Relative randomness for martin-löf random sets

  • Authors:

  • NingNing Peng;Kojiro Higuchi;Takeshi Yamazaki;Kazuyuki Tanaka

  • Affiliations:

  • Mathematical Institute, Tohoku University, Sendai, Japan;Mathematical Institute, Tohoku University, Sendai, Japan;Mathematical Institute, Tohoku University, Sendai, Japan;Mathematical Institute, Tohoku University, Sendai, Japan

  • Venue:
  • CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
  • Year:
  • 2012

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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 {Gn}n∈ω of all c.e. open sets. For any f∈Γ, {Gf(n)}n∈ω is called a Γ-indexed test if μ(Gf(n))≤2−n for all n. We prove that weak n-randomness is strictly stronger than semi $\Delta^0_n$-randomness, for n2. Moreover, we investigate the relationships between various definitions of randomness.