Abstract
Ordinal computability uses ordinals instead of natural numbers in abstract machines like register or Turing machines. We give an overview of the computational strengths of α-β-machines, where α and β bound the time axis and the space axis of some machine model. The spectrum ranges from classical Turing computability to ∞-∞-computability which corresponds to Gödel’s model of constructible sets. To illustrate some typical techniques we prove a new result on Infinite Time Register Machines (= ∞-ω-register machines) which were introduced in [6]: a real number x ∈ ω2 is computable by an Infinite Time Register Machine iff it is Turing computable from some finitely iterated hyperjump 0(n).
The author wants to thank Joel Hamkins and Philip Welch for a very inspiring discussion at the EMU 2008 workshop at New York in which the techniques and results bounding the strength of Infinite Time Register Machines were suggested and conjectured.
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Koepke, P. (2009). Ordinal Computability. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_29
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DOI: https://doi.org/10.1007/978-3-642-03073-4_29
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