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∞.Comment: Technical Report, 57 page