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39.2 Optimal Compression, Kolmogorov Complexity
We define the Kolmogorov complexity of a bitstream
B to be the shortest bitstream that outputs
B. Let the Java-Kolmogorov complexity be the shortest Java program that generates
B.Note that for any bitstream
,
definitely exists. However, finding and proving
might be difficult or even impossible.
An important thing to note is that Kolmogorov complexity is language-independent. To run any program in one language in another, all I have to do is write an interpreter. For example, if I want to run a Python program that is not easily translatable to Java, I could instead just write a Java interpreter to read the text of the Python program and run it. In this case,
, where
is the length of the interpreter (a constant value).
This highlights a very deep fact about Kolmogorov complexity: most bitstreams are fundamentally incompressible no matter which language we choose for our compression algorithm.
Consider a bitstream of 1,000,000 bits. Out of all compression algorithms possible, only 1 in
bitstreams have a change of being compressed by more than 50% (499,999 bits or less).
Another important fact regarding Kolmogorov complexity is that it is impossible to compute. A proof of this fact is provided here.
Practically, this means that it is impossible to write a "perfect" (optimal) compression algorithm, since we can't even compute the length of the shortest program!
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