Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of random elements coincide. Elaborating on work of Bienvenu [1], we determine all the implications that hold between the equivalence relations induce...

Randomness is an incredibly useful phenomenon, with applications in many fields. We can solve some problems using randomness exponentially faster than we know how to do deterministically. Do we need truly random bits for these algorithms? What if we don’t have access to perfect randomness? Using results from the theory of computational pseudorandomness, we find that many algorithms can be made ...

Do completely unpredictable events exist? Classical physics excludes fundamental randomness. Although quantum theory makes probabilistic predictions, this does not imply that nature is random, as randomness should be certified without relying on the complete structure of the theory being used. Bell tests approach the question from this perspective. However, they require prior perfect randomness...

Randomness is crucial to computer science, both in theory and applications. In complexity theory, randomness augments computers to offer more powerful models. In cryptography, randomness is essential for seed generation, where the computational model used is generally probabilistic. However, ideal randomness, which is usually assumed to be available in computer science theory and applications, ...

When testing a set of data for randomness according to a probability distribution that depends on a parameter, access to this parameter can be considered as a computational resource. We call a randomness test Hippocratic if it is not permitted to access this resource. In these terms, we show that for Bernoulli measures μp, 0 6 p 6 1 and the Martin-Löf randomness model, Hippocratic randomness of...

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computabl...

The second law of thermodynamics states that entropy, as a measure of randomness in a system, increases over time. Although studies have investigated biological sequence randomness from different aspects, it remains unknown whether sequence randomness changes over time and whether this change consists with the second law of thermodynamics. To capture the dynamics of randomness in molecular sequ...

2 What we have learned? A personal pick 4 2.1 From randomness to complexity . . . . . . . . . . . . . . . . . . . . . 4 2.2 Formalization of randomness: infinite strings . . . . . . . . . . . . . 5 2.3 Random versus lawless . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Randomness and finite strings: incompressibility . . . . . . . . . . . 7 2.5 Representation and Kolmogorov complexi...

We observe that the randomness-complexity of an algorithm effects the time-complexity of implementing a version of it that utilizes a weak source of randomness (through a randomness-extractor). This provides an additional motivation for the study of the randomness complexity of randomized algorithms. We note that this motivation applies especially in the case that derandomization is prohibiting...

The Griffiths inequalities for Ising spin glasses are proved on the Nishimori line with various bond randomness which includes Gaussian and ±J bond randomness. The proof for Ising systems with Gaussian bond randomness has already been carried out by Morita et al, which uses not only the gauge theory but also the properties of the Gaussian distribution, so that it cannot be directly applied to t...