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Resolution and the Binary Encoding of Combinatorial Principles

Authors Stefan Dantchev, Nicola Galesi, Barnaby Martin

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Author Details

Stefan Dantchev
  • Department of Computer Science University of Durham, UK
Nicola Galesi
  • Dipartimento di Informatica, Sapienza Università di Roma, Italy
Barnaby Martin
  • Department of Computer Science University of Durham, UK

Acknowledgements

We are grateful to Ilario Bonacina for reading a preliminary version of this work and addressing us some useful comments and observations. We are further grateful to several anonymous reviewers for further corrections and comments.

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Stefan Dantchev, Nicola Galesi, and Barnaby Martin. Resolution and the Binary Encoding of Combinatorial Principles. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 6:1-6:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.CCC.2019.6

Abstract

Res(s) is an extension of Resolution working on s-DNFs. We prove tight n^{Omega(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. [Massimo Lauria et al., 2017] 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 [Nathan Segerlind et al., 2004], 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-PHP^m_n for m>n. Using the the same recursive approach we prove the new result that for any delta>0, Bin-PHP^m_n requires proofs of size 2^{n^{1-delta}} in Res(s) for s=o(log^{1/2}n). Our lower bound is almost optimal since for m >= 2^{sqrt{n log n}} there are quasipolynomial size proofs of Bin-PHP^m_n 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 Pi_2-form and with no finite model.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
Keywords
  • Proof complexity
  • k-DNF resolution
  • binary encodings
  • Clique and Pigeonhole principle

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