Degrees of difficulty of generalized

Important examples of Π 1 classes of functions f ∈ ω are the classes of sets (elements of 2) which separate a given pair of disjoint r.e. sets: S2(A0, A1) := { f ∈ 2 : (∀i < 2)(∀x ∈ Ai)f(x) 6= i }. A wider class consists of the classes of functions f ∈ k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: Sk(A0, . . . , Ak−1) := { f ∈ k : (∀i < k)(∀x ∈ Ai)f(x) 6= i }. 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 denote the Medvedev degrees of those Sk(A0, . . . , Ak−1) such that no m+ 1 sets among A0, . . . , Ak−1 have a nonempty intersection. It is shown that each S k is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions DNRk is the greatest element of S k . If 2 ≤ l < k, then 0M is the only degree in S l which is below a member of S k . Each S k is densely ordered and has the splitting property and the same holds for the lattice Lk it generates. The elements of S k are exactly the joins of elements of S i for d k me ≤ i ≤ k.