The use of Turing machines for calculating finite binary sequences is studied from the point of view of information theory and the theory of recursive functions. Various results are obtained concerning the number of instructions in programs. A modified form of Turing machine is studied from the same point of view. An application to the problem of defining a patternless sequence is proposed in terms of the concepts here 2 G. J. Chaitin developed. Introduction In this paper the Turing machine is regarded as a general purpose computer and some practical questions are asked about programming it. Given an arbitrary finite binary sequence, what is the length of the shortest program for calculating it? What are the properties of those binary sequences of a given length which require the longest programs? Do most of the binary sequences of a given length require programs of about the same length? The questions posed above are answered in Part 1. In the course of answering them, the logical ...

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We discuss some aspects of algorithmic randomness and state some open problems in this area. The first part is devoted to the question "What is a computably random sequence?" Here we survey some of the approaches to algorithmic randomness and address some questions on these concepts. In the second part we look at the Turing degrees of Martin-Lof random sets. Finally, in the third part we deal with relativized randomness. Here we look at oracles which do not change randomness. 1980 Mathematics Subject Classification. Primary 03D80; Secondary 03D28. 1 Introduction Formalizations of the intuitive notions of computability and randomness are among the major achievements in the foundations of mathematics in the 20th century. It is commonly accepted that various equivalent formal computability notions -- like Turing computability or -recursiveness -- which were introduced in the 1930s and 1940s adequately capture computability in the intuitive sense. This belief is expressed in the w...

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It is shown that the class of Kolmogorov-Loveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the Kolmogorov-Loveland stochastic sequences are closed under selecting subsequences by KolmogorovLoveland selection rules, i.e., by not necessarily monotonic partially computable selection rules. As a corollary, we obtain an easy proof for the previously known result that the Kolmogorov-Loveland stochastic sequences form a proper subclass of the Mises-Wald-Church stochastic sequences.

For every binary sequence A, there is an infinite binary sequence S such that A P tt S and S is stochastic in the sense of Kolmogorov and Loveland.

This paper appeared in T. Ferguson et al (eds.): Probability, statistics and game theory, papers in honor of David Blackwell, Institute for Mathematical Statistics 1996

Introduction Investigation of the measure-theoretic structure of complexity classes began with the development of resource-bounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resource-bounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], Ambos-Spies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resource-bounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resource-bounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their

Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated two-person zero-sum games in which the stage games have no pure strategy equilibrium. Computational complexity considerations are introduced to restrict players ’ strategy sets. The use of Kolmogorov complexity allows us to obtain a sufficient condition for equilibrium existence. The resulting theory has implications for the empirical literature that tests the equilibrium hypothesis in a similar context. In particular, the failure of some tests for randomness does not justify rejection of equilibrium play.

This article is a 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 philosophy. Intuitively, randomness is a lack of order or predictability. If randomness is the opposite of determinism, then algorithmic randomness is the opposite of computability. Besides many other things, these concepts have been used to quantify Ockham’s razor, solve the induction problem, and define intelligence. Contents 1 Why were you initially drawn to the study of computation