Optimal Short-Circuit Resilient Formulas

TL;DR

This paper enhances fault-tolerance in boolean formulas, increasing resilience from 1/6 to 1/5 of short-circuit faults with polynomial size growth, and establishes the optimality of this resilience level using interactive coding schemes inspired by Blockchain.

Contribution

It improves the resilience of boolean formulas against short-circuit faults from 1/6 to 1/5 and proves this is the maximum possible with sub-exponential size increase, using novel coding schemes.

Findings

Resilience against 1/5 faults achieved with polynomial size increase

Maximum resilience of 1/5 established for sub-exponential size formulas

Coding schemes inspired by Blockchain technology developed for fault tolerance

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.