Circuits Resilient to Short-Circuit Errors

SIAM Journal on Computing, Ahead of Print.
Abstract. Given a Boolean circuit [math], we wish to convert it to a circuit [math] that computes the same function as [math], even if some of its gates suffer from adversarial short circuit errors, i.e., their output is replaced by the value of one of their inputs [D. J. Kleitman, F. T. Leighton, and Y. Ma, J. Comput. System Sci., 55 (1997), pp. 385–401]. Can we design such a resilient circuit [math] whose size is roughly comparable to that of [math]? Prior work [T. Kalai, A. B. Lewko, and A. Rao, Formulas resilient to short-circuit errors, in Foundations of Computer Science (FOCS), 2012, pp. 490–499; M. Braverman et al., Optimal short-circuit resilient formulas, in Computational Complexity Conference (CCC), Vol. 137, 2019, pp. 10:1–10:22] gave a positive answer for the special case where [math] is a formula. We study the general case and show that any Boolean circuit [math] of size [math] can be converted to a new circuit [math] of quasi-polynomial size [math] that computes the same function as [math], even if a [math] fraction of the gates on any root-to-leaf path in [math] are short circuited. Moreover, if the original circuit [math] is a formula, the resilient circuit [math] is of near-linear size [math]. The construction of our resilient circuits utilizes the connection between circuits and dag-like communication protocols [A. Razborov, Izvestiya of the RAN, 59 (1995), pp. 201–224; P. Pudlák, On extracting computations from propositional proofs (a survey), in Foundations of Software Technology and Theoretical Computer Science (FSTTCS) Vol. 8, 2010, pp. 30–41; D. Sokolov, Dag-like communication and its applications, in Computer Science Symposium in Russia (CSR), Springer, 2017, pp. 294–307], originally introduced in the context of proof complexity.