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Optimal Short-Circuit Resilient Formulas

Authors Mark Braverman, Klim Efremenko , Ran Gelles , Michael A. Yitayew

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Interactive computation
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Circuit Complexity
  • Noise-Resilient Circuits
  • Interactive Coding
  • Coding Theory
  • Karchmer-Wigderson Games

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Abstract

We consider fault-tolerant boolean formulas in which the output of a faulty gate is short-circuited to one of the gate’s inputs. A recent result by Kalai et al. [FOCS 2012] converts any boolean formula into a resilient formula of polynomial size that works correctly if less than a fraction 1/6 of the gates (on every input-to-output path) are faulty. We improve the result of Kalai et al., and show how to efficiently fortify any boolean formula against a fraction 1/5 of short-circuit gates per path, with only a polynomial blowup in size. We additionally show that it is impossible to obtain formulas with higher resilience and sub-exponential growth in size.
Towards our results, we consider interactive coding schemes when noiseless feedback is present; these produce resilient boolean formulas via a Karchmer-Wigderson relation. We develop a coding scheme that resists up to a fraction 1/5 of corrupted transmissions in each direction of the interactive channel. We further show that such a level of noise is maximal for coding schemes with sub-exponential blowup in communication. Our coding scheme takes a surprising inspiration from Blockchain technology.

Cite As Get BibTex

Mark Braverman, Klim Efremenko, Ran Gelles, and Michael A. Yitayew. Optimal Short-Circuit Resilient Formulas. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.CCC.2019.10

Author Details

Mark Braverman
  • Department of Computer Science, Princeton University, USA
Klim Efremenko
  • Computer Science Department, Ben-Gurion University, Beer Sheba, Israel
Ran Gelles
  • Faculty of Engineering, Bar-Ilan University, Ramat Gan, Israel
Michael A. Yitayew
  • Department of Computer Science, Princeton University, USA

Funding

  • Braverman, Mark: Supported in part by an NSF CAREER award (CCF-1149888), NSF CCF-1525342, a Packard Fellowship in Science and Engineering, and the Simons Collaboration on Algorithms and Geometry.
  • Efremenko, Klim: Supported in part by the Israel Science Foundation (ISF) through grant No. 1456/18.
  • Gelles, Ran: Supported in part by the Israel Science Foundation (ISF) through grant No. 1078/17.

Acknowledgements

The authors would like to thank Raghuvansh Saxena and the anonymous reviewer for spotting an error in a preliminary version of this manuscript.

References

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