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 FOE1∞ that is the extension of first-order logic with a generalized quantifier ∃∞, where ∃∞x.φ means that there are infinitely many objects satisfying φ. An important part of our work consists of a model-theoretic analysis of FOE1∞.