Noise Threshold for Universality of Two-Input Gates

It is known that epsi-noisy gates with two inputs are universal for arbitrary computation (i.e., can compute any function with bounded error), if all gates fail independently with probability epsi and epsi < beta
₂
= (3 - radic7)/4 ap 8.856%. In this paper, it is shown that this bound is tight for formulas, by proving that gates with two inputs, in which each gate fails with probability at least beta
₂
cannot be universal. Hence, there is a threshold on the tolerable noise for formulas with two-input gates and it is beta
₂
. It is conjectured that the same threshold also holds for circuits.