Abstract

We provide several effective equivalent characterizations of EF (the modal logic of the descendant relation) on arbitrary trees. More specifically, we prove that, for EF-bisimulation invariant properties of trees, being definable by an EF formula, being a Borel set, and being definable in weak monadic second order logic, all coincide. The proof builds upon a known algebraic characterization of EF for the case of finitely branching trees due to Bojanczyk and Idziaszek. We furthermore obtain characterizations of modal logic on transitive Kripke structures as a fragment of weak monadic second order logic and of the µ-calculus.

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