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

We continue an investigation into resource-bounded Kolmogorov complexity (Allender et al., 2006 [4], ), 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) (Allender et al., 2006 [4]). 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 (Buhrman et al., 2002 [15]) 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⊆nonuniformNC1” 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.