Expressive completeness of temporal logic of trees

Many temporal and modal logic languages can be regarded as subsets of rst order logic i e the semantics of a temporal logic formula is given as a rst order condition on points of the underlying models Kripke structures Often the set of possible models is restricted to models which are trees A temporal logic language is rst order expressively complete if for every rst order condition for a node of a tree there exists an equivalent temporal formula which expresses the same condition In this paper expressive completeness of the temporal logic language with the set of operators U until S since and Xk k next is proved and the result is extended to various other tree like structures