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We consider a specific class of tree structures that can represent basicstructures in linguistics and computer science such as XML documents, parsetrees, and treebanks, namely, finite node-labeled sibling-ordered trees. Wepresent axiomatizations of the monadic second-order logic (MSO), monadictransitive closure logic (FO(TC1)) and monadic least fixed-point logic(FO(LFP1)) theories of this class of structures. These logics can expressimportant properties such as reachability. Using model-theoretic techniques, weshow by a uniform argument that these axiomatizations are complete, i.e., eachformula that is valid on all finite trees is provable using our axioms. As abackdrop to our positive results, on arbitrary structures, the logics that westudy are known to be non-recursively axiomatizable.

Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees