Complete axiomatizations of MSO, FO(TC¹) and FO(LFP¹) on finite trees

We propose axiomatizations of monadic second-order logic (MSO), monadic transitive closure logic (FO(TC¹)) and monadic least fixpoint logic (FO(LFP¹)) on finite node-labeled sibling-ordered trees. We show by a uniform argument, that our axiomatizations are complete, i.e., in each of our logics, every formula which is valid on the class of finite trees is provable using our axioms. We are interested in this class of structures because it allows to represent basic structures of computer science such as XML documents, linguistic parse trees and treebanks. The logics we consider are rich enough to express interesting properties such as reachability. On arbitrary structures, they are well known to be not recursively axiomatizable.