Document Open Access Logo

Polynomial Size, Short-Circuit Resilient Circuits for NC

Authors Yael Tauman Kalai, Raghuvansh R. Saxena

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2025.90.pdf
  • Filesize: 0.79 MB
  • 18 pages

Document Identifiers

Author Details

Yael Tauman Kalai
  • Microsoft Research, Cambridge, MA, USA
  • MIT, Cambridge, MA, USA
Raghuvansh R. Saxena
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

  • Saxena, Raghuvansh R.: Supported by the Department of Atomic Energy, Government of India, under project no. RTI4001.

Cite As Get BibTex

Yael Tauman Kalai and Raghuvansh R. Saxena. Polynomial Size, Short-Circuit Resilient Circuits for NC. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 90:1-90:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.90

Abstract

We show how to convert any circuit of poly-logarithmic depth and polynomial size into a functionally equivalent circuit of polynomial size (and polynomial depth) that is resilient to adversarial short-circuit errors. Specifically, the resulting circuit computes the same function even if up to ε d gates on every root-to-leaf path are short-circuited, i.e., their output is replaced with the value of one of its inputs, where d is the depth of the circuit and ε > 0 is a fixed constant. 
Previously, such a result was known for formulas (Kalai-Lewko-Rao, FOCS 2012). It was also known how to convert general circuits to error resilient ones whose size is quasi-polynomial in the size of the original circuit (Efremenko et al. STOC 2022). The reason both these works do not extend to our setting is that there may be many paths from the root to a given gate, and the resilient circuits needs to "remember" a lot of information about these paths, which causes it to be large. Our main idea is to reduce the amount of this information at the cost of increasing the depth of the resilient circuit.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Circuit complexity
Keywords
  • Error-resilient computation
  • short-circuit errors

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Mirza Ahad Baig, Suvradip Chakraborty, Stefan Dziembowski, Małgorzata Gałązka, Tomasz Lizurej, and Krzysztof Pietrzak. Efficiently testable circuits. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), 2023. URL: https://doi.org/h10.4230/LIPIcs.ITCS.2023.10.
  2. Mirza Ahad Baig, Suvradip Chakraborty, Stefan Dziembowski, Małgorzata Gałązka, Tomasz Lizurej, and Krzysztof Pietrzak. Efficiently testable circuits without conductivity. In Theory of Cryptography Conference, 2023. URL: https://doi.org/10.1007/978-3-031-48621-0_5.
  3. Maria Luisa Bonet and Samuel R. Buss. Size-depth tradeoffs for boolean fomulae. Information Processing Letters, 49(3):151-155, 1994. URL: https://doi.org/10.1016/0020-0190(94)90093-0.
  4. Mark Braverman, Klim Efremenko, Ran Gelles, and Michael A. Yitayew. Optimal short-circuit resilient formulas. In Computational Complexity Conference (CCC), volume 137, pages 10:1-10:22, 2019. URL: https://doi.org/10.4230/LIPICS.CCC.2019.10.
  5. T.-H. Hubert Chan, Zhibin Liang, Antigoni Polychroniadou, and Elaine Shi. Small memory robust simulation of client-server interactive protocols over oblivious noisy channels. In Symposium on Discrete Algorithms (SODA), pages 2349-2365, 2020. URL: https://doi.org/10.1137/1.9781611975994.144.
  6. Roland L'vovich Dobrushin and SI Ortyukov. Upper bound on the redundancy of self-correcting arrangements of unreliable functional elements. Problemy Peredachi Informatsii, 13(3):56-76, 1977. Google Scholar
  7. Klim Efremenko, Bernhard Haeupler, Yael Tauman Kalai, Pritish Kamath, Gillat Kol, Nicolas Resch, and Raghuvansh R. Saxena. Circuits resilient to short-circuit errors. In 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, 2022. Google Scholar
  8. William S. Evans and Nicholas Pippenger. On the maximum tolerable noise for reliable computation by formulas. IEEE Transactions on Information Theory, 44(3):1299-1305, 1998. URL: https://doi.org/10.1109/18.669417.
  9. William S. Evans and Leonard J. Schulman. Signal propagation and noisy circuits. IEEE Transactions on Information Theory, 45(7):2367-2373, 1999. URL: https://doi.org/10.1109/18.796377.
  10. William S. Evans and Leonard J. Schulman. On the maximum tolerable noise of k-input gates for reliable computation by formulas. IEEE Transactions on Information Theory, 49:3094-3098, 2003. URL: https://doi.org/10.1109/TIT.2003.818405.
  11. Tomás Feder. Reliable computation by networks in the presence of noise. IEEE Transactions on Information Theory, 35(3):569-571, 1989. URL: https://doi.org/10.1109/18.30978.
  12. Péter Gács and Anna Gál. Lower bounds for the complexity of reliable boolean circuits with noisy gates. IEEE Transactions on Information Theory, 40(2):579-583, 1994. URL: https://doi.org/10.1109/18.312190.
  13. Anna Gál. Lower bounds for the complexity of reliable boolean circuits with noisy gates. In Foundations of Computer Science (FOCS), pages 594-601, 1991. URL: https://doi.org/10.1109/SFCS.1991.185424.
  14. Anna Gál and Mario Szegedy. Fault tolerant circuits and probabilistically checkable proofs. In Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference, pages 65-73, 1995. URL: https://doi.org/10.1109/SCT.1995.514728.
  15. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. In Symposium on Theory of Computing (STOC), pages 902-911, 2018. URL: https://doi.org/10.1145/3188745.3188838.
  16. 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.
  17. Bruce E. Hajek and Timothy Weller. On the maximum tolerable noise for reliable computation by formulas. IEEE Transactions on Information Theory, 37(2):388-391, 1991. URL: https://doi.org/10.1109/18.75261.
  18. Yael Tauman Kalai, Allison B. Lewko, and Anup Rao. Formulas resilient to short-circuit errors. In Foundations of Computer Science (FOCS), pages 490-499, 2012. URL: https://doi.org/10.1109/FOCS.2012.69.
  19. Mauricio Karchmer and Avi Wigderson. Monotone circuits for connectivity require super-logarithmic depth. In Symposium on Theory of Computing (STOC), pages 539-550, 1988. URL: https://doi.org/10.1145/62212.62265.
  20. Daniel J. Kleitman, Frank Thomson Leighton, and Yuan Ma. On the design of reliable boolean circuits that contain partially unreliable gates. Journal of Computer and System Sciences, 55(3):385-401, 1997. URL: https://doi.org/10.1006/JCSS.1997.1531.
  21. Jan Krajícek. Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. Journal of Symbolic Logic, 62(2):457-486, 1997. URL: https://doi.org/10.2307/2275541.
  22. Nicholas Pippenger. On networks of noisy gates. In Foundations of Computer Science (FOCS), pages 30-38, 1985. URL: https://doi.org/10.1109/SFCS.1985.41.
  23. Nicholas Pippenger. Reliable computation by formulas in the presence of noise. IEEE Transactions on Information Theory, 34(2):194-197, 1988. URL: https://doi.org/10.1109/18.2628.
  24. Pavel Pudlák. On extracting computations from propositional proofs (a survey). In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010), volume 8, pages 30-41, 2010. URL: https://doi.org/10.4230/LIPICS.FSTTCS.2010.30.
  25. Alexander Razborov. Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic. Izvestiya of the RAN, pages 201-224, 1995. Google Scholar
  26. Leonard J Schulman. Communication on noisy channels: A coding theorem for computation. In Foundations of Computer Science (FOCS), pages 724-733. IEEE, 1992. URL: https://doi.org/10.1109/SFCS.1992.267778.
  27. Leonard J. Schulman. Deterministic coding for interactive communication. In Symposium on Theory of Computing (STOC), pages 747-756, 1993. URL: https://doi.org/10.1145/167088.167279.
  28. 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.
  29. Dmitry Sokolov. Dag-like communication and its applications. In Proceedings of the 12th Computer Science Symposium in Russia (CSR), pages 294-307. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-58747-9_26.
  30. John von Neumann. Probabilistic logics and the synthesis of reliable organisms from unreliable components. Automata Studies, pages 43-98, 1956. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact