CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
| Hi-index | 0.00 |
Algorithmic randomness in the Cantor space, 2ω, has recently become the subject of intense study. Originally defined in terms of the fair coin measure, algorithmic randomness has since been extended, for example in Reimann and Slaman [22, 23], to more general measures. Others have meanwhile developed definitions of algorithmic randomness for different spaces, for example the space of continuous functions on the unit interval (Fouché [8, 9]), more general topological spaces (Hertling and Weihrauch [12]), and the closed subsets of 2ω (Barmpalias et al. [1], Kjos-Hanssen and Diamondstone [14]). Our work has also been to develop a definition of algorithmically random closed subsets. We take a very different approach, however, from that taken by Barmpalias et al. [1] and Kjos-Hanssen and Diamondstone [14].One of the central definitions of algorithmic randomness in Cantor space is Martin-Löf randomness. We use the probability theory of random closed sets (RACS) to prove that Martin-Löf randomness can be defined in the space of closed subsets of any locally compact, Hausdorff, second countable space. We then explore the Martin-Löf random closed subsets of the spaces N , 2ω, and R under different measures. In the case of 2ω we prove that the definitions of Barmpalias et al. [1] and Kjos-Hanssen and Diamondstone [14] are compatible with our approach. In the case of N we prove that the Martin-Löf random subsets are exactly those with Martin-Löf random characteristic functions. In the case of R we investigate the Martin-Löf random closed sets under generalized Poisson processes. This leads to a characterization of the Martin-Löf random elements of R as exactly the reals contained in some Martin-Löf random closed subset of R .