Abstract
AbstractWe investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal.
Publisher
Cambridge University Press (CUP)
Reference45 articles.
1. Lebesgue Convergence Theorems and Reverse Mathematics
2. Algorithmic randomness, reverse mathematics, and the dominated convergence theorem
3. [35] Nies, A. , Stephan, F. , and Terwijn, S. A. , Randomness, relativization and Turing degrees, this Journal, vol. 70 (2005), no. 2, pp. 515–535.
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献