Weak MSO: Automata and Expressiveness Modulo Bisimilarity

We prove that the bisimulation-invariant fragment of weak monadic second-order logic (WMSO) is equivalent to the fragment of the modal $μ$-calculus where the application of the least fixpoint operator $μp.φ$ is restricted to formulas $φ$ that are continuous in $p$. 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 $\mathrm{FOE}_1^\infty$ that is the extension of first-order logic with a generalized quantifier $\exists^\infty$, where $\exists^\infty x. ϕ$ means that there are infinitely many objects satisfying $ϕ$. An important part of our work consists of a model-theoretic analysis of $\mathrm{FOE}_1^\infty$.

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