There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field. Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, KolmogorovLoveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.

An Introduction to Kolmogorov Complexity and Its Applications

The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity. The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional interpretation can be deduced. The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.

"relative State" Formulation of Quantum Mechanics

Matrix Factorization Techniques for Recommender Systems

Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

/pdf/the-knowledge-complexity-of-interactive-proof-systems-31apre0ecf.pdf

The knowledge complexity of interactive proof-systems

The mathematical theory of information

The book presents in a mathematical clear way the fundamentals of algorithmic information theory and a few selected applications. This 2nd edition presents new and important results obtained in recent years: the characterization of computable enumerable random reals, the construction of an Omega Number for which ZFC cannot determine any digits, and the first successful attempt to compute the exact values of 64 bits of a specific Omega Number. Finally, the book contains a discussion of some interesting philosophical questions related to randomness and mathematical knowledge. "Professor Calude has produced a first-rate exposition of up-to-date work in information and randomness." D.S. Bridges, Canterbury University, co-author, with Errett Bishop, of Constructive Analysis; "The second edition of this classic work is highly recommended to anyone interested in algorithmic information and randomness." G.J. Chaitin, IBM Research Division, New York, author of Conversations with a Mathematician; "This book is a must for a comprehensive introduction to algorithmic information theory and for anyone interested in its applications in the natural sciences." K. Svozil, Technical University of Vienna, author of Randomness & Undecidability in Physics

Information and Randomness: An Algorithmic Perspective

Very large databases are a major opportunity for science and data analytics is a remarkable new field of investigation in computer science. The effectiveness of these tools is used to support a “philosophy” against the scientific method as developed throughout history. According to this view, computer-discovered correlations should replace understanding and guide prediction and action. Consequently, there will be no need to give scientific meaning to phenomena, by proposing, say, causal relations, since regularities in very large databases are enough: “with enough data, the numbers speak for themselves”. The “end of science” is proclaimed. Using classical results from ergodic theory, Ramsey theory and algorithmic information theory, we show that this “philosophy” is wrong. For example, we prove that very large databases have to contain arbitrary correlations. These correlations appear only due to the size, not the nature, of data. They can be found in “randomly” generated, large enough databases, which—as we will prove—implies that most correlations are spurious. Too much information tends to behave like very little information. The scientific method can be enriched by computer mining in immense databases, but not replaced by it.

https://hal.archives-ouvertes.fr/hal-01380626/document

The Deluge of Spurious Correlations in Big Data

Information and Randomness

From the Publisher:
This book contains invited lectures and refereed talks presented at the First International Conference on Unconventional Models of Computation, Auckland, New Zealand, January 1998, organised by the Centre for Discrete Mathematics and Theoretical Computer Science, New Zealand, and the Santa Fe Institute, USA. It covers all major areas of unconventional computation, especially quantum computing, computing using organic molecules (DNA), and various proposals for computations that go beyond the Turing model.

Cristian S. Calude

Papers

The mathematical theory of information

Information and Randomness: An Algorithmic Perspective

The Deluge of Spurious Correlations in Big Data

Information and Randomness

Unconventional Models of Computation