Optimal Short-Circuit Resilient Formulas

Mark Braverman, Klim Efremenko, Ran Gelles, Michael Yitayew

Research output: Contribution to journalArticlepeer-review


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 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 of 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 corruptions in up to a fraction of 1/5 of the transmissions in each direction of the interactive channel. We further show that such a level of noise is maximal for coding schemes whose communication blowup is sub-exponential. Our coding scheme has taken a surprising inspiration from Blockchain technology.

Original languageEnglish (US)
Article number26
JournalJournal of the ACM
Issue number4
StatePublished - Aug 23 2022

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Systems Engineering
  • Information Systems
  • Hardware and Architecture
  • Artificial Intelligence


  • Circuit complexity
  • Karchmer-Wigderson games
  • coding theory
  • interactive coding
  • noise-resilient circuits


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