## Abstract

We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [14], as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and (n^1.4) clauses. Such small size refutations would improve the state of the art (with respect to the clause density) efficient refutation algorithm, which works only for (n^1.5) many

clauses [15].

We demonstrate polynomial-size refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations with small integer coefficients, denoted R(quad); this is a weak extension of cutting planes with small coefficients. We show that R(quad) is weakly automatizable iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with linear instead of quadratic equations (introduced in [28]). This reduces the problem of refuting random 3CNF with n variables and (n^1.4) clauses to the interpolation problem

of R(quad) and to the weak automatizability of R(lin).

clauses [15].

We demonstrate polynomial-size refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations with small integer coefficients, denoted R(quad); this is a weak extension of cutting planes with small coefficients. We show that R(quad) is weakly automatizable iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with linear instead of quadratic equations (introduced in [28]). This reduces the problem of refuting random 3CNF with n variables and (n^1.4) clauses to the interpolation problem

of R(quad) and to the weak automatizability of R(lin).

Original language | English |
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Title of host publication | Automata, Languages, and Programming - 41st International Colloquium, (ICALP) 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part {I} |

Subtitle of host publication | -- |

Pages | 1015-1026 |

Number of pages | 12 |

Publication status | Published - 8 Jul 2014 |

### Publication series

Name | LNCS |
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