Ordinal computability

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 ε ω 2 is computable by an Infinite Time Register Machine iff it is Turing computable from some finitely iterated hyperjump 0 (n). © 2009 Springer Berlin Heidelberg.