Related papers: Some Theorems on the Algorithmic Approach to Proba…
The book A Treatise on Probability was published by John Maynard Keynes in 1921. It contains a critical assessment of the foundations of probability and of the current statistical methodology. As a modern reader, we review here the aspects…
This paper looks at Turing's postulations about Artificial Intelligence in his paper 'Computing Machinery and Intelligence', published in 1950. It notes how accurate they were and how relevant they still are today. This paper notes the…
The notion of an individual random sequence goes back to von Mises. We describe the evolution of this notion, especially the use of martingales (suggested by Ville), and the development of algorithmic information theory in 1960s and 1970s…
The Bayesian statistical paradigm uses the language of probability to express uncertainty about the phenomena that generate observed data. Probability distributions thus characterize Bayesian analysis, with the rules of probability used to…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
Two philosophical applications of the concept of program-size complexity are discussed. First, we consider the light program-size complexity sheds on whether mathematics is invented or discovered, i.e., is empirical or is a priori. Second,…
This paper has been withdrawn by the author; a revised version is part of the author's phd-thesis "Quasi-logarithmic structures" (Zurich, 2007).
The familiar theories of physics have the feature that the application of the theory to make predictions in specific circumstances can be done by means of an algorithm. We propose a more precise formulation of this feature --- one based on…
This abstract extends on the previous work (arXiv:1407.2646, arXiv:1606.00075) on program induction using probabilistic programming. It describes possible further steps to extend that work, such that, ultimately, automatic probabilistic…
Belief updating in Bayes nets, a well known computationally hard problem, has recently been approximated by several deterministic algorithms, and by various randomized approximation algorithms. Deterministic algorithms usually provide…
In this paper we study subsequences of random numbers. In Kamae (1973), selection functions that depend only on coordinates are studied, and their necessary and sufficient condition for the selected sequences to be normal numbers is given.…
Some formulas and speculations are presented relative to integrable systems and quantum mechanics.
This article is a brief guide to the field of algorithmic information theory (AIT), its underlying philosophy, and the most important concepts. AIT arises by mixing information theory and computation theory to obtain an objective and…
In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has…
In this article we demonstrate how algorithmic probability theory is applied to situations that involve uncertainty. When people are unsure of their model of reality, then the outcome they observe will cause them to update their beliefs. We…
This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. This is camera-ready copy prepared for publication as a book, but at the last minute I decided to publish it…
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…
This is a typeset version of Alan Turing's Second World War research paper \textit{The Applications of Probability to Cryptography}. A companion paper \textit{Paper on Statistics of Repetitions} is also available in typeset form from arXiv…
We survey concepts at the frontier of research connecting artificial, animal and human cognition to computation and information processing---from the Turing test to Searle's Chinese Room argument, from Integrated Information Theory to…
This is a draft version of an invited article for a forthcoming book `The genesis of Langlands Program', eds. Julia Mueller and Freydoon Shahidi, which will be published in the London Mathematics Society Lecture Notes Series. It gives a…