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- W2034673847 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 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 n>1, there exists a Σn0 non-zero degree which does not bound any Δn0 non-zero degree. An application of this result is that in the K-degrees there exists a Σ20 degree which forms a minimal pair with all Σ10 degrees. This extends the work of Csima and Montalbán (2006) [8] and Merkle and Stephan (2007) [17]. Our main result is that, given any Δ20 family C of sequences, there is a Δ20 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." @default.
- W2034673847 created "2016-06-24" @default.
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- W2034673847 date "2011-09-01" @default.
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- W2034673847 title "Kolmogorov complexity of initial segments of sequences and arithmetical definability" @default.
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- W2034673847 doi "https://doi.org/10.1016/j.tcs.2011.06.006" @default.
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