Resource-bounded kolmogorov complexity revisited

  • Harry Buhrman
  • Lance Fortnow
Structural Complexity I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1200)

Abstract

We take a fresh look at CD complexity, where CDt(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.

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Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Harry Buhrman
    • 1
  • Lance Fortnow
    • 2
  1. 1.CWIGB AmsterdamThe Netherlands
  2. 2.Department of Computer ScienceCWI & University of ChicagoChicago

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