computational complexity

, Volume 21, Issue 3, pp 479–497 | Cite as

Low-Depth Witnesses are Easy to Find

  • Luís Antunes
  • Lance Fortnow
  • Alexandre Pinto
  • André Souto


Kolmogorov Complexity measures the amount of information in a string by the size of the smallest program that generates that string. Antunes, Fortnow, van Melkebeek, and Vinodchandran captured the notion of useful information by computational depth, the difference between the polynomial-time-bounded Kolmogorov complexity and traditional Kolmogorov complexity.

We show unconditionally how to probabilistically find satisfying assignments for formulas that have at least one assignment of logarithmic depth. The converse holds under a standard hardness assumption though fails if BPP = FewP = EXP. We also prove that assuming the existence of good pseudorandom generators one cannot increase the depth of a string efficiently.


Computational depth SAT formulas Kolmogorov complexity Pseudorandom generators 

Subject classification

68Q17 68Q30 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Luís Antunes
    • 1
  • Lance Fortnow
    • 2
  • Alexandre Pinto
    • 3
  • André Souto
    • 4
  1. 1.Faculty of Sciences, Instituto de TelecomunicaçõesUniversity of PortoPortoPortugal
  2. 2.Northwestern UniversityEvanstonUSA
  3. 3.Instituto Superior da Maia (ISMAI), Centro de Ciências e Tecnologias de Computação (CCTC)Universidade do MinhoAvioso S. PedroPortugal
  4. 4.Instituto de TelecomunicaçõesFaculty of Sciences University of PortoPortoPortugal

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