The Lost Melody Phenomenon

Abstract

A typical phenomenon for machine models of transfinite computations is the existence of so-called lost melodies, i.e. real numbers $x$ such that the characteristic function of the set $\{x\}$ is computable while $x$ itself is not (a real having the first property is called recognizable). This was first observed by J. D. Hamkins and A. Lewis for infinite time Turing machine, then demonstrated by P. Koepke and the author for $ITRM$s. We prove that, for unresetting infinite time register machines introduced by P. Koepke, recognizability equals computability, i.e. the lost melody phenomenon does not occur. Then, we give an overview on our results on the behaviour of recognizable reals for $ITRM$s. We show that there are no lost melodies for ordinal Turing machines or ordinal register machines without parameters and that this is, under the assumption that $0^{\sharp}$ exists, independent of $ZFC$. Then, we introduce the notions of resetting and unresetting $\alpha$-register machines and give some information on the question for which of these machines there are lost melodies