Abstract
We propose axiomatizations of monadic second-order logic (MSO), monadic transitive closure logic (FO(TC 1 )) and monadic least fixpoint logic (FO(LFP 1 )) 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.
We are grateful to Jouko Väänänen for helpful comments on an earlier draft. A full version including proofs is available online http://www.illc.uva.nl/Publications/ResearchReports/PP-2008-44.text.pdf. The authors are supported by a GLoRiClass fellowship of the European Commission (Research Training Fellowship MEST-CT-2005-020841) and by the Netherlands Organization for Scientific Research (NWO) grant 639.021.508, respectively.
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Gheerbrant, A., ten Cate, B. (2008). Complete Axiomatizations of MSO, FO(TC 1 ) and FO(LFP 1 ) on Finite Trees. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2009. Lecture Notes in Computer Science, vol 5407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92687-0_13
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