the formal semantics of natural languages following the tradition of Montague [19]. Using techniques pertaining to the denotational semantics of the simply typed λ-calculus, we show that the emptiness and membership problems for both types of grammars are decidable. In the course of the proof of the decidability results for OI, we identify a decidable variant of the λ-definability problem, and prove a stronger form of Statman’s finite completeness Theorem [28].


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  1. 1.
    Aehlig, K., de Miranda, J.G., Ong, C.-H.L.: Safety is not a restriction at level 2 for string languages. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 490–504. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Aho, A.V.: Indexed grammars - an extension of context-free grammars. J. ACM 15(4), 647–671 (1968)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Amadio, R.M., Curien, P.-L.: Domains and Lambda-Calculi. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press (1998)Google Scholar
  4. 4.
    Blum, W., Ong, C.-H.L.: The safe lambda calculus. Logical Methods in Computer Science 5(1:3), 1–38 (2009)MathSciNetGoogle Scholar
  5. 5.
    Bourreau, P., Salvati, S.: A datalog recognizer for almost affine λ-cfgs. In: Kanazawa, M., Kornai, A., Kracht, M., Seki, H. (eds.) MOL 12. LNCS, vol. 6878, pp. 21–38. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Broadbent, C.H.: The limits of decidability for first order logic on cpda graphs. In: STACS, pp. 589–600 (2012)Google Scholar
  7. 7.
    Damm, W.: The IO- and OI-hierarchies. Theor. Comput. Sci. 20, 95–207 (1982)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    de Groote, P.: Towards abstract categorial grammars. In: ACL (ed.) Proceedings 39th Annual Meeting of ACL, pp. 148–155 (2001)Google Scholar
  9. 9.
    de Groote, P., Lebedeva, E.: On the dynamics of proper names. Technical report, INRIA (2010)Google Scholar
  10. 10.
    de Groote, P., Lebedeva, E.: Presupposition accommodation as exception handling. In: SIGDIAL, pp. 71–74. ACL (2010)Google Scholar
  11. 11.
    Engelfriet, J.: Iterated stack automata and complexity classes. Inf. Comput. 95(1), 21–75 (1991)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fischer, M.J.: Grammars with macro-like productions. PhD thesis, Harvard University (1968)Google Scholar
  13. 13.
    Haddad, A.: IO vs OI in higher-order recursion schemes. In: FICS. EPTCS, vol. 77, pp. 23–30 (2012)Google Scholar
  14. 14.
    Huet, G.: Résolution d’équations dans des langages d’ordre 1,2,...,ω. Thèse de doctorat en sciences mathématiques, Université Paris VII (1976)Google Scholar
  15. 15.
    Kanazawa, M.: Parsing and generation as datalog queries. In: Proceedings of the 45th Annual Meeting of ACL, pp. 176–183. ACL (2007)Google Scholar
  16. 16.
    Knapik, T., Niwiński, D., Urzyczyn, P.: Higher-order pushdown trees are easy. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 205–222. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  17. 17.
    Lebedeva, E.: Expressing Discourse Dynamics Through Continuations. PhD thesis, Université de Lorraine (2012)Google Scholar
  18. 18.
    Loader, R.: The undecidability of λ-definability. In: Logic, Meaning and Computation: Essays in Memory of Alonzo Church, pp. 331–342. Kluwer (2001)Google Scholar
  19. 19.
    Montague, R.: Formal Philosophy: Selected Papers of Richard Montague. Yale University Press, New Haven (1974)Google Scholar
  20. 20.
    Moschovakis, Y.: Sense and denotation as algorithm and value. In: Logic Colloquium 1990: ASL Summer Meeting in Helsinki, vol. 2, p. 210. Springer (1993)Google Scholar
  21. 21.
    Muskens, R.: Lambda Grammars and the Syntax-Semantics Interface. In: Proceedings of the Thirteenth Amsterdam Colloquium, pp. 150–155 (2001)Google Scholar
  22. 22.
    Ong, C.-H.L.: On model-checking trees generated by higher-order recursion schemes. In: LICS, pp. 81–90 (2006)Google Scholar
  23. 23.
    Parys, P.: On the significance of the collapse operation. In: LICS, pp. 521–530 (2012)Google Scholar
  24. 24.
    Salvati, S.: Recognizability in the Simply Typed Lambda-Calculus. In: Ono, H., Kanazawa, M., de Queiroz, R. (eds.) WoLLIC 2009. LNCS, vol. 5514, pp. 48–60. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  25. 25.
    Salvati, S.: On the membership problem for non-linear acgs. Journal of Logic Language and Information 19(2), 163–183 (2010)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Salvati, S., Manzonetto, G., Gehrke, M., Barendregt, H.: Loader and Urzyczyn are logically related. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 364–376. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  27. 27.
    Salvati, S., Walukiewicz, I.: Recursive schemes, Krivine machines, and collapsible pushdown automata. In: Finkel, A., Leroux, J., Potapov, I. (eds.) RP 2012. LNCS, vol. 7550, pp. 6–20. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  28. 28.
    Statman, R.: Completeness, invariance and λ-definability. Journal of Symbolic Logic 47(1), 17–26 (1982)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Terui, K.: Semantic evaluation, intersection types and complexity of simply typed lambda calculus. In: RTA, pp. 323–338 (2012)Google Scholar
  30. 30.
    van Rooij, I.: The tractable cognition thesis. Cognitive Science 32, 939–984 (2008)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Gregory M. Kobele
    • 1
  • Sylvain Salvati
    • 2
  1. 1.University of ChicagoUSA
  2. 2.INRIA, LaBRIUniversité de BordeauxFrance

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