Arity hierarchy for temporal logics

Abstract

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 C T L ∗ and E C T L + have no finite basis.