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

TL;DR

This paper establishes syntactic characterizations of semantic properties of modal μ-calculus formulas, enabling effective decision procedures and linking automata theory with model theory.

Contribution

It provides new syntactic fragments for semantic properties of μ-calculus formulas, with automata-theoretic proofs and elementary-time decidability results.

Findings

Syntactic characterizations of semantic properties are effectively obtainable.

Decidability of properties is achievable in elementary time.

Automata transformations induce these semantic characterizations.

Abstract

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.