On initial segment complexity and degrees of randomness

@article{Miller2008OnIS,
  title={On initial segment complexity and degrees of randomness},
  author={J. Miller and L. Yu},
  journal={Transactions of the American Mathematical Society},
  year={2008},
  volume={360},
  pages={3193-3210}
}
  • J. Miller, L. Yu
  • Published 2008
  • Mathematics
  • Transactions of the American Mathematical Society
One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define X ≤K Y to mean that (Vn) K(X | n) < K(Y | n) + 0(1). The equivalence classes under this relation are the K-degrees. We prove that if X ⊕ Y is 1-random, then X and Y have no upper bound in the K-degrees (hence, no join). We also prove that n-randomness is closed upward in the K-degrees. Our main tool is another structure intended to measure the degree of… Expand
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