### Abstract

We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff's pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications.

Original language | English |
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Title of host publication | STACS 2009 - 26th International Symposium on Theoretical Aspects of Computer Science |

Pages | 469-480 |

Number of pages | 12 |

Volume | 3 |

Publication status | Published - 2009 |

Event | 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009 - Freiburg, Germany Duration: 26 Feb 2009 → 28 Feb 2009 |

### Other

Other | 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009 |
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Country | Germany |

City | Freiburg |

Period | 26/02/09 → 28/02/09 |

### Keywords

- Birkhoff's ergodic theorem
- Computable measures
- Schnorr randomness

### ASJC Scopus subject areas

- Software

## Cite this

Gacs, P., Hoyrup, M., & Rojas, C. (2009). Randomness on computable probability spaces - A dynamical point of view. In

*STACS 2009 - 26th International Symposium on Theoretical Aspects of Computer Science*(Vol. 3, pp. 469-480)