The famous asynchronous computability theorem (ACT) relates the existence of an asynchronous wait-free shared memory protocol for solving a task with the existence of a simplicial map from a subdivision of the simplicial complex representing the inputs to the simplicial complex representing the allowable outputs. The original theorem relies on a correspondence between protocols and simplicial maps in round-structured models of computation that induce a compact topology. This correspondence, however, is far from obvious for computation models that induce a non-compact topology, and indeed previous attempts to extend the ACT have failed. This paper shows that in every non-compact model, protocols solving tasks correspond to simplicial maps that need to be continuous. It first proves a generalized ACT for sub-IIS models, some of which are non-compact, and applies it to the set agreement task. Then it proves that in general models too, protocols are simplicial maps that need to be continuous, hence showing that the topological approach is universal. Finally, it shows that the approach used in ACT that equates protocols and simplicial complexes actually works for every compact model. Our study combines, for the first time, combinatorial and point-set topological aspects of the executions admitted by the computation model.

翻译：著名的异步可计算性定理(ACT)将求解任务的无等待异步共享内存协议的存在性，等价于存在从表示输入的单纯复形之细分到表示允许输出的单纯复形的单纯映射。原始定理依赖于在诱导紧致拓扑的轮次结构计算模型中协议与单纯映射之间的对应关系。然而，对于诱导非紧致拓扑的计算模型而言，这种对应关系远非显然——事实上，先前扩展ACT的尝试均已失败。本文证明：在每个非紧致模型中，求解任务的协议对应于需要满足连续性的单纯映射。首先证明子-IIS模型（其中部分模型是非紧致的）的广义ACT，并将其应用于集合协议任务。继而证明在一般模型中，协议同样是需要满足连续性的单纯映射，从而表明拓扑方法的普适性。最后证明，ACT中用于等价协议与单纯复形的方法对每个紧致模型均有效。本研究首次将计算模型所允许执行过程的组合拓扑与点集拓扑特性相结合。