## Abstract

Let f be a computable function from finite sequences of 0's and 1's to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to a PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree.

Original language | English |
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Pages (from-to) | 742-758 |

Number of pages | 17 |

Journal | Annals of Pure and Applied Logic |

Volume | 165 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 Feb |

Externally published | Yes |

## Keywords

- Effective Hausdorff dimension
- Kolmogorov complexity
- Martin-Löf randomness
- Models of arithmetic
- Partial randomness

## ASJC Scopus subject areas

- Logic