Depth as Randomness Deficiency

Sophistication as Randomness Deficiency

The sophistication of a string measures how much structural information it contains. We introduce naive sophistication, a variant of sophistication based on randomness deficiency. Naive sophistication measures the minimum number of bits needed to specify a set in which the string is a typical element. Thanks to Vereshchagin and Vitányi, we know that sophistication and naive sophistication are e...

Universality, optimality, and randomness deficiency

A Martin-Löf test U is universal if it captures all non-Martin-Löf random sequences, and it is optimal if for every ML-test V there is a c ∈ ω such that ∀n(Vn+c ⊆ Un). We study the computational differences between universal and optimal ML-tests as well as the effects that these differences have on both the notion of layerwise computability and the Weihrauch degree of LAY, the function that pro...

Randomness as a Constraint

Running head : RANDOMNESS AS INFERENCE 1 Subjective randomness as statistical inference

Some events seem more random than others. For example, when tossing a coin, a sequence of eight heads in a row does not seem very random. Where do these intuitions about randomness come from? We argue that subjective randomness can be understood as the result of a statistical inference assessing the evidence that an event provides for having been produced by a random generating process. We show...

On Bitcoin as a public randomness source

1 Stanford University 2 Concordia University 3 Princeton University Abstract. We formalize the use of Bitcoin as a source of publiclyverifiable randomness. As a side-effect of Bitcoin’s proof-of-work-based consensus system, random values are broadcast every time new blocks are mined. We can derive strong lower bounds on the computational min-entropy in each block: currently, at least 68 bits of...