On closure ordinals for the modal mu-calculus

The closure ordinal of a formula of modal μ-calculus μXφ is the least ordinal κ, if it exists, such that the denotation of the formula and the κ-th iteration of the monotone operator induced by φ coincide across all transition systems (finite and infinite). It is known that for every α < ω there is a formula φ of modal logic such that μXφ has closure ordinal α [3]. We prove that the closure ordinals arising from the alternation-free fragment of modal μcalculus (the syntactic class capturing Σ2∩Π2) are bounded by ω. In this logic satisfaction can be characterised in terms of the existence of tableaux, trees generated by systematically breaking down formulæ into their constituents according to the semantics of the calculus. To obtain optimal upper bounds we utilise the connection between closure ordinals of formulæ and embedded order-types of the corresponding tableaux.