Degrees of randomized computability: decomposition into atoms

Abstract: In this paper we study structural properties of LV-degrees of the algebra of collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. We construct atoms and infinitely divisible elements of this algebra generated by sequences, which cannot be Martin-L\"of random and, moreover, these sequences cannot be Turing equivalent to random sequences. The constructions are based on the corresponding templates which can be used for defining the special LV-degrees. In particular, we present the template for defining atoms of the algebra of LV-degrees and obtain the decomposition of the maximal LV-degree into a countable sequence of atoms and their non-zero complement -- infinitely divisible LV-degree. We apply the templates to establish new facts about specific LV-degrees, such as the LV-degree of the collection of sequences of hyperimmune degree. We construct atoms defined by collections of hyperimmune sequences, moreover, a representation of LV-degree of the collection of all hyperimmune sequences will be obtained in the form of a union of an infinite sequence of atoms and an infinitely divisible element.