On Array Noncomputable Degrees, Maximal Pairs and Simplicity Properties

In this thesis, we give contributions to topics which are related to array noncomputable (a.n.c.) Turing degrees, maximal pairs and to simplicity properties. The outline is as follows. In Chapter 2, we introduce a subclass of the a.n.c. Turing degrees, the so called completely array noncomputable (c.a.n.c. for short) Turing degrees. Here, a computably enumerable (c.e.) Turing degree a is c.a.n.c. if any c.e. set A ∈ a is weak truth-table (wtt) equivalent to an a.n.c. set. We show in Section 2.3 that these degrees exist (indeed, there exist infinitely many low c.a.n.c. degrees) and that they cannot be high. Moreover, we apply some of the ideas used to show the existence of c.a.n.c. Turing degrees to show the stronger result that there exists a c.e. Turing degree whose c.e. members are halves of maximal pairs in the c.e. computably Lipschitz (cl) degrees, thereby solving the first part of the first open problem given in the paper by Ambos-Spies, Ding, Fan and Merkle [ASDFM13]. In Chapter 3, we present an approach to extending the notion of array noncomputability to the setting of almost-c.e. sets (these are the sets which correspond to binary representations of left-c.e. reals). This approach is initiated by the Heidelberg Logic Group and it is worked out in detail in an upcoming paper by Ambos-Spies, Losert and Monath [ASLM18], in the thesis of Losert [Los18] and in [ASFL+]. In [ASLM18], the authors introduce the class of sets with the universal similarity property (u.s.p. for short; throughout this thesis, sets with the u.s.p. will shortly be called u.s.p. sets) which is a strong form of array noncomputability in the setting of almost-c.e. sets and they show that sets with this property exist precisely in the c.e. not totally ω-c.e. degrees. Then it is shown that, using u.s.p. sets, one obtains a simplified method for showing the existence of almost-c.e. sets with a property P (for certain properties P) that are contained in c.e. not totally ω-c.e. degrees, namely by showing that u.s.p. sets have property P. This is demonstrated by showing that u.s.p. sets are computably bounded random (CB-random), thereby extending a result from Brodhead, Downey and Ng [BDN12]. Moreover, it is shown that the c.e. not totally ω-c.e. degrees can be characterized as those c.e. degrees which contain an almost-c.e. set which is not cl-reducible to any complex almost-c.e. set. This affirmatively answers a conjecture by Greenberg. For the if-direction of the latter result, we prove a new result on maximal pairs in the almost-c.e. sets by showing the existence of locally almost-c.e. sets which are halves of maximal pairs in the almost-c.e. sets such that the second half can be chosen to be c.e. and arbitrarily sparse. This extends Yun Fan’s result on maximal pairs [Fan09]. By our result, we also get a new proof of one of the main results in Barmpalias, Downey and Greenberg [BDG10], namely that in any c.e. a.n.c. degree there is a left-c.e. real which is not cl-reducible to any ML-random left-c.e. real. In this thesis, we give an overview of some of the results from [ASLM18] and sketch some of the proofs to illustrate this new methodology and, subsequently, we give a detailed proof of the above maximal pair result. In Chapter 4, we look at the interaction between a.n.c. wtt-degrees and the most commonly known simplicity properties by showing that there exists an a.n.c. wtt-degree which contains an r-maximal set. By this result together with the result by Ambos-Spies [AS18] that no a.n.c. wtt-degree contains a dense simple set, we obtain a complete characterization which of the classical simplicity properties may hold for a.n.c. wtt-degrees. The guiding theme for Chapter 5 is a theorem by Barmpalias, Downey and Greenberg [BDG10] in which they characterize the c.e. not totally ω-c.e. degrees as the c.e. degrees which contain a c.e. set which is not wtt-reducible to any hypersimple set. So Ambos-Spies asked what the above characterization would look like if we replaced hypersimple sets by maximal sets in the above theorem. In other words, what are the c.e. Turing degrees that contain c.e. sets which are not wtt-reducible to any maximal set. We completely solve this question on the set level by introducing the new class of eventually uniformly wtt-array computable (e.u.wtt-a.c.) sets and by showing that the c.e. sets with this property are precisely those c.e. sets which are wtt-reducible to maximal sets. Indeed, this characterization can be extended in that we can replace wtt-reducible by ibT-reducible and maximal sets by dense simple sets. By showing that the c.e. e.u.wtt-a.c. sets are closed downwards under wtt-reductions and under the join operation, it follows that the c.e. wtt-degrees containing e.u.wtt-a.c. sets form an ideal in the upper semilattice of the c.e. wtt-degrees and, further, we obtain a characterization of the c.e. wtt-degrees which contain c.e. sets that are not wtt-reducible to any maximal set. Moreover, we give upper and lower bounds (with respect to ⊆) for the class of the c.e. e.u.wtt-a.c. sets. For the upper bound, we show that any c.e. e.u.wtt-a.c. set has array computable wtt-degree. For the lower bound, we introduce the notion of a wtt-superlow set and show that any wtt-superlow c.e. set is e.u.wtt-a.c. Besides, we show that the wtt-superlow c.e. sets can be characterized as the c.e. sets whose bounded jump is ω-computably approximable (ω-c.a. for short); hence, they are precisely the bounded low sets as introduced in the paper by Anderson, Csima and Lange [ACL17]. Furthermore, we prove a hierarchy theorem for the wtt-superlow c.e. sets and we show that there exists a Turing complete set which lies in the intersection of that hierarchy. Finally, it is shown that the above bounds are strict, i.e., there exist c.e. e.u.wtta. c. sets which are not wtt-superlow and that there exist c.e. sets whose wtt-degree is array computable and which are not e.u.wtt-a.c. (where here, we obtain the separation even on the level of Turing degrees). The results from Chapter 5 will be included in a paper which is in preparation by Ambos-Spies, Downey and Monath [ASDM19]