An infinite hierarchy of temporal logics over branching time

Many temporal logics have been suggested as branching time specification formalisms during the past 20 years. These logics were compared against each other for their expressive power, model checking complexity, and succinetness. Yet, unlike the case for linear time logics, no canonical temporal logic of branching time was agreed upon. We offer an explanation for the multiplicity of temporal logics over branching time and provide an objective quantified yardstick to measure these logics. We define an infinite hierarchy BTL_k of temporal logics and prove its strictness. We examine the expressive power of commonly used branching time temporal logics. We show that CTL* has no finite base, and that almost all of its many sublogics suggested in the literature are inside the second level of our hierarchy. We introduce new Ehrenfeucht-Fraissé games on trees and use them as our main tool to prove inexpressibility.