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Resolution and the binary encoding of combinatorial principles

Dantchev, Stefan; Galesi, Nicola; Martin, Barnaby

Resolution and the binary encoding of combinatorial principles Thumbnail


Nicola Galesi


Amir Shpilka


Res(s) is an extension of Resolution working on s-DNFs. We prove tight n (k) lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [27] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [35], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHPmn for m > n. Using the the same recursive approach we prove the new result that for any > 0, Bin-PHPmn requires proofs of size 2n1− in Res(s) for s = o(log1/2 n). Our lower bound is almost optimal since for m 2 p n log n there are quasipolynomial size proofs of Bin-PHPmn in Res(log n). Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in 2-form and with no finite model.


Dantchev, S., Galesi, N., & Martin, B. (2019). Resolution and the binary encoding of combinatorial principles. In A. Shpilka (Ed.), 34th Computational Complexity Conference (CCC 2019) (6:1-6:25).

Conference Name Computational Complexity Conference
Conference Location New Brunswick, New Jersey, USA
Acceptance Date Apr 30, 2019
Online Publication Date Jul 16, 2019
Publication Date Jul 31, 2019
Deposit Date May 19, 2019
Publicly Available Date Jul 16, 2019
Volume 137
Pages 6:1-6:25
Series Title Leibniz International Proceedings in Informatics (LIPIcs)
Series ISSN 1868-8969
Book Title 34th Computational Complexity Conference (CCC 2019).
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