Complete Axiomatizations of $MSO$, $FO(TC^1)$, $FO(LFP^1)$ on Finite Trees
AmÃ©lie Gheerbrant, Balder ten Cate
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.