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Short Proofs Are Hard to Find

Authors Ian Mertz, Toniann Pitassi, Yuanhao Wei

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ACM Subject Classification
  • Theory of computation → Proof complexity
  • Hardware → Theorem proving and SAT solving
Keywords
  • automatizability
  • Resolution
  • SAT solvers
  • proof complexity

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Abstract

We obtain a streamlined proof of an important result by Alekhnovich and Razborov [Michael Alekhnovich and Alexander A. Razborov, 2008], showing that it is hard to automatize both tree-like and general Resolution. Under a different assumption than [Michael Alekhnovich and Alexander A. Razborov, 2008], our simplified proof gives improved bounds: we show under ETH that these proof systems are not automatizable in time n^f(n), whenever f(n) = o(log^{1/7 - epsilon} log n) for any epsilon > 0. Previously non-automatizability was only known for f(n) = O(1). Our proof also extends fairly straightforwardly to prove similar hardness results for PCR and Res(r).

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Ian Mertz, Toniann Pitassi, and Yuanhao Wei. Short Proofs Are Hard to Find. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 84:1-84:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.84

Author Details

Ian Mertz
  • University of Toronto, Canada
Toniann Pitassi
  • University of Toronto, Canada
  • Institute for Advanced Study, Princeton, NJ, USA
Yuanhao Wei
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Mertz, Ian: Research supported by NSERC.
  • Pitassi, Toniann: Research supported by NSERC.
  • Wei, Yuanhao: Work done while a student at University of Toronto. Research supported by NSERC.

Acknowledgements

The authors thank Noah Fleming, Pravesh Kothari, Denis Pankratov, Robert Robere, Mika Göös, and Avi Wigderson for helpful conversations. We also thank Yijia Chen for providing more details on the construction in [Yijia Chen and Bingkai Lin, 2016], and Pasin Manurangsi for giving feedback on the parameters in [Parinya Chalermsook et al., 2017].

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