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)


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.


Turing Machine Kolmogorov Complexity Random String Satisfying Assignment Small Program 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 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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