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 finite 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 first that in every non-compact model, protocols solving tasks correspond to simplicial maps that need to be continuous. This correspondence is then used to prove that the approach used in ACT that equates protocols and simplicial complexes actually works for every compact model, and to show a generalized ACT, which applies also to non-compact computation models. Finally, the generalized ACT is applied to the set agreement task. Our study combines 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）将异步无等待共享内存协议解决任务的存在性，与从表示输入的单纯复形细分到表示允许输出的单纯复形的单纯映射的存在性联系起来。该原始定理依赖于诱导紧致拓扑的有限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，这种对应关系远非显而易见，且此前扩展ACT的尝试均未成功。本文首先证明：在每一个非紧致模型中，解决任务的协议对应于需要满足连续性的单纯映射。随后，利用这一对应关系证明了ACT中用于等价协议与单纯复形的方法实际上适用于所有紧致模型，并给出了一个也适用于非紧致计算模型的广义ACT。最后，将广义ACT应用于集合共识任务。本研究结合了计算模型所允许执行的组合拓扑与点集拓扑特性。