Sub-computabilities

  • Fabien Givors
  • Gregory Lafitte
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6914)

Abstract

Every recursively enumerable set of integers (r.e. set) is enumerable by a primitive recursive function. But if the enumeration is required to be one-one, only a proper subset of all r.e. sets qualify. Starting from a collection of total recursive functions containing the primitive recursive functions, we thus define a sub-computability as an enumeration of the r.e. sets that are themselves one-one enumerable by total functions of the given collection. Notions similar to the classical computability ones are introduced and variants of the classical theorems are shown. We also introduce sub-reducibilities and study the related completeness notions. One of the striking results is the existence of natural (recursive) sets which play the role of low (non-recursive) solutions to Post’s problem for these sub-reducibilities. The similarity between sub-computabilities and (complete) computability is surprising, since there are so many missing r.e. sets in sub-computabilities. They can be seen as toy models of computability.

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References

  1. 1.
    Friedman, H., Sheard, M.: Elementary descent recursion and proof theory. Annals of Pure and Applied Logic 71(1), 1–45 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Koz’minyh, V.V.: On a presentation of partial recursive functions by compositions. Algebra i Logika 11(3), 270–294 (1972) (in Russian)MathSciNetGoogle Scholar
  3. 3.
    Kristiansen, L.: Papers on subrecursion theory. Ph.D. thesis, Department of Informatics, University of Oslo (1996)Google Scholar
  4. 4.
    Kristiansen, L.: A jump operator on honest subrecursive degrees. Archive for Mathematical Logic 37(2), 105–125 (1998)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kristiansen, L.: Lown, highn, and intermediate subrecursive degrees. In: Calude, D. (ed.) Combinatorics, Computation and Logic, pp. 286–300. Springer, Singapore (1999)Google Scholar
  6. 6.
    Kristiansen, L., Schlage-Puchta, J.C., Weiermann, A.: Streamlined subrecursive degree theory. Annals of Pure and Applied Logic ( to appear, 2011)Google Scholar
  7. 7.
    Nies, A.: Computability and Randomness. Oxford University Press, Oxford (2009)CrossRefMATHGoogle Scholar
  8. 8.
    Odifreddi, P.: Classical Recursion Theory. Elsevier, North-Holland, Amsterdam (1988)MATHGoogle Scholar
  9. 9.
    Odifreddi, P.: Classical Recursion Theory, vol. II. North Holland - Elsevier, Amsterdam (1999)MATHGoogle Scholar
  10. 10.
    Rathjen, M.: The realm of ordinal analysis. In: Cooper, S.B., Truss, J. (eds.) Sets and Proofs, pp. 219–279. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  11. 11.
    Rose, H.E.: Subrecursion: Functions and Hierarchies, Oxford Logic Guides, vol. 9. Oxford University Press, USA (1984)MATHGoogle Scholar
  12. 12.
    Veblen, O.: Continuous increasing functions of finite and transfinite ordinals. Trans. Amer. Math. Soc. 9, 280–292 (1908)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Fabien Givors
    • 1
  • Gregory Lafitte
    • 1
  1. 1.Laboratoire d’Informatique Fondamentale de Marseille (LIF)CNRS – Aix-Marseille UniversitéMarseille Cedex 13France

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