3 research outputs found
The Lost Melody Phenomenon
A typical phenomenon for machine models of transfinite computations is the
existence of so-called lost melodies, i.e. real numbers such that the
characteristic function of the set is computable while 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 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 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 exists, independent of
. Then, we introduce the notions of resetting and unresetting
-register machines and give some information on the question for which
of these machines there are lost melodies