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Two different strong normalization proofs?

Computability versus functional of finite type

  • Conference paper
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Book cover Higher-Order Algebra, Logic, and Term Rewriting (HOA 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1074))

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Abstract

A proof of ∀t∃nSN(t, n) (term t performs at most n reduction steps) is given, based on strong computability predicates. Using modified realizability, a bound on reduction lengths is extracted from it. This upper bound is compared with the one Gandy defines, using strictly monotonic functionals. This reveals a remarkable connection between his proof and Tait's. We show the details for simply typed λ-calculus and Gödel's T. For the latter system, program extraction yields considerably sharper upper bounds.

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References

  1. U. Berger. Program extraction from normalization proofs. In M. Bezem and J.F. Groote ed., Proc. of TLCA '93, Utrecht, volume 664 of LNCS, pages 91–106. Springer Verlag, 1993.

    Google Scholar 

  2. R.O. Gandy. Proofs of strong normalization. In J.R. Hindley and J.P. Seidin ed., To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 457–477. Academic Press, London, 1980.

    Google Scholar 

  3. J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types, volume 7 of Cambridge tracts in theoretical computer science. Cambridge University Press, 1989.

    Google Scholar 

  4. G. Kreisel. Interpretation of analysis by means of constructive functionals of finite types. In A. Heyting ed., Constructivity in Mathematics, pages 101–128. North-Holland, 1959.

    Google Scholar 

  5. R. Loader. Normalisation by translation. http://sable.ox.ac.uk/ loader/, 1995.

    Google Scholar 

  6. J.C. van de Pol. Termination proofs for higher-order rewrite systems. In J. Heering et al ed., Proc. of HOA '93, volume 816 of LNCS, pages 305–325. Springer Verlag, 1994.

    Google Scholar 

  7. J.C. van de Pol and H. Schwichtenberg. Strict functionals for termination proofs. In M. Dezani-Ciancaglini and G. Plotkin ed., Proc. of TLCA'95, volume 902 of LNCS, pages 350–364. Springer Verlag, 1995.

    Google Scholar 

  8. D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad ed., Proc. of the 2nd Scandinavian Logic Symposium, pages 235–307, Amsterdam, 1971. North-Holland.

    Google Scholar 

  9. W.W. Tait. Intensional interpretation of functionals of finite types I. JSL, 32:198–212, 1967.

    Google Scholar 

  10. A.S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Number 344 in LNM. Springer Verlag, Berlin, 1973. A 2nd corrected edition appeared as ILLC X-93-05, University of Amsterdam.

    Google Scholar 

  11. R. de Vrijer. Exactly estimating functionals and strong normalization. Proc. of the Koninklijke Nederlandse Akademie van Wetenschappen, 90(4):479–493, Dec 1987.

    Google Scholar 

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Authors and Affiliations

  1. Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584, CS Utrecht, The Netherlands

    Jaco van de Pol

Authors
  1. Jaco van de Pol
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Editor information

Gilles Dowek Jan Heering Karl Meinke Bernhard Möller

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© 1996 Springer-Verlag Berlin Heidelberg

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van de Pol, J. (1996). Two different strong normalization proofs?. In: Dowek, G., Heering, J., Meinke, K., Möller, B. (eds) Higher-Order Algebra, Logic, and Term Rewriting. HOA 1995. Lecture Notes in Computer Science, vol 1074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61254-8_27

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  • DOI: https://doi.org/10.1007/3-540-61254-8_27

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61254-4

  • Online ISBN: 978-3-540-68389-6

  • eBook Packages: Springer Book Archive

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