Codes for Metastability-Containing Addition

TL;DR

This paper addresses the challenge of performing addition with uncertain, metastability-affected inputs by designing specialized codes that prevent imprecision amplification, providing theoretical bounds and an optimal code construction.

Contribution

It establishes an upper bound on the rate of metastability-containing codes and introduces an asymptotically optimal code for preserving addend uncertainty.

Findings

Proved an upper bound on code rate for metastability containment.

Designed an asymptotically optimal code for uncertain addition.

Discussed implementation of adders compatible with the proposed codes.

Abstract

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.