Optimal Short-Circuit Resilient Formulas

07/13/2018
by   Mark Braverman, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
11/06/2018

Interactive coding resilient to an unknown number of erasures

We consider distributed computations between two parties carried out ove...
research
07/03/2022

Efficient Interactive Coding Achieving Optimal Error Resilience Over the Binary Channel

Given a noiseless protocol π_0 computing a function f(x, y) of Alice and...
research
05/02/2020

On P Versus NP

I generalize a well-known result that P = NP fails for monotone polynomi...
research
02/20/2020

Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates

The class FORMULA[s] ∘𝒢 consists of Boolean functions computable by size...
research
12/14/2018

Teleportation-based quantum homomorphic encryption scheme with quasi-compactness and perfect security

This article defines encrypted gate, which is denoted by EG[U]:|α〉→((a,b...
research
09/08/2023

On Interactive Coding Schemes with Adaptive Termination

In interactive coding, Alice and Bob wish to compute some function f of ...
research
07/13/2023

Short Boolean Formulas as Explanations in Practice

We investigate explainability via short Boolean formulas in the data mod...

Please sign up or login with your details

Forgot password? Click here to reset