Two-Way Unary Temporal Logic over Trees

We consider a temporal logic EF+F^-1 for unranked, unordered finite trees. The logic has two operators: EF\phi, which says "in some proper descendant \phi holds", and F^-1\phi, which says "in some proper ancestor \phi holds". We present an algorithm for deciding if a regular language of unranked finite trees can be expressed in EF+F^-1. The algorithm uses a characterization expressed in terms of forest algebras.