10.23638/LMCS-14(1:14)2018 Fontaine, Gaëlle Gaëlle Fontaine Venema, Yde Yde Venema Some model theory for the modal $\mu$-calculus: syntactic characterisations of semantic properties episciences.org 2018 Computer Science - Logic in Computer Science F.4.1 contact@episciences.org episciences.org 2018-01-22T15:40:03+01:00 2018-02-06T09:28:12+01:00 2018-02-06 eng Journal article https://lmcs.episciences.org/4225 arXiv:1801.05994 1860-5974 PDF 1 Logical Methods in Computer Science ; Volume 14, Issue 1 ; 1860-5974 This paper contributes to the theory of the modal $\mu$-calculus by proving some model-theoretic results. More in particular, we discuss a number of semantic properties pertaining to formulas of the modal $\mu$-calculus. For each of these properties we provide a corresponding syntactic fragment, in the sense that a $\mu$-formula $\xi$ has the given property iff it is equivalent to a formula $\xi'$ in the corresponding fragment. Since this formula $\xi'$ will always be effectively obtainable from $\xi$, as a corollary, for each of the properties under discussion, we prove that it is decidable in elementary time whether a given $\mu$-calculus formula has the property or not. The properties that we study all concern the way in which the meaning of a formula $\xi$ in a model depends on the meaning of a single, fixed proposition letter $p$. For example, consider a formula $\xi$ which is monotone in $p$; such a formula a formula $\xi$ is called continuous (respectively, fully additive), if in addition it satisfies the property that, if $\xi$ is true at a state $s$ then there is a finite set (respectively, a singleton set) $U$ such that $\xi$ 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.