Abstract
We present two complete systems for subtyping polymorphic types. One system is in the style of natural deduction, while another is a Gentzen style sequent calculus system. We prove several metamathematical properties for these systems including cut elimination, subject reduction, coherence, and decidability of type reconstruction. Following the approach by J.Mitchell, the sequents are given a simple semantics using logical relations over applicative structures. The systems are complete with respect to this semantics. The logic which emerges from this paper can be seen as a successor to the original Hilbert style system proposed by J. Mitchell in 1988, and to the “half way” sequent calculus of G. Longo, K. Milsted and S. Soloviev proposed in 1995.
This work is partly supported by NSF Grants CCR-9417382, CCR-9304144, and by Polish KBN Grant 2 P301 031 06.
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E.S. Bainbridge, P.J. Freyd, A. Scedrov, and P.J. Scott. Functorial polymorphism. Theoretical Computer Science, 70:35–64, 1989. Corrigendum ibid., 71:431, 1990. Preliminary report in Logical Foundations of Functional Programming, ed. G. Huet, Addison-Wesley (1990) 315–327.
K. Bruce and G. Longo. A modest model of records, inheritance and bounded quantification. Information and Computation, 87(1/2):196–240, 1990.
K. Bruce and J.C. Mitchell. PER models of subtyping, recursive types and higher-order polymorphism. In Proc. 19th ACM Symp. on Principles of Programming Languages, pages 316–327, January 1992.
P.-L. Curien and G. Ghelli. Coherence of subsumption, minimum typing and type-checking in F≤. Math. Struct. in Comp. Sci., 2:55–91, 1992.
L. Cardelli and G. Longo. A semantic basis for Quest. Technical Report 55, DEC Systems Research Center, 1990. To appear in J. Functional Programming.
L. Cardelli, S. Martini, J. Mitchell, and A. Scedrov. An extension of System F with subtyping. Information and Computation, 109:4–56, 1994. Preliminary version appeared Proc. Theor. Aspects of Computer Software, Springer LNCS 526, September 1991, pages 750–770.
L. Cardelli and P. Wegner. On understanding types, data abstraction, and polymorphism. Computing Surveys, 17(4):471–522, 1985.
D.M. Gabbay. On 2nd order intuitionistic propositional calculus with full comprehension. Arch. Math. Logik, 16:177–186, 1974.
J.-Y. Girard. Une extension de l'interpretation de Gödel à l'analyse, et son application à l'élimination des coupures dans l'analyse et la théorie des types. In J.E. Fenstad, editor, 2nd Scandinavian Logic Symposium, pages 63–92. North-Holland, Amsterdam, 1971.
J.-Y. Girard. Interpretation fonctionelle et elimination des coupures de l'arithmetique d'ordre superieur. These D'Etat, Universite Paris VII, 1972.
J.-Y. Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201–217, 1993.
G. Longo, K. Milsted, and S. Soloviev. A logic of subtyping. In Proc. IEEE Symp. on Logic in Computer Science, pages 292–299, 1995.
M.H. Löb. Embedding first order predicate logic in fragments of intuitionistic logic. Journal of Symbolic Logic, 41:705–718, 1976.
J.C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76(2/3):211–249, 1988. Reprinted in Logical Foundations of Functional Programming, ed. G. Huet, Addison-Wesley (1990) 153–194.
J.C. Reynolds. Towards a theory of type structure. In Paris Colloq. on Programming, pages 408–425, Berlin, 1974. Springer-Verlag LNCS 19.
J. Tiuryn. Equational axiomatization of bicoercibility for polymorphic types. In Ed. P.S. Thiagarajan, editor, Proc. 15th Conference Foundations of Software Technology and Theoretical Computer Science, volume 1026 of Lecture Notes in Computer Science, pages 166–179. Springer Verlag, 1995.
J. Tiuryn and P. Urzyczyn. The subtyping problem for second-order types is undecidable. In Proc. LICS 96, to appear, 1996.
A.S. Troelstra and D. van Dalen. Constructivism in Mathematics. An Introduction. Volume 1. North-Holland, 1988. ISBN-0-444-705066.
J. Wells. Typability and type checking in the second-order λ-calculus are equivalent and undecidable. In Proc. 9th Ann. IEEE Symp. Logic in Comput. Sci., pages 176–185, 1994.
J. Wells. The undecidability of Mitchell's subtyping relation. Technical report, Computer Sci. Dept., Boston University, December 1995.
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Tiuryn, J. (1996). A sequent calculus for subtyping polymorphic types. In: Penczek, W., Szałas, A. (eds) Mathematical Foundations of Computer Science 1996. MFCS 1996. Lecture Notes in Computer Science, vol 1113. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61550-4_144
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DOI: https://doi.org/10.1007/3-540-61550-4_144
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