Impure Simplicial Complex and Term-Modal Logic with Assignment Operators
Yuanzhe Yang
TL;DR
The paper develops a term-modal language with assignment operators to reason about epistemic states on impure simplicial complexes where agents may be dead. It defines both simplicial and first-order Kripke semantics, proves their expressivity alignment on local epistemic models, and provides a strongly complete axiomatization along with an assignment-normal form. It further analyzes intensional distributed knowledge and demonstrates how living-agent content can be captured while avoiding problematic dead-agent expressions. The work lays groundwork for future connections to three-valued semantics and broader structural generalizations beyond simplicial frameworks.
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.