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We study 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… (More)

For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognise this language with a non-deterministic or alternating parity automaton. These questions are known as, respectively, the non-deterministic and the alternating Rabin-Mostowski index problems. Whether they can be answered effectively is a… (More)

We show that the modal µ-calculus over GL collapses to the modal fragment by showing that the fixpoint formula is reached after two iterations and answer to a question posed by van Benthem in [vBe06]. Further, we introduce the modal µ ∼-calculus by allowing fixpoint con-structors for any formula where the fixpoint variable appears guarded but not… (More)

Alternating automata on infinite trees induce operations on languages which do not preserve natural equivalence relations, like having the same Mostowski–Rabin index, the same Borel rank, or being continuously reducible to each other (Wadge equivalence). In order to prevent this, alternation needs to be restricted to the choice of direction in the tree. For… (More)

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… (More)

Recently, Mikołaj Bojá nczyk introduced a class of max-regular languages, an extension of regular languages of infinite words preserving many of its usual properties. This new class can be seen as a different way of generalizing the notion of regularity from finite to infinite words. This paper compares regular and max-regular languages in terms of… (More)

For deterministic tree automata, classical hierarchies, like Mostowski-Rabin (or index) hierarchy, Borel hierarchy, or Wadge hierarchy, are known to be decidable. However, when it comes to non-deterministic tree automata, none of these hierarchies is even close to be understood. Here we make an attempt in paving the way towards a clear understanding of tree… (More)

We prove that the bisimulation-invariant fragment of weak monadic second-order logic (WMSO) is equivalent to the fragment of the modal <i>μ</i>-calculus where the application of the least fixpoint operator <i>μp</i>.φ is restricted to formulas φ that are continuous in <i>p</i>. Our proof is automata-theoretic in nature; in particular, we… (More)

We show that it is decidable whether a given a regular tree language belongs to the class ∆ 0 2 of the Borel hierarchy, or equivalently whether the Wadge degree of a regular tree language is countable.

—We provide a characterization theorem, in the style of van Benthem and Janin-Walukiewicz, for the alternation-free fragment of the modal µ-calculus. For this purpose we introduce a variant of standard monadic second-order logic (MSO), which we call well-founded monadic second-order logic (WFMSO). When interpreted in a tree model, the second-order… (More)