2011/03/08 by Laurent Bienvenu, Peter Gacs, Bienvenu, Laurent +7
Computer Science · Mathematics · #03D32 #68Q30 #F.1 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Logic (math.LO) #Probability (math.PR) #acm:03D32 #acm:68Q30 #cs.IT #math.IT #math.LO #math.PR #msc:03D32 #msc:68Q30
paper · pdf · doi:10.48550/arxiv.1103.1529
arxiv created 2011/05/26 · arxiv updated 2011/05/27
The paper considers quantitative versions of different randomness notions: algorithmic test measures the amount of non-randomness (and is infinite for non-random sequences). We start with computable measures on Cantor space (and Martin-Lof randomness), then consider uniform randomness (test is a function of a sequence and a measure, not necessarily computable) and arbitrary constructive metric spaces. We also consider tests for classes of measures, in particular Bernoulli measures on Cantor space, and show how they are related to uniform tests and original Martin-Lof definition. We show that Hyppocratic (blind, oracle-free) randomness is equivalent to uniform randomness for measures in an effectively orthogonal effectively compact class. We also consider the notions of sparse set and on-line randomness and show how they can be expressed in our framework.