Simplicial complexes are a convenient semantic primitive to reason about processes (agents) communicating with each other in synchronous and asynchronous computation. Impure simplicial complexes distinguish active processes from crashed ones, in other words, agents that are alive from agents that are dead. In order to rule out that dead agents reason about themselves and about other agents, three-valued epistemic semantics have been proposed where, in addition to the usual values true and false, the third value stands for undefined: the knowledge of dead agents is undefined and so are the propositional variables describing their local state. Other semantics for impure complexes are two-valued where a dead agent knows everything. Different choices in designing a semantics produce different three-valued semantics, and also different two-valued semantics. In this work, we categorize the available choices by discounting the bad ones, identifying the equivalent ones, and connecting the non-equivalent ones via a translation. The main result of the paper is identifying the main relevant distinction to be the number of truth values and bridging this difference by means of a novel embedding from three- into two-valued semantics. This translation also enables us to highlight quite fundamental modeling differences underpinning various two- and three-valued approaches in this area of combinatorial topology. In particular, pure complexes can be defined as those invariant under the translation.

翻译：单纯复形是一种方便的语义原语，用于推理在同步和异步计算中相互通信的进程（智能体）。不纯单纯复形区分活跃进程与崩溃进程，即存活智能体与失效智能体。为了排除失效智能体对自身及其他智能体的推理，研究者提出了三值认知语义，其中除了通常的真与假之外，第三个值表示未定义：失效智能体的知识及其描述局部状态的命题变量均为未定义。而不纯复形的其他语义采用二值形式，其中失效智能体知晓一切。不同语义设计选择会产生不同的三值语义以及不同的二值语义。本研究通过排除不良选择、识别等价选择，并通过翻译连接非等价选择，对现有选择进行分类。本文的主要成果是识别出真值数量为主要相关区别，并通过一种新颖的从三值语义到二值语义的嵌入来弥合这种差异。该翻译还使我们能够突出该组合拓扑领域中各种二值与三值方法背后的基本建模差异。特别地，纯复形可定义为对翻译保持不变的那些复形。