# Turing and randomness

@inproceedings{Downey2017TuringAR, title={Turing and randomness}, author={R. Downey}, booktitle={The Turing Guide}, year={2017} }

In an unpublished manuscript, Turing anticipated the basic ideas behind the theory of algorithmic randomness. He did so by nearly 30 years. Turing used a computationally constrained version of “measure theory” to answer a question of Borel in number theory. This question concerned constructing what are called “absolutely normal” numbers. In this article, I will try to explain what these mysterious terms mean, and what Turing did. 1 Borel, number theory and normality 1.1 Repeated decimals in… Expand

#### Topics from this paper

#### 2 Citations

Foundations of Online Structure Theory II; The Operator Approach

- Mathematics
- 2020

We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying… Expand

FOUNDATIONS OF ONLINE STRUCTURE THEORY

- Computer Science
- The Bulletin of Symbolic Logic
- 2019

Abstract The survey contains a detailed discussion of methods and results in the new emerging area of online “punctual” structure theory. We also state several open problems.

#### References

SHOWING 1-10 OF 57 REFERENCES

Turing's Normal Numbers: Towards Randomness

- Mathematics, Computer Science
- CiE
- 2012

An algorithm that produces real numbers normal to every integer base is given that proves the existence of computable normal numbers and it is the best solution to date to Borel's problem on giving examples of normal numbers. Expand

An Introduction to Kolmogorov Complexity and Its Applications

- Computer Science, Psychology
- Texts and Monographs in Computer Science
- 1993

The book presents a thorough treatment of the central ideas and their applications of Kolmogorov complexity with a wide range of illustrative applications, and will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. Expand

Turing's unpublished algorithm for normal numbers

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 2007

In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for… Expand

A Formal Theory of Inductive Inference. Part II

- Computer Science, Mathematics
- Inf. Control.
- 1964

1. Summary In Part I, four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably… Expand

Randomness Through Computation: Some Answers, More Questions

- Computer Science
- 2011

This review volume consists of a set of chapters written by leading scholars, most of them founders of their fields. The volume is intended to explain the phenomenon of Randomness through the use of… Expand

On the Random Character of Fundamental Constant Expansions

- Computer Science, Mathematics
- Exp. Math.
- 2001

A theory to explain random behavior for the digits in the expansions of fundamental mathematical constants and proofs of base-2 normality for a collection of celebrated constants, including π, log 2, ζ(3), and others are proposed. Expand

Computability and randomness

- Mathematics
- 2009

The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The… Expand

The Definition of Random Sequences

- Computer Science, Mathematics
- Inf. Control.
- 1966

It is shown that the random elements as defined by Kolmogorov possess all conceivable statistical properties of randomness and can equivalently be considered as the elements which withstand a certain universal stochasticity test. Expand

Algorithmic Randomness and Complexity

- Mathematics, Computer Science
- Theory and Applications of Computability
- 2010

This chapter discusses Randomness-Theoretic Weakness, Omega as an Operator, Complexity of C.E. Sets, and other Notions of Effective Randomness. Expand

Computing Machinery and Intelligence.

- Computer Science, Sociology
- 1980

The question, “Can machines think?” is considered, and the question is replaced by another, which is closely related to it and is expressed in relatively unambiguous words. Expand