## The Pervasive Reach of Resource-Bounded Kolmogorov Complexity in Computational Complexity Theory

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@MISC{Allender_thepervasive,

author = {Eric Allender and Michal Koucky and Detlef Ronneburger and Sambuddha Roy},

title = {The Pervasive Reach of Resource-Bounded Kolmogorov Complexity in Computational Complexity Theory },

year = {}

}

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### Abstract

We continue an investigation into resource-bounded Kolmogorov complexity [ABK + 06], 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) [ABK + 06]. 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.