On Two- and Three-valued Semantics for Impure Simplicial Complexes

TL;DR

This paper compares two- and three-valued semantics for impure simplicial complexes, categorizes different approaches, and introduces a translation bridging the two, clarifying fundamental modeling differences in topological semantics.

Contribution

It identifies the key distinction between two- and three-valued semantics and provides a novel translation from three- to two-valued semantics, clarifying their differences.

Findings

Main distinction is the number of truth values used.

A translation from three-valued to two-valued semantics is introduced.

Pure complexes are characterized as invariant under the translation.

Abstract

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.