On minimal wtt-degrees and computably enumerable Turing degrees

Computability theorists have studied many different reducibilities between sets of natural numbers including one reducibility (≤1), many-one reducibility (≤m), truth table reducibility (≤tt), weak truth table reducibility (≤wtt) and Turing reducibility (≤T ). The motivation for studying reducibilities stronger that Turing reducibility stems from internally motivated questions about varying the access mechanism to the oracle, and the fact that most natural reducibilities arising in classical mathematics tend to be stronger than ≤T . For instance consider the reduction of, say, the word problem to the conjucacy problem in combinatorial group theory. Deeper examples include Downey and Remmel’s [3] proof that if V is a enumerable subspace of V∞, then the degrees of computably enumerable (c.e.) bases of V are precisely the weak truth table (wtt-)degrees below the degree of V . Similarly, wtt-reducibility proved fundamental in the work on differential geometry Nabutovsky and Weinberger [13], as studied by Csima [2] and Soare [21]. A final motivation is a technical one: results about strong reducibilities and their interactions with Turing reducibility can lead to significant insight into the structure of (for example) the Turing (T -)degrees. There are innumerable examples of this phenomenon and a good example is the first paper of Ladner and Sasso [11] in which they construct locally distributive parts of the c.e. T -degrees using the wtt-degrees and their interactions with the T -degrees. For general information concerning these reducibilities, we refer the reader to the survey article by Odifreddi [14] as well as the books by Rogers [17], Odifreddi [15] and Soare [20]. The concern of this paper is the interaction of minimality and enumerability, two of the basic objects of classical computability. All constructions of minimal degrees are basically effective forcing arguments of one kind or another and such constructions are relatively incompatible with the construction of effective objects. In particular, by Sacks Splitting Theorem, no c.e. T -degree can be a minimal T -degree. On the other hand, it is known that there can be c.e. sets of minimal m-degree and of minimal tt-degree. Since wtt-reducibility is intermediate between ≤tt and ≤T , it is natural to wonder what happens here. Again, Sacks Splitting Theorem shows that no wtt-degree of a c.e. set can have minimal wtt-degree, but this leaves open