A set of infinite binary sequences C ⊆ 2 N 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 deep 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 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, 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.1991 Mathematics Subject Classification. 03D32, 68Q30, 03D80. Key words and phrases. computability theory, algorithmic randomness, Π 0 1 classes, probabilistic computation.Both authors acknowledge the support of the John Templeton Foundation through the grant "Structure and Randomness in the Theory of Computation."Depth can be viewed as a strong form of negligibility. A deep Π 0 1 class C has the property that producing an initial segment of some member of C is in some sense maximally difficult: the probability of obtaining such an initial segment not only converges to 0 as we consider longer initial segments of members of C, but this convergence is fast, as we can effectively bound the rate of convergence. That initial segments of members of deep classes are so difficult to produce via probabilistic computation reflects the fact that members of deep classes are highly structured: initial segments of these sequences cannot be successfully produced by any combination of Turing machine and random oracle.Although the notion of depth is isolated for the first time in the present study, it is implicitly used in the work of both Levin [22] and Stephan [32]. In fact, in each of these papers one can extract a proof that the consistent completions of Peano arithmetic form a deep class, a result that we reprove in Section 7.The primary goals of this paper are (1) to prove a number of basic results about deep Π 0 1 classes, (2) to determine the exact relationship between negligibility and depth (and a related notion we call tt-negligibility), and (3) to provide a number of examples of deep Π 0 1 classes that occur naturally in computability theory and algorithmic randomness.The outline of this paper is as follows: In Section 2, we provide the technical background...
Osvald Demuth studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later.In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the interactions between constructive analysis, algorithmic randomness, and computability theory. We will focus specifically on (i) Demuth's work on the differentiability of Markov computable functions and his study of constructive versions of the Denjoy alternative, (ii) Demuth's independent discovery of the main notions of algorithmic randomness, as well as the development of Demuth randomness, and (iii) the interactions of truth-table reducibility, algorithmic randomness, and semigenericity in Demuth's work. §1. Introducing Demuth. The mathematician Osvald Demuth worked primarily on constructive analysis in the Russian style, which was initiated by Markov,Šanin, Ceȋtin, and others in the 1950s. Born in 1936 in Prague, Demuth graduated from the Faculty of Mathematics and Physics at Charles University in Prague in 1959 with the equivalent of a master's degree. Thereafter he studied constructive mathematics under the supervision of A.A. Markov Jr. in Moscow, where he successfully defended his doctoral thesis in 1964. After completing his doctoral studies with Markov, he returned to Charles University, completing his Habilitation in 1968. He remained at Charles University until the end of his life in 1988. During this period of time, he faced intense persecution for his political views. After the Russian invasion of the Czech Republic in 1968, Demuth left the Communist Party in 1969, as he was opposed to the invasion. The consequences of this decision for Demuth's career were dire. From 1972From -1978, he was forbidden to lecture at the university, although he continued his scientific work during this period. Moreover, he was not permitted to travel abroad until 1987. Lastly, he never achieved the rank of full professor, even though he clearly deserved this rank.Despite these hardships, Demuth made a number of contributions to constructive analysis. His most significant work revealed deep and interesting connections between notions of typicality naturally occurring in constructive analysis and various notions of algorithmic randomness. These connections
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