Randomness, Computation and Mathematics

Abstract

Abstract. This article examines some of the recent advances in our understanding of algorithmic randomness. It also discusses connections with various areas of mathematics, computer science and other areas of science. Some questions and speculations will be discussed.

Monotonous betting strategies in warped casinos

“…The study of the power of strategies in (ii) is quite interesting from the point of view of stochastic processes, as it relates to key concepts such as martingale decompositions, variation and various forms of boundedness or integrability. Questions (1) and ( 2) under (ii) are also directly relevant to a question about the separation of two randomness notions in algorithmic information theory, asked by Kastermans (see [Downey, 2012] and [Downey and Hirschfeldt, 2010, §7.9]). As we point out in §5, a positive answer of (1) or (2) for the case of strategies under (ii) would give a very simple and elegant positive answer to Kasterman's question.…”