On the expressive power of modal logics on trees

Abstract

Various logical languages are compared regarding their expressive power with respect to models consisting of finitely bounded branching infinite trees. The basic multimodal logic with backward- and forward necessity operators is equivalent to restricted first order logic; by adding the binary temporal operators ”since” and ”until” we get the expressive power of first order logic on trees. Hence (restricted) propositional quantifiers in temporal logic correspond to (restricted) set quantifiers in predicate logic. Adding the CTL path modality ”E” to temporal logic gives the expressive power of path logic. Tree grammar operators give a logic as expressive as weak second order logic, whereas adding fixed point quantifiers (in the so-called propositional μ-calculus) results in a logic expressivly equivalent to monadic second order logic on trees.