Arity hierarchy for temporal logics

A major result concerning temporal logics is Kamp’s Theorem which states that the pair of modalities “until” and “since” is expressively complete for the first-order fragment of the monadic logic over the linear-time canonical model of naturals.The paper concerns the expressive power of temporal logics over trees. The main result states that in contrast to Kamp’s Theorem, for every n there is a modality of arity n definable by a monadic logic formula, which is not equivalent over trees to any temporal logic formula which uses modalities of arity less than n. Its proof takes advantage of an instance of Shelah’s composition theorem.This result has interesting corollaries, for instance reproving that CTL∗ and ECTL+ have no finite basis.