The document discusses Kolmogorov complexity, which is defined as the length of the shortest computer program needed to generate a given object as output. It proves that for any length n, there exists an incompressible string of that length. While Kolmogorov complexity is not computable, it demonstrates that the complexity of strings can be arbitrarily large. The document also discusses how low-complexity or minimal art aligns with principles of algorithmic information theory and how an observer's perception of beauty may relate to improved data compression.

1. Kolmogorov Complexity, Art, and all that
Aleksandar Bradic
CTO, Supplyframe
April 19 2017

2. Definition
The Kolmogorov complexity of an object, such as a piece of text,
is the length of the shortest computer program (in a predetermined
programming language) that produces the object as output.

3. Which is more complex?
1111111111111111111111111111111111111111
vs.
0000100000101101111101100111101111101000

5. The Kolmogorov complexity of a string x is the length of the
smallest program that outputs x, relative to some model of
computation. That is
Cf (x) = minp{|p| : f (p) = x}
for some computer f.
A string is incompressible if C(x) |x|

6. Are there incompressible strings?
Theorem: For all n, there exists an incompressible string of
length n
Proof: There are 2n strings of length n and fewer than 2n
descriptions that are shorter than n:
n−1
i=0 2i = 2n − 1 < 2n

7. Incompressibility Theorem
A string x is c-incompressible if C(x) ≥ |x| − c, for some constant
c.
The number of strings of length n that are c-incompressible is at
least
2n − 2n−c+1 + 1
Example (c=10): The fraction of all strings of length n with
complexity less than n − 10 is smaller than
2n−11+1
2n = 1
1024

8. Uncomputability of Kolmogorov complexity
Theorem: There exists strings of arbitrary large Kolmogorov
complexity. Formally, for each n ∈ N, there is a string s with
C(s) ≥ n.
Proof: Otherwise all of the infinitely many possible finite strings
could be generated by the finitely many programs with a
complexity below n bits.

9. Uncomputability of Kolmogorov complexity
C(s) is not a computable function

10. Low-Complexity Art
Schmidhuber characterizes low-complexity art as the computer age
equivalent of minimal art. He also describes an algorithmic theory
of beauty and aesthetics based on the principles of algorithmic
information theory and minimum description length. It explicitly
addresses the subjectivity of the observer and postulates that
among several input data classified as comparable by a given
subjective observer, the most pleasing one has the shortest
description, given the observers previous knowledge and his or her
particular method for encoding the data.

12. Example
Initialization: Draw a circle of arbitrary radius and center
position. Arbitrary select a point on the first circle and use it as a
center of a second circle and use it as the center of a second circle
with equal radius. The first two circles are defined as legal circles.
Rule 1: Whenever two legal circles of equal radius touch or
intersect, draw another legal circle of equal radius with the
intersection point as its center.
Rule 2: Within every legal circle with center point p and radius r,
draw another legal circle whose center point is also p but whose
radius is r/2.

13. Schmidhuber explicitly distinguishes between beauty and
interestingness. He assumes that any observer continually tries to
improve the predictability and compressibility of the observations
by discovering regularities such as repetitions and symmetries and
fractal self-similarity. When the observer’s learning process (which
may be a predictive neural network) leads to improved data
compression the number of bits required to describe the data
decreases. The temporary interestingness of the data corresponds
to the number of saved bits, and thus (in the continuum limit) to
the first derivative of subjectively perceived beauty.

14. Lightpen: Simple (repl-friendly) DSL for SVG

19. Thanks!