Once and for all

Orna Kupferman*, Amir Pnueli, Moshe Y. Vardi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


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.

Original languageAmerican English
Pages (from-to)981-996
Number of pages16
JournalJournal of Computer and System Sciences
Issue number3
StatePublished - May 2012


  • Alternating automata
  • Decision procedures
  • Expressive power
  • Past-time operators
  • Temporal logic


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