254 research outputs found
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
Translating the Cantor set by a random
We determine the constructive dimension of points in random translates of the
Cantor set. The Cantor set "cancels randomness" in the sense that some of its
members, when added to Martin-Lof random reals, identify a point with lower
constructive dimension than the random itself. In particular, we find the
Hausdorff dimension of the set of points in a Cantor set translate with a given
constructive dimension
Redundancy of information: lowering dimension
Let At denote the set of infinite sequences of effective dimension t. We
determine both how close and how far an infinite sequence of dimension s can be
from one of dimension t, measured using the Besicovitch pseudometric. We also
identify classes of sequences for which these infima and suprema are realized
as minima and maxima. When t < s, we find d(X,At) is minimized when X is a
Bernoulli p-random, where H(p)=s, and maximized when X belongs to a class of
infinite sequences that we call s-codewords. When s < t, the situation is
reversed.Comment: 28 pages, 3 figure
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