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The Modal μ-Calculus Hierarchy on Restricted Classes of Transition Systems
The Journal of Symbolic Logic
We discuss the strictness of the modal µ-calculus hierarchy over some restricted classes of transition systems. First, we show that the hierarchy is strict over reﬂexive frames. By proving the ﬁnite model theorem for reﬂexive systems the same results holds for ﬁnite models. Second, we prove that over transitive systems the hierarchy collapses to the alternation-free fragment. In order to do this the ﬁnite model theorem for transitive transition systems is also proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment.
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