On Approximate Uncomputability of the Kolmogorov Complexity Function

Abstract

Kolmogorov complexity C(x) of a string x is the length of its shortest possible description. It is well known that C(x) is not computable. Moreover, any computable lower estimate of C(x) is bounded by a constant. We study the following question: suppose that we want to compute C with some precision and some amount of errors. For which parameters is it possible? Our main result is the following: the error must be at least an inverse exponential function of the precision. It gives two striking implications. Firstly, no computable function approximate Kolmogorov complexity much better than the length function does. Secondly, time-bounded Kolmogorov complexity is sufficiently far from unbounded Kolmogorov complexity for any particular computable time bound.