## A New Proof of the Weak Pigeonhole Principle (2000)

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### BibTeX

@MISC{Maciel00anew,

author = {Alexis Maciel and Toniann Pitassi and Alan R. Woods},

title = {A New Proof of the Weak Pigeonhole Principle},

year = {2000}

}

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### Abstract

The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [16] showed how to prove the weak pigeonhole principle with bounded-depth, quasipolynomial-size proofs. Their argument was further refined by Kraj'icek [9]. In this paper, we present a new proof: we show that the the weak pigeonhole principle has quasipolynomial-size LK proofs where every formula consists of a single AND/OR of polylog fan-in. Our proof is conceptually simpler than previous arguments, and is optimal with respect to depth. 1 Introduction The pigeonhole principle is a fundamental axiom of mathematics, stating that there is no one-to-one mapping from m pigeons to n holes when m ? n. It expresses Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 136995815, U.S.A. alexis@clarkson.edu. Research supported by NSF grant CCR-9877150. y Department of Computer Science, University o...

### Citations

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72 | Polynomial size proofs of the propositional pigeonhole principle
- Buss
- 1987
(Show Context)
Citation Context ...g m. Moreover, the depth of the proof is dependent on the the exponent in the polylog m. Lastly, by formalizing circuits that count, one can prove PHP~ for any n < m with polynomial-size Frege proofs =-=[2]-=-. There are many interesting open problems that are raised by this work. Most importantly, are there polynomial-size, constant-depth proofs of either the weak pigeonhole principle, or the onto weak pi... |

67 | Exponential lower bounds to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms - Kraj́ıček, Pudlák, et al. - 1995 |

57 | Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic - Razborov - 1995 |

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- Buss, Turán
- 1988
(Show Context)
Citation Context ...circuits computing a function (in this case approximate counting), but where any proof of correctness of the function cannot be carried out in an equally feasible way. Lastly, a question left open in =-=[4]-=-, are there polynomial-size Resolution proofs of PHP~, when m > n2/logn? We conjecture that the answer is no. 7. ACKNOWLEDGEMENTS We would like to thank Jan Kraji~ek for stimulating discussions that l... |

46 | Bounded arithmetic and lower bounds in Boolean complexity - Razborov - 1995 |

37 | Resolution and the weak pigeonhole principle
- Buss, Pitassi
- 1997
(Show Context)
Citation Context ...lly simpler than the previous upper bound due to Paris, Wilkie and Woods: it is a simple divide and conquer, along the lines of the upper bounds for Resolution proofs of the weak pigeonhole principle =-=[3]-=-, combined with an amplification phase which allows us to speed up the induction. 369 The outline for the remainder of the paper is as follows. In Section 2, we give precise definitions of the pigeonh... |

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(Show Context)
Citation Context ... We now repeat these two steps to obtain a sequence of injective functions from In] to In/4], from In 2] to [n/4], from [n] to [n/8], from [n 2] to [n/8], ..., until an injective function from [n] to =-=[1]-=- is obtained. This is the desired contradiction, which proves PHP~ 2 . Now we prove PHP~ " . Again by contradiction, suppose that there is an injective function f from [2n] to [n]. We define a functio... |

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1 |
Lower bounds to the size of constant-depth propositional proofs
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(Show Context)
Citation Context ...diagonalization argument, Paris, Wilkie and Woods [9] showed how to prove the weak pigeonhole principle with bounded-depth, quasipolynomial-size proofs. Their argument was further refined by Krajf~ek =-=[5]-=-. In this paper, we present a new proof: we show that the the weak pigeonhole principle has quasipolynomial-size proofs where every formula consists of a single AND/OR. of polylog fan-in. Our proof is... |

1 | Resolution and the weak pigeonhole principle, Computer Science Logic - Buss, Pitassi - 1997 |

1 | ak, Ramsey's theorem in bounded arithmetic, Computer Science Logic - Pudl' - 1992 |

1 | Seminar notes (15.11.99). Typeset - Kraj'icek - 1999 |