An introduction to Kolmogorov Complexity and Its Applications
An introduction to Kolmogorov Complexity and Its Applications
Algorithmic Information Theory Reveals Knowledge and Compression Boundaries
Algorithmic Information Theory (AIT), a cornerstone of theoretical computer science and mathematics, changes our views on data, complexity, and randomness. Kolmogorov Complexity (KC), the foundation of AIT, is often considered the most accurate indicator of the information in a single item, such as a number, string of text, or set of bits.
AIT: Algorithmic Information Theory
Algorithmic information theory studies Kolmogorov complexity and other complexity metrics applied to strings and data structures in computer science and mathematics. It measures the descriptive complexity, or computational complexity, of a single object, unlike classical Shannon entropy, which measures the average information content over a probability distribution.
AIT measures the absolute information needed to specify an item. AIT relies on Ray Solomonoff's complexity and probability theorem, which Andrey Kolmogorov independently published. The greater field of probability and descriptive complexity is called Kolmogorov complexity.
Kolmogorov complexity?
In algorithmic information theory, Kolmogorov complexity assesses the computational or descriptive complexity of an item, such as a string of text or a sequence of bits.
The Kolmogorov complexity of an item x is the length of the shortest computer program (using a fixed, universal Turing machine U) that generates x as its only output before quitting.
K(x)=shortest program p with U(p)=x.
where U is a universal Turing machine.
KC quantifies descriptive complexity mathematically. It intuitively depicts situations' compressibility. A pattern string can be constructed with a small program, lowering Kolmogorov complexity. For instance, “x = 0101010101010101” can be simplified to “print 01 eight times.”
Complex or algorithmically random items have a smallest program roughly as long as the item. Kolmogorov complexity is high because “print y exactly” is the only short program that works for a random string y.
Kolmogorov complexity is the shortest lossless data compression “zip file” for any data. The Invariance Theorem limits differences with a fixed, additive constant, regardless of universal programming language choices.
Example
Take two strings:
x = 01010101010101 y = 110010000110000111011... (random string)
Simple description of x: “print 01 eight times.”
The program is brief and has low Kolmogorov complexity.
No straightforward pattern exists for y. The shortest program may be “print y exactly.”
Long program = high Kolmogorov complexity.
Proof and Computational Limits
Kolmogorov complexity is elegantly defined but has severe theoretical restrictions.
A key discovery is that the Kolmogorov complexity function K(x) cannot be computed. No universal method can determine the complexity K(x) of any string x. The Halting Problem's non-computability discourages naive attempts to construct a function by iterating over all conceivable programs because certain programs won't terminate. The complexity function K solves the halting issue Turing-equivalently.
Incompleteness Theorem: Chaitin's incompleteness theorem adds to the theoretical restrictions by showing that a string cannot be formally proven if its complexity surpasses a threshold L, which varies by formal axiomatic system. Even if most strings are complex and cannot be compressed, the axiomatic technique does not prove that a string is exceedingly intricate. This impossible result resembles the Berry paradox and Gödel's incompleteness theorem.
Applications of Kolmogorov complexity
Gregory Chaitin, who presented the theorem, Andrey Kolmogorov, who is associated with the complexity measure, and Ray Solomonoff, who invented algorithmic probability in 1960, all contributed to Kolmogorov complexity.
AIT measures individual item information content, unlike Shannon entropy, which determines the average across a probability distribution.
Many fields are supported by AIT:
A string is algorithmically random if all computer programs that can create it are as least as long as the string.
Minimum Description Length (MDL): This statistical and inductive inference principle states that the best hypothesis for a dataset is the one that minimises the total description length—the hypothesis and the data encoded using it.
Universal Probability: P(x)-free probability. The equation log 1/P(x)=K(x)+O(1), expressing the likelihood of a universal Turing machine producing x from a random input stream, is closely related to the prefix-free complexity K(x).
Biology: KC may favour evolutionary processes with simpler “programs” (genomes), such as the insect head direction circuit, which have low Kolmogorov complexity.
Research is also ongoing on Conditional Kolmogorov Complexity (K(x|y)), which determines the length of the shortest program to compute x given y, and Time-Bounded Complexity, which limits the program search.
