Algorithmic information theory (nonfiction)

Algorithmic information theory (AIT) is a "merger of information theory and computer science" that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as strings or any other data structure. In other words, it is shown within algorithmic information theory that computational incompressibility "mimics" (except for a constant that only depends on the chosen universal programming language) the relations or inequalities found in information theory. According to Gregory Chaitin, it is "the result of putting Claude Shannon's information theory and Alan Turing's computability theory into a cocktail shaker and shaking vigorously."

Algorithmic information theory principally studies measures of irreducible information content of strings (or other data structures). Because most mathematical objects can be described in terms of strings, or as the limit of a sequence of strings, it can be used to study a wide variety of mathematical objects, including integers. Indeed, one of the main motivations behind AIT is the very study of the information carried by mathematical objects as in the field of metamathematics, e.g., as shown by the incompleteness results mentioned below. Other main motivations came from: surpassing the limitations of classical information theory for single and fixed objects; formalizing the concept of randomness; and finding accurate probabilistic inference without prior knowledge of the probability distribution (e.g., whether it is independent and identically distributed, markovian, or even stationary).

In this way, AIT is known to be basically found upon three main mathematical concepts and the relations between them:

• algorithmic complexity
• algorithmic randomness
• algorithmic probability

Besides the formalization of a universal measure for irreducible information content of computably generated objects, some main achievements of AIT were to show that: in fact algorithmic complexity follows (in the self-delimited case) the same inequalities (except for a constant) that entropy does, as in classical information theory; randomness is incompressibility; and, within the realm of randomly generated software, the probability of occurrence of any data structure is of the order of the shortest program that generates it when running on a universal machine.

See also

• Algorithmic probability (nonfiction) - a mathematical method of assigning a prior probability to a given observation.
• Algorithmically random sequence (nonfiction) - an infinite sequence of binary digits that appears[clarification needed] random to any algorithm. The notion can be applied analogously to sequences on any finite alphabet (e.g. decimal digits). Random sequences are key objects of study in algorithmic information theory.
• Chaitin's constant (nonfiction) - a real number representing the probability that a randomly constructed program will halt. Although there are infinitely many halting probabilities, one for each method of encoding programs, it is common to use the letter Ω to refer to them as if there were only one. Each halting probability is a normal and transcendental real number that is not computable, which means that there is no algorithm to compute its digits. Indeed, each halting probability is Martin-Löf random, meaning there is not even any algorithm which can reliably guess its digits.
• Chaitin–Kolmogorov randomness (nonfiction)
• Computationally indistinguishable (nonfiction)
• Distribution ensemble (nonfiction)
• Epistemology (nonfiction)
• Inductive inference (nonfiction)
• Inductive probability (nonfiction)
• Invariance theorem (nonfiction)
• Kolmogorov complexity (nonfiction) - given an object, such as a piece of text, Kolmogorov complexity the length of the shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources needed to specify the object.
• Limits of knowledge (nonfiction)
• Minimum description length (nonfiction)
• Minimum message length (nonfiction)
• Pseudorandom ensemble (nonfiction)
• Pseudorandom generator (nonfiction)
• Claude Shannon (nonfiction)
• Simplicity theory (nonfiction)
• Solomonoff's theory of inductive inference (nonfiction)
• Alan Turing (nonfiction)
• Uniform ensemble (nonfiction)

• Algorithmic information theory @ Wikipedia