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Optimality of size-degree tradeoffs for polynomial calculus

Optimality of size-degree tradeoffs for polynomial calculus Optimality of Size-Degree Tradeoffs for Polynomial Calculus NICOLA GALESI and MASSIMO LAURIA Sapienza ”Universita di Roma ` 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 š m variables which requires PC and PCR refutations of degree ( 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. Categories and Subject Descriptors: F.2 [Analysis of Algorithms and Problem Complexity] General Terms: Theory Additional Key Words and http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computational Logic (TOCL) Association for Computing Machinery

Optimality of size-degree tradeoffs for polynomial calculus

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Publisher
Association for Computing Machinery
Copyright
Copyright © 2010 by ACM Inc.
ISSN
1529-3785
DOI
10.1145/1838552.1838556
Publisher site
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Abstract

Optimality of Size-Degree Tradeoffs for Polynomial Calculus NICOLA GALESI and MASSIMO LAURIA Sapienza ”Universita di Roma ` 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 š m variables which requires PC and PCR refutations of degree ( 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. Categories and Subject Descriptors: F.2 [Analysis of Algorithms and Problem Complexity] General Terms: Theory Additional Key Words and

Journal

ACM Transactions on Computational Logic (TOCL) – Association for Computing Machinery

Published: Oct 1, 2010

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