Algorithmic Information Theory: Fundamentals and Applications

By Fouad Sabry

What Is Algorithmic Information Theory

The field of theoretical computer science known as algorithmic information theory, or AIT for short, is concerned with the relationship between computation and information of computably generated things (as opposed to stochastically generated objects), such as strings or any other data structure. In other words, algorithmic information theory demonstrates that computational incompressibility "mimics" (with the exception of a constant that solely depends on the universal programming language that was selected) the relations or inequalities that are present in information theory. Gregory Chaitin explains that it is "the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking them vigorously."

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Algorithmic Information Theory

Chapter 2: Kolmogorov Complexity

Chapter 3: Chaitin's Constant

Chapter 4: Gregory Chaitin

Chapter 5: Algorithmic Probability

Chapter 6: Solomonoff's Theory of Inductive Inference

Chapter 7: Minimum Description Length

Chapter 8: Random Sequence

Chapter 9: Algorithmically Random Sequence

Chapter 10: Incompressibility Method

(II) Answering the public top questions about algorithmic information theory.

(III) Real world examples for the usage of algorithmic information theory in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of algorithmic information theory' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of algorithmic information theory.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 27, 2023

Book preview

Chapter 1: Algorithmic information theory

Computably created objects, such as strings or any other data structure, are of interest to the field of theoretical computer science known as algorithmic information theory (AIT), which studies the connection between computation and information in this context. In other words, the relations or inequalities established by information theory are mimicked (with the exception of a constant that depends exclusively on the universal programming language of choice) through computational incompressibility.

When it comes to complexity metrics, strings are where Algorithmic Information Theory really shines (or other data structures). It can be used to explore a wide range of mathematical objects, including integers, because most mathematical objects can be defined in terms of strings, or as the limit of a sequence of strings.

The information content of a string is, informally speaking, the same as the length of the most compact conceivable self-contained representation of that string from the perspective of algorithmic information theory. A self-contained representation is a program that, when run, produces the original string, written in a universally-accepted but arbitrary programming language.

In this light, a three-thousand-page encyclopedia has less data than the same number of pages filled with random letters, despite the encyclopedia's obvious advantages. This is because it requires knowledge of every letter's identity in order to rebuild the full sequence of random letters. One could probably recreate the line Ths sntnc hs lw nfrmtn cntnt from the context and the consonants present even if all the vowels were deleted from it, as is the case with the encyclopedia.

Algorithmic information theory provides formal, logical definitions of a random string and a random infinite sequence, unlike classical information theory, which relied on physical and philosophical intuitions about nondeterminism and likelihood. The Kolmogorov complexity of a string is invariant up to an additive constant dependent only on the choice of universal Turing machine, but the set of random strings relies on the choice of the universal Turing machine used to define it. This means that the collection of infinite random sequences does not depend on which universal machine is used.

Theoretical insights from algorithmic information retrieval, including the theorem of Chaitin on incompleteness, seem to go against the grain of conventional mathematical and philosophical thinking.

Most notable among these is the construction of Chaitin's constant Ω, a real integer representing the likelihood that a universal Turing machine would terminate when fed random outcomes from a fair coin toss as input (sometimes thought of as the probability that a random computer program will eventually halt).

Although Ω is easily defined, in any consistent axiomatizable theory one can only compute finitely many digits of Ω, therefore it is, in a way, unknown, providing an absolute limit on knowledge that is reminiscent of Gödel's incompleteness theorems.

Although the digits of Ω cannot be determined, many properties of Ω are known; for example, the binary digits are uniformly dispersed since it is a random sequence generated by an algorithm (in fact it is normal).

Ray Solomonoff established the field of algorithmic information theory in the early 1960s; Andrey Kolmogorov (1965) and Gregory Chaitin (1966) each made significant contributions to the field independently.

Kolmogorov complexity, often known as algorithmic information complexity, comes in a number of flavors; Leonid Levin is mostly responsible for the most popular one, which is based on self-delimiting programs (1974).

Per Martin-Löf also contributed significantly to the information theory of infinite sequences.

Mark Burgin, in a manuscript submitted to a journal edited by Andrey Kolmogorov, established the Blum axioms (Blum, 1967), an axiomatic approach to algorithmic information theory (Burgin 1982).

In the field of algorithmic information theory, the axiomatic approach incorporates alternative methods.

Various algorithmic information measures can be viewed as special cases of axiomatically specified measures.

Rather than reiterating previous theorems,, including the fundamental invariance theorem, in respect to every criterion, All of these results can be simply derived from a single corresponding theorem stated in an axiomatic context.

The axiomatic method in mathematics has this benefit in general.

The book expands on the axiomatic method of algorithmic information theory (Burgin 2005) and applies it to the field of software metrics (Burgin and Debnath 2010), 2003; It was Debnath and Burgin who, 2003).

If the Kolmogorov complexity of a binary string is at least the string length, then the string can be considered random. Some strings of a particular length are truly random, and nearly all strings are extremely close to being truly random, as shown by a simple counting argument. The collection of random strings does depend on a fixed choice of universal Turing machine (informally, a fixed description language in which the descriptions are delivered), just as Kolmogorov complexity does. One can (and does) discuss the qualities of random strings as a group without first specifying a universal machine because the collection of random strings as a whole has similar properties independent of the fixed machine.

We say that a binary sequence infinity long is random if and only if, for some fixed value of c, for every n, The first n-element section of the sequence has a Kolmogorov complexity of at most n c.

From the perspective of the usual fair coin or Lebesgue measure on the space of infinite binary sequences, it can be proved that practically all sequences are random.

Also, Given that it is possible to demonstrate that the difference in Kolmogorov complexity between any two universal machines is at most constant,, It makes no difference which universal machine you use to compile random infinite sequences (in contrast to finite strings).

This definition of randomness is usually called Martin-Löf randomness, after Per Martin-Löf, in order to set it apart from related concepts of randomness.

To differentiate it from more robust ideas of randomness (2-randomness), it is