Chinese translation by Cheng Qi: ``Description Complexity and
Applications,'' China Science
Press, Beijing, December 1998
(1999 National 1st Prize for Excellent Books in Science
and Technology published in the Peoples Republic of China.)
Translation of parts in Russian
by A.K. Shen and N.K. Vereshchagin in ``Uspekhi Mat. Nauk''
and in Japanese by O. Watanabe in ``Handbook of Theoretical
Computer Science.''
Achievements of the area:
Pointwise Randomness. Kolmogorov
complexity is a modern notion of randomness dealing with the
quantity of information in individual objects; that is, pointwise randomness
rather than average randomness as produced by a random source.
It was proposed by
A.N. Kolmogorov
in 1965 to quantify the randomness of individual
objects in an objective and absolute manner. This is impossible
by classical probability theory (a branch of measure
theory satisfying the so-called Kolmogorov axioms formulated
in 1933).
Kolmogorov complexity is known variously as `algorithmic
information', `algorithmic entropy', `Kolmogorov-Chaitin
complexity', `descriptional complexity', `shortest program length',
`algorithmic randomness', and others.
Incompressibility Method.
In some parts of the research interests mentioned
on Paul
Vitanyi's home page a new mathematical
proof technique was developed,
now known as the `incompressibility method'.
The incompressibility method is a basic
general technique such as the `pigeon hole' argument,
`the counting method' or the `probabilistic method'.
The new method is based on Kolmogorov complexity.
Absolute Information.
The Kolmogorov complexity
of an object is a form of absolute information of the individual
object. This is not possible to do by
C.E. Shannon's information theory. Unlike Kolmogorov complexity,
information theory is only concerned with the average
information of a random source.
Universal Induction and Data Compression.
Traditional wisdom has it
that the better a theory compresses the learning data
concerning some phenomenon under investigation, the better we
learn, generalize, and the better the theory predicts unknown data.
This belief is vindicated in practice but before the advent
of Kolmogorov complexity has not been rigorously
proved in a general setting. Making
these ideas rigorous involves the length of the shortest effective description
of an individual object: its Kolmogorov complexity.
Ray Solomonoff invented the notion of universal prediction
using the Kolmogorov complexity based universal distribution.
Universal prediction is related to optimal effective compression.
The latter is almost always a best
strategy in hypotheses identification
(an ideal form of the minimum description length (MDL)
principle). Whereas the single best hypothesis does
not necessarily give the best prediction, it can be demonstrated
that nonetheless compression is almost always
the best strategy in prediction methods in the style of R. Solomonoff.
A
short biography
of Kolmogorov is available.
`Kolmogorov' complexity was earlier proposed by
Ray Solomonoff
in a Zator Technical Report in 1960 and in a long paper
in Information and Control in 1964. Also
Gregory Chaitin
proposed this idea in a paper in J. ACM in 1969.
This paper continues a paper by G. Chaitin in J.ACM in 1966,
which however was concerned with
a complexity measure based on Turing machine state-symbol
product following ideas of
C.E. Shannon. Unlike Kolmogorov complexity,
those complexity measures are not objective and absolute (recursively invariant).
After we and others pioneered several successful applications
of Kolmogorov complexity in the theory of computation,
the general pattern of the incompressibility method emerged.
The incompressibility method and Kolmogorov complexity
is truly a versatile mathematical tool. It is a sharper relative of
classical information theory (absolute information of individual object rather
than average information over a random ensemble) and yet
satisfies many of the laws of classical information theory---although
with a slight error term.
Applications of Kolmogorov
complexity by us and others have been
given in a plethora of areas, including
randomness of individual finite objects or infinite sequences,
Martin-Loef tests for randomness,
Goedel's incompleteness result, information theory of individual objects,
universal probability, general inductive reasoning,
inductive inference, prediction, mistake bounds, computational
learning theory, inference in statistics,
the incompressibility method, combinatorics,
graph theory,
Kolmogorov 0-1 Laws,
probability theory,
theory of parallel and distributed computation,
time and space complexity of computations,
average case analysis of algorithms such as HEAPSORT,
language recognition, string matching,
routing in computer networks,
circuit theory,
formal language and automata theory,
parallel computation, Turing machine complexity,
complexity of tapes, stacks, queues,
average complexity,
lower bound proof techniques,
structural complexity theory,
oracles,
logical depth, universal optimal search, physics and computation,
dissipationless reversible computing, information distance
and picture similarity, thermodynamics of computing, statistical
thermodynamics and Boltzmann entropy.
