Resource-Bounded Complexity

Abstract

Recursion theory has a resource-bounded version in computational complexity theory. Similarly, Kolmogorov complexity has resource-bounded Kolmogorov complexity (also known as generalized Kolmogorov complexity). Several authors suggested early on the possibility of restricting the power of the device used to compress strings. Says Kolmogorov:

“The concept discussed ... does not allow for the ‘difficulty’ of preparing a program p for passing from an object x to an object y. ... [some] object permitting a very simple program, i.e., with very small complexity K(x) can be restored by short programs only as the result of computations of a thoroughly unreal nature. ... [this concerns] the relationship between the necessary complexity of a program and its permissible difficulty t. The complexity K(x) that was obtained [before] is, in this case, the minimum of K ^t(x) on the removal of the constraints on t.” [Kolmogorov]