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Editorial Team
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Asked: 2026-05-15T17:49:36+00:00

This is a theoretical question, so expect that many details here are not computable

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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 want to compress. The result should be a self-extracting binary (can be x86 assembler, but it can also be some other hypothetical Turing-complete low level language) which outputs s.

Now, we can easily iterate through all possible such binaries and programs, ordered by size. Let B_s be the sub-list of these binaries who output s (of course B_s is uncomputable).

As every set of positive integers must have a minimum, there must be a smallest program b_min_s in B_s.

For what languages (i.e. set of strings) do we know something about the size of b_min_s? Maybe only an estimation. (I can construct some trivial examples where I can always even calculate B_s and also b_min_s, but I am interested in more interesting languages.)

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  1. Editorial Team

    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.

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