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Computable randomness and its properties

Monday 2nd July 2012 - 17:30 to 18:00
INI Seminar Room 1
Computable randomness at first does not seem as natural of a randomness notion as Schnorr and Martin-Löf randomness. However, recently Brattka, Miller, and Nies [1] have shown that computable randomness is closely linked to differentiability. Why is this so? What are the chances that, say, computable randomness will also be linked to the ergodic theorem? In this talk I will explain how computable randomness is similar to and how it is different from other notions of randomness.

Unlike other notions of randomness, computable randomness is closely linked to the Borel sigma-algebra of a space. This has a number of interesting implications:

  1. Computable randomness can be extended to other computable probability spaces, but this extension is more complicated to describe [2].
  2. Computable randomness is invariant under isomorphisms, but not morphisms (a.e.-computable measure-preserving maps) [2].
  3. Computable randomness is connected more with differentiability than with the ergodic theorem.
  4. Dyadic martingales and martingales whose filtration converges to a "computable" sigma-algebra characterize computable randomness, while more general computable betting strategies do not.

However, this line of research still leaves many open questions about the nature of computable randomness and the nature of randomness in general. I believe the tools used to explore computable randomness may have other applications to algorithmic randomness and computable analysis.

[1] Vasco Brattka, Joseph S. Miller, and André Nies. "Randomness and differentiability." Submitted.

[2] Jason Rute. "Computable randomness and betting for computable probability spaces." In preparation.

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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons