论文信息 - Effective Randomness of Unions and Intersections

Effective Randomness of Unions and Intersections

We investigate the μ-randomness of unions and intersections of random sets under various notions of randomness corresponding to different probability measures. For example, the union of two relatively Martin-Löf random sets is not Martin-Löf random but is random with respect to the Bernoulli measure $\lambda_{\frac{3}{4}}$ under which any number belongs to the set with probability $\frac{3}{4}$. Conversely, any $\lambda_{\frac{3}{4}}$ random set is the union of two Martin-Löf random sets. Unions and intersections of random closed sets are also studied.

Douglas A. Cenzer | Rebecca Weber

[1] Douglas A. Cenzer,et al. Random Closed Sets , 2006, CiE.

[2] W. Fitch. Random sequences. , 1983, Journal of molecular biology.

[3] Jeffrey B. Remmel,et al. Chapter 13 Π10 classes in mathematics , 1998 .

[4] Douglas A. Cenzer,et al. Algorithmic Randomness and Capacity of Closed Sets , 2011, Log. Methods Comput. Sci..

[5] R. Soare. Recursively enumerable sets and degrees , 1987 .

[6] Bjørn Kjos-Hanssen,et al. Martin-Löf randomness and Galton-Watson processes , 2012, Ann. Pure Appl. Log..

[7] Per Martin-Löf,et al. The Definition of Random Sequences , 1966, Inf. Control..

[8] George Barmpalias,et al. Algorithmic Randomness of Closed Sets , 2007, J. Log. Comput..

[9] Vassilios Gregoriades. The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function , 2011 .

[10] Ulrich Berger,et al. Logical Approaches to Computational Barriers , 2006 .

[11] R. Mises. Grundlagen der Wahrscheinlichkeitsrechnung , 1919 .

[12] I︠U︡riĭ Leonidovich Ershov. Recursive algebra, analysis and combinatorics , 1998 .

[13] Douglas A. Cenzer,et al. Effectively closed sets and enumerations , 2008, Arch. Math. Log..

[14] A. Nies. Computability and randomness , 2009 .