9,378 research outputs found
On the information carried by programs about the objects they compute
In computability theory and computable analysis, finite programs can compute
infinite objects. Presenting a computable object via any program for it,
provides at least as much information as presenting the object itself, written
on an infinite tape. What additional information do programs provide? We
characterize this additional information to be any upper bound on the
Kolmogorov complexity of the object. Hence we identify the exact relationship
between Markov-computability and Type-2-computability. We then use this
relationship to obtain several results characterizing the computational and
topological structure of Markov-semidecidable sets
The "paradox" of computability and a recursive relative version of the Busy Beaver function
In this article, we will show that uncomputability is a relative property not
only of oracle Turing machines, but also of subrecursive classes. We will
define the concept of a Turing submachine, and a recursive relative version for
the Busy Beaver function which we will call Busy Beaver Plus function.
Therefore, we will prove that the computable Busy Beaver Plus function defined
on any Turing submachine is not computable by any program running on this
submachine. We will thereby demonstrate the existence of a "paradox" of
computability a la Skolem: a function is computable when "seen from the
outside" the subsystem, but uncomputable when "seen from within" the same
subsystem. Finally, we will raise the possibility of defining universal
submachines, and a hierarchy of negative Turing degrees.Comment: 10 pages. 0 figures. Supported by the National Council for Scientific
and Technological Development (CNPq), Brazil. Book chapter published in
Information and Complexity, Mark Burgin and Cristian S. Calude (Editors),
World Scientific Publishing, 2016, ISBN 978-981-3109-02-5, available at
http://www.worldscientific.com/worldscibooks/10.1142/10017. arXiv admin note:
substantial text overlap with arXiv:1612.0522
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