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We describe the motivation for the construction of a general framework for priority arguments, the ideas incorporated into the construction of the framework, and the use of the framework to prove theorems in computability theory which require priority arguments.
We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years.
A completeness theory for parameterized computational complexity has been studied in a series of recent papers, and has been shown to have many applications in diverse problem domains including familiar graph-theoretic problems, VLSI layout, games, computational biology, cryptography, and...
A Cantor singleton is the unique nonrecursive member of some
class. In this paper we investigate the relationships between the following three notions: Cantor singletons, Cantor-Bendixson rank, and recursive join. Among other results, we show that the rank of
We prove that every countable relation on the enumeration degrees,
, is uniformly definable from parameters in
. Consequently, the first order theory of
is recursively isomorphic to the second order theory of arithmetic. By an effective version of...
We show that the length of a hierarchy of domains with totality, based on the standard domain for the natural numbers
and closed under dependent products of continuously parameterised families of domains will be the first ordinal not recursive in
and any real. As a part of...
I consider the projectum of a
-r.e. set. It is shown that there are tame r.e. sets with small projecta and that there are tame r.e. sets with large projecta.
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