Map of Common Knowledge Logics

In order to capture the concept of common knowledge, various extensions of multi-modal epistemic logics, such as xed-point type extensions and in nitary ones, have been proposed. Although we have now a good list of such proposed extensions, the relationships among these extensions are still unclear. The purpose of this paper is to draw a map showing the relationships among them. In the propositional case, these logics turn to be all Kripke complete and could be comparable in a meaningful manner. In the predicate case, there is the gap shown by F. Wolter that the predicate extension of the Halpern-Moses xedpoint type common knowledge logic is Kripke incomplete. However, if we go further to an in nitary extension, Kripke completeness would be recovered. In drawing the map, we focus on what are happening around the gap in the predicate case. The map enables us to better understand the common knowledge logics as a whole.