Admissibles in Gaps

  • Merlin CarlEmail author
  • Bruno DurandEmail author
  • Grégory LafitteEmail author
  • Sabrina Ouazzani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)


We consider clockable ordinals for Infinite Time Turing Machines (ITTMs), i.e., halting times of ITTMs on the empty input. It is well-known that, in contrast to the writable ordinals, the set of clockable ordinals has ‘gaps’. In this paper, we show several results on gaps, mainly related to the admissible ordinals they may properly contain. We prove that any writable ordinal can occur as the order type of the sequence of admissible ordinals in such a gap. We give precise information on their ending points. We also investigate higher rank ordinals (recursively inaccessible, etc.). Moreover, we show that those gaps can have any reasonably effective length (in the sense of ITTMs) compared to their starting point.


Infinite Time Turing Machines (ITTMs) Admissible Ordinals Clockable Ordinals Ordinate Writer Empty Input 
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.


  1. 1.
    Barwise, J.: Admissible Sets and Structures: An Approach to Definability Theory. Perspectives in Mathematical Logic, vol. 7. Springer, Heidelberg (1975)CrossRefzbMATHGoogle Scholar
  2. 2.
    Durand, B., Lafitte, G.: A constructive swiss knife for infinite time turing machines (2016)Google Scholar
  3. 3.
    Hamkins, J.D., Lewis, A.: Infinite time turing machines. J. Symbolic Log. 65(2), 567–604 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Welch, P.D.: Eventually infinite time turing degrees: Infinite time decidable reals. J. Symbolic Log. 65(3), 1193–1203 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Welch, P.D.: The length of infinite time turing machine computations. Bull. London Math. Soc. 32(2), 129–136 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Welch, P.D.: The transfinite action of 1 tape turing machines. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 532–539. Springer, Heidelberg (2005). doi: 10.1007/11494645_65 CrossRefGoogle Scholar
  7. 7.
    Welch, P.D.: Characteristics of discrete transfinite time turing machine models: Halting times, stabilization times, and normal form theorems. Theoret. Comput. Sci. 410, 426–442 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Fachbereich Mathematik und StatistikUniversität KonstanzKonstanzGermany
  2. 2.Lehrstuhl für Theoretische InformatikUniversität PassauPassauGermany
  3. 3.LIRMM, CNRS, Université de MontpellierMontpellierFrance
  4. 4.LACL, Université Paris-EstCréteilFrance

Personalised recommendations