Randomness on Computable Probability Spaces—A Dynamical Point of View

Authors:
Peter Gács;Mathieu Hoyrup;Cristóbal Rojas
Affiliations:
Boston University, Computer Science Department, 111 Cummington Street, 02215-2411, Boston, MA, USA;INRIA Nancy, 615, Rue du Jardin Botanique, BP 239, 54506, Vandœuvre-lès-Nancy, France;The Fields Institute for Research in Mathematical Sciences, 222 College Street, M5T 3J, Toronto, Ontario, Canada
Venue:
Theory of Computing Systems
Year:
2011

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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.