Once and for all

It has long been known that past-time operators add no expressive power to linear temporal logics. In this paper, we consider the extension of branching temporal logics with past-time operators. Two possible views regarding the nature of past in a branching-time model induce two different such extensions. In the first view, past is branching and each moment in time may have several possible futures and several possible pasts. In the second view, past is linear and each moment in time may have several possible futures and a unique past. Both views assume that past is finite. We discuss the practice of these extensions as specification languages, characterize their expressive power, and examine the complexity of their model-checking and satisfiability problems.

► We consider the extension of branching temporal logics with past-time operators.
► We examine two possible semantics: one in which past is linear and one in which past is branching.
► For both semantics, we study their practice as specification languages and characterize their expressive power.
► We study the complexity of the model-checking and satisfiability problems for the extended logics.