computational complexity

, Volume 19, Issue 4, pp 501–519 | Cite as

Random Cnf’s are Hard for the Polynomial Calculus

  • Eli Ben-SassonEmail author
  • Russell Impagliazzo


We prove linear lower bounds on the Polynomial Calculus (PC) refutation-degree of random CNF whenever the underlying field has characteristic greater than 2. Our proof follows by showing the PC refutation-degree of a unsatisfiable system of linear equations modulo 2 is equivalent to its Gaussian width, a concept defined by the late Mikhail Alekhnovich.

The equivalence of refutation-degree and Gaussian width which is the main contribution of this paper, allows us to also simplify the refutation-degree lower bounds of Buss et al. (2001) and additionally prove non-trivial upper bounds on the resolution and PC complexity of refuting unsatisfiable systems of linear equations.


Propositional proof complexity polynomial calculus Groebner basis random CNF formulae 

Subject classification.

03F20 03B05 68Q17 


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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Computer ScienceTechnion – Israel Institute of TechnologyHaifaIsrael
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  3. 3.Computer Science and EngineeringUniversity of California, San DiegoLa JollaUSA

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