### Abstract

Let Γ be a set of functions on the natural numbers. We introduce a new randomness notion called semi Γ-randomness, which is associated with a Γ-indexed test. Fix a computable sequence {G _{n}} _{nεω} of all c.e. open sets. For any f ε Γ, {G _{f(n)}} _{nεω} is called a Γ-indexed test if μ(G _{f(n)}) ≤ 2 ^{-n} for all n. We prove that weak n-randomness is strictly stronger than semi Δ _{n} ^{0}-randomness, for n > 2. Moreover, we investigate the relationships between various definitions of randomness.

Original language | English |
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Title of host publication | How the World Computes - Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012, Proceedings |

Pages | 581-588 |

Number of pages | 8 |

DOIs | |

Publication status | Published - 2012 Jun 18 |

Event | Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012 - Cambridge, United Kingdom Duration: 2012 Jun 18 → 2012 Jun 23 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7318 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012 |
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Country | United Kingdom |

City | Cambridge |

Period | 12/6/18 → 12/6/23 |

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

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## Cite this

Peng, N., Higuchi, K., Yamazaki, T., & Tanaka, K. (2012). Relative randomness for Martin-Löf random sets. In

*How the World Computes - Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012, Proceedings*(pp. 581-588). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7318 LNCS). https://doi.org/10.1007/978-3-642-30870-3_58