Reliable Computation by Large-Alphabet Formulas in the Presence of Noise

We present two new positive results for reliable computation using formulas over physical alphabets of size $q \gt 2$ . First, we show that for logical alphabets of size $\ell = q$ the threshold for denoising using gates subject to q-ary symmetric noise with error probability $\varepsilon $ is strictly larger than that for Boolean computation, and we show that reliable computation is possible as long as signals remain distinguishable, i.e. $\epsilon \lt (q - 1) / q$ , in the limit of large fan-in $k \rightarrow \infty $ . We also determine the point at which generalized majority gates with bounded fan-in fail, and show in particular that reliable computation is possible for $\epsilon \lt (q - 1) / (q (q + 1))$ in the case of q prime and fan-in $k = 3$ . Secondly, we provide an example where $\ell \lt q$ , showing that reliable Boolean computation, $\ell = 2$ , can be performed using 2-input ternary, $q = 3$ , logic gates subject to symmetric ternary noise of strength $\varepsilon \lt 1/6$ by using the additional alphabet element for error signaling.