## Polynomial size proofs of the propositional pigeonhole principle (1987)

Venue: | Journal of Symbolic Logic |

Citations: | 72 - 7 self |

### BibTeX

@ARTICLE{Buss87polynomialsize,

author = {Samuel R. Buss},

title = {Polynomial size proofs of the propositional pigeonhole principle},

journal = {Journal of Symbolic Logic},

year = {1987},

volume = {52},

pages = {916--927}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic. $1. Introduction. The motivation for this paper comes primarily from two sources. First, Cook and Reckhow [2] and Statman [7] discussed connections between lengths of proofs in propositional logic and open questions in com-putational complexity such as whether NP = co-NP. Cook and Reckhow used the propositional pigeonhole principle as an example of a family of true formulae which

### Citations

221 |
Parity, circuits and the polynomial-time hierarchy
- Furst, Saxe, et al.
- 1984
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Citation Context ...ic. The situation described below is somewhat analogous to the relationship between constant depth, polynomial size circuits and the relativized polynomial hierarchy as discussed by Furst-Saxe-Sipser =-=[3]-=-, Yao [12] and others. DEFINITION. The Ck- and &-formulae are defined inductively as follows: (1) A propositional variable is a Co-formula and a 170-formula. (2) If A is a Ci-formula (Ui-formula) then... |

185 |
Separating the polynomial-time hierarchy by oracles
- Yao
- 1985
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Citation Context ...ituation described below is somewhat analogous to the relationship between constant depth, polynomial size circuits and the relativized polynomial hierarchy as discussed by Furst-Saxe-Sipser [3], Yao =-=[12]-=- and others. DEFINITION. The Ck- and &-formulae are defined inductively as follows: (1) A propositional variable is a Co-formula and a 170-formula. (2) If A is a Ci-formula (Ui-formula) then i A is a ... |

83 |
Models of Computation
- Savage
- 1998
(Show Context)
Citation Context ...tead we must use a technique called "carry-save-addition". Carrysave-addition is a well-known technique for computing the summation of a vector of numbers with a logarithmic depth circuit (see Savage =-=[6]-=-). As we see below, it allows us to define counting with polynomial size propositional formulae; without the use of carry-save addition formulae of size O(n'Og('Ogn)) would be required.s922 SAMUEL R. ... |

78 |
Counting problems in bounded arithmetic
- Paris, Wilkie
- 1985
(Show Context)
Citation Context ...n of the pigeonhole principle for functions defined by bounded formulae. This question is related to the size of Frege proofs of the propositional pigeonhole principle by a result of Paris and Wilkie =-=[5]-=-; namely, if Id, proves a relativized version of the pigeonhole principle then there are constant formula-depth, polynomial size Frege proofs of the propositional pigeonhole Received July 4, 1986. Res... |

74 | R.: Bounded arithmetic - Buss |

37 |
The intractability of Resolution. Theoret
- Haken
- 1985
(Show Context)
Citation Context ...al logic) were exponential size. The main result of this paper is that the propositional pigeonhole principle also has polynomial size Frege proofs, contrary to expectations. On the other hand, Haken =-=[4]-=- has shown that any resolution proof of the propositional pigeonhole principle must be of exponential size. It follows that a Frege proof system has an exponential speedup over resolution (this was or... |

23 |
Some problems in logic and number theory, and their connections
- Woods
- 1981
(Show Context)
Citation Context ...xponential speedup over resolution (this was originally proved by Urquhart [ll] with a different set of formulae). The second motivation is from research in theories of bounded arithmetic. Alan Woods =-=[lo]-=- showed that Id, could prove the existence of an infinite number of primes if it were the case that Id, could prove the pigeonhole principle for functions definable by a bounded formula. Alex Wilkie [... |

1 |
The relative eficiency of propositional proof systems
- COOK, RECKHOW
- 1979
(Show Context)
Citation Context ...ver resolution. We also discuss connections to provability in theories of bounded arithmetic. $1. Introduction. The motivation for this paper comes primarily from two sources. First, Cook and Reckhow =-=[2]-=- and Statman [7] discussed connections between lengths of proofs in propositional logic and open questions in computational complexity such as whether NP = co-NP. Cook and Reckhow used the proposition... |

1 |
Complexity of derivations from quantijier-free Horn formulae, mechanical introduction of explicit definitions, and refinement of completeness theorems, Logic Colloquium '76
- STATMAN
- 1977
(Show Context)
Citation Context ...We also discuss connections to provability in theories of bounded arithmetic. $1. Introduction. The motivation for this paper comes primarily from two sources. First, Cook and Reckhow [2] and Statman =-=[7]-=- discussed connections between lengths of proofs in propositional logic and open questions in computational complexity such as whether NP = co-NP. Cook and Reckhow used the propositional pigeonhole pr... |

1 |
On the complexity of derivation in propositional calculus, Studies in constructiue mathematics and mathematical logic
- TSETIN
- 1968
(Show Context)
Citation Context ...ke of definiteness, we define the extended Frege system eF to have the language, axioms and rules of F plus a new rule called the extension rule. (The extension rule was originally defined by Tseitin =-=[8]-=-.) A sequence of formulae A,, . . . , A, is an e8-proof iff each Ai is an axiom or is deduced by modus ponens or by the extension rule. Ai is deduced by the extension rule iff Ai is of the form (pi + ... |

1 |
Talk presented at Logic Colloquium '84
- WILKIE
- 1984
(Show Context)
Citation Context ...] showed that Id, could prove the existence of an infinite number of primes if it were the case that Id, could prove the pigeonhole principle for functions definable by a bounded formula. Alex Wilkie =-=[9]-=- discovered that a weak form of the pigeonhole principle is provable in Id, + 0, and that this implies that Id, + 0, can prove the existence of an infinite number of primes; however, it is still open ... |