## Abstract

The proof system resolution over parities (Res(⊕)) operates with disjunctions of linear equations (linear clauses) over GF(2); it extends the resolution proof system by incorporating linear algebra over GF(2). Over the years, several exponential lower bounds on the size of tree-like refutations have been established. However, proving a superpolynomial lower bound on the size of dag-like Res(⊕) refutations remains a highly challenging open question. We prove an exponential lower bound for regular Res(⊕). Regular Res(⊕) is a subsystem of dag-like Res(⊕) that naturally extends regular resolution. This is the first known superpolynomial lower bound for a fragment of dag-like Res(⊕) which is exponentially stronger than tree-like Res(⊕). In the regular regime, resolving linear clauses C_{1} and C_{2} on a linear form f is permitted only if, for both i∈ {1,2}, the linear form f does not lie within the linear span of all linear forms that were used in resolution rules during the derivation of C_{i}. Namely, we show that the size of any regular Res(⊕) refutation of the binary pigeonhole principle BPHP_{n}^{n+1} is at least 2^{Ω(3√n/logn)}. A corollary of our result is an exponential lower bound on the size of a strongly read-once linear branching program solving a search problem. This resolves an open question raised by Gryaznov, Pudlak, and Talebanfard (CCC 2022). As a byproduct of our technique, we prove that the size of any tree-like Res(⊕) refutation of the weak binary pigeonhole principle BPHP_{n}^{m} is at least 2^{Ω(n)} using Prover-Delayer games. We also give a direct proof of a width lower bound: we show that any dag-like Res(⊕) refutation of BPHP_{n}^{m} contains a linear clause C with Ω(n) linearly independent equations.

Original language | English |
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Title of host publication | STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing |

Editors | Bojan Mohar, Igor Shinkar, Ryan O�Donnell |

Publisher | Association for Computing Machinery |

Pages | 640-651 |

Number of pages | 12 |

ISBN (Electronic) | 9798400703836 |

DOIs | |

State | Published - 10 Jun 2024 |

Event | 56th Annual ACM Symposium on Theory of Computing, STOC 2024 - Vancouver, Canada Duration: 24 Jun 2024 → 28 Jun 2024 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 56th Annual ACM Symposium on Theory of Computing, STOC 2024 |
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Country/Territory | Canada |

City | Vancouver |

Period | 24/06/24 → 28/06/24 |

## Keywords

- binary pigeonhole principle
- lower bounds
- proof complexity
- regular resolution
- resolution over linear equations

## ASJC Scopus subject areas

- Software