  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.
