The arithmetical hierarchy over the reals

Abstract

An usual classification of undecidable problems in computability theory is the one given by the arithmetical hierarchy introduced by Kleene. Very recently. L. Blum, M. Shub and S. Smale devised a model of computation able to work with real numbers and showed the main properties of a computability theory for that model. In this paper we introduce the arithmetical hierarchy for it. A syntactical characterization is given, together with some complete problems. Also, it is shown that if nondeterministic machines are considered instead of deterministic ones, a different set of classes is obtained (unlike the classical case) and all these classes are related with the classification of sets of reals done in descriptive set theory. © 1992 Oxford University Press.