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The emptiness problem for intersection types

Published online by Cambridge University Press:  12 March 2014

Paweł Urzyczyn
Paweł Urzyczyn*
Affiliation:
Institute of Informatics, University of Warsaw, Ul. Banacha 2, 02-097 Warszawa, Poland E-mail: urzy@mimuw.edu.pl
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Abstract

We study the intersection type assignment system as defined by Barendregt, Coppo and Dezani. For the four essential variants of the system (with and without a universal type and with and without subtyping) we show that the emptiness (inhabitation) problem is recursively unsolvable. That is, there is no effective algorithm to decide if there is a closed term of a given type. It follows that provability in the logic of “strong conjunction” of Mints and Lopez-Escobar is also undecidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1195 - 1215
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Alessi, F. and Barbanera, F., Strong conjunction and intersection types, Proceedings of the MFCS 1991 (Tarlecki, A., editor), LNCS 520, Springer, Berlin, 1991, pp. 64–73.Google Scholar
[2] Barendregt, H., Coppo, M., and Dezani-Ciancaglini, M., A filter lambda model and the completeness of type assignment, this Journal, vol. 48 (1983), pp. 931–940.Google Scholar
[3] Cardone, F. and Coppo, M., Two extensions of Curry's type inference system, Logic and Computer Science (Odifreddi, P., editor), Academic Press, 1990, pp. 19–75.Google Scholar
[4] Coppo, M. and Dezani-Ciancaglini, M., An extension of basic functionality theory for lambda-calculus, Notre Dame Journal of Formal Logic, vol. 21 (1980), pp. 685–693.Google Scholar
[5] Coppo, M., Dezani-Ciancaglini, M., and Venneri, B., Functional character of solvable terms, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 27 (1981), pp. 45–58.Google Scholar
[6] Dezani-Ciancaglini, M., Ghilezan, S., and Venneri, B., The “relevance” of intersection and union types, Notre Dame Journal of Formal Logic (1997), to appear.Google Scholar
[7] Dowek, G., The undecidability of type assignment in the λII-calculus, Proceedings of Typed Lambda Calculi and Applications (Bezem, M. and Groote, J.F., editors), LNCS 664, Springer-Verlag, Berlin, 1993, pp. 139–145.CrossRefGoogle Scholar
[8] Gabbay, D.M., Semantical Investigations in Heyting's Intuitionistic Logic, D. Reidel Publ. Co., 1981.CrossRefGoogle Scholar
[9] Girard, J.-Y., Lafont, Y., and Taylor, P., Proofs and Types, Cambridge University Press, 1989.Google Scholar
[10] Hindley, J.R., The principal type-scheme of an object in combinatory logic, Transactions of the American Mathematical Society, vol. 146 (1969), pp. 29–60.Google Scholar
[11] Howard, W.A., The formulae-as-types notion of construction, in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism (Seldin, J.P. and Hindley, J.R., editors), Academic Press, 1980, pp. 479–490.Google Scholar
[12] Kfoury, A.J. and Tiuryn, J., Type reconstruction in finite-rank fragments of the second-order λ-calculus, Information and Computation, vol. 98 (1989), no. 2, pp. 228–257.Google Scholar
[13] Kfoury, A.J. and Wells, J.B., A direct algorithm for type inference in the rank-2 fragment of the second-order λ-calculus, Proceedings of LISP and Functional Programming ACM (1994), pp. 196–207.Google Scholar
[14] Kurata, T. and Takahashi, M., Decidable properties of intersection type systems, Proceedings of Typed Lambda Calculi and Applications (Dezani-Ciancaglini, M. and Plotkin, G., editors), LNCS 902, Springer-Verlag, Berlin, 1995, pp. 297–311.CrossRefGoogle Scholar
[15] Leivant, D., Polymorphic type inference, Proceedings of the 10th ACM Symposium on Principles of Programming Languages, Austin, Texas, 1983, pp. 88–98.Google Scholar
[16] Löb, M. H., Embedding first order predicate logic in fragments of intuitionistic logic, this Journal, vol. 41 (1976), no. 4, pp. 705–718.Google Scholar
[17] Lopez-Escobar, E.G.K., Proof Functional Connectives, Proceedings of Methods in Mathematical Logic, 1993, LNMath 1130, Springer-Verlag, Berlin, 1985, pp. 208–221.Google Scholar
[18] Milner, R., A theory of type polymorphism in programming, Journal of Computer and System Sciences, vol. 17 (1978), pp. 348–375.Google Scholar
[19] Mints, G.E., The completeness of provable readability, Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 420–441.Google Scholar
[20] Pottinger, G., A type assignment for the strongly normalizable λ-terms, To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism (Seldin, J.P. and Hindley, J.R., editors), Academic Press, London, 1980, pp. 561–577.Google Scholar
[21] Statman, R., Intuitionistic propositional logic is polynomial-space complete, Theoretical Computer Science, vol. 9 (1979), pp. 67–72.Google Scholar
[22] Urzyczyn, P., The emptiness problem for intersection types, Proceedings of the 9th IEEE Symposium on Logic in Computer Science, Paris, France, 1994, pp. 300–309.Google Scholar
[23] Urzyczyn, P., Type inhabitation in typed lambda calculi (a syntactic approach), Proceedings of Typed Lambda Calculi and Applications (de Groote, P. and Hindley, J.R., editors), LNCS 1210, Springer-Verlag, Berlin, 1997, pp. 373–389.Google Scholar
[24] Urzyczyn, P., Type reconstruction in Fomega; , Mathematical Structures in Computer Science, vol. 7 (1997), pp. 329–358, (Preliminary version in Proceedings of TLCA , 1993.).Google Scholar
[25] van Bakel, S., Intersection Type Disciplines in Lambda Calculus and Applicative Term Rewriting Systems, Mathematisch Centrum, Amsterdam, 1993.Google Scholar
[26] Venneri, B., Intersection types as logical formulae, Journal of Logic and Computation, vol. 4 (1994), no. 2, pp. 109–124.Google Scholar
[27] Wells, J.B., Typability and type checking in the second-order λ-calculus calculus are equivalent and undecidable, Proceedings of the 9th IEEE Symposium on Logic in Computer Science, Paris, France, 1994, (To appear in Annals of Pure and Applied Logic. ), pp. 176–185.Google Scholar
[28] Wells, J.B., Dimock, A., Muller, R., and Turbak, F., A typed intermediate language for flow-directed compilation, Proceedings of the 7th International Joint Conference on the Theory and Practice of Software Development, Lille, France, LNCS 1214, Springer-Verlag, Berlin, 1997, pp. 757–771.Google Scholar