Impure Simplicial Complex and Term-Modal Logic with Assignment Operators

TL;DR

This paper introduces a new term-modal language with assignment operators for modeling multi-agent epistemic scenarios involving dead agents, providing semantics, axiomatization, and expressivity analysis to improve conceptual clarity.

Contribution

It develops a novel term-modal language with assignment operators, along with semantics, axiomatization, and expressivity results for modeling epistemic situations with dead agents.

Findings

Semantic equivalence between simplicial and Kripke models for local epistemic models

Complete axiomatization of the proposed epistemic logic

Language features a normal form and effectively models distributed knowledge

Abstract

Impure simplicial complexes are a powerful tool to model multi-agent epistemic situations where agents may die, but it is difficult to define a satisfactory semantics for the ordinary propositional modal language on such models, since many conceptually dubious expressions involving dead agents can be expressed in this language. In this paper, we introduce a term-modal language with assignment operators, in which such conceptually dubious expressions are syntactically excluded. We define both simplicial semantics and first-order Kripke semantics for this language, characterize their respective expressivity through notions of bisimulation, and show that the two semantics are equivalent when we consider a special class of first order Kripke models called local epistemic models. We also offer a complete axiomatization for the epistemic logic based on this language, and show that our language has a notion of assignment normal form. Finally, we discuss the behavior of a kind of intensional distributed knowledge that can be naturally expressed in our language.