Impure Simplicial Complexes: Complete Axiomatization

Combinatorial topology is used in distributed computing to model concurrency and asynchrony. The basic structure in combinatorial topology is the simplicial complex, a collection of subsets called simplices of a set of vertices, closed under containment. Pure simplicial complexes describe message passing in asynchronous systems where all processes (agents) are alive, whereas impure simplicial complexes describe message passing in synchronous systems where processes may be dead (have crashed). Properties of impure simplicial complexes can be described in a three-valued multi-agent epistemic logic where the third value represents formulas that are undefined, e.g., the knowledge and local propositions of dead agents. In this work we present the axiomatization called $\mathsf{S5}^{\bowtie}$ and show that it is sound and complete for the class of impure complexes. The completeness proof involves the novel construction of the canonical simplicial model and requires a careful manipulation of undefined formulas.