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Why so many temporal logics climb up the trees.pdf

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Why so many temporal logics climb up the trees

Why so many Temporal Logics Climb up the Trees?Alexander Rabinovich and Shahar MaozSchool of Mathematical ScienceTel Aviv UniversityTel Aviv, Israel 69978frabino, maozg@math.tau.ac.ilAbstract. Many temporal logics were suggested as branching time spec-i cation formalisms during the last 20 years. These logics were comparedagainst each other for their expressive power, model checking complexityand succinctness. Yet, unlike the case for linear time logics, no canonicaltemporal logic of branching time was agreed upon. We oer an expla-nation for the multiplicity of temporal logics over branching time andprovide an objective quanti ed `yardstick to measure these logics.We de ne an in nite hierarchy BTLk of temporal logics and prove itsstrictness. We show that CTL has no nite base, and that almost allof its many sub-logics suggested in the literature are inside the secondlevel of our hierarchy. We show that for every logic based on a nite setof modalities, the complexity of model checking is linear both in the sizeof structure and the size of formula.1 IntroductionVarious temporal logics have been proposed for reasoning about so-called \re-active systems, computer hardware or software systems which exhibit (poten-tially) non-terminating and non-deterministic behavior. Such a system is typi-cally represented by the (potentially) in nite sequences of computation statesthrough which it may evolve, where we associate with each state the set of atomicpropositions which are true in that state, along with the possible next state tran-sitions to which it may evolve. Thus its behavior is denoted by a (potentially)in nite rooted tree, with the initial state of the system represented by the rootof the tree.Temporal Logic (TL) is a convenient framework for the speci cation prop-erties of systems. This made TL a popular subject in the Computer Sciencecommunity and it enjoyed extensive research during the last 20 years. In tem-poral logic the relevant properties of the system are d