Abstract

Recall that E T is the upper semilattice of recursively enumerable Turing degrees. Two basic, classical, unresolved issues concerning E T are: Issue 1: To find a specific, natural, r.e. Turing degree a ∈ E T which is> 0 and < 0 ′. Issue 2: To find a “smallness property ” of an infinite co-r.e. set A ⊆ ω which insures that deg T(A) = a ∈ E T is> 0 and < 0 ′. These unresolved issues go back to Post’s 1944 paper, Recursively enumerable sets of positive integers and their decision problems. Mass Problems to the Rescue! We address Issues 1 and 2 by passing from decision problems to mass problems. 2 Outline of this talk: We embed E T into a slightly larger structure, Pw, which is much better behaved. In the Pw context, we obtain satisfactory, positive answers to Issues 1 and 2. What is this wonderful structure Pw? Briefly, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. In order to explain Pw, we must first explain: • mass problems, • weak degrees, and • nonempty Π 0 1 subsets of 2ω. 3 Mass problems (informal discussion): A “decision problem ” is the problem of deciding whether a given n ∈ ω belongs to a fixed set A ⊆ ω or not. To compare decision problems, we use Turing reducibility. A ≤ T B means that A can be computed using an oracle for B. A “mass problem ” is a problem with a not necessarily unique solution. (By contrast, a “decision problem ” has only one solution.) The “mass problem ” associated with a set P ⊆ ω ω is the “problem ” of computing an element of P. The “solutions ” of P are the elements of P. One mass problem is said to be “reducible” to another if, given any solution of the second problem, we can use it as an oracle to compute a solution of the first problem. 4 Rigorous definition: Let P and Q be subsets of ω ω. We view P and Q as mass problems. We say that P is weakly reducible to Q if (∀Y ∈ Q) (∃X ∈ P) (X ≤ T Y). This is abbreviated P ≤w Q.

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