Weak MSO: Automata and Expressiveness Modulo Bisimilarity

TL;DR

This paper characterizes the bisimulation-invariant fragment of weak monadic second-order logic using automata and a restricted modal $$-calculus, with automata based on an extended first-order logic with infinity quantifiers.

Contribution

It introduces automata characterizing WMSO's expressive power over trees and analyzes a logic with a generalized quantifier, linking automata theory and logic in bisimulation invariance.

Findings

Automata characterize WMSO over arbitrary trees.

A logic with  quantifier extends first-order logic.

Bisimulation-invariant WMSO equals a restricted modal -calculus.

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