What is the maximum theoretically possible compression rate?

Question

This is a theoretical question, so expect that many details here are not computable in practice or even in theory.

Let\'s say I have a string s that I w

Answer 1

Basically, you need enough information to rebuild your original information. I guess the other answers are more helpful for your theoretical discussion, but just keep this in mind.

Answer 2

This is Kolmogorov complexity, and you are correct that it's not computable. If it were, you could create a paradoxical program of length n that printed a string with Kolmogorov complexity m > n.

Clearly, you can bound b_min_s for given inputs. However, as far as I know most of the efforts to do so have been existence proofs. For instance, there is an ongoing competition to compress English Wikipedia.

Answer 3

The maximal (avarage) compression rate possible is 1:1.
The number of possible inputs is equal to the number of outputs.
It has to be to be able to map the output back to the input.
To be able to store the output you need container at the same size as the minimal container for the input - giving 1:1 compression rate.

Answer 4

Claude Shannon estimated the information density of the English language to be somewhere between 0.6 and 1.3 bits per character in his 1951 paper Prediction and Entropy of Printed English (PDF, 1.6 MB. Bell Sys. Tech. J (3) p. 50-64).