Axioms for Logics of Knowledge and Past Time: Synchrony and Unique Initial States

Sound and complete axiomatizations are provided for two different logics involving modalities for knowledge and both past and future time modalities. The logics considered allow for multiple agents with unique initial state and synchrony. The synchrony restriction gives every agent access to a system clock. Such semantic restrictions are of particular interest in the context of past time modalities since both synchrony and unique initial state restrictions are not expressible using future time modalities.