Randomness conservation inequalities; information and independence in mathematical theories
Information and Control
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
Computable analysis: an introduction
Computable analysis: an introduction
ICALP '98 Proceedings of the 25th International Colloquium on Automata, Languages and Programming
Exact Expressions for some Randomness Tests
Proceedings of the 4th GI-Conference on Theoretical Computer Science
Computability on subsets of metric spaces
Theoretical Computer Science - Topology in computer science
Notions of Probabilistic Computability on Represented Spaces
Electronic Notes in Theoretical Computer Science (ENTCS)
An effective ergodic theorem and some applications
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
ALT '08 Proceedings of the 19th international conference on Algorithmic Learning Theory
Supermartingales in Prediction with Expert Advice
ALT '08 Proceedings of the 19th international conference on Algorithmic Learning Theory
A computable approach to measure and integration theory
Information and Computation
A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties
Theoretical Computer Science
Computability of probability measures and Martin-Löf randomness over metric spaces
Information and Computation
Applications of Effective Probability Theory to Martin-Löf Randomness
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
An Application of Martin-Löf Randomness to Effective Probability Theory
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Effective symbolic dynamics, random points, statistical behavior, complexity and entropy
Information and Computation
Prequential randomness and probability
Theoretical Computer Science
Supermartingales in prediction with expert advice
Theoretical Computer Science
Randomness and the ergodic decomposition
CiE'11 Proceedings of the 7th conference on Models of computation in context: computability in Europe
Computability of the Radon-Nikodym derivative
CiE'11 Proceedings of the 7th conference on Models of computation in context: computability in Europe
A constructive version of Birkhoff's ergodic theorem for Martin-Löf random points
Information and Computation
Noise vs computational intractability in dynamics
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
On-Line regression competitive with reproducing kernel hilbert spaces
TAMC'06 Proceedings of the Third international conference on Theory and Applications of Models of Computation
Von Neumann's Biased Coin Revisited
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
Hi-index | 5.23 |
The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Löf) and to arbitrary distributions (by Levin). We recall the main ideas and problems of Levin's theory, and report further progress in the same framework. The issues are the following: • Allow non-compact spaces (like the space of continuous functions, underlying the Brownian motion). • The uniform test (deficiency of randomness) dp(x) (depending both on the outcome x and the measure P) should be defined in a general and natural way. • See which of the old results survive: existence of universal tests, conservation of randomness, expression of tests in terms of description complexity, existence of a universal measure, expression of mutual information as "deficiency of independence". • The negative of the new randomness test is shown to be a generalization of complexity in continuous spaces; we show that the addition theorem survives.The paper's main contribution is introducing an appropriate framework for studying these questions and related ones (like statistics for a general family of distributions).