Reliable computation by large-alphabet formulas in the presence of noise

TL;DR

This paper demonstrates that reliable computation is achievable with large-alphabet formulas under noise, surpassing Boolean limits, by analyzing thresholds for denoising and gate failure in noisy, multi-valued logic systems.

Contribution

It introduces new thresholds for noise tolerance in large-alphabet formulas and shows how to perform reliable Boolean computation using ternary logic with signaling.

Findings

Threshold for denoising exceeds Boolean case for large alphabets.

Reliable computation possible with noise level below specific thresholds.

Using additional alphabet elements enables Boolean computation under noise.

Abstract

We present two new positive results for reliable computation using formulas over physical alphabets of size $q > 2$. 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 is possible as long as signals remain distinguishable, i.e. $\epsilon < (q - 1) / q$, 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 $\epsilon < (q - 1) / (q (q + 1))$ in the case of $q$ prime and fan-in $k = 3$. Secondly, we provide an example where $\ell < q$, showing that reliable Boolean computation can be performed using $2$-input ternary logic gates subject to symmetric ternary noise of strength $\varepsilon < 1/6$ by using the additional alphabet element for error signaling.