February 15, 2007

Algorithmic Information Theory

Randomness and Proof” is a nice, introductory-level article on
algorithmic information theory. It’s by G. J. Chaitin, who
has personally done a lot of the foundational work in this
area. A random, interesting snippet:

Solomonoff represented a scientist’s observations as a
series of binary digits. The scientist seeks to explain
these observations through a theory, which can be regarded
as an algorithm capable of generating the series and
extending it, that is, predicting future observations. For
any given series of observations there are always several
competing theories, and the scientist must choose among
them. The model demands that the smallest algorithm, the one
consisting of the fewest bits, be selected. Stated another
way, this rule is the familiar formulation of Occam’s razor:
Given differing theories of apparently equal merit, the
simplest is to be preferred.

Thus in the Solomonoff model a theory that enables one to
understand a series of observations is seen as a small
computer program that reproduces the observations and makes
predictions about possible future observations. The smaller
the program, the more comprehensive the theory and the
greater the degree of understanding. Observations that are
random cannot be reproduced by a small program and therefore
cannot be explained by a theory. In addition the future
behavior of a random system cannot be predicted. For random
data the most compact way for the scientist to communicate
his observations is for him to publish them in their entirety.

I’m writing a paper on Kolmogorov complexity for a class in complexity theory I’m taking this semester — if I get a chance, I’ll try to blog about some of the fascinating ideas in this field.

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