Safety for bisimulation in monadic second-order logic

We characterize those formulas of MSO monadic second order logic that are safe for bisim ulation formulas de ning binary relations such that any bisimulation is also a bisimulation with respect to these de ned relations Every such formula is equivalent to one constructed from calculus tests atomic actions and the regular operations The proof uses a characterization of completely additive calculus formulas formulas p that distribute over arbitrary unions It turns out that complete additivity is equivalent to distributivity over countable unions For FOL rst order logic a similar theorem is shown giving an alternative proof to the original of Here though distributivity over nite unions is su cient This enables us to show that the characterization of safe FOL formulas carries over to the setting of nite models