Abstract.
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.
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Manuscript received 20 October 2008
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Ben-Sasson, E., Impagliazzo, R. Random Cnf’s are Hard for the Polynomial Calculus. comput. complex. 19, 501–519 (2010). https://doi.org/10.1007/s00037-010-0293-1
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DOI: https://doi.org/10.1007/s00037-010-0293-1