Related papers: Randomness and Differentiability

This article is a brief personal account of the past, present, and future of algorithmic randomness, emphasizing its role in inductive inference and artificial intelligence. It is written for a general audience interested in science and…

We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue differentiation theorem. We generalize Ko's framework for polynomial space computability in $\mathbb{R}^n$ to define…

We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that…

We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically…

We study the question, ``For which reals $x$ does there exist a measure $\mu$ such that $x$ is random relative to $\mu$?'' We show that for every nonrecursive $x$, there is a measure which makes $x$ random without concentrating on $x$. We…

We investigate the role of continuous reductions and continuous relativisation in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with…

In the study of random access machines (RAMs) it has been shown that the availability of an extra input integer, having no special properties other than being sufficiently large, is enough to reduce the computational complexity of some…

This paper explores a novel definition of Schnorr randomness for noncomputable measures. We say $x$ is uniformly Schnorr $\mu$-random if $t(\mu,x)<\infty$ for all lower semicomputable functions $t(\mu,x)$ such that $\mu\mapsto\int…

We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n, all but countably many reals are n-random for such a measure,…

Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically computable if there is a partial computable function f whose domain has…

Schnorr showed that a real is Martin-Loef random if and only if all of its initial segments are incompressible with respect to prefix-free complexity. Fortnow and independently Nies, Stephan and Terwijn noticed that this statement remains…

Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of…

Since their appearance in the 1950s, computational models capable of performing probabilistic choices have received wide attention and are nowadays pervasive in almost every areas of computer science. Their development was also inextricably…

We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in…

We study the computational power of randomized computations on infinite objects, such as real numbers. In particular, we introduce the concept of a Las Vegas computable multi-valued function, which is a function that can be computed on a…

We solve the covering problem for Demuth randomness, showing that a computably enumerable set is computable from a Demuth random set if and only if it is strongly jump-traceable. We show that on the other hand, the class of sets which form…

We elaborate the notions of Martin-L\"of and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform…

In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…

Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the…

A real \alpha is called recursively enumerable ("r.e." for short) if there exists a computable, increasing sequence of rationals which converges to \alpha. It is known that the randomness of an r.e. real \alpha can be characterized in…