Deep Π^0_1 Classes [article]

Laurent Bienvenu, Christopher P. Porter
2017 arXiv   pre-print
A set of infinite binary sequences C⊆2^ω is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of Π^0_1 classes. In this paper, we introduce the notion of depth for Π^0_1 classes, which is a stronger form of negligibility. Whereas a negligible Π^0_1 class C has the property that one cannot probabilistically compute a member of C with positive probability, a
more » ... eep Π^0_1 class C has the property that one cannot probabilistically compute an initial segment of a member of C with high probability. That is, the probability of computing a length n initial segment of a deep Π^0_1 class converges to 0 effectively in n. We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We also provide a number of examples of deep Π^0_1 classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, we examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin's Ω.
arXiv:1403.0450v3 fatcat:i63bvw7llnhodbdw6tn5sjcswq