Monadic Second-Order Logic and Transitive Closure Logics over Trees

Model theoretic syntax is concerned with studying the descriptive complexity of grammar formalisms for natural languages by defining their derivation trees in suitable logical formalisms. The central tool for model theoretic syntax has been monadic second-order logic (MSO). Much of the recent research in this area has been concerned with finding more expressive logics to capture the derivation trees of grammar formalisms that generate non-context-free languages. The motivation behind this search for more expressive logics is to describe formally certain mildly context-sensitive phenomena of natural languages. Several extensions to MSO have been proposed, most of which no longer define the derivation trees of grammar formalisms directly, while others introduce logically odd restrictions. We therefore propose to consider first-order transitive closure logic. In this logic, derivation trees can be defined in a direct way. Our main result is that transitive closure logic, even deterministic transitive closure logic, is more expressive in defining classes of tree languages than MSO. (Deterministic) transitive closure logics are capable of defining non-regular tree languages that are of interest to linguistics.