353 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
Infinite computations with random oracles
We consider the following problem for various infinite time machines. If a
real is computable relative to large set of oracles such as a set of full
measure or just of positive measure, a comeager set, or a nonmeager Borel set,
is it already computable? We show that the answer is independent from ZFC for
ordinal time machines (OTMs) with and without ordinal parameters and give a
positive answer for most other machines. For instance, we consider, infinite
time Turing machines (ITTMs), unresetting and resetting infinite time register
machines (wITRMs, ITRMs), and \alpha-Turing machines for countable admissible
ordinals \alpha
Towards a Church-Turing-Thesis for Infinitary Computations
We consider the question whether there is an infinitary analogue of the
Church-Turing-thesis. To this end, we argue that there is an intuitive notion
of transfinite computability and build a canonical model, called Idealized
Agent Machines (s) of this which will turn out to be equivalent in
strength to the Ordinal Turing Machines defined by P. Koepke
Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals
We describe the basic theory of infinite time Turing machines and some recent
developments, including the infinite time degree theory, infinite time
complexity theory, and infinite time computable model theory. We focus
particularly on the application of infinite time Turing machines to the
analysis of the hierarchy of equivalence relations on the reals, in analogy
with the theory arising from Borel reducibility. We define a notion of infinite
time reducibility, which lifts much of the Borel theory into the class
in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference,
200
Generalized Effective Reducibility
We introduce two notions of effective reducibility for set-theoretical
statements, based on computability with Ordinal Turing Machines (OTMs), one of
which resembles Turing reducibility while the other is modelled after Weihrauch
reducibility. We give sample applications by showing that certain (algebraic)
constructions are not effective in the OTM-sense and considerung the effective
equivalence of various versions of the axiom of choice
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