We study two fundamental problems of distributed computing, consensus and approximate agreement, through a novel approach for proving lower bounds and impossibility results, that we call the asynchronous speedup theorem. For a given $n$-process task $\Pi$ and a given computational model $M$, we define a new task, called the closure of $\Pi$ with respect to $M$. The asynchronous speedup theorem states that if a task $\Pi$ is solvable in $t\geq 1$ rounds in $M$, then its closure w.r.t. $M$ is solvable in $t-1$ rounds in $M$. We prove this theorem for iterated models, as long as the model allows solo executions. We illustrate the power of our asynchronous speedup theorem by providing a new proof of the wait-free impossibility of consensus using read/write registers, and a new proof of the wait-free impossibility of solving consensus using registers and test\&set objects for $n>2$. The proof is merely by showing that, in each case, the closure of consensus (w.r.t. the corresponding model) is consensus itself. Our main application is the study of the power of additional objects, namely test\&set and binary consensus, for wait-free solving approximate agreement faster. By analyzing the closure of approximate agreement w.r.t. each of the two models, we show that while these objects are more powerful than read/write registers from the computability perspective, they are not more powerful as far as helping solving approximate agreement faster is concerned.
翻译:我们研究的是分配计算、共识和大致协议这两个根本问题,即通过新颖的方法来证明低限值和不可能的结果,即我们称之为无序超速理论。对于一个给定的美元处理任务和一个给定的计算模型,我们定义了一个新的任务,即关闭美元对美元。无序超速理论指出,如果一个任务用美元1回合以1美元计算,以美元计算,以美元计算,以美元计算较低的界限和不可能的结果,而以美元计算超速超速的超速理论。对于一个给定的美元处理任务和一个给定的计算模型,我们用美元计算超速理论来计算超速理论。只要该模型允许单独处决,我们就能证明这个超速模型。我们通过提供一个新的证据,证明我们无法等待的共识,使用免费登记册和测试目标以美元计算,以美元计算,以美元计算,以美元计算,以美元计算超速计算,以美元计算,以美元计算,以美元计算,以美元计算超速的超速计算。我们每次的登记册,以更接近的精确的精确的推算方式证明,我们最接近的“超速的协议”的“超速协议是“超速协议”。