Philosophy of Everything

Please check my informal introduction to "Kolmogorov complexity" if you are not familiar with the subject. In this article I will approach some less discussed ramifications of it and hopefully some new ideas: an example of a Kolmogorov compression that would require an arbitrary high amount of working memory before shrinking to the compressed string (conjecture). 1. Root cause why "Kolmogorov complexity" is uncomputable Kolmogororov complexity is uncomputable because Turing machines have a halting problem :) Let's start with the observation the the Halting problem of a Turing machine having a limited band is actually decidable, unlike the infinite band one. A "finite band Turing machine" is actually a finite state machine that will either stop without going through the same state, or will cycle infinitely. Having a limited number of possible states, it cannot run too much without repeating a previous state. As soon as the machine returns to...

I want to give you an intuition about  Kolmogorov complexity , without being very formal. Basically, Kolmogorov complexity defines a measure of how small can you compress a certain string of characters. You can think about it like measuring how small can you make a file (string of characters) by compressing it with ZIP or similar. A file storing a very redundant information (like "aaaaaaaaaaaaaaaa....") can be compressed in a much smaller size compared with a random one. For calculating the "Kolmogorov complexity", the "compressed" form is considered to be something like a self-extract archive: the shortest program that can output the initial string of characters (the "file/string to be compressed"). The Kolmogorov complexity is more general and includes classical compressing algorithms. For example a Huffman code is not very powerful in compressing a string like "123456789101112131415161718192021222324..." (concatenated consecut...