Abstract

The set A is low for Martin-Lof random if each random set is already random relative to A. A is K-trivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of Ambos-Spies and Kucera [2], showing that each low for Martin-Lof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering

Citations

411	Recursively Enumerable Sets and Degrees - Soare - 1987
294	The definition of random sequences - Martin-Löf
274	A theory of program size formally identical to information theory - Chaitin - 1975
133	Zufälligkeit und Wahrscheinlichkeit - Schnorr - 1971
55	Information-theoretic characterizations of recursive infinite strings - Chaitin - 1976
54	Draft of a paper (or series of papers) on Chaitin’s work. Unpublished notes - Solovay - 1975
50	Schnorr randomness - Downey, Griffiths
49	Trivial reals - Downey, Hirschfeldt, et al. - 2003
32	An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees - Ambos-Spies, Jockusch, et al. - 1984
28	Lowness for the class of random sets - Kučera, Terwijn - 1999
27	Randomness in computability theory - Ambos-Spies, Kučera - 1999
26	Reals which compute little - Nies - 2006
14	An almost deep degree - Cholak, Groszek, et al. - 2001
14	The degrees of hyperimmune sets - Martin, Miller - 1968
11	On relative randomness - Kucera - 1993
10	Program-size complexity of initial segments and domination reducibility - Calude, Coles - 1999
10	Defining the Turing jump - Shore, Slaman - 1999
9	Algorithmic randomness and lowness - Terwijn, Zambella
9	On sequences with simple initial segments. ILLC technical report - Zambella - 1990
8	Lowness properties of reals and hyper-immunity - Bedregal
8	Mathematical metaphysics of randomness. Theoret - Muchnik, Semenov, et al. - 1998
8	When oracles do not help - Slaman, Solovay - 1991
6	Pseudo jump operators I: the r.e. case - Shore - 1983
4	Information and randomness. Monographs in Theoretical Computer Science. An EATCS Series - Calude - 1994
4	On a question of Ambos-Spies and Kučera - Kjos-Hanssen, Stephan, et al. - 1997
3	Degrees of Unsolvability, volume 55 of Annals of Mathematical Studies - Sacks - 1963
2	Ideals in the recursively enumerable degrees - Nies
2	Variations on low for random and K-trivial - Nies, Stephan