110 research outputs found
Relatively computably enumerable reals
A real X is defined to be relatively c.e. if there is a real Y such that X is
c.e.(Y) and Y does not compute X. A real X is relatively simple and above if
there is a real Y <_T X such that X is c.e.(Y) and there is no infinite subset
Z of the complement of X such that Z is c.e.(Y). We prove that every nonempty
Pi^0_1 class contains a member which is not relatively c.e. and that every
1-generic real is relatively simple and above.Comment: 5 pages. Significant changes from earlier versio
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
Computably enumerable Turing degrees and the meet property
Working in the Turing degree structure, we show that those degrees which contain computably enumerable sets all satisfy the meet property, i.e. if a is c.e. and b < a, then there exists non-zero m < a with b ^m = 0. In fact, more than this is true: m may always be chosen to be a minimal degree. This settles a conjecture of Cooper and Epstein from the 80s
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
Recommended from our members
Computability Theory
Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
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