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.

翻译：我们针对使用大小为 $q > 2$ 的物理字母表上的公式进行可靠计算，提出了两个新的正面结果。首先，我们证明，对于大小为 $\ell = q$ 的逻辑字母表，在受错误概率为 $\varepsilon$ 的 $q$ 元对称噪声干扰的门电路中进行去噪的阈值严格大于布尔计算的情况，并且在大扇入极限 $k \rightarrow \infty$ 下，只要信号保持可区分（即 $\epsilon < (q - 1) / q$），计算是可能的。我们还确定了具有有界扇入的广义多数门失效的点，并特别证明，在 $q$ 为素数且扇入 $k = 3$ 的情况下，当 $\epsilon < (q - 1) / (q (q + 1))$ 时，可靠计算是可能的。其次，我们给出了一个 $\ell < q$ 的示例，展示了通过利用额外的字母表元素进行错误信号传递，可以在受强度为 $\varepsilon < 1/6$ 的对称三元噪声干扰的 $2$ 输入三元逻辑门上进行可靠的布尔计算。