Inhabitation in typed lambda-calculi (a syntactic approach)

  • Pawel Urzyczyn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1210)


A type is inhabited (non-empty) in a typed calculus iff there is a closed term of this type. The inhabitation (emptiness) problem is to determine if a given type is inhabited. This paper provides direct, purely syntactic proofs of the following results: the inhabitation problem is PSPACE-complete for simply typed lambda-calculus and undecidable for the polymorphic second-order and higher-order lambda calculi (systems F and Fω).


Type Variable Atomic Formula Intuitionistic Logic Propositional Variable Kripke Model 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arts, T., Embedding first order predicate logic in second order prepositional logic, Master Thesis, Katholieke Universiteit Nijmegen, 1992.Google Scholar
  2. 2.
    Arts, T., Dekkers, W., Embedding first order predicate logic in second order propositional logic, Technical report 93-02, Katholieke Universiteit Nijmegen, 1993.Google Scholar
  3. 3.
    Barendregt, H.P., Lambda calculi with types, chapter in: Handbook of Logic in Computer Science (S. Abramsky, D.M. Gabbay, T. Maibaum eds.), Oxford University Press, 1990, pp. 117–309.Google Scholar
  4. 4.
    Gabbay, D.M., On 2nd order intuitionistic propositional calculus with full comprehension, Arch. math. Logik, 16 (1974), 177–186.Google Scholar
  5. 5.
    Gabbay, D.M., Semantical Investigations in Heyting's Intuitionistic Logic, D. Reidel Publ. Co., 1981.Google Scholar
  6. 6.
    Girard, J.-Y., Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7, Cambridge University Press, Cambridge, 1989.Google Scholar
  7. 7.
    Leivant, D., Polymorphic type inference, Proc. 10-th ACM Symposium on Principles of Programming Languages, ACM, 1983.Google Scholar
  8. 8.
    Löb, M. H., Embedding first order predicate logic in fragments of intuitionistic logic, Journal Symb. Logic, 41, No. 4 (1976), 705–718.Google Scholar
  9. 9.
    Prawitz, D., Some results for intuitionistic logic with second order quantification rules, in: Intuitionism and Proof Theory (A. Kino, J. Myhill, R.E. Vesley eds.), North-Holland, Amsterdam, 1970, pp. 259–270.Google Scholar
  10. 10.
    Rasiowa, H., Sikorski, R., The Mathematics of Metamathematics, PWN, Warsaw, 1963.Google Scholar
  11. 11.
    Statman, R., Intuitionistic propositional logic is polynomial-space complete, Theoretical Computer Science 9, 67–72 (1979)Google Scholar
  12. 12.
    Urzyczyn, P., Type reconstruction in Fω, to appear in Mathematical Structures in Computer Science. Preliminary version in Proc. Typed Lambda Calculus and Applications (M. Bezem and J.F. Groote, eds.), LNCS 664, Springer-Verlag, Berlin, 1993, pp. 418–432.Google Scholar
  13. 13.
    Urzyczyn, P., The emptiness problem for intersection types, Proc. 9th IEEE Symposium on Logic in Computer Science, 1994, pp. 300–309. (To appear in Journal of Symbolic Logic.)Google Scholar
  14. 14.
    Wells, J.B., Typability and type checking in the second-order λ-calculus calculus are equivalent and undecidable, Proc. 9th IEEE Symposium on Logic in Computer Science, 1994, pp. 176–185. (To appear in Annals of Pure and Applied Logic.)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Pawel Urzyczyn
    • 1
  1. 1.Institute of InformaticsUniversity of WarsawWarszawaPoland

Personalised recommendations