Computability, in the presence of asynchrony and failures, is one of the central questions in distributed computing. The celebrated asynchronous computability theorem (ACT) charaterizes the computing power of the read-write shared-memory model through the geometric properties of its protocol complex: a combinatorial structure describing the states the model can reach via its finite executions. This characterization assumes that the memory is of unbounded capacity, in particular, it is able to store the exponentially growing states of the full-information protocol. In this paper, we tackle an orthogonal question: what is the minimal memory capacity that allows us to simulate a given number of rounds of the full-information protocol? In the iterated immediate snapshot model (IIS), we determine necessary and sufficient conditions on the number of bits an IIS element should be able to store so that the resulting protocol is equivalent, up to isomorphism, to the full-information protocol. Our characterization implies that $n\geq 3$ processes can simulate $r$ rounds of the full-information IIS protocol as long as the bit complexity per process is within $\Omega(r n)$ and $O(r n \log n)$. Two processes, however, can simulate any number of rounds of the full-information protocol using only $2$ bits per process, which implies, in particular, that just $2$ bits per process are sufficient to solve $\varepsilon$-agreement for arbitrarily small $\varepsilon$.

翻译：在异步与故障存在条件下的可计算性是分布式计算的核心问题之一。著名的异步可计算定理（ACT）通过协议复形的几何性质——描述模型在有限执行中可达状态的组合结构——刻画了读写共享内存模型的计算能力。该刻画假设内存容量无上限，特别地，它能够存储全信息协议中呈指数增长的状态。本文探讨了一个正交问题：模拟全信息协议给定轮次所需的最小内存容量是多少？在迭代即时快照模型（IIS）中，我们确定了IIS元素所需存储比特数达到与全信息协议同构等价的充分必要条件。该刻画表明：当每个进程的比特复杂度处于$\Omega(r n)$与$O(r n \log n)$之间时，$n\geq 3$个进程可模拟$r$轮全信息IIS协议。然而，两个进程仅需每个进程$2$比特即可模拟任意轮次的全信息协议，这特别意味着每个进程仅需$2$比特便足以解决任意小$\varepsilon$的$\varepsilon$-共识问题。