Approximation and small depth Frege proofs

S. Bellantoni, T. Pitassi, A. Urquhart
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引用次数: 74

Abstract

M. Ajtai (1988) recently proved that if, for some fixed d, every formula in a Frege proof of the propositional pigeonhole principle PHP/sub n/ has depth at most d, then the proof size is not less than any polynomial in n. By introducing the notion of an approximate proof the authors demonstrate how to eliminate the nonstandard model theory, including the nonconstructive use of the compactness theorem, from Ajtai's lower bound. An approximate proof is one in which each inference is sound on a subset of the possible truth assignments-possibly a different subset for each inference. The authors also improve the lower bound, giving a specific superpolynomial function bounding the proof size from below.<>
近似和小深度弗雷格证明
M. Ajtai(1988)最近证明,如果对于某个固定的d,命题鸽子洞原理PHP/sub n/的Frege证明中的每个公式的深度最多为d,则证明的大小不小于n中的任何多项式。通过引入近似证明的概念,作者演示了如何从Ajtai的下界中消除非标准模型理论,包括紧性定理的非构造性使用。近似证明是这样一种证明:在可能的真值分配的一个子集上,每个推理都是合理的——每个推理可能是不同的子集。作者还改进了下界,给出了一个特定的超多项式函数来限定证明大小
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