Degrees of Total Algorithms versus Degrees of Honest Functions

  • Lars Kristiansen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)


We prove a few theorems elucidating the relationship between to different approaches to subrecursive degree theory. One approach has its roots in the theory of algorithms and Turing degrees. The other approach has its roots in subrecursive hierarchies of fast-growing functions.


Reducibility Relation Computable Function Total Function Jump Operator Proof Method 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blankertz, B., Weiermann, A.: How to Characterize Provably Total Functions. In: Hajek (ed.) Gödel 1996. LNL, vol. 6, pp. 205–213. Springer, Heidelberg (1996)Google Scholar
  2. 2.
    Buchholz, W., Cichon, A., Weiermann, A.: A Uniform Approach to Fundamental Sequences and Hierarchies. Mathematical Logic Quarterly 40, 273–286 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cai, M.: Degrees of Relative Provability. Accepted for publication in the Notre Dame Journal of Formal Logic (manuscript)Google Scholar
  4. 4.
    Cai, M.: Elements of Classical Recursion Theory: Degree-Theoretic Properties and Combinatorial Properties. PhD Thesis, Department of Mathematics, Cornell University (2011)Google Scholar
  5. 5.
    Kaye, R.: Models of Peano Arithmetic. Clarendon Press, Oxford (1991)zbMATHGoogle Scholar
  6. 6.
    Kristiansen, L.: Information Content and Computational Complexity of Recursive Sets. In: Hajek (ed.) Gödel 1996. LNL, vol. 6, pp. 235–246. Springer, Heidelberg (1996)Google Scholar
  7. 7.
    Kristiansen, L.: A Jump Operator on Honest Subrecursive Degrees. Archive for Mathematical Logic 37, 105–125 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Kristiansen, L.: Subrecursive Degrees and Fragments of Peano Arithmetic. Archive for Mathematical Logic 40, 365–397 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kristiansen, L.: Papers on Subrecursion Theory. Dr Scient Thesis, Department of Informatics, University of Oslo (1996) ISBN 82-7368-130-0Google Scholar
  10. 10.
    Kristiansen, L.: \(\mbox{Low}_n\), \(\mbox{High}_n\), and Intermediate Subrecursive Degrees. In: Calude, Dinneen (eds.) Combinatorics, Computation and Logic, pp. 286–300. Springer, Singapore (1999)Google Scholar
  11. 11.
    Kristiansen, L., Lubarsky, R.S., Weiermann, A., Schlage-Puchta, J.-C.: On the Structure of Honest Elementary Degrees. In: Friedman, Koerwien, Müller (eds.) Accepted for Publication in the Proceedings of the Infinity ProjectGoogle Scholar
  12. 12.
    Kristiansen, L., Weiermann, A., Schlage-Puchta, J.-C.: Streamlined Subrecursive Degree Theory. Annals of Pure and Applied Logic 163, 698–716 (2012)zbMATHCrossRefGoogle Scholar
  13. 13.
    Lindström, P.: Aspects of Incompleteness. LNL, vol. 10. Springer, Berlin (1997)zbMATHGoogle Scholar
  14. 14.
    Machtey, M.: Augmented Loop Languages and Classes of Computable Functions. Journal of Computer and System Sciences 6, 603–624 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Machtey, M.: The Honest Subrecursive Classes are a Lattice. Information and Control 24, 247–263 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Machtey, M.: On the Density of Honest Subrecursive Classes. Journal of Computer and System Sciences 10, 183–199 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Meyer, A.R., Ritchie, D.M.: A Classification of the Recursive Functions. Zeitschr. f. Math. Logik und Grundlagen d. Math. Bd. 18, 71–82 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Odifreddi, P.: Classical Recursion Theory. North-Holland (1989)Google Scholar
  19. 19.
    Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw Hill (1967)Google Scholar
  20. 20.
    Rose, H.E.: Subrecursion. Functions and Hierarchies. Clarendon Press, Oxford (1984)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Lars Kristiansen
    • 1
  1. 1.Department of MathematicsOsloNorway

Personalised recommendations