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Structures definable in polymorphism

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Abstract

Encodings in polymorphism with finite product types are considered. These encodings are given in terms ofI-algebras. They have the property that the ground terms are precisely the closed normal terms of the encoded types. The proof of a well-known result is transplanted to the setting and it is shown why weak recursion is admissible. The paper also shows how to carry out the dual encodings using the existential quantifier.

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References

  1. Barendregt H. The Lambda Calculus—Its Syntax and Semantics. volume 103 of Studies in Logics. North-Holland, 1984.

  2. Böhm C, Berarducci A. Automatic synthesis of typed λ-program on term algebras.Theoretical Computer Science, 1985, 39: 135–154.

    Article  MATH  MathSciNet  Google Scholar 

  3. Cartmell J. Generalized algebraic theories and contextual categories.Annals of Pure and Applied Logic, 1986, 32: 209–243.

    Article  MATH  MathSciNet  Google Scholar 

  4. Coquand T, Huet G. Constructions: A higher order proof system for mechanizing mathematics. InEUROCAL85, LNCS203, 1985, pp. 151–184, Springer-Verlag.

  5. Di Cosmo R. Isomorphisms of Types.PhD thesis, Dipartimento di Informatica, Università di Pisa, 1993.

  6. Curien P-L, Di Cosmo R. A confluent reduction system for the λ-calculus with subjective pairing and terminal object. InICALP'91, Leach, Monien, and Artalejo (eds.), pp.291–302, Springer-Verlag.

  7. Dybjer P. Inductively defined sets in Martin-Löf's set theory. InProceedings of the Workshop on General Logic, Edinburgh, 1988. ECS-LFCS-88-52.

  8. Freyd P. POLYNAT in PER. InCategories in Computer Science and Logic, Gray J and Scedrov A (eds.), June 1989, pp.67–68, Boulder, A.M.S.

    Google Scholar 

  9. Fu Y. Some semantic issues in type theory. PhD thesis, Department of Computer Science, University of Manchester, May 1992.

  10. Fu Y. Understanding inductive types in constructions.Technical Report UMCS-93-6-5, Department of Computer Science, University of Manchester, June 1993.

  11. Fu Y. Recursive models of general inductive types.Fundamenta Informatiae, 1996, 26: 115–131.

    MATH  Google Scholar 

  12. Fu Y. Categorical properties of logical frameworks.Mathematical Structures in Computer Science, 1997, 7: 1–47.

    Article  MATH  Google Scholar 

  13. Fu Y. Constructive sets in computable sets.Journal of Computer Science and Technology, 1997, 12: 425–440.

    Article  MATH  MathSciNet  Google Scholar 

  14. Girard J, Lafont Y, Taylor P. Proofs and Types. Volume 7 ofCambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

  15. Goguen H, Luo Z. Inductive data types: Well-ordering types revisited.Technical report, LFCS, University of Edinburgh, April 1992.

  16. Hagino T. A Categorical programming language. PhD thesis, LFCS, University of Edinburgh, 1987.

  17. Hagino T. A typed lambda calculus with categorical type constructions. InCategory Theory and Computer Science, Pitt D, Poigné A, Rydeheard D (eds.), LNCS 283, 1987, pp. 140–157, Springer-Verlag.

  18. Hyland J, Pitts A. The theory of constructions: Categorical semantics and topos-theoretic models. InCategories in Computer Science and Logic, Gray J and Scedrov A (eds.), June 1989, pp.137–199, Boulder, A.M.S.

    Google Scholar 

  19. Hyland M, Robinson E, Rosolini G. Algebraic types in per models. InMathematical Foundations of Programming Semantics, Main Met al. (eds.), LNS 442, 1990, pp.333–350, Springer-Verlag.

  20. Mendler N. Inductive definition in type theory. PhD thesis, Cornell University, 1987.

  21. Mendler N. Inductive types and type constraints in the second-order lambda calculus.Annals of Pure and Applied Logic, 1991, 51: 159–172.

    Article  MATH  MathSciNet  Google Scholar 

  22. Mendler N. Predicative type universes and primitive recursion. InProceedings of the Sixth Symposium on Logic in Computer Science, IEEE Computer Science Press, July 1991, pp.173–184.

  23. Mitchell J, Plotkin G. Abstract types have existential type. InACM TOPLAS, 1988, pp.470–502.

  24. Ore C. The extended calculus of constructions (ECC) with inductive types.Information and Computation, August 1992, 99: 231–264.

    Article  MATH  MathSciNet  Google Scholar 

  25. Pfenning F, Paulin-Mohring C. Inductively defined types in the calculus of constructions. InMathematical Foundations of Programming Semantics, Main Met al. (eds.), LNCS442, 1991, pp.209–228. Springer-Verlag.

  26. Phoa W. Two results on set-theoretic polymorphism. InCategory Theory and Computer Science, Pitt D, Curien Pet al. (eds.), LNCS 530, September 1991, pp. 219–235, Paris, Springer-Verlag.

    Chapter  Google Scholar 

  27. Prawitz D. Natural Deduction. Almqvist and Wiksell, 1965.

  28. Wraith G. A note on catetorical datatypes. InCategory Theory and Computer Science, Pitt D, Rydeheard Det al. (eds.), LNCS 389, September 1989, pp.213–223, Manchester, Springer-Verlag.

    Google Scholar 

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Supported by NNSFC, grant number 69503006.

For the biography ofFu Yuxi please refer to P.208, No.3, Vol.13 of this Journal.

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Fu, Y. Structures definable in polymorphism. J. of Comput. Sci. & Technol. 13, 579–587 (1998). https://doi.org/10.1007/BF02946501

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  • DOI: https://doi.org/10.1007/BF02946501

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