Finite Convergence of Mu-Calculus Fixpoints on Genuinely Infinite Structures

The modal μ-calculus can only express bisimulation-invariant properties. It is a simple consequence of Kleene’s Fixpoint Theorem that on structures with finite bisimulation quotients, the fixpoint iteration of any formula converges after finitely many steps. We show that the converse does not hold: we construct a word with an infinite bisimulation quotient that is locally regular so that the iteration for any fixpoint formula of the modal μ-calculus on it converges after finitely many steps. This entails decidability of μ-calculus model-checking over this word. We also show that the reason for the discrepancy between infinite bisimulation quotients and trans-finite fixpoint convergence lies in the fact that the μ-calculus can only express regular properties.