We investigate the fundamental task of addition under uncertainty, namely, addends that are represented as intervals of numbers rather than single values. One potential source of such uncertainty can occur when obtaining discrete-valued measurements of analog values, which are prone to metastability. Naturally, unstable bits impact gate-level and, consequently, circuit-level computations. Using Binary encoding for such an addition produces a sum with an amplified imprecision. Hence, the challenge is to devise an encoding that does not amplify the imprecision caused by unstable bits. We call such codes recoverable. While this challenge is easily met for unary encoding, no suitable codes of high rates are known. In this work, we prove an upper bound on the rate of preserving and recoverable codes for a given bound on the addends' combined uncertainty. We then design an asymptotically optimal code that preserves the addends' combined uncertainty. We then discuss how to obtain adders for our code. The approach can be used with any known or future construction for containing metastability of the inputs. We conjecture that careful design based on existing techniques can lead to significant latency reduction.

翻译：本研究探讨不确定性条件下的基本加法任务，即加数以数值区间而非单一值的形式表示。此类不确定性的潜在来源之一可能出现在获取模拟值的离散化测量结果时，这些测量结果容易受到亚稳态的影响。自然，不稳定比特会影响门级运算，进而影响电路级计算。对此类加法使用二进制编码会产生精度损失被放大的求和结果。因此，挑战在于设计一种不会放大由不稳定比特引起的精度损失的编码方案，我们将此类编码称为可恢复编码。虽然单热编码能轻易满足此要求，但目前尚未已知具有高编码率且适用的编码方案。本工作中，我们首先证明了在给定加数总不确定度约束下，保持性可恢复编码的编码率存在上界。随后我们设计了一种能保持加数总不确定度的渐近最优编码方案。接着我们讨论了如何实现该编码的加法器结构。该方法可与任何已知或未来提出的输入亚稳态控制技术结合使用。我们推测基于现有技术的精心设计能够显著降低运算延迟。