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 $\mu p.\phi$ is restricted to formulas $\phi$ 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 ${\exists }^{\infty }x.\varphi$ means that there are infinitely many objects satisfying $\varphi$. An important part of our work consists of a model-theoretic analysis of ${\mathrm{FOE}}_1^{\infty }$.

Technical Report, 57 pages

Cited by in corpus (1)

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