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Lambda calculus characterizations of poly-time

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Typed Lambda Calculi and Applications (TLCA 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 664))

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Abstract

We consider typed λ-calculi with pairing over the algebra W of words over {0,1}, with a destructor and discriminator function. We show that the poly-time functions are precisely the functions (1) λ-representable using simple types, with abstract input (represented by Church-like terms) and concrete output (represented by algebra terms); (2) λ-representable using simple types, with abstract input and output, but with the input and output representations differing slightly; (3) λ-representable using polymorphic typing with type quantification ranging over multiplicative types only; (4) λ-representable using simple and list types (akin to ML style) with abstract input and output; and (5) λ-representable over the algebra of flat lists (in place of W), using simple types, with abstract input and output.

This preliminary report contains only a few selected proofs. The full paper is to appear in Fundamenta Informaticae.

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References

  1. Hendrick P. Barendregt, The Lambda-Calculus: Its Syntax and Semantics, North-Holland, 1980.

    Google Scholar 

  2. Corrado Böhm and Allessandro Berarducci, Automatic synthesis of typed λ-programs on term algebras, Theoretical Computer Science 39 (1985) 135–154.

    Google Scholar 

  3. Stephen Bellantoni and Stephen Cook, A new recursion-theoretic characterization of the poly-time functions, to appear in Computational Complexity 1992.

    Google Scholar 

  4. Stephen Bloch, Functional characterizations of uniform log-depth and polylog-depth circuit families, to appear in the Proceedings of the 1992 IEEE Conference on Structure in Complexity.

    Google Scholar 

  5. Samuel Buss, Bounded Arithmetic, Bibliopolis, Naples, 1986.

    Google Scholar 

  6. A. Cobham, The intrinsic computational difficulty of functions, in Y. Bar-Hillel (ed.), Proceedings of the International Conference on Logic, Methodology, and Philosophy of Science, North-Holland, Amsterdam (1962) 24–30.

    Google Scholar 

  7. Steven Fortune, Daniel Leivant, and Michael O'Donnell, The expressiveness of simple and second order type structures, Journal of the ACM 30 (1983), pp 151–185.

    Google Scholar 

  8. Steven Fortune, Topics in Computational Complexity, PhD Dissertation, Cornell University, Ithaca, NY, 1979.

    Google Scholar 

  9. Jean-Yves Girard, Interprétation fonctionelle et élimination des coupures dans l'arithmétique d'ordre superieur, Thèse de Doctorat d'Etat, 1972, Paris.

    Google Scholar 

  10. A. Grzegoczyk, Some classes of recursive functions, Rozprawy Mate. IV, Warsaw, 1953.

    Google Scholar 

  11. Yuri Gurevich and Saharon Shelah, Fixed-point extensions of first-order logic, Annals of Pure and Applied Logic 32 (1986) 265–280.

    Google Scholar 

  12. Yuri Gurevich, Algebras of feasible functions, Twenty Fourth Symposium on Foundations of Computer Science, IEEE Computer Society Press, 1983, 210–214.

    Google Scholar 

  13. Neil Immerman, Relational queries computable in polynomial time, Information and Control 68 (1986) 86–104. Preliminary report in Fourteenth ACM Symposium on Theory of Computing (1982) 147–152.

    Google Scholar 

  14. Neil Immerman, Languages which capture complexity classes, SIAM Journal of Computing 16 (1987) 760–778.

    Google Scholar 

  15. Daniel Leivant, Subrecursion and lambda representation over free algebras (Preliminary Summary), in Samuel Buss & Philip Scott (eds.), Feasible Mathematics, Perspectives in Computer Science, Birkhauser-Boston, New York (1990) 281–291.

    Google Scholar 

  16. Daniel Leivant, Inductive definitions over finite structures, Information and Computation 89 (1990) 95–108.

    Google Scholar 

  17. Daniel Leivant, A foundational delineation of computational feasiblity, in Proceedings of the Sixth IEEE Conference on Logic in Computer Science (Amsterdam), IEEE Computer Society Press, Washington, D.C., 1991.

    Google Scholar 

  18. Daniel Leivant, Finitely stratified polymorphism, Information and Computation, 1991.

    Google Scholar 

  19. Daniel Leivant, A foundational delineation of poly-time, Information and Computation, 1993.

    Google Scholar 

  20. Daniel Leivant, Stratified functional programs and computational complexity, Conference Record of the Twentieth Annual ACM Symposium on Principles of Programming Languages, 1993.

    Google Scholar 

  21. Harry Mairson, A simple proof of a theorem of Statman, to appear in Theoretical Computer Science, 1992.

    Google Scholar 

  22. Michael O'Donnell, A programming language theorem which is independent of Peano Arithemtic, Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, ACM, New York, 1979, 176–188.

    Google Scholar 

  23. Christos Papadimitriou, A note on the expressive power of PROLOG, Bull. EATCS 26 (June 1985) 21–23.

    Google Scholar 

  24. Rósza Péter, Rekursive Funktionen, Akadémiai Kiadó, Budapest, 1966. English translation: Recursive Functions, Academic Press, New York, 1967.

    Google Scholar 

  25. Dag Prawitz, Natural Deduction, Almqvist and Wiskel, Uppsala, 1965.

    Google Scholar 

  26. John Reynolds, Towards a theory of type structures, in J. Loeckx (ed.), Conference on Porgramming, Springer-Verlag (LNCS #19), Berlin, 1974, pp. 408–425.

    Google Scholar 

  27. Vladimir Sazonov, Polynomial computability and recursivity in finite domains, Electronische Informationsverarbeitung und Kybernetik 7 (1980) 319–323.

    Google Scholar 

  28. Helmut Schwichtenberg, Definierbare Funktionen im Lambda-Kalkul mit Typen, Archiv f. Logik u. Grundlagenforsch. 17 (1976) 113–114.

    Google Scholar 

  29. Richard Statman, The typed λ-calculus is not elementary recursive, Theoretical Computer Science 9 (1979) 73–81.

    Google Scholar 

  30. Moshe Vardi, Complexity and relational query languages, Fourteenth ACM Symposium on Theory of Computing (1982) 137–146.

    Google Scholar 

  31. Marek Zaionc, Word operations definable in typed λ-calculus, Theoretical Computer Science 52 (1987).

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Indiana University, 47405, Bloomington, IN

    Daniel Leivant & Jean-Yves Marion

Authors
  1. Daniel Leivant
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  2. Jean-Yves Marion
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Marc Bezem Jan Friso Groote

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© 1993 Springer-Verlag Berlin Heidelberg

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Leivant, D., Marion, JY. (1993). Lambda calculus characterizations of poly-time. In: Bezem, M., Groote, J.F. (eds) Typed Lambda Calculi and Applications. TLCA 1993. Lecture Notes in Computer Science, vol 664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0037112

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  • DOI: https://doi.org/10.1007/BFb0037112

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56517-8

  • Online ISBN: 978-3-540-47586-6

  • eBook Packages: Springer Book Archive

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