## Abstract

We investigate the space complexity of refuting 3-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random 3-CNF φ in n variables requires, with high probability, Ω(n) distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation of φ requires, with high probability, Ω(n) clauses each of width Ω(n) to be kept at the same time in memory. This gives a Ω(n^{2}) lower bound for the total space needed in Resolution to refute φ. These results are best possible (up to a constant factor) and answer questions about space complexity of 3-CNFs. The main technical innovation is a variant of Hall's Lemma. We show that in bipartite graphs with bipartition (L,R) and left-degree at most 3, L can be covered by certain families of disjoint paths, called VW-matchings, provided that L expands in R by a factor of (2−ϵ), for ϵ<1/5.

Original language | English |
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Pages (from-to) | 165-176 |

Number of pages | 12 |

Journal | Information and Computation |

Volume | 255 |

DOIs | |

Publication status | Published - 1 Aug 2017 |

Externally published | Yes |

## Keywords

- Monomial space
- Polynomial calculus
- Proof complexity
- Random CNFs
- Resolution
- Total space