2026-07-22T21:35:13Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10832 2024-03-06T10:46:49Z ddc:004 open_access

Optimal Short-Circuit Resilient Formulas Braverman, Mark Efremenko, Klim Gelles, Ran Yitayew, Michael A. Circuit Complexity Noise-Resilient Circuits Interactive Coding Coding Theory Karchmer-Wigderson Games 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Mark Braverman and Klim Efremenko and Ran Gelles and Michael A. Yitayew 2019 Is Part Of LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2019.10 urn:nbn:de:0030-drops-108326 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.10 eng https://creativecommons.org/licenses/by/3.0/legalcode