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Kolmogorov Complexity

Computational Complexity Course Report

Henry Xiao

Queen’s University

School of Computing

Kingston, Ontario, Canada

March 2004

In computer science, the concepts of algorithm and information are fundamental. So the
measurement of information or algorithms is crucial in sense of describing. In 1965
Andrey Nikolaevich Kolmogorov [O'Connor, and Robertson, 1999sian
mathematician, established the algorithmic theory of randomness via a measure of
complexity, now referred to Kolmogorov complexity. According to Kolmogorov, the
complexity of an object is the length of the shortest computer program that can reproduce
the object. All algorithms can be expressed in programming language based on Turing
machine models equally succinctly, up to a fixed additive constant term. The remarkable
usefulness and inherent rightness of the theory of Kolmogorov complexity or so called
Descriptive complexity, stems from this independence of the description method.
The idea of Kolmogorov complexity first appeard in the 1960’s in papers by
Kolmogorov, Solomonoff and Chaitin. As specified by Schöning and Randall, an
algorithm can exhibit very different complexity behavior in the worst case and in the
average case. The Kolmogorov complexity is defined a probability distribution under
which worst-case and average-case running time (for all algorithm simultaneously) are
the same (up to constant factors). Quick sort algorithm has been widely taken as an
example to show the applicability of Kolmogorov complexity since the algorithm takes

O(n logn) time in average but Ω(n
) time at worst case. Later, the Kolmogorov
complexity was connected with Information Theory and proved to be closely related to
Claude Shannon's entropy rate of an information source. The theory base of Kolmogorov
complexity has also be extended to data compression and communication for the sake of
true information measure.

Kolmogorov Complexity Theory
We will briefly take a look at Kolmogorov Complexity definition and some main related
results at this section. For details, please refer to [Cover and Thomas, 1991

[Schöning and Pruim 1998

Definition: The Kolmogorov Complexity Ku(x) of a string x with respect to a universal
computer U is defined as:

Ku(x) = min l(p),

p : U(p) = x
the minimum length over all programs that print x and halt. Thus Ku(x) is the shortest
description length of x over all descriptions interpreted by computer U. (Note Turing
machine is regarded as universal computer in computer science.)
The concept of Kolmogorov Complexity asks for the minimal unambiguous
description of a sequence. It can be used to prove complexity lower bounds. [Schöning

and Pruim 1998
ples of using this technique. And indeed, as mentioned,
the proofs obtained in this way are much more “elegant”, or at least shorter, than the
original proofs. In a few cases, the lower bounds were first achieved by means of
Kolmogorov Complexity. Another quite important definition is the conditional
Kolmogorov complexity which is based on the knowledge of the length of x, denoted as

Ku(x| l(x)) = min l(p),

p : U(p, l(x)) = x

3 This is the shortest description length if the computer U has the length of x made
available to it. Quite a few different results have been shown for conditional Kolmogorov
complexity too.
We look at some basic and interesting properties of Kolmogorov complexity and
then consider some examples. The proofs of the theorems are eliminated because of the
purpose of this report. However, for your interests, please refer to [Cover and Thomas,
1991] for complete proofs.
Both the lower and upper bounds of Kolmogorov complexity of a given sequence
have been derived early. There are some choices of both bounds from different aspects.
The followings are all established theorems for bounds.

Universality of Kolmogorov complexity: If U is a universal computer, then for any other

computer A: Ku(x) ≤ K

(x) + c
for all string x ∈ {0, 1}*, where the constant c

not depend on x.
Conditional complexity is less than the length of sequence: Ku(x| l(x)) ≤ l(x)+ c.
Upper bound on Kolmogorov complexity: Ku(x) ≤ Ku(x| l(x)) + 2log (l(x)) + c.
Lower bound on Kolmogorov complexity: The number of strings x with complexity
Ku(x) ≤ k satisfies: |{ x ∈ {0, 1}*: Ku(x) ≤ k } | < 2
The Kolmogorov complexity of a binary string x is bounded by:



|n) ≤ nH


) + 2log n + c,

where H
(p) = -p log p – (1-p) log (1-p) is the binary entropy function.

4 The above five theorems are basically considered to be very important facts of
Kolmogorov complexity. Many other interesting results have been derived applying these
theorems. We consider some examples of Kolmogorov complexity here to show the
usability of the theory with its properties. First, we will look at some intuitive ideas
directly coming from the definition. A sequence of n zeros has a constant Kolmogorov
complexity, i.e. K(000…0|n) = c, since if we assume n is known, the a short program can
directly print out n zeros. The same case can be applied to

π, where the first n bits of
πcan be calculated using a simple series expression. A somewhat surprising result for
fractal is that regarding its complex calculations, it is still essentially very simple in terms
of Kolmogorov complexity which is nearly zero. An integer on the other hand, has higher
complexity even it looks very straight forward. It is obviously true that the complexity of
describing an integer will be constant if we know the length of the integer, i.e. K(n|l(n)) =
c. However, in general, the computer does not know the length of binary representation
of the integer. So we must inform the computer in some way when the description ends.
We can bound the description using the upper bound we got so far: K(n) ≤ 2log n + c.
We can also prove there are an infinite number of integers n such that K(n) > log n. This
is probably less intuitive than we thought. From above examples, it is not hard to see the
true measurement of information or algorithm would be rather hard without the
development of Kolmogorov complexity. With the support of the Kolmogorov
complexity theory, one can describe information more accurate in computer science.
Kolmogorov complexity also applied to algorithmically random, incompressible
sequences, the halting problem, etc. We can not go into details of those examples; instead,
we briefly look at those problems here just to get some taste. Many literatures have been
5 published on Kolmogorov complexity related questions. One can refer to [Li, and Vitanyi,
1999eading. The algorithmically random and
incompressible sequences are defined based on the Kolmogorov complexity properties
that some sequences hold. We say a sequence x

, x

, x

… x

algorithmically random if


| n) ≥ n. And we can say a string x incompressible if lim K(x


| n)/n =
1. The definitions seem very intuitive with respect of Kolmogorov complexity. In fact, if
every element of the sequence is completely generated in random, we can not predict any
later elements from current. Indeed, we will need to describe each element separately.
However, if the Kolmogorov complexity verse n approaching 1 for a string in probability,
then we can actually interpret this as the proportions of 0’s and 1’s in this string are
almost equal, which is ½. By this meaning, it is true that we can not compress the string
since any bit will be a critical contribution to the whole, which also specifies the
randomness of the string. The next significant application is on the halting problem.
Using Kolmogorov complexity, we can actually demonstrate that the problem can not be
solved by an algorithm because of the non-computability of Kolmogorov complexity. It
is rather a surprising fact of the non-computability. However, practically speaking, one
may never be able to tell the shortest program since there are infinite many programs for
a given sequence. We can only estimate the complexity by running more and more
programs, as we know the bound will converge to the Kolmogorov complexity. Many
other results can also be found on published literatures related with probability theory and
information theory. At the following section, we take a look at one of the most important
results related with the central idea of information theory – Entropy.

Kolmogorov Complexity and Entropy
As mentioned at introduction, the Kolmogorov complexity and entropy of a sequence of
random variables are highly related. In general, the expected value of the Kolmogorov
complexity of a random sequence is to its entropy.
From information theory founded by Shannon, the true measure of information on
random variables is entropy. This relationship actually proves the correctness of the
Kolmogorov complexity as a measurement of information and algorithms. Kolmogorov
complexity states the shortest description (program) for a random variable. Then
complexity of the sequence constructed by random variables will approach to the expect
value of the set of Kolmogorov complexities for each variable, i.e. E[1/nK(X

|n)] →H(X),
supported by the law of large numbers in probability. Respectively, the information
measure of the sequence is indeed entropy. Actually, one can always show the program
lengths satisfy prefix condition, since if the computer halts on any program, it does not
look any further for input. The relationship can be shown further with Kraft inequality.
The relationship provides us to two ways of complexity measures, which either
takes after Kolmogorov complexity, involving finding some computer or abstract
automaton which will produce the pattern of interest, or take after information theory and
produce something like the entropy, which, while in principle computable, can be very
hard to calculate reliably for experimental systems. This is actually very powerful
principle in physics (see [Li, and Vitanyi, 1999ore interesting idea
about “Occam’s Razor” as a general principle governing scientific research can be
derived from Kolmogorov theory as we will see at the next section.

Occam’s Razor
I decide to include this topic mainly because its importance plus the idea really amazed
me. At 14th century, William of Occam (Ockham in some literatures), a logician, said
“Nunquam ponenda est pluralitas sine necessitate”, i.e., explanations should not be
multiplied beyond necessity, which forms the basis of methodological reductionism. Here,
our argument will be a special case of it.
Recent papers have suggested a connection between Occam's Razor and
Kolmogorov complexity. Many literatures take Laplace’s sun rising problem as an
example to explain the connection. Laplace considered the probability that the sun will
rise again tomorrow, given that it has risen every day in recorded history. He solved it
before Kolmogorov complexity was introduced. However, the problem can be
reconsidered through Kolmogorov complexity. If we use 1 to represent the sun rise, then
the probability that the next symbol is a 1 given n 1’s in the sequence so far is: ∑


1y) ≈ p(1

) = c > 0. And the probability that the next symbol is 0, which means that

the sun will not rise: ∑


0y) ≈ p(1

0) ≈ 2

-log n

, since any 1

0… yields a description
of n with length at least K(n), i.e. about log n + O(log log n). Hence the conditional
probability of observing a 0 next is: p(0|1

) = p(1

0) / (p(1

0) + p(1

))≈ 1/(cn+1). The
result is very similar to 1/(n+1) derived by Laplace. The Kolmogorv complexity solution
to this question is actually following the Occam’s Razor by weighting possible
explanations by their complexity.
It often happens that the best explanation is much more complicated than the
simplest possible explanation because it requires fewer assumptions. Albert Einstein
8 wrote in 1933 “Theories should be as simple as possible, but no simpler.” In our case,
Kolmogorov complexity does provide us an alternative approach to explain things in
many science fields.

Kolmogorov complexity is a profound theory for information and algorithm measure.
The theory is somehow different from others that we have studied in computational
complexity so far. I feel the theory tries to observe the complexity from a new approach.
It is worth reading something about Kolmogorov [O'Connor, and Robertson, 1999
understand his original idea, which he developed from mathematical perspective. It is in
high level of abstraction, but closely related with many things in practice. Vladimir
V’yugin presents some applications of Kolmogorov complexity in his review [V’yugin,



atically. In com
puting, I found many ongoing topics related with
Kolmogorov complexity are on information process ([Levin, 1999
Dowe, 1999a]). Generally, the application of Kolmogorv complexity is based on
framework of the Minimum Description Length (MDL) principle and Minimum Message
Length (MML) principle, which are out the scope of this report, but refer to [Wallace, and
Dowe, 1999b] for introductions.
I strongly believe the Kolmogorov complexity should have more appearances in
future research topics. Along with information theories, we need to deal with information
as well as problems coming with it. Like the Shannon’s entropy theory established
today’s communication system, the closely related Kolmogorov complexity shall also be
of great potential to be applied to future researches.


Cover, M. Thomas, and Thomas, A. Joy. Elements of information theory. John Wiley &
Sons, Inc. 1991.

Gammerman, Alexander, and Vovk, Vladimir. Kolmogorov complexity: sources, theory
and applications. The Computer Journal, vol. 42, no. 4, 1999.

Levin, A. Leonid. Robust Measures of Information. The Computer Journal, vol. 42, no. 4,

Li, Ming, and Vitanyi, Paul. An Introduction to Kolmogorov Complexity and Its
Applications. Springer-Verlag, New York, 1999.

O'Connor, J J, and Robertson, E F. “Andrey Nikolaevich Kolmogorov”. School of
Mathematics and Statistics, University of St. Andrews, Scotland, 1999.

Rissanen, Jorma. Discussion of paper `Minimum Message Length and Kolmogorov
Complexity' by C. S. Wallace and D. L. Dowe. The Computer Journal, vol. 42, no. 4,

Schöning, Uwe, and Pruim, Randall. Gems of theoretical computer science. Springer-
Verlag Berlin Heidelberg 1998.


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