Algorithmic randomness and computability

Abstract

Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. Mathematics Subject Classification (2000). Primary 68Q30, 68Q15, 03D15, 03D25, 03D28, 03D30.

Randomness, Computation and Mathematics

“…In particular, we have seen significant clarification as to the mathematical relationship between algorithmic computational power of infinite random sources and algorithmic randomness. Much of this material has been reported in the short surveys Downey [27], Nies [53] and long surveys [26,30] …”

Computability theory, algorithmic randomness and Turing's anticipation

“…In particular, we have seen significant clarification as to the mathematical relationship between algorithmic computational power of infinite random sources and level algorithmic randomness. Much of this material has been reported in the short surveys Downey [41,42], Nies [95] and long surveys [40,47] and long monographs Downey and Hirschfeldt [46] and Nies [94]. Also the book edited by Hector Zenil [152] has a lot of discussion of randomness of varying levels of technicality, many aimed at a general audience.…”