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