The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory

We continue an investigation into resource-bounded Kolmogorov complexity [ABK06], which highlights the close connections between circuit complexity and Levin’s time-bounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity. The Kolmogorov measures that have been introduced have many advantages over other approaches to defining resource-bounded Kolmogorov complexity (such as much greater independence from the underlying choice of universal machine that is used to define the measure) [ABK06]. Here, we study the properties of other measures that arise naturally in this framework. The motivation for introducing yet more notions of resource-bounded Kolmogorov complexity are two-fold: • to demonstrate that other complexity measures such as branching-program size and formula size can also be discussed in terms of Kolmogorov complexity, and • to demonstrate that notions such as nondeterministic Kolmogorov complexity and distinguishing complexity [BFL02] also fit well into this framework. The main theorems that we provide using this new approach to resource-bounded Kolmogorov complexity are: • A complete set (RKNt) for NEXP/poly defined in terms of strings of high Kolmogorov complexity. • A lower bound, showing that RKNt is not in NP ∩ coNP. • New conditions equivalent to the conditions “NEXP ⊆ nonuniform NC” and “NEXP ⊆ L/poly”. • Theorems showing that “distinguishing complexity” is closely connected to both FewEXP and to EXP. • Hardness results for the problems of approximating formula size and branching program size.