Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree

Abstract

In this paper we study the expressive power of the full branching time logic CTL* (of Clarke, Emerson, Halpern and Sistla), by comparing it with fragments of monadic second-order logic over the binary tree. We show that over binary tree models the system CTL* has the same expressive power as monadic second-order logic in which set quantification is restricted to infinite paths (in particular, the full strength of first-order logic is captured here by CTL*). This generalizes Kamp's theorem on the equivalence between propositional linear time logic and first-order logic over the ordering of the natural numbers. For the proof an extension of CTL* by past operators is introduced. The transition from monadic formulas to CTL* rests on a model-theoretic decomposition lemma for trees that is justified by an application of the Ehrenfeucht-Fraissé game. The paper concludes by discussing variants of the main result, dealing for example with trees of higher branching index and the extended branching time logic ECTL*.