Abstract
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).
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This work was partly supported by NSF grant CCR-9417382, KBN Grant 8 T11C 034 10 and by ESPRIT BRA7232 “Gentzen”.
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References
Arts, T., Embedding first order predicate logic in second order prepositional logic, Master Thesis, Katholieke Universiteit Nijmegen, 1992.
Arts, T., Dekkers, W., Embedding first order predicate logic in second order propositional logic, Technical report 93-02, Katholieke Universiteit Nijmegen, 1993.
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.
Gabbay, D.M., On 2nd order intuitionistic propositional calculus with full comprehension, Arch. math. Logik, 16 (1974), 177–186.
Gabbay, D.M., Semantical Investigations in Heyting's Intuitionistic Logic, D. Reidel Publ. Co., 1981.
Girard, J.-Y., Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7, Cambridge University Press, Cambridge, 1989.
Leivant, D., Polymorphic type inference, Proc. 10-th ACM Symposium on Principles of Programming Languages, ACM, 1983.
Löb, M. H., Embedding first order predicate logic in fragments of intuitionistic logic, Journal Symb. Logic, 41, No. 4 (1976), 705–718.
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.
Rasiowa, H., Sikorski, R., The Mathematics of Metamathematics, PWN, Warsaw, 1963.
Statman, R., Intuitionistic propositional logic is polynomial-space complete, Theoretical Computer Science 9, 67–72 (1979)
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.
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.)
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.)
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Urzyczyn, P. (1997). Inhabitation in typed lambda-calculi (a syntactic approach). In: de Groote, P., Roger Hindley, J. (eds) Typed Lambda Calculi and Applications. TLCA 1997. Lecture Notes in Computer Science, vol 1210. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62688-3_47
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DOI: https://doi.org/10.1007/3-540-62688-3_47
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