268 research outputs found
The Complexity of Orbits of Computably Enumerable Sets
The goal of this paper is to announce there is a single orbit of the c.e.
sets with inclusion, \E, such that the question of membership in this orbit
is -complete. This result and proof have a number of nice
corollaries: the Scott rank of \E is \wock +1; not all orbits are
elementarily definable; there is no arithmetic description of all orbits of
\E; for all finite , there is a properly
orbit (from the proof).
A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi
Arithmetic complexity via effective names for random sequences
We investigate enumerability properties for classes of sets which permit
recursive, lexicographically increasing approximations, or left-r.e. sets. In
addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably,
Schnorr, and Kurtz random sets, weakly 1-generics and their complementary
classes, we find that there exist characterizations of the third and fourth
levels of the arithmetic hierarchy purely in terms of these notions.
More generally, there exists an equivalence between arithmetic complexity and
existence of numberings for classes of left-r.e. sets with shift-persistent
elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz
non-randoms) have left-r.e. numberings, there is no canonical, or acceptable,
left-r.e. numbering for any class of left-r.e. randoms.
Finally, we note some fundamental differences between left-r.e. numberings
for sets and reals
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
The hierarchy of equivalence relations on the natural numbers under computable reducibility
The notion of computable reducibility between equivalence relations on the
natural numbers provides a natural computable analogue of Borel reducibility.
We investigate the computable reducibility hierarchy, comparing and contrasting
it with the Borel reducibility hierarchy from descriptive set theory.
Meanwhile, the notion of computable reducibility appears well suited for an
analysis of equivalence relations on the c.e.\ sets, and more specifically, on
various classes of c.e.\ structures. This is a rich context with many natural
examples, such as the isomorphism relation on c.e.\ graphs or on computably
presented groups. Here, our exposition extends earlier work in the literature
concerning the classification of computable structures. An abundance of open
questions remains.Comment: To appear in Computabilit
A constructive version of Birkhoff's ergodic theorem for Martin-L\"of random points
A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite
binary sequence {\omega} and an effectively open set A of measure less than 1,
some tail of {\omega} is not in A. We first prove several results in the same
spirit and generalize them via an effective version of a weak form of
Birkhoff's ergodic theorem. We then use this result to get a stronger form of
it, namely a very general effective version of Birkhoff's ergodic theorem,
which improves all the results previously obtained in this direction, in
particular those of V'Yugin, Nandakumar and Hoyrup, Rojas.Comment: Improved version of the CiE'10 paper, with the strong form of
Birkhoff's ergodic theorem for random point
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