Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
The Medvedev lattice of degrees of difficulty
Computability, enumerability, unsolvability
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Important examples of @P"1^0 classes of functions f@?@w@w are the classes of sets (elements of 2@w) which separate a given pair of disjoint r.e. sets: S"2(A"0,A"1):={f@?2@w:(@?ii}. A wider class consists of the classes of functions f@?k@w which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k@?@w: S"k(A"0,...,A"k"-"1):={f@?k@w:(@?ii}. We study the structure of the Medvedev degrees of such classes and show that the set of degrees realized depends strongly on both k and the extent to which the r.e. sets intersect. Let S"k^m denote the Medvedev degrees of those S"k(A"0,...,A"k"-"1) such that no m+1 sets among A"0,...,A"k"-"1 have a nonempty intersection. It is shown that each S"k^m is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions DNR"k is the greatest element of S"k^1. If 2@?l