Noise threshold for universality of 2-input gates

TL;DR

This paper establishes a noise threshold of approximately 8.856% for 2-input gates in formulas to maintain computational universality, showing that higher noise levels prevent universal computation.

Contribution

It proves that formulas with 2-input gates cannot be universal if the gate failure probability exceeds the threshold, extending previous results on noise tolerance.

Findings

Threshold for universality is approximately 8.856%

Formulas with higher noise cannot perform universal computation

Conjecture that the same threshold applies to circuits

Abstract

Evans and Pippenger showed in 1998 that noisy gates with 2 inputs are universal for arbitrary computation (i.e. can compute any function with bounded error), if all gates fail independently with probability epsilon and epsilon<theta, where theta is roughly 8.856%. We show that formulas built from gates with 2 inputs, in which each gate fails with probability at least theta cannot be universal. Hence, there is a threshold on the tolerable noise for formulas with 2-input gates and it is theta. We conjecture that the same threshold also holds for circuits.