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

https://doi.org/10.1016/j.jcss.2010.06.004Get rights and content
Under an Elsevier user license
open archive

Abstract

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 NPcoNP.

  • New conditions equivalent to the conditions “NEXPnonuniform NC1” and “NEXPL/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.

Keywords

Circuit complexity
Distinguishing complexity
FewEXP
Formula size
Kolmogorov complexity
NEXP

Cited by (0)

Much of this material appeared in preliminary form as “Derandomization and Distinguishing Complexity” in the Proceedings of the 2003 IEEE Conference on Computational Complexity (Allender et al., 2003 [6]).

1

Partially supported by NSF grants DMS-0652582, CCF-0830133, and CCF-0832787.

2

Supported in part by NSF grant CCF-0514703 and by grants GA ČR 201/07/P276 and P202/10/0854, project No. 1M0021620808 of MŠMT ČR, Institutional Research Plan No. AV0Z10190503 and grant IAA100190902 of GA AV ČR.

3

This work was done while at York College, City University of New York.