Computability and Randomness

Abstract

Historical Roots Von Mises. Around 1930, Kolmogorov and others founded the theory of probability, basing it on measure theory. Probability theory is concerned with the distribution of outcomes in sample spaces. It does not seek to give any meaning to the notion of an individual object, such as a single real number or binary string, being random, but rather studies the expected values of random variables.

Extending the Reach of the Point-to-Set Principle

“…This principle has been used to obtain strengthened lower bounds on the Hausdorff dimensions of generalized Furstenberg sets [27], extend the fractal intersection formula for Hausdorff dimension from Borel sets to arbitrary sets [25], and prove that Marstrand's projection theorem for Hausdorff dimension holds for any set E whose Hausdorff and packing dimensions coincide, whether or not E is analytic [26]. 2 (See [5,6,23,24] for reviews of these developments.) More recently, the point-to-set principle has been used to prove that V = L implies that the maximal thin co-analytic set has Hausdorff dimension 1 [40] and that the Continuum Hypothesis implies that every s ∈ (0, 1] is the Hausdorff dimension of a Hamel basis of the vector space R over the field Q [21].…”

Complexity-based permutation entropies: from deterministic time series to white noise

“…In computer science, algorithmic (or Kolmogorov) complexity refers to the shortest computer code that generates the sequence at hand [4]. In arithmetic there are also a number of proposals, some of them going deep into the concepts of randomness, compressibility and typicality [5,6,7]. This paper deals with the concept of permutation complexity of real-valued time series introduced in [8,9,10], so our symbols will be ordinal patterns or permutations of length L [11].…”