修正模拟：证明空间下界的新方法
Revisionist Simulations: A New Approach to Proving Space Lower Bounds
摘要：
对于$x\leq k$，确定$x$ -无障碍$k$ -集合一致性的空间复杂度是一个开放的问题。在$x$ -无障碍协议中，要求进程在执行中返回，其中至多$x$进程采取步骤。在$n>k$进程中，解决这个问题所需的寄存器数量的最佳上限是$n-k+x$寄存器。没有比$2$更好的一般下限。我们证明，任何$x$ -无障碍协议解决$n>k$进程之间的$k$ -设置协议使用至少$\lfloor(n-x)/(k+1-x)\rfloor+1$寄存器。我们的主要工具是一个模拟，作为减少从不可能的确定性等待自由$k$ -集协议：如果一个协议使用较少的寄存器，那么它是可能的$k+1$进程模拟协议和确定性解决$k$ -集协议在一个等待自由的方式，这是不可能的。模拟的一个关键组成部分是模拟过程的能力，以修改过去的模拟过程。我们引入了一个新的增强快照对象，这有助于这一点。我们还证明了任何空间上的无障碍协议所使用的寄存器的数量下界适用于协议，满足非确定性独奏终止。因此，我们针对无障碍（$x=1$）情况的$\lfloor(n-1)/k\rfloor+1$下限也适用于随机无等待无协议。特别是，这给出了一个严格的下限$n$寄存器，用于解决无障碍和随机等待的共识。最后，我们的新技术可以应用于获得一个空间较低的$\lfloor n/2\rfloor+1$的$ε$ -近似协议，足够小的$ε$。它要求参与的进程在彼此的$ε$中返回值。最著名的上限是$\lceil\log(1/ε)\rceil$和$n$，而没有一般的下限是已知的。
英文摘要：
Determining the space complexity of $x$-obstruction-free $k$-set agreement for $x\leq k$ is an open problem. In $x$-obstruction-free protocols, processes are required to return in executions where at most $x$ processes take steps. The best known upper bound on the number of registers needed to solve this problem among $n>k$ processes is $n-k+x$ registers. No general lower bound better than $2$ was known. We prove that any $x$-obstruction-free protocol solving $k$-set agreement among $n>k$ processes uses at least $\lfloor(n-x)/(k+1-x)\rfloor+1$ registers. Our main tool is a simulation that serves as a reduction from the impossibility of deterministic wait-free $k$-set agreement: if a protocol uses fewer registers, then it is possible for $k+1$ processes to simulate the protocol and deterministically solve $k$-set agreement in a wait-free manner, which is impossible. A critical component of the simulation is the ability of simulating processes to revise the past of simulated processes. We introduce a new augmented snapshot object, which facilitates this. We also prove that any space lower bound on the number of registers used by obstruction-free protocols applies to protocols that satisfy nondeterministic solo termination. Hence, our lower bound of $\lfloor(n-1)/k\rfloor+1$ for the obstruction-free ($x=1$) case also holds for randomized wait-free free protocols. In particular, this gives a tight lower bound of exactly $n$ registers for solving obstruction-free and randomized wait-free consensus. Finally, our new techniques can be applied to get a space lower of $\lfloor n/2\rfloor+1$ for $ε$-approximate agreement, for sufficiently small $ε$. It requires participating processes to return values within $ε$ of each other. The best known upper bounds are $\lceil\log(1/ε)\rceil$ and $n$, while no general lower bounds were known.
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