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Optimal Short-Circuit Resilient Formulas

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



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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

Acknowledgements

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

Cite AsGet 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

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.

Subject Classification

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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References

  1. Shweta Agrawal, Ran Gelles, and Amit Sahai. Adaptive Protocols for Interactive Communication. In 2016 IEEE International Symposium on Information Theory (ISIT), pages 595-599, 2016. URL: https://doi.org/10.1109/ISIT.2016.7541368.
  2. Elwyn R. Berlekamp. Block coding with noiseless feedback. PhD thesis, Massachusetts Institute of Technology, 1964. Google Scholar
  3. Zvika Brakerski, Yael T. Kalai, and Moni Naor. Fast Interactive Coding Against Adversarial Noise. J. ACM, 61(6):35:1-35:30, December 2014. URL: https://doi.org/10.1145/2661628.
  4. M. Braverman, R. Gelles, J. Mao, and R. Ostrovsky. Coding for Interactive Communication Correcting Insertions and Deletions. IEEE Transactions on Information Theory, 63(10):6256-6270, October 2017. URL: https://doi.org/10.1109/TIT.2017.2734881.
  5. M. Braverman and A. Rao. Toward Coding for Maximum Errors in Interactive Communication. Information Theory, IEEE Transactions on, 60(11):7248-7255, November 2014. URL: https://doi.org/10.1109/TIT.2014.2353994.
  6. Mark Braverman and Klim Efremenko. List and Unique Coding for Interactive Communication in the Presence of Adversarial Noise. SIAM Journal on Computing, 46(1):388-428, 2017. URL: https://doi.org/10.1137/141002001.
  7. Mark Braverman, Klim Efremenko, Ran Gelles, and Michael A. Yitayew. Optimal Short-Circuit Resilient Formulas. CoRR, arXiv:1807.05014, 2018. URL: http://arxiv.org/abs/1807.05014.
  8. K. Efremenko, R. Gelles, and B. Haeupler. Maximal Noise in Interactive Communication Over Erasure Channels and Channels With Feedback. IEEE Transactions on Information Theory, 62(8):4575-4588, August 2016. URL: https://doi.org/10.1109/TIT.2016.2582176.
  9. Matthew Franklin, Ran Gelles, Rafail Ostrovsky, and Leonard J. Schulman. Optimal Coding for Streaming Authentication and Interactive Communication. Information Theory, IEEE Transactions on, 61(1):133-145, January 2015. URL: https://doi.org/10.1109/TIT.2014.2367094.
  10. R. Gelles, B. Haeupler, G. Kol, N. Ron-Zewi, and A. Wigderson. Explicit Capacity Approaching Coding for Interactive Communication. IEEE Transactions on Information Theory, 64(10):6546-6560, October 2018. URL: https://doi.org/10.1109/TIT.2018.2829764.
  11. Ran Gelles. Coding for Interactive Communication: A Survey. Foundations and Trends® in Theoretical Computer Science, 13(1–2):1-157, 2017. URL: https://doi.org/10.1561/0400000079.
  12. Ran Gelles and Bernhard Haeupler. Capacity of Interactive Communication over Erasure Channels and Channels with Feedback. SIAM Journal on Computing, 46(4):1449-1472, 2017. URL: https://doi.org/10.1137/15M1052202.
  13. Ran Gelles, Ankur Moitra, and Amit Sahai. Efficient Coding for Interactive Communication. Information Theory, IEEE Transactions on, 60(3):1899-1913, March 2014. URL: https://doi.org/10.1109/TIT.2013.2294186.
  14. Mohsen Ghaffari and Bernhard Haeupler. Optimal Error Rates for Interactive Coding II: Efficiency and List Decoding. In Proceedings of the IEEE Symposium on Foundations of Computer Science, FOCS '14, pages 394-403, 2014. URL: https://doi.org/10.1109/FOCS.2014.49.
  15. Mohsen Ghaffari, Bernhard Haeupler, and Madhu Sudan. Optimal Error Rates for Interactive Coding I: Adaptivity and Other Settings. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC '14, pages 794-803, 2014. URL: https://doi.org/10.1145/2591796.2591872.
  16. Bernhard Haeupler. Interactive Channel Capacity Revisited. In Proceedings of the IEEE Symposium on Foundations of Computer Science, FOCS '14, pages 226-235, 2014. URL: https://doi.org/10.1109/FOCS.2014.32.
  17. Bernhard Haeupler, Pritish Kamath, and Ameya Velingker. Communication with Partial Noiseless Feedback. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015), volume 40 of Leibniz International Proceedings in Informatics (LIPIcs), pages 881-897. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.881.
  18. M. Horstein. Sequential transmission using noiseless feedback. IEEE Transactions on Information Theory, 9(3):136-143, July 1963. URL: https://doi.org/10.1109/TIT.1963.1057832.
  19. Y. T. Kalai, A. Lewko, and A. Rao. Formulas Resilient to Short-Circuit Errors. In Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, pages 490-499, October 2012. URL: https://doi.org/10.1109/FOCS.2012.69.
  20. Mauricio Karchmer and Avi Wigderson. Monotone Circuits for Connectivity Require Super-Logarithmic Depth. SIAM Journal on Discrete Mathematics, 3(2):255-265, 1990. URL: https://doi.org/10.1137/0403021.
  21. Dan Kleitman, Tom Leighton, and Yuan Ma. On the Design of Reliable Boolean Circuits That Contain Partially Unreliable Gates. J. Comput. Syst. Sci., 55(3):385-401, December 1997. URL: https://doi.org/10.1006/jcss.1997.1531.
  22. Gillat Kol and Ran Raz. Interactive channel capacity. In STOC '13: Proceedings of the 45th annual ACM Symposium on theory of computing, pages 715-724, 2013. URL: https://doi.org/10.1145/2488608.2488699.
  23. Satoshi Nakamoto. Bitcoin: A peer-to-peer electronic cash system, 2008. URL: https://bitcoin.org/bitcoin.pdf.
  24. Denis Pankratov. On the Power of Feedback in Interactive Channels. [Online:] http://people.cs.uchicago.edu/~pankratov/papers/feedback.pdf, 2013.
  25. Sridhar Rajagopalan and Leonard Schulman. A coding theorem for distributed computation. In STOC '94: Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 790-799, 1994. URL: https://doi.org/10.1145/195058.195462.
  26. Leonard J. Schulman. Communication on noisy channels: a coding theorem for computation. Foundations of Computer Science, Annual IEEE Symposium on, pages 724-733, 1992. URL: https://doi.org/10.1109/SFCS.1992.267778.
  27. Leonard J. Schulman. Coding for interactive communication. IEEE Transactions on Information Theory, 42(6):1745-1756, 1996. URL: https://doi.org/10.1109/18.556671.
  28. C. Shannon. The zero error capacity of a noisy channel. IRE Transactions on Information Theory, 2(3):8-19, September 1956. URL: https://doi.org/10.1109/TIT.1956.1056798.
  29. John von Neumann. Probabilistic logics and the synthesis of reliable organisms from unreliable components. In C. E. Shannon and J. McCarthy, editors, Automata studies, volume 34 of Annals of Mathematics Studies, pages 43-98. Princeton University Press, Princeton, 1956. Google Scholar
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