Randomness, relativization and Turing degrees

Abstract

Abstract. We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅ (n−1) . We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) ≥ |x| − c. The 'only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity.Next we prove some results on lowness. Among other things, we characterize the 2-random sets as those 1-random sets that are low for Chaitin's Ω. Also, 2-random sets form minimal pairs with 2-generic sets. The r.e. low for Ω sets coincide with the r.e. K-trivial ones.Finally we show that the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree. We make some remarks about hyperimmune-free and PA-complete degrees.Mathematical Subject Classification: 68Q30, 03D15, 03D28, 03D80, 28E15.

Kolmogorov Complexity and Algorithmic Randomness

“…128 Докажите, что В самом деле, априорная вероятность на дереве не меньше меры (и даже любой другой перечислимой снизу полумеры) с точностью до ограниченного множителя, что после логарифмирования и даёт требуемое неравенство.…”

“…См. [103,128]; простое доказательство приведено в [9]. Аналогичный критерий 2-случайности можно сформулировать и в терминах префиксной сложности: надо потребовать, чтобы KP (( ) ) было не меньше + KP ( ) − для некоторого и для бесконечно многих ( [197], простое доказательство в [198]).…”

Computability theory, algorithmic randomness and Turing's anticipation

“…For instance, we have seen that it is impossible for a real to have C(X n) ≥ + n for all n, but Martin-Löf showed in his original paper [101] that there are reals X with C(X n) ≥ + n for infinitely many n, and that these are all 1-random. Joe Miller [103] and later Nies, Stephan and Terwijn [97] showed that such randoms are precisely the 2-randoms, and later Miller [104] showed that the 2-randoms are exactly those that achieve maximal prefix-free complexity (which is n + K(n)) infinitely often. Also Becher and Gregorieff [10] have a kind of index set characterizations of higher notions of randomness.…”

“…On the other hand, it is possible to show that within the high degrees the separations between computable, Schnorr, and Martin-Löf randomness all occuri ( [97]). In the hyperimmune-free degrees, weak randomness coincides with all of these as well as weak 2-randomness.…”

Joining non-low C.E. sets with diagonally non-computable functions

“…Nies, Stephan and Terwijn [14] showed that Proposition 2.7 holds if we assume that X is random relative to A.…”