Post’s Problem for Ordinal Register Machines

  • Joel D. Hamkins
  • Russell G. Miller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)


We study Post’s Problem for the ordinal register machines defined in [6], showing that its general solution is positive, but that any set of ordinals solving it must be unbounded in the writable ordinals. This mirrors the results in [3] for infinite-time Turing machines, and also provides insight into the different methods required for register machines and Turing machines in infinite time.


computability ordinal computability ordinal register machine Post’s Problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Nat. Acad. Sci (USA) 43, 236–238 (1957)CrossRefMATHGoogle Scholar
  2. 2.
    Hamkins, J.D., Lewis, A.: Infinite-time Turing machines. Journal of Symbolic Logic 65(2), 567–604 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Hamkins, J.D., Lewis, A.: Post’s problem for supertasks has both positive and negative solutions. Arch. Math. Logic 41, 507–523 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Harrington, L., Soare, R.I.: Post’s Program and Incomplete Recursively Enumerable Sets. Proc. Nat. Acad. Sci (USA) 88, 10242–10246 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Koepke, P.: Turing computations on ordinals. Bulletin of Symbolic Logic 11(3), 377–397 (2005)CrossRefMATHGoogle Scholar
  6. 6.
    Koepke, P., Bissell-Siders, R.: Ordinal Register Machines (to appear)Google Scholar
  7. 7.
    Koepke, P.: α-Recursion theory and ordinal computability, electronic manuscriptGoogle Scholar
  8. 8.
    Muchnik, A.A.: On the Unsolvability of the Problem of Reducibility in the Theory of Algorithms. Dokl. Akad. Nauk SSSR, N.S (Russian) 109, 194–197 (1956)MathSciNetMATHGoogle Scholar
  9. 9.
    Sacks, G.E., Simpson, S.G.: The α-finite injury method. Annals of Mathematical Logic 4, 343–367 (1972)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, New York (1987)CrossRefMATHGoogle Scholar
  11. 11.
    Welch, P.: The lengths of infinite time Turing machine computations. Bulletin of the London Mathematical Society 32(2), 129–136 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Joel D. Hamkins
    • 1
  • Russell G. Miller
    • 2
  1. 1.Department of Mathematics, College of Staten Island, and Doctoral Program in Mathematics, The CUNY Graduate CenterUSA
  2. 2.Department of Mathematics, Queens College, and Doctoral Program in Computer Science, The CUNY Graduate CenterUSA

Personalised recommendations