The transfinite action of 1 tape turing machines

Abstract

We produce a classification of the pointclasses of sets of reals produced by infinite time turing machines with 1-tape. The reason for choosing this formalism is that it apparently yields a smoother classification of classes defined by algorithms that halt at limit ordinals. We consider some relations of such classes with other similar notions, such as arithmetical quasi-inductive definitions. It is noted that the action of ω many steps of such a machine can correspond to the double jump operator (in the usual Turing sense): a→a″. The ordinals beginning gaps in the "clockable" ordinals are admissible ordinals, and the length of such gaps corresponds to the degree of reflection those ordinals enjoy. © Springer-Verlag Berlin Heidelberg 2005.