Resource-bounded kolmogorov complexity revisited

Abstract

We take a fresh look at CD complexity, where CD ^t(x) is the smallest program that distinguishes x from all other strings in time t(¦x¦). We also look at a CND complexity, a new nondeterministic variant of CD complexity.

We show several results relating time-bounded C, CD and CND complexity and their applications to a variety of questions in computational complexity theory including:

• Showing how to approximate the size of a set using CD complexity avoiding the random string needed by Sipser. Also we give a new simpler proof of Sipser's lemma.

• A proof of the Valiant-Vazirani lemma directly from Sipser's earlier CD lemma.

• A relativized lower bound for CND complexity.

• Exact characterizations of equivalences between C, CD and CND complexity.

• Showing that a satisfying assignment can be found in output polynomial time if and only if a unique assignment can be found quickly. This answers an open question of Papadimitriou.

• New Kolmogorov-based constructions of the following relativized worlds:

• There exists an infinite set in P with no sparse infinite subsets in NP.

• EXP=NEXP but there exists a nondeterministic exponential time Turing machine whose accepting paths cannot be found in exponential time.

• Satisfying assignment cannot be found with nonadaptive queries to SAT.

Part of this research was done while visiting The University of Chicago. Partially supported by the Dutch foundation for scientific research (NWO) through NFI Project ALADDIN, under contract number NF 62-376 and SION project 612-34-002, and by the European Union through NeuroCOLT ESPRIT Working Group Nr. 8556, and HC&M grant nr. ERB4050PL93-0516.

Supported in part by NSF grant CCR 92-53582 and the Fulbright scholar program and NWO.

At http://www.cs.uchicago.edu/^∼fortnow/papers/ a full version of this paper including complete proofs can be found.