The Kolmogorov complexity of a string is the length of the shortest program that generates it. A binary string is said to have trivial Kolmogorov complexity if its complexity is at most the complexity of its length. Intuitively, such strings carry no more information than the information that is inevitably coded into their length (which is the same as the information coded into a sequence of 0s...

The notion of Kolmogorov complexity (=the minimal length of a program that generates some object) is often useful as a kind of language that allows us to reformulate some notions and therefore provide new intuition. In this survey we provide (with minimal comments) many different examples where notions and statements that involve Kolmogorov complexity are compared with their counterparts not in...

We use the connection of Hausdorff dimension and Kolmogorov complexity to describe a geometry on the Cantor set including concepts of angle, projections and scalar multiplication. A question related to compressibility is addressed using these geometrical ideas.

The very first Kolmogorov’s paper on algorithmic information theory [1] was entitled “Three approaches to the definition of the quantity of information”. These three approaches were called combinatorial, probabilistic and algorithmic. Trying to establish formal connections between combinatorial and algorithmic approaches, we prove that every linear inequality including Kolmogorov complexities c...

Conditional Kolmogorov complexity can be understood as the complexity of the problem “ ”, where is the problem “construct ” and is the problem “construct ”. Other logical operations ( ) can be interpreted in a similar way, extending Kolmogorov interpretation of intuitionistic logic and Kleene realizability. This leads to interesting problems in algorithmic information theory. Some of these ques...

The Kolmogorov complexity theory emerged in sixties; main definitions were independently found by Ray Solomonoff, A.N. Kolmogorov and G. Chaitin. The motivations for the definition were quite different. For Solomonoff the main goal was inductive inference theory. Kolmogorov's work was closely connected with foundations of probability theory and information theory. Chaitin's studied the length o...

It is known that dimension of a set in a metric space can be characterized in information-related terms – in particular, in terms of Kolmogorov complexity of different points from this set. The notion of Kolmogorov complexity K(x) – the shortest length of a program that generates a sequence x – can be naturally generalized to conditional Kolmogorov complexity K(x : y) – the shortest length of a...