Explicit Generic Common Knowledge

Abstract

The name Generic Common Knowledge (GCK) was suggested by Artemov to capture a state of a multi-agent epistemic system that yields iterated knowledge I(ϕ): ‘any agent knows that any agent knows that any agent knows…ϕ’ for any number of iterations. The generic common knowledge of ϕ, $\mbox{\em GCK}(\varphi)$, yields I(ϕ),

$$ \mbox{\em GCK}(\varphi)\rightarrow I(\varphi) $$

but is not necessarily logically equivalent to I(ϕ). Modal logics with GCK were suggested by McCarthy and Artemov. It has been shown that in the usual epistemic scenarios, GCK can replace the conventional common knowledge. Artemov noticed that such epistemic actions as public announcements of atomic sentences, generally speaking, yield GCK rather than the conventional common knowledge. In this paper we introduce logics with explicit GCK and show that they realize corresponding modal systems, i.e., GCK, along with the individual knowledge modalities, can be always made explicit.