Related papers: Leibniz, Randomness and the Halting Probability
In 1686 in his Discours de Metaphysique, Leibniz points out that if an arbitrarily complex theory is permitted then the notion of "theory" becomes vacuous because there is always a theory. This idea is developed in the modern theory of…
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…
The ideas of aleatoric and epistemic uncertainty are widely used to reason about the probabilistic predictions of machine-learning models. We identify incoherence in existing discussions of these ideas and suggest this stems from the…
We introduce a notion of computable randomness for infinite sequences that generalises the classical version in two important ways. First, our definition of computable randomness is associated with imprecise probability models, in the sense…
Since the Turing test was first proposed by Alan Turing in 1950, the primary goal of artificial intelligence has been predicated on the ability for computers to imitate human behavior. However, the majority of uses for the computer can be…
The article contains an outline of a possible new direction for Computability Logic (see www.csc.villanova.edu/~japaridz/CL/ ), focused on computability without infinite memory or other impossible-to-possess computational resources. The new…
In this paper the author presents some non-conventional thoughts on the complexity of the Universe and the algorithmic reproducibility of the human brain, essentially sparked off by the notion of algorithmic complexity. We must warn that…
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
The 20th century has revealed two important limitations of scientific knowledge. On the one hand, the combination of Poincar\'e's nonlinear dynamics and Heisenberg's uncertainty principle leads to a world picture where physical reality is,…
The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the…
The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic…
We discuss the possibility of constructing a function that validates the definition or not definition of the partial recursive functions of one variable. This is a topic in computability theory, which was first approached by Alan M. Turing…
Due to common misconceptions about the Church-Turing thesis, it has been widely assumed that the Turing machine provides an upper bound on what is computable. This is not so. The new field of hypercomputation studies models of computation…
Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then…
The present paper gives an account for the general mathematical reader of the life and work of Martin Davis. Since two rather comprehensive autobiographical accounts and two long biographical interviews already exist, the present work…
This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Godel's proof, the first modern incompleteness…
This expository paper advocates an approach to physics in which ``typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions…
This year is the 100th birth anniversary of Richard Philips Feynman. This article commemorates his scientific contributions and lasting legacy.
In this paper, we study Bernoulli random sequences, i.e., sequences that are Martin-L\"of random with respect to a Bernoulli measure $\mu_p$ for some $p\in[0,1]$, where we allow for the possibility that $p$ is noncomputable. We focus in…
The primary aim of this chapter is, commemorating the 150th anniversary of Riemann's death, to explain how the idea of {\it Riemann sum} is linked to other branches of mathematics. The materials I treat are more or less classical and…