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An Abstract Strong Normalization Theorem

  • Ulrich Berger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3634)

Abstract

We prove a strong normalization theorem for abstract term rewriting systems based on domain-theoretic models. The theorem applies to extensions of Gödel’s system T by various forms of recursion related to bar recursion for which strong normalization was hitherto unknown.

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References

  1. 1.
    Plotkin, G.: LCF considered as a programming language. Theoretical Computer Science 5, 223–255 (1977)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Berger, U.: Continuous semantics for strong normalization. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 23–34. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Berger, U.: Strong normalization for applied lambda calculi. Submitted to: Logical Methods in Computer Science, January 2005 (2005)Google Scholar
  4. 4.
    Spector, C.: Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In: Dekker, F.D.E. (ed.) Recursive Function Theory: Proc. Symposia in Pure Mathematics, vol. 5, pp. 1–27. American Mathematical Society, Providence (1962)Google Scholar
  5. 5.
    Howard, W.A.: Functional interpretation of bar induction by bar recursion. Composito Mathematicae 20, 107–124 (1968)zbMATHGoogle Scholar
  6. 6.
    Berardi, S., Bezem, M., Coquand, T.: On the computational content of the axiom of choice. Journal of Symbolic Logic 63, 600–622 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Berger, U., Oliva, P.: Modified bar recursion and classical dependent choice. In: Logic Colloquium 2001, Springer, Heidelberg (2005)Google Scholar
  8. 8.
    Berger, U.: A computational interpretation of open induction. In: Titsworth, F. (ed.) Proceedings of the Ninetenth Annual IEEE Symposium on Logic in Computer Science, pp. 326–334. IEEE Computer Society, Los Alamitos (2004)CrossRefGoogle Scholar
  9. 9.
    Scott, D.S.: Outline of a mathematical theory of computation. In: 4th Annual Princeton Conference on Information Sciences and Systems, pp. 169–176 (1970)Google Scholar
  10. 10.
    Griffor, E., Lindström, I., Stoltenberg-Hansen, V.: Mathematical theory of domains. Cambridge University Press, Cambridge (1993)Google Scholar
  11. 11.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Clarendon Press, Oxford (1994)Google Scholar
  12. 12.
    Raoult, J.C.: Proving open properties by induction. Information processing letters 29, 19–23 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Coquand, T.: A note on the open induction principle (1997)Google Scholar
  14. 14.
    Mahboubi, A.: An induction principle over real numbers. Submitted to Archive for Mathematical Logic (2004)Google Scholar
  15. 15.
    Nash-Williams, C.: On well-quasi-ordering finite trees. Proc. Cambridge Phil. Soc. 59, 833–835 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Tait, W.: Normal form theorem for barrecursive functions of finite type. In: Fenstad, J. (ed.) Proceedings of the Second Scandinavian Logic Symposium, pp. 353–367. North–Holland, Amsterdam (1971)CrossRefGoogle Scholar
  17. 17.
    Girard, J.Y.: Interprétation functionelle et élimination des coupures de l’arithmétique d’ordre supérieur. PhD thesis, Université Paris VII (1972)Google Scholar
  18. 18.
    Bezem, M.: Strong normalization of barrecursive terms without using infinite terms. Archive for Mathematical Logic 25, 175–181 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ong, L., Ritter, E.: A generic normalisation argument: Application to the calculus of constructions. In: Meinke, K., Börger, E., Gurevich, Y. (eds.) CSL 1993. LNCS, vol. 832, pp. 261–279. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  20. 20.
    Hyland, J., Ong, C.H.: Modified realizability semantics and strong normalization proofs. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 179–194. Springer, Heidelberg (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ulrich Berger
    • 1
  1. 1.University of Wales Swansea 

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