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We consider the satisfiability problem for the two-variable fragment offirst-order logic over finite unranked trees. We work with signaturesconsisting of some unary predicates and the binary navigational predicateschild, right sibling, and their respective transitive closures. We prove thatthe satisfiability problem for the logic containing all these predicates isEXPSPACE-complete. Further, we consider the restriction of the class ofstructures to singular trees, i.e., we assume that at every node precisely oneunary predicate holds. We observe that the full logic and even for unorderedtrees remain EXPSPACE-complete over finite singular trees, but the complexitydecreases for some weaker logics. Namely, the logic with one binary predicate,descendant is NEXPTIME-complete, and its guarded version is PSPACE-completeover finite singular trees, even though both these logics are EXPSPACE-completeover arbitrary finite trees.

Satisfiability of the Two-Variable Fragment of First-Order Logic over Trees