## Abstract

We construct quasipolynomial-size proofs of the propositional pigeonhole principle in the deep inference system KS, addressing an open problem raised in previous works and matching the best known upper bound for the more general class of monotone proofs.

We make significant use of monotone formulae computing boolean threshold functions, an idea previously considered in works of Atserias et al. The main construction, monotone proofs witnessing the symmetry of such functions, involves an implementation of merge-sort in the design of proofs in order to tame the structural behaviour of atoms, and so the complexity of normalization. Proof transformations from previous work on atomic flows are then employed to yield appropriate KS proofs.

As further results we show that our constructions can be applied to provide quasipolynomial-size KS proofs of the parity principle and the generalized pigeonhole principle. These bounds are inherited for the class of monotone proofs, and we are further able to construct nO(log log n)-size monotone proofs of the weak pigeonhole principle with (1 + ε)n pigeons and n holes for ε = 1/logk n, thereby also improving the best known bounds for monotone proofs.

We make significant use of monotone formulae computing boolean threshold functions, an idea previously considered in works of Atserias et al. The main construction, monotone proofs witnessing the symmetry of such functions, involves an implementation of merge-sort in the design of proofs in order to tame the structural behaviour of atoms, and so the complexity of normalization. Proof transformations from previous work on atomic flows are then employed to yield appropriate KS proofs.

As further results we show that our constructions can be applied to provide quasipolynomial-size KS proofs of the parity principle and the generalized pigeonhole principle. These bounds are inherited for the class of monotone proofs, and we are further able to construct nO(log log n)-size monotone proofs of the weak pigeonhole principle with (1 + ε)n pigeons and n holes for ε = 1/logk n, thereby also improving the best known bounds for monotone proofs.

Original language | English |
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Title of host publication | CSL-LICS '14 Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) |

Publisher | Association for Computing Machinery (ACM) |

Number of pages | 10 |

ISBN (Print) | 9781450328869 |

DOIs | |

Publication status | Published - 14 Jul 2014 |

Event | Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) - Vienna, Austria Duration: 14 Jul 2014 → 18 Jul 2014 |

### Conference

Conference | Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) |
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Country/Territory | Austria |

City | Vienna |

Period | 14/07/14 → 18/07/14 |

## Keywords

- Pigeonhole Principle
- Deep Inference
- Monotone Proofs
- Atomic Flows