Mathematical Logic

The partial ordering of Medvedev reducibility restricted to the family of 1 classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a 1 class, which we call a “c.e. separating class”. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes. The Medvedev lattice was introduced in [5] to classify problems according to their degree of difficulty. A mass problem is a set of functions f mapping natural numbers to natural numbers and is thought of as representing the set of solutions to some problem. For example, we might consider the set of 4-colorings of a given countably infinite graph G as a set of functions each mapping ω into {1, 2, 3, 4}. One such set P is reducible to another set Q (written P ≤M Q) iff there is a partial computable functional which maps Q into P . Thus if we have a solution in Q, then we can use to compute a solution in P . As usual, P ≡M Q means that both P ≤M Q and Q ≤M P , P <M Q means P ≤M Q but not Q ≤M P , and the Medvedev degree dgM(P ) of P is the class of all sets Q such that P ≡M Q. We will see below that the set of Medvedev degrees is a lattice with meet and join given by the natural operations of direct product and disjoint union. For more on the general notion of Medvedev degrees, see the survey by Sorbi [9]. In this paper, we will examine the sublatticePM of degrees of 1 classes of sets, that is, nonempty subclasses of {0, 1}ω. (We will refer to elements of {0, 1}ω simply as sets.) The main result of this paper is that the partial ordering ≤M restricted to this sublattice is dense. We first introduce some notation. For a finite sequence σ ∈ {0, 1}n, we let |σ | = n denote the length of σ . For σ ∈ {0, 1}n and X ∈ {0, 1}ω, we say that σ is an initial segment of X (written σ ≺ X) if X(i) = σ(i) for all i ≤ |σ |. The interval I (σ ) determined by σ is {X ∈ {0, 1}ω : σ ≺ X}. These intervals form a basis for D. Cenzer: Department of Mathematics, 358 Little Hall, P.O. Box 118105, University of Florida, Gainesville, Fl 32611-8105, USA. e-mail: [email protected] P.G. Hinman: Department of Mathematics, 2072 East Hall, University of Michigan, Ann Arbor, MI 48109-1109, USA. e-mail: [email protected] Mathematics Subject Classification (2000): 03D30, 03D25