1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) machines, ϕe, can be used; access to

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. Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching. Here, I would like to concentrate on just one instance of that influence that I feel has been of special significance to the study of the degrees of unsolvability in general and on my own work in particular--- the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals

this paper.

G. E. Sacks [3; p. 171, q. 4] asked whether there is a uniformly recursively enumerable (r.e.) ascending sequence of r.e. degrees which has as one of its minimal upper bounds another r.e. degree. We show (see the Corollary below) that the answer is &quot; yes &quot;. a is said to be a string if it is the restriction A[n] of a characteristic function A to the first n +1 non-negative integers for some number n. a is said to be a beginning of A of length n +1. We use (j> to denote the empty string. Let {<De} be a standard enumeration of the partial recursive functionals. We will need a uniformly recursive double sequence {Oc> s} of finite approximations to {Oc} such that for each e, s, and such that for each s, Oe s is empty for all but a finite number of e's. Define, for each e, Then {Fe} is a standard list of the partial recursive functions with suitably well-behaved set of approximations {Fe< J. THEOREM. There is a sequence of simultaneously r.e. degrees {bj and a r.e. degree a such that a is recursive in no finite subset o/{b,}, b { is recursive in a for each i, and, for each c < a, c is not an upper bound for {bf}. Proof. We enumerate at stages 0, 1, 2,..., s,... finite sets As, Bt s,i ^ 0, such that, if A = (J As, B i = U Bi>s, then the following objectives are satisfied:

Abstract. We show that there is a low T-upper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 T-degrees for which there is a low T-upper bound. 1.

We study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We call a class S of computably enumerable sets bounded iff there is an m-in- complete computably enumerable set A such that every set in S is m-reducible to A. For example, we show that the class of eectively simple sets is bounded; but the class of maximal sets is not. Furthermore, the class of computably enumerable sets Turing reducible to a computably enumerable set B is bounded iff B is low 2 . For r = bwtt, tt, wtt and T , there is a bounded class intersecting every computably enumerable r-degree; for r = c, d and p, no such class exists. AMS Classication: 03D30; 03D25 Keywords: Computably enumerable sets (= Recursively enumerable sets); Simple sets; m-reducibility; Strong reducibilities; 3 classes; Ideals; Exact pairs 1 Introduction With a typical priority argument, one can show that for any simple set A, there is a simple set B such that B m A. Carl ...

review articles doi:10.1145/2093548.2093569 Embodied and disembodied computing at the Turing Centenary.