In this article, I consider the status of several statements analogous to the ChurchTuring thesis that assert that some definition of algorithmic randomness captures the intuitive conception of randomness. I argue that we should not only reject the theses that have appeared in the algorithmic randomness literature, but more generally that we ought not evaluate the adequacy of a definition of ra...

A set A is a base for Schnorr randomness if it is Turing reducible to a set R which is Schnorr random relative to A, and the notion of a base for weak 1-genericity can be defined similarly. We show that A is a base for Schnorr randomness if and only if A is a base for weak 1-genericity if and only if the halting set K is not Turing reducible to A. Furthermore, we define a set A to be high for S...

We study weak 2 randomness, weak randomness relative to ∅′ and Schnorr randomness relative to ∅′. One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the Martin-Löf random reals relative to A. We discuss the connections with LR-reducibility and also study the reducibility associated with weak 2randomness.

The goal of this paper is to examine randomness sources available in current mobile phones or other mobile devices. We identify potential sources of randomness and perform an analysis focused on the camera and the microphone input noise as promising sources of randomness. We also perform statistical tests and analyse quality of these sources of randomness including estimation of entropy in the ...

To test for “randomness” in spatial point patterns, we propose two test statistics that are obtained by “reducing” two-dimensional point patterns to the one-dimensional one. Also the exact and asymptotic distribution of these statistics are drawn.

Photonic band gaps in disordered two-dimensional photonic crystals are investigated for two typical types of randomness: cylinder site displacements ~site randomness! and cylinder radius variations ~size randomness!. The plane-wave expansion method with a supercell technique is applied to calculate the density-of-states ~DOS! for the disordered photonic crystals. In particular, numerical simula...

Algorithmic randomness over general spaces has been considered such as an effective topological space and a computable metric space. In this paper we generalize algorithmic randomness to a computable topological space. First we define computable measures on a computable topological space and study computability of the evaluation. Next we define randomnesses via three approaches. Measure randomn...

Randomness is an extremely useful resource in diverse fields in computer science, such as randomized algorithms, distributed algorithms, and cryptography. In these fields, we often assume that perfect randomness are available. However, the assumption may not be realistic, since random sources available in our physical world tend to be correlated and biased. Furthermore, there is no way to ensur...

Martin-Lof de ned in nite random sequences over a nite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After studying basic results and constructions for such randomness spaces a general invariance result is proved...