Frame Definability for Classes of Trees in the µ-calculus

We are interested in frame definability of classes of trees, using formulas of the μ-calculus. In this set up, the proposition letters (or in other words, the free variables) in the μ-formulas correspond to second order variables over which universally quantify. Our main result is a semantic characterization of the MSO definable classes of trees that are definable by a μ-formula. We also show that it is decidable whether a given MSO formula corresponds to a μ-formula, in the sense that they define the same class of trees. Basic modal logic and μ-calculus can be seen as logical languages for talking about Kripke models and Kripke frames. On Kripke models every modal formula is equivalent to a first order formula in one free variable and every μ-calculus formula is equivalent to a monadic second order formula in one free first order variable. On Kripke frames, we universally quantify over the free propositional variables occurring in the formulas and each modal formula or μ-formula is equivalent to a sentence of monadic second order logic. For example, the modal formula p→ 3p corresponds locally on Kripke models to the first order formula α(u, P ) = P (u)→ ∃v(uRv ∧P (v)) (where P is a unary predicate corresponding to p, R is the binary relation of the model and u is a point of the model). The same modal formula corresponds globally on Kripke frames to the second order sentence ∀P∀uα(u, P ), which happens to be equivalent to the first order sentence ∀u, uRu. The expressive power of modal logic from both perspectives (models and frames) has been extensively studied. For Kripke models, Johan van Benthem characterized modal logic semantically as the bisimulation invariant fragment of first order logic [vB76]. The problem whether a formula of first order logic in one free variable has a modal correspondent on the level of models, is undecidable [vB96]. The expressive power of modal logic on Kripke frames has been studied since the 1970s and this study gave rise to many key results in the modal logic area. When interpreted on frames, modal logic corresponds to a fragment of monadic ? The research of the first author has been made possible by VICI grand 639.073.501 of the NWO. We would like to thank Balder ten Cate for inspiring us and for helping us continuously during the research and redaction of this paper. We also thank Johan van Benthem, Diego Figueira, Luc Segoufin and Yde Venema for helpful insights. second order logic, because the definition of validity involves quantifying over all the proposition letters in the formulas. However, most work has concentrated on the first order aspect of modal definability. A landmark result is the Goldblatt-Thomason Theorem [GT75] which gives a characterization of the first order definable classes of frames that are modally definable, in terms of semantic criteria. It is undecidable whether a given first order sentence corresponds to a modal formula, in the sense that they define the same class of frames. On the level of Kripke models, the expressive power of the μ-calculus is well understood. In [JW96], David Janin and Igor Walukiewicz showed that the μ-calculus is the bisimulation invariant fragment of MSO. It is undecidable whether a class of Kripke models definable in MSO is definable by a formula of the μ-calculus. For classes of trees, the problem becomes decidable (see [JW96]). About the expressive power of the μ-calculus on the level of Kripke frames, nothing is known. This paper contributes to a partial solution of this question by giving a characterization of theMSO definable classes of trees that are definable by a μ-formula. Our main result states that an MSO definable class of trees is definable in the μ-calculus iff it is closed for subtrees and p-morphic images. We also show that given an MSO formula, it is decidable whether there exists a μ-formula which defines the same class of trees as the MSO formula. The proof is in three steps. First, we use the connection between MSO and the graded μ-calculus proved by Igor Walukiewicz [Wal02] and establish a correspondence between the MSO formulas that are preserved under p-morphic images and a fragment that is between the μ-calculus and the graded μ-calculus (the fragment with a counting 2 operator and a usual 3 operator). We call this fragment the 2-graded μ-calculus. The second step consists in showing that each 2-graded μ-formula φ can be translated into a μ-formula ψ such that locally, the truth of φ (on trees seen as Kripke models) corresponds to the validity of ψ (on trees seen as Kripke frames). So this step is a move from the model perspective to the frame perspective. The last step consists in shifting from the local perspective to the global one (that is, we are interested in validity at all points, not at a given point).