Kolmogorov complexity of initial segments of sequences and arithmetical definability

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 Σn nonzero degree which does not bound any ∆n nonzero degree. An application of this result is that in the K-degrees there exists a Σ2 degree which forms a minimal pair with all Σ1 degrees. This extends work of Csima/Montalbán [CM06] and Merkle/Stephan [MS07]. Our main result is that, given any ∆2 family C of sequences, there is a ∆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.