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In algorithmic information theory (a subfield of computer science), the Kolmogorov complexity of an object, such as a piece of text, is a measure of the computational resources needed to specify the object. It is named after Soviet Russian mathematician Andrey Kolmogorov.
For example, consider the following two strings of length 64, each containing only lowercase letters, digits, and spaces:
The first string has a short English-language description, namely "ab 32 times", which consists of 11 characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, which has 64 characters.
More formally, the complexity of a string is the length of the string's shortest description in some fixed universal description language. The sensitivity of complexity relative to the choice of description language is discussed below. It can be shown that the Kolmogorov complexity of any string cannot be more than a few bytes larger than the length of the string itself. Strings whose Kolmogorov complexity is small relative to the string's size are not considered to be complex. The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.
To define Kolmogorov complexity, we must first specify a description language for strings. Such a description language can be based on any programming language, such as Lisp, Pascal, or Java Virtual Machine bytecode. If P is a program which outputs a string x, then P is a description of x. The length of the description is just the length of P as a character string, multiplied by the number of bits in a character (e.g. 7 for ASCII).
We could alternatively choose an encoding for Turing machines, where an encoding is a function which associates to each Turing Machine M a bitstring <M>. If M is a Turing Machine which on input w outputs string x, then the concatenated string <M> w is a description of x. For theoretical analysis, this approach is more suited for constructing detailed formal proofs and is generally preferred in the research literature. The binary lambda calculus may provide the simplest definition of complexity yet. In this article we will use an informal approach.
Any string s has at least one description, namely the program
function GenerateFixedString() return s
If a description of s, d(s), is of minimal length—i.e. it uses the fewest number of characters—it is called a minimal description of s. Then the length of d(s)—i.e. the number of characters in the description—is the Kolmogorov complexity of s, written K(s). Symbolically,
We now consider how the choice of description language affects the value of K and show that the effect of changing the description language is bounded.
Theorem. If K1 and K2 are the complexity functions relative to description languages L1 and L2, then there is a constant c (which depends only on the languages L1 and L2) such that
Proof. By symmetry, it suffices to prove that there is some constant c such that for all bitstrings s,
Now, suppose there is a program in the language L1 which acts as an interpreter for L2:
function InterpretLanguage(string p)
where p is a program in L2. The interpreter is characterized by the following property:
Thus if P is a program in L2 which is a minimal description of s, then InterpretLanguage(P) returns the string s. The length of this description of s is the sum of
This proves the desired upper bound.
See also invariance theorem.
The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff who published it in 1960, describing it in "A Preliminary Report on a General Theory of Inductive Inference" as part of his invention of algorithmic probability. He gave a more complete description in his 1964 publications, "A Formal Theory of Inductive Inference," Part 1 and Part 2 in Information and Control.
Andrey Kolmogorov later independently published this theorem in Problems Inform. Transmission, Gregory Chaitin also presents this theorem in J. ACM; Chaitin's paper was submitted October 1966, revised in December 1968 and cites both Solomonoff's and Kolmogorov's papers.
The theorem says that among algorithms that decode strings from their descriptions (codes) there exists an optimal one. This algorithm, for all strings, allows codes as short as allowed by any other algorithm up to an additive constant that depends on the algorithms, but not on the strings themselves. Solomonoff used this algorithm, and the code lengths it allows, to define a string's `universal probability' on which inductive inference of a string's subsequent digits can be based. Kolmogorov used this theorem to define several functions of strings: complexity, randomness, and information.
When Kolmogorov became aware of Solomonoff's work, he acknowledged Solomonoff's priority For several years, Solomonoff's work was better known in the Soviet Union than in the Western World. The general consensus in the scientific community, however, was to associate this type of complexity with Kolmogorov, who was concerned with randomness of a sequence while Algorithmic Probability became associated with Solomonoff, who focused on prediction using his invention of the universal a priori probability distribution.
In the following discussion let K(s) be the complexity of the string s.
It is not hard to see that the minimal description of a string cannot be too much larger than the string itself: the program GenerateFixedString above that outputs s is a fixed amount larger than s.
Theorem. There is a constant c such that
The first result is that there is no way to compute K.
Theorem. K is not a computable function.
In other words, there is no program which takes a string s as input and produces the integer K(s) as output. We show this by contradiction by making a program that creates a string that should only be able to be created by a longer program. Suppose there is a program
function KolmogorovComplexity(string s)
that takes as input a string s and returns K(s). Now consider the program
function GenerateComplexString(int n) for i = 1 to infinity: for each string s of length exactly i if KolmogorovComplexity(s) >= n return s quit
This program calls KolmogorovComplexity as a subroutine. This program tries every string, starting with the shortest, until it finds a string with complexity at least n, then returns that string. Therefore, given any positive integer n, it produces a string with Kolmogorov complexity at least as great as n. The program itself has a fixed length U. The input to the program GenerateComplexString is an integer n; here, the size of n is measured by the number of bits required to represent n which is log2(n). Now consider the following program:
function GenerateParadoxicalString() return GenerateComplexString(n0)
This program calls GenerateComplexString as a subroutine and also has a free parameter n0. This program outputs a string s whose complexity is at least n0. By an auspicious choice of the parameter n0 we will arrive at a contradiction. To choose this value, note s is described by the program GenerateParadoxicalString whose length is at most
where C is the "overhead" added by the program GenerateParadoxicalString. Since n grows faster than log2(n), there exists a value n0 such that
But this contradicts the definition of s having a complexity at least n0. That is, by the definition of K(s), the string s returned by GenerateParadoxicalString is only supposed to be able to be generated by a program of length n0 or longer, but GenerateParadoxicalString is shorter than n0. Thus the program named "KolmogorovComplexity" cannot actually computably find the complexity of arbitrary strings.
This is proof by contradiction where the contradiction is similar to the Berry paradox: "Let n be the smallest positive integer that cannot be defined in fewer than twenty English words." It is also possible to show the uncomputability of K by reduction from the uncomputability of the halting problem H, since K and H are Turing-equivalent.
In the programming languages community there is a corollary known as the full employment theorem, stating there is no perfect size-optimizing compiler.
The chain rule for Kolmogorov complexity states that
It states that the shortest program that reproduces X and Y is no more than a logarithmic term larger than a program to reproduce X and a program to reproduce Y given X. Using this statement one can define an analogue of mutual information for Kolmogorov complexity.
It is straightforward to compute upper bounds for : simply compress the string with some method, implement the corresponding decompressor in the chosen language, concatenate the decompressor to the compressed string, and measure the resulting string's length.
A string s is compressible by a number c if it has a description whose length does not exceed . This is equivalent to saying . Otherwise s is incompressible by c. A string incompressible by 1 is said to be simply incompressible; by the pigeonhole principle which applies because every compressed string maps to only one uncompressed string, incompressible strings must exist, since there are bit strings of length n but only 2n − 1 shorter strings, that is strings of length less than n, i.e. with length 0,1,...,n-1.
For the same reason, most strings are complex in the sense that they cannot be significantly compressed: is not much smaller than , the length of s in bits. To make this precise, fix a value of n. There are bitstrings of length n. The uniform probability distribution on the space of these bitstrings assigns exactly equal weight to each string of length n.
Theorem. With the uniform probability distribution on the space of bitstrings of length n, the probability that a string is incompressible by c is at least .
To prove the theorem, note that the number of descriptions of length not exceeding is given by the geometric series:
There remain at least
many bitstrings of length n that are incompressible by c. To determine the probability divide by .
We know that, in the set of all possible strings, most strings are complex in the sense that they cannot be described in any significantly "compressed" way. However, it turns out that the fact that a specific string is complex cannot be formally proved, if the string's complexity is above a certain threshold. The precise formalization is as follows. First fix a particular axiomatic system S for the natural numbers. The axiomatic system has to be powerful enough so that to certain assertions A about complexity of strings one can associate a formula FA in S. This association must have the following property: if FA is provable from the axioms of S, then the corresponding assertion A is true. This "formalization" can be achieved either by an artificial encoding such as a Gödel numbering or by a formalization which more clearly respects the intended interpretation of S.
Theorem. There exists a constant L (which only depends on the particular axiomatic system and the choice of description language) such that there does not exist a string s for which the statement
(as formalized in S) can be proven within the axiomatic system S.
Note that by the abundance of nearly incompressible strings, the vast majority of those statements must be true.
The proof of this result is modeled on a self-referential construction used in Berry's paradox. The proof is by contradiction. If the theorem were false, then
We can find an effective enumeration of all the formal proofs in S by some procedure
function NthProof(int n)
which takes as input n and outputs some proof. This function enumerates all proofs. Some of these are proofs for formulas we do not care about here (since every possible proof in the language of S is produced for some n). Some of these are complexity formulas of the form K(s) ≥ n where s and n are constants in the language of S. There is a program
function NthProofProvesComplexityFormula(int n)
which determines whether the nth proof actually proves a complexity formula K(s) ≥ L. The strings s and the integer L in turn are computable by programs:
function StringNthProof(int n)
function ComplexityLowerBoundNthProof(int n)
Consider the following program
function GenerateProvablyComplexString(int n) for i = 1 to infinity: if NthProofProvesComplexityFormula(i) and ComplexityLowerBoundNthProof(i) ≥ n return StringNthProof(i) quit
Given an n, this program tries every proof until it finds a string and a proof in the formal system S of the formula K(s) ≥ L for some L ≥ n. The program terminates by our Assumption (X). Now this program has a length U. There is an integer n0 such that U + log2(n0) + C < n0, where C is the overhead cost of
function GenerateProvablyParadoxicalString() return GenerateProvablyComplexString(n0) quit
The program GenerateProvablyParadoxicalString outputs a string s for which there exists an L such that K(s) ≥ L can be formally proved in S with L ≥ n0. In particular K(s) ≥ n0 is true. However, s is also described by a program of length U + log2(n0) + C so its complexity is less than n0. This contradiction proves Assumption (X) cannot hold.
Similar ideas are used to prove the properties of Chaitin's constant.
The minimum message length principle of statistical and inductive inference and machine learning was developed by C.S. Wallace and D.M. Boulton in 1968. MML is Bayesian (it incorporates prior beliefs) and information-theoretic. It has the desirable properties of statistical invariance (the inference transforms with a re-parametrisation, such as from polar coordinates to Cartesian coordinates), statistical consistency (even for very hard problems, MML will converge to any underlying model) and efficiency (the MML model will converge to any true underlying model about as quickly as is possible). C.S. Wallace and D.L. Dowe (1999) showed a formal connection between MML and algorithmic information theory (or Kolmogorov complexity).
Kolmogorov randomness (also called algorithmic randomness) defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string. This definition of randomness is critically dependent on the definition of Kolmogorov complexity. To make this definition complete, a computer has to be specified, usually a Turing machine. According to the above definition of randomness, a random string is also an "incompressible" string, in the sense that it is impossible to give a representation of the string using a program whose length is shorter than the length of the string itself. However, according to this definition, most strings shorter than a certain length end up being (Chaitin-Kolmogorovically) random because the best one can do with very small strings is to write a program that simply prints these strings.
It can be shown that for the output of Markov information sources, Kolmogorov complexity is related to the entropy of the information source. More precisely, the Kolmogorov complexity of the output of a Markov information sources, normalized by the length of the output, converges almost surely (as the length of the output goes to infinity) to the entropy of the source.
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