Formulas Resilient to Short-Circuit Errors

We show how to efficiently convert any boolean formula F into a boolean formula E that is resilient to short-circuit errors (as introduced by Kleitman et al. [KLM94]). A gate has a short-circuit error when the value it computes is replaced by the value of one of its inputs. We guarantee that E computes the same function as F, as long as at most (1/10 – e) of the gates on each path from the output to an input have been corrupted in E. The corruptions may be chosen adversarially, and may depend on the formula E and even on the input. We obtain our result by extending the Karchmer-Wigderson connection between formulas and communication protocols to the setting of adversarial error. This enables us to obtain error-resilient formulas from error-resilient communication protocols.