Randomness for non-computable measures

Randomness for Non-computable Measures

Different approaches have been taken to defining randomness for non-computable probability measures. We will explain the approach of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gács, Hoyrup and Rojas. We will show that these approaches are fundamentally equivalent. Having clarified what it means to be random for a non-computable probabilit...

Effective Randomness for Computable Probability Measures

Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of random elements coincide. Elaborating on work of Bienvenu [1], we determine all the implications that hold between the equivalence relations induce...

Lowness for Computable and Partial Computable Randomness

Ambos-Spies and Kucera [1, Problem 4.8] asked whether there is a non-computable set which is low for the computably random sets. We show that no such set exists. The same result holds for partial computable randomness. Each tally language that is low for polynomial randomness is on a polynomial time tree of bounded width. This research was done in 2003 but has not been published so far.

Sub-computable Boundedness Randomness

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-Löf tests, (2) Kolmogorov complexity, and (3)...

Fairness, Computable Fairness, and Randomness