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Logical Relations and Inductive/Coinductive Types

  • Thorsten Altenkirch
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1584)

Abstract

We investigate a λ calculus with positive inductive and coinductive types , which we call λ μ,ν , using logical relations. We show that parametric theories have the strong categorical properties, that the representable functors and natural transformations have the expected properties. Finally we apply the theory to show that terms of functorial type are almost canonical and that monotone inductive definitions can be reduced to positive in some cases.

Keywords

Natural Transformation Parametric Theory Logical Relation Extensional Theory Combinatory Logic 
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.

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References

  1. AP93.
    Abadi, M., Plotkin, G.: A logic for parametric polymorphism. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 361–375. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  2. Buc81.
    Buchholz, W.: The ω μ + 1-rule. In: Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Lecture Notes in Mathematics, vol. 897, pp. 188–233 (1981)Google Scholar
  3. Geu92.
    Geuvers, H.: Inductive and coinductive types with iteration and recursion. In: Workshop on Types for Proofs and Programs, Båstad, pp. 193–217 (1992)Google Scholar
  4. Has90.
    Hasegawa, R.: Categorical datatypes in parametric polymorphism. In: Fourth Asian Logic Conference (1990)Google Scholar
  5. Loa97.
    Loader, R.: Equational theories for inductive types. Annals of Pure and Applied Logic 84, 175–217 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  6. Mat98.
    Matthes, R.: Extensions of System F by Iteration And Primitive Recursion on Monotone Inductive Types. PhD thesis, University of Munich (1998) (to appear)Google Scholar
  7. Mog.
    Moggi, E.: Email to the Type mailing listGoogle Scholar
  8. MR91.
    Ma, Q., Reynolds, J.C.: Types, abstraction and parametric polymorphis, part 2. In: Proceedings of the 1991 Mathematics of Programming Semantics Conference (1991)Google Scholar
  9. Rey83.
    Reynolds, J.C.: Types, abstraction and parametric polymorphism. In: Mason, R.E.A. (ed.) Information Processing, vol. 83 (1983)Google Scholar
  10. RR94.
    Robinson, E.P., Rosolini, G.: Reflexive graphs and parametric polymorphism. In: Abramsky, S. (ed.) Proc. 9th Symposium in Logic in Computer Science, Paris, pp. 364–371. IEEE Computer Society, Los Alamitos (1994)CrossRefGoogle Scholar
  11. Wad89.
    Wadler, P.: Theorems for free! Revised version of a paper appearing. In: 4 th International Symposium on Functional Programming Languages and Computer Architecture (June 1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Thorsten Altenkirch
    • 1
  1. 1.Ludwig-Maximilians-UniversitätMünchenGermany

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