2 research outputs found
Multiple Permitting and Bounded Turing Reducibilities
We look at various properties of the computably enumerable (c.e.) not totally ω-c.e. Turing degrees.
In particular, we are interested in the variant of multiple permitting given by those degrees. We
define a property of left-c.e. sets called universal similarity property which can be viewed as a
universal or uniform version of the property of array noncomputable c.e. sets of agreeing with any
c.e. set on some component of a very strong array. Using a multiple permitting argument, we
prove that the Turing degrees of the left-c.e. sets with the universal similarity property coincide
with the c.e. not totally ω-c.e. degrees. We further introduce and look at various notions of socalled
universal array noncomputability and show that c.e. sets with those properties can be found
exactly in the c.e. not totally ω-c.e. Turing degrees and that they guarantee a special type of
multiple permitting called uniform multiple permitting. We apply these properties of the c.e. not
totally ω-c.e. degrees to give alternative proofs of well-known results on those degrees as well as
to prove new results. E.g., we show that a c.e. Turing degree contains a left-c.e. set which is not
cl-reducible to any complex left-c.e. set if and only if it is not totally ω-c.e. Furthermore, we prove
that the nondistributive finite lattice S7 can be embedded into the c.e. Turing degrees precisely
below any c.e. not totally ω-c.e. degree.
We further look at the question of join preservation for bounded Turing reducibilities r and r′
such that r is stronger than r′. We say that join preservation holds for two reducibilities r and
r′ if every join in the c.e. r-degrees is also a join in the c.e. r′-degrees. We consider the class of
monotone admissible (uniformly) bounded Turing reducibilities, i.e., the reflexive and transitive
Turing reducibilities with use bounded by a function that is contained in a (uniformly computable)
family of strictly increasing computable functions. This class contains for example identity bounded
Turing (ibT-) and computable Lipschitz (cl-) reducibility. Our main result of Chapter 3 is that join
preservation fails for cl and any strictly weaker monotone admissible uniformly bounded Turing
reducibility. We also look at the dual question of meet preservation and show that for all monotone
admissible bounded Turing reducibilities r and r′ such that r is stronger than r′, meet preservation
holds. Finally, we completely solve the question of join and meet preservation in the classical
reducibilities 1, m, tt, wtt and T
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]