12 research outputs found
Computability Theory
Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science
Kolmogorov complexity and computably enumerable sets
We study the computably enumerable sets in terms of the: (a) Kolmogorov
complexity of their initial segments; (b) Kolmogorov complexity of finite
programs when they are used as oracles. We present an extended discussion of
the existing research on this topic, along with recent developments and open
problems. Besides this survey, our main original result is the following
characterization of the computably enumerable sets with trivial initial segment
prefix-free complexity. A computably enumerable set is -trivial if and
only if the family of sets with complexity bounded by the complexity of is
uniformly computable from the halting problem
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]
Use-Bounded Strong Reducibilities
We study the degree structures of the strong reducibilities and , as well as and . We show that any noncomputable c.e. set is part of a uniformly c.e. copy of (\BQ,\leq) in the c.e. cl-degrees within a single wtt-degree; that there exist uncountable chains in each of the degree structures in question; and that any countable partially-ordered set can be embedded into the cl-degrees, and any finite partially-ordered set can be embedded into the ibT-degrees. We also offer new proofs of results of Barmpalias and Lewis-Barmpalias concerning the non-existence of cl-maximal sets
Randomness and Computability
This thesis establishes significant new results in the area of algorithmic randomness.
These results elucidate the deep relationship between randomness
and computability.
A number of results focus on randomness for finite strings. Levin introduced
two functions which measure the randomness of finite strings. One
function is derived from a universal monotone machine and the other function
is derived from an optimal computably enumerable semimeasure. Gacs
proved that infinitely often, the gap between these two functions exceeds the
inverse Ackermann function (applied to string length). This thesis improves
this result to show that infinitely often the difference between these two functions
exceeds the double logarithm. Another separation result is proved for
two different kinds of process machine.
Information about the randomness of finite strings can be used as a computational
resource. This information is contained in the overgraph. Muchnik
and Positselsky asked whether there exists an optimal monotone machine
whose overgraph is not truth-table complete. This question is answered in the
negative. Related results are also established.
This thesis makes advances in the theory of randomness for infinite binary
sequences. A variant of process machines is used to characterise computable
randomness, Schnorr randomness and weak randomness. This result is extended
to give characterisations of these types of randomness using truthtable
reducibility. The computable Lipschitz reducibility measures both the
relative randomness and the relative computational power of real numbers. It
is proved that the computable Lipschitz degrees of computably enumerable
sets are not dense.
Infinite binary sequences can be regarded as elements of Cantor space.
Most research in randomness for Cantor space has been conducted using the
uniform measure. However, the study of non-computable measures has led to
interesting results. This thesis shows that the two approaches that have been
used to define randomness on Cantor space for non-computable measures:
that of Reimann and Slaman, along with the uniform test approach first introduced
by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent.
Levin established the existence of probability measures for which all infinite sequences are random. These measures are termed neutral measures. It is
shown that every PA degree computes a neutral measure. Work of Miller is
used to show that the set of atoms of a neutral measure is a countable Scott set
and in fact any countable Scott set is the set of atoms of some neutral measure.
Neutral measures are used to prove new results in computability theory. For
example, it is shown that the low computable enumerable sets are precisely
the computably enumerable sets bounded by PA degrees strictly below the
halting problem.
This thesis applies ideas developed in the study of randomness to computability
theory by examining indifferent sets for comeager classes in Cantor
space. A number of results are proved. For example, it is shown that there
exist 1-generic sets that can compute their own indifferent sets
Joins and Meets in the Partial Orders of the Computably Enumerable ibT- and cl-Degrees
A bounded reducibility is a preorder on the power set of the integers which is obtained from Turing reducibility by the additional requirement that, for a reduction of A to B, for every input x the oracle B is only asked oracle queries y < f(x)+1, where f is from some given set F of total computable functions.
The most general example of a bounded reducibility is weak-truth-table reducibility, where F is just the set of all computable functions. In this thesis we study the so-called strongly bounded reducibilites, which are obtained by choosing F={id} and F={id+c: c constant}, respectively (where id is the identity function).
We start by giving a machine-independent characterisation of these reducibilities, define the degree structures of the computably enumerable ibT- and cl-degrees and review some important properties of ibT- and cl-reducibility concerning strictly increasing computable functions (called shifts) and the permitting method.
Then we turn to the degree structures mentioned above, and in particular to existence and nonexistence of joins and meets of a finite set of degrees. As Barmpalias and independently Fan and Lu have shown, these structures are not upper semi-lattices; it is also known that they are not lower semi-lattices. We extend these results by showing that the existence of a join or meet of n degrees does in general not imply the existence of a join or meet, respectively, of any subset containining more than one element of these degrees. We also show that even if deg(A) and deg(B) have a join, there is no uniform way to compute a member of this join from A and B, contrasting the join in the Turing degrees. We conclude this part by looking at the substructure which consists of the degrees of simple sets and show that this structure is not closed with respect to the join operation. This is the dual of a theorem of Ambos-Spies stating that the simple degrees are not closed with respect to meets.
Next, we investigate lattice embeddings into the c.e. r-degrees. Due to an observation of Ambos-Spies, the proof that every finite distributive lattice can be embedded into the computably enumerable Turing degrees carries over to the c.e. r-degrees. We show that the smallest nondistributive lattices N5 and M3 can also be embedded, but only the N5 can be embedded preserving the least element. Since every nondistributive lattice contains at least one of these two lattices as a sublattice, this motivates the conjecture that every finite lattice can be embedded. We show this for two other nondistributive lattices, the S7 und S8.
Finally, we compare the c.e. ibT- and c.e. cl-degrees and prove that these are not elementarily equivalent. To show this, we study under which conditions on two degrees a and c with a<c it holds that there exists a degree b<c such that c is the join of a and b. In this context we also show that, while shifts provide a simple method to produce a lesser r-degree a to some given noncomputable r-degree c, there is no computable shift which uniformly produces such an a with the additional property that no degree b as above exists
Optimal asymptotic bounds on the oracle use in computations from Chaitin’s Omega
Chaitin’s number is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real �, there exists a Turing functional via which computes �, and such that the number of bits of that are needed for the computation of the first n bits of � (i.e. the use on argument n) is bounded above by a computable function h(n) = n + o (n). We characterise the asymptotic upper bounds on the use of Chaitin’s in oracle computations of halting probabilities (i.e. c.e. reals). We show that the following two conditions are equivalent for any computable function h such that h(n)