Weak MSO: Automata and Expressiveness Modulo Bisimilarity

ABSTRACT. We prove that the bisimulation-invariant fragment of weak monadic second-order logic (WMSO) is equivalent to the fragment of the modal $\mu$-calculus where the application of the least fixpoint operator $\mu x.\varphi$ is restricted to formulas $\varphi$ that are continuous in $x$. Our proof is automata-theoretic in nature; in particular, we introduce a class of automata characterizing the expressive power of WMSO over tree models of arbitrary branching degree. The transition map of these automata is defined in terms of a logic $\olque$ that is the extension of first-order logic with a generalized quantifier $\qu$, where $\qu x. \phi$ means that there are infinitely many objects satisfying $\phi$. An important part of our work consists of a model-theoretic analysis of $\olque$.