Kolmogorov complexity of initial segments of sequences and arithmetical definability
George Barmpalias, Charlotte Vlek
Abstract:
The structure of the K-degrees provides a way to classify sets of
natural numbers or infinite binary sequences with respect to the level
of randomness of their initial segments. In the K-degrees of infinite
binary sequences, X is below Y if the prefix-free Kolmogorov
complexity of the first n bits of X is less than the complexity of the
first n bits of Y , for each n. Identifying infinite binary sequences
with subsets of N, we study the K-degrees of arithmetical sets and
explore the interactions between arithmetical definability and prefix
free Kolmogorov complexity.
We show that in the K-degrees, for each n > 1 there exists a ~^0_n
nonzero degree which does not bound any ~^0_n nonzero degree. An
application of this result is that in the K-degrees there exists a
~^0_2 degree which forms a minimal pair with all ~^0_1 degrees. This
extends work of Csima/Montalbán [CM06] and Merkle/Stephan [MS07]. Our
main result is that, given any ~^0_2 family C of sequences, there is a
~^0_2 sequence of non-trivial initial segment complexity which is not
larger than the initial segment complexity of any non-trivial member
of C. This general theorem has the following surprising
consequence. There is a 0'-computable sequence of nontrivial initial
segment complexity which is not larger than the initial segment
complexity of any nontrivial computably enumerable set.
Our analysis and results demonstrate that, examining the extend to
which arithmetical definability interacts with the K reducibility (and
in general any ~weak reducibility~) is a fruitful way of studying the
induced structure.
Keywords: Kolmogorov complexity; K-degrees; Relative complexity; arithmetical complexity