Synchronous Counting is the task of reaching agreement on a common round counter in a synchronous system of $n$ nodes with up to $t$ Byzantine faults in a self-stabilizing manner. That is, after transient faults may have arbitrarily corrupted the system state and ceased, the at least $n-t$ non-faulty nodes need to (re-)establish that (i) their local outputs are identical and (ii) increase by $1$ modulo $C$ in each round. An overhead-free reduction from consensus shows that all known lower bounds and impossibilities for consensus carry over to the counting problem. In the other direction, prior work has established that a consensus algorithm $\mathcal{A}$ can be turned into a counting algorithm at small overhead relative to the running time and bit complexity of $\mathcal{A}$, without losing resilience. Taking inspiration from early-stopping consensus protocols, in this work we introduce the concept of early stabilization. That is, if there are $0\le f\le t$ (persistent) faults in an execution, the algorithm should stabilize in a number of rounds that depends on $f$ only. Likewise, we seek to achieve an amortized bit complexity that is adaptive in the number of actual faults $f$. By developing a number of modular building blocks suitable to these goals, we develop a $C$-counting algorithm that stabilizes within asymptotically optimal $O(f+1)$ rounds, has message size $O(\log^2 n + \log C)$, and has amortized bit complexity $O(n(f\log C +\log^2 n))$.

翻译：同步计数任务是在一个包含$n$个节点、最多$t$个拜占庭故障的同步系统中，以自稳定方式就公共轮次计数器达成一致。即，在瞬时故障可能已任意破坏系统状态并停止后，至少$n-t$个无故障节点需要（重新）建立：（i）它们的本地输出一致，且（ii）每个轮次按模$C$增加$1$。从共识问题的无开销归约表明，共识的所有已知下界和不可能性均适用于计数问题。另一方面，先前工作已证实，可将共识算法$\mathcal{A}$以相对于$\mathcal{A}$的运行时间和比特复杂度较小的开销转化为计数算法，且不损失弹性。受早期停止共识协议的启发，本文引入早期稳定的概念。即，若一次执行中存在$0\le f\le t$个（持续）故障，算法应在仅依赖于$f$的轮数内稳定。类似地，我们寻求实现一个适应实际故障数$f$的摊销比特复杂度。通过开发若干适用于这些目标的模块化构建模块，我们提出一种$C$-计数算法，该算法在渐近最优的$O(f+1)$轮内稳定，消息大小为$O(\log^2 n + \log C)$，且摊销比特复杂度为$O(n(f\log C +\log^2 n))$。