Kupferman, OrnaPnueli, AmirVardi, Moshe Y.2014-11-212014-11-212012Kupferman, Orna, Pnueli, Amir and Vardi, Moshe Y.. "Once and For All." <i>Journal of Computer and System Sciences,</i> 78, no. 3 (2012) Elsevier: 981-996. http://dx.doi.org/10.1016/j.jcss.2011.08.006.https://hdl.handle.net/1911/78494It 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.engThis is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier.Journal articletemporal logicpast-time operatorsexpressive powerdecision proceduresalternating automatahttp://dx.doi.org/10.1016/j.jcss.2011.08.006