23 research outputs found
Algorithmic Randomness and Capacity of Closed Sets
We investigate the connection between measure, capacity and algorithmic
randomness for the space of closed sets. For any computable measure m, a
computable capacity T may be defined by letting T(Q) be the measure of the
family of closed sets K which have nonempty intersection with Q. We prove an
effective version of Choquet's capacity theorem by showing that every
computable capacity may be obtained from a computable measure in this way. We
establish conditions on the measure m that characterize when the capacity of an
m-random closed set equals zero. This includes new results in classical
probability theory as well as results for algorithmic randomness. For certain
computable measures, we construct effectively closed sets with positive
capacity and with Lebesgue measure zero. We show that for computable measures,
a real q is upper semi-computable if and only if there is an effectively closed
set with capacity q
Lines Missing Every Random Point
We prove that there is, in every direction in Euclidean space, a line that
misses every computably random point. We also prove that there exist, in every
direction in Euclidean space, arbitrarily long line segments missing every
double exponential time random point.Comment: Added a section: "Betting in Doubly Exponential Time.
Covering the Recursive Sets
Contains fulltext :
147448.pdf (preprint version ) (Open Access
Covering the recursive sets
Contains fulltext :
173462.pdf (preprint version ) (Open Access
Lowness for the class of schnorr random reals
10.1137/S0097539704446323SIAM Journal on Computing353647-657SMJC
Higher Kurtz randomness
10.1016/j.apal.2010.04.001Annals of Pure and Applied Logic161101280-1290APAL
Selection by recursively enumerable sets
10.1007/978-3-642-38236-9_14Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)7876 LNCS144-15