Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
On C-Degrees, H-Degrees and T-Degrees
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Computability and Randomness
Time-Bounded Kolmogorov Complexity and Solovay Functions
MFCS '09 Proceedings of the 34th International Symposium on Mathematical Foundations of Computer Science 2009
Kolmogorov complexity and the recursion theorem
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
On the number of infinite sequences with trivial initial segment complexity
Theoretical Computer Science
Effective strong nullness and effectively closed sets
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
Universal computably enumerable sets and initial segment prefix-free complexity
Information and Computation
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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 theK-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 n1, there exists a @S"n^0 non-zero degree which does not bound any @D"n^0 non-zero degree. An application of this result is that in the K-degrees there exists a @S"2^0 degree which forms a minimal pair with all @S"1^0 degrees. This extends the work of Csima and Montalban (2006) [8] and Merkle and Stephan (2007) [17]. Our main result is that, given any @D"2^0 family C of sequences, there is a @D"2^0 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 non-trivial initial segment complexity, which is not larger than the initial segment complexity of any non-trivial 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.