Related papers: The Infinity of Randomness
We are now witnessing a rapid growth of a new part of group theory which has become known as "statistical group theory". A typical result in this area would say something like ``a random element (or a tuple of elements) of a group G has a…
According to quantum theory, randomness is a fundamental property of the universe yet classical physics is mostly deterministic. In this article I show that it is possible for deterministic systems to arise from random ones and discuss the…
Artificial intelligence commonly refers to the science and engineering of artificial systems that can carry out tasks generally associated with requiring aspects of human intelligence, such as playing games, translating languages, and…
Several methods are available in the literature to stochastically compare random variables and random vectors. We introduce the notion of asymptotic stochastic order for random processes and define four such orders. Various properties and…
We examine the extent to which random samplings from the values of a random set, determine the distribution of the random set itself. We also comment on how, given the statistics of the sampling, to detect the distribution. Several methods…
This paper proposes an alternative language for expressing results of the algorithmic theory of randomness. The language is more precise in that it does not involve unspecified additive or multiplicative constants, making mathematical…
Despite the impressive performance of Artificial Intelligence (AI) systems, their robustness remains elusive and constitutes a key issue that impedes large-scale adoption. Robustness has been studied in many domains of AI, yet with…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
In recent years there has been a tremendous surge in the general capabilities of AI systems, mainly fuelled by training foundation models on internetscale data. Nevertheless, the creation of openended, ever self-improving AI remains…
Transcript of G.J. Chaitin's 2 March 2000 Carnegie Mellon University School of Computer Science Distinguished Lecture. The notion of randomness is taken from physics and applied to pure mathematics in order to shed light on the…
We have introduced in former work the concept of Deep Randomness and its interest to design Unconditionally Secure communication protocols. We have in particular given an example of such protocol and introduced how to design a Deep Random…
Quasirandomness is a general mathematical concept meant to encapsulate several characteristics usually satisfied by random combinatorial objects, and which we regard as describing when a given object 'looks random'. In this survey we…
The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to…
Since human randomness production has been studied and widely used to assess executive functions (especially inhibition), many measures have been suggested to assess the degree to which a sequence is random-like. However, each of them…
We consider a generalized model of repeated quantum interactions, where a system $\mathcal{H}$ is interacting in a random way with a sequence of independent quantum systems $\mathcal{K}_n, n \geq 1$. Two types of randomness are studied in…
In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure $\lambda $, a choice needs to be made. One approach is to allow randomness tests to access the measure $\lambda $ as an…
Abstract Contextuality is a property of systems of random variables. The identity of a random variable in a system is determined by its joint distribution with all other random variables in the same context. When context changes, a variable…
An increasing number of decisions regarding the daily lives of human beings are being controlled by artificial intelligence (AI) algorithms in spheres ranging from healthcare, transportation, and education to college admissions,…
This paper questions the generally accepted assumption that one can make a random choice that is independent of the rest of the universe. We give a general description of any setup that could be conceived to generate random numbers. Based…
Here we present in a single essay a combination and completion of the several aspects of the problem of randomness of individual objects which of necessity occur scattered in our texbook "An Introduction to Kolmogorov Complexity and Its…