Optimality of size-degree tradeoffs for polynomial calculus

  • Authors:

  • Nicola Galesi;Massimo Lauria

  • Affiliations:

  • Sapienza—Università di Roma, Rome, Italy;Sapienza—Università di Roma, Rome, Italy

  • Venue:
  • ACM Transactions on Computational Logic (TOCL)
  • Year:
  • 2010

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Abstract

There are methods to turn short refutations in polynomial calculus (Pc) and polynomial calculus with resolution (Pcr) into refutations of low degree. Bonet and Galesi [1999, 2003] asked if such size-degree tradeoffs for Pc [Clegg et al. 1996; Impagliazzo et al. 1999] and Pcr [Alekhnovich et al. 2004] are optimal. We answer this question by showing a polynomial encoding of the graph ordering principle on m variables which requires Pc and Pcr refutations of degree Ω(&sqrt; m). Tradeoff optimality follows from our result and from the short refutations of the graph ordering principle in Bonet and Galesi [1999, 2001]. We then introduce the algebraic proof system Pcrk which combines together polynomial calculus and k-DNF resolution (Resk). We show a size hierarchy theorem for Pcrk: Pcrk is exponentially separated from Pcrk+1. This follows from the previous degree lower bound and from techniques developed for Resk. Finally we show that random formulas in conjunctive normal form (3-CNF) are hard to refute in Pcrk.