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Is there a machine learning task (or any task/problem) that one can by construction know the Kolmogorov Complexity (or minimum description length)? I know the Kolmogorov Complexity is uncomputable but I was curious if there existed a set of tasks that the Kolmogorov Complexity can be known apriori and one can vary the Kolmogorov Complexity of those related tasks. I know Kolmogorov Complexity is uncomputable but I thought perhaps if it was by construction known then the task described -- perhaps it was doable.

Some ideas I thought of:

  1. Random bitstrings: As mentioned before, the Kolmogorov complexity of a fair coin flip bitstring of length n is exactly n. No shorter description is possible.
  2. Lookup tables: If you construct a lookup table that maps n-bit input binary strings to arbitrary outputs, the table requires 2^n entries, each of b bits. So the total complexity is 2^n * b.
  3. Parity functions: As you mentioned, an n-bit parity function requires describing each of the n input bits, so the complexity is n. More formally, you need n + c bits where c is the size of the program that computes XOR. For large n, c is negligible.
  4. Decision tree models: If you construct a full binary decision tree of depth d, it has 2^d leaves. To specify it requires 2^d output class labels + internal node conditions. Each condition can be specified in c bits. So total complexity is roughly c2^d.

Ideally I want to implement it in a computer to create this family of known Kolmogorov Complexity problems.

fyi: uncomputable means that there is no algorithm to compute it in general and I know KC is uncomputable in the general case.

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  • $\begingroup$ The Kolmogorov complexity is uncomputable but "... von oben rekursiv aufzählbar." -- This hint found in German Wikipedia is missing in the French and Enghlish version. So, good luck. BTW, what the hell will you do with this information? What does it help in real world? $\endgroup$
    – m-stgt
    Feb 3 at 7:16
  • $\begingroup$ @m-stgt uncomputable means that there is no algorithm to compute it in general. But it's easy to compute it for say a totally uniform random string of size n. It's n. specifying the string. I'd like to correlate it with computable measures of complexity of tasks. $\endgroup$ Feb 5 at 5:43
  • $\begingroup$ @m-stgt fields like formal methods deal with uncomputable problems all the time. They reason by specific case usually and they have drawbacks. I'm not looking for a general algorithm :) $\endgroup$ Feb 5 at 5:44
  • $\begingroup$ "know the Kolmogorov Complexity" of what? I'm a bit lost by what you're asking, what string you want to know the KC of, and how that is supposed to be related to a ML problem. What do you mean by the KC of a "task"? Can you edit your post to formulate the problem statement more clearly/explicitly? $\endgroup$
    – D.W.
    Feb 10 at 8:17
  • $\begingroup$ "the Kolmogorov complexity of a fair coin flip bitstring of length n is exactly n" - This is not correct. The KC of a string depends on the string, not on how it was chosen. The KC of a string chosen by flipping a fair coin n times is no greater than n, but it could be smaller. For instance, TTTTTT..TTT is a possible outcome, and its KC is a constant. There seem to be some misconceptions in the question about KC. $\endgroup$
    – D.W.
    Feb 10 at 8:19

1 Answer 1

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Not really.

To establish the Kolmogorov complexity of a string, we'd have to know the halting status and output of all Turing machines with a shorter representation. Thus we can't have exact Kolmogorov complexities for anything with length longer than that for which we have a value for the busy beaver function. That's not a lot.

As m-stgt found, the complexity is "recursively enumerable from above," which is to say there is a recursively enumerable decreasing sequence of functions that approaches the complexity function $K(s)$. The method, due to Kolmogorov, is however not enlightening and terminates on no inputs: Let the sequence $K_t(s)$ be indexed by natural $t$, and let $K_0(s)$ be a naïve upper bound on the complexity of $s$. Then, to find $K_{t+1}(s)$, run all Turing machines with descriptions shorter than $K_t$ for $t$ steps, and take $K_{t+1}(s)$ to be the minimum of $K_t(s)$ and the description lengths of any Turing machine that outputs $s$. This is obviously decreasing in $t$, and as $t\rightarrow\infty$ you have $K_t(s)\rightarrow K(s)$. This computes the complexity in the same sense that running all Turing machines of a certain length increasingly longer computes the busy beaver function: eventually you'll have the answer, but the tricky bit is knowing when you have it.

All of the examples you give are formulations of asking what the maximum complexity of a string of a given length is, and this will always be at least the length: you can't possibly describe all strings with descriptions of shorter lengths, as there are more strings of a given length than descriptions shorter. Meanwhile, there is a trivial upper bound on this given by the length of the program that hardcodes an output of a given length, which will be some negligible constant overhead on the string length.

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