Some model theory for the modal $μ$-calculus: syntactic characterisations of semantic properties

Abstract

This paper contributes to the theory of the modal $μ$-calculus by proving some model-theoretic results. More in particular, we discuss a number of semantic properties pertaining to formulas of the modal $μ$-calculus. For each of these properties we provide a corresponding syntactic fragment, in the sense that a $μ$-formula $ξ$ has the given property iff it is equivalent to a formula $ξ'$ in the corresponding fragment. Since this formula $ξ'$ will always be effectively obtainable from $ξ$, as a corollary, for each of the properties under discussion, we prove that it is decidable in elementary time whether a given $μ$-calculus formula has the property or not. The properties that we study all concern the way in which the meaning of a formula $ξ$ in a model depends on the meaning of a single, fixed proposition letter $p$. For example, consider a formula $ξ$ which is monotone in $p$; such a formula a formula $ξ$ is called continuous (respectively, fully additive), if in addition it satisfies the property that, if $ξ$ is true at a state $s$ then there is a finite set (respectively, a singleton set) $U$ such that $ξ$ remains true at $s$ if we restrict the interpretation of $p$ to the set $U$. Each of the properties that we consider is, in a similar way, associated with one of the following special kinds of subset of a tree model: singletons, finite sets, finitely branching subtrees, noetherian subtrees (i.e., without infinite paths), and branches. Our proofs for these characterization results will be automata-theoretic in nature; we will see that the effectively defined maps on formulas are in fact induced by rather simple transformations on modal automata. Thus our results can also be seen as a contribution to the model theory of modal automata.