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

We consider the satisfiability problem for the two-variable fragment of first-order logic over finite unranked trees. We work with signatures consisting of some unary predicates and the binary navigational predicates ↓ (child), → (right sibling), and their respective transitive closures ↓+, → . We prove that the satisfiability problem for the logic containing all these predicates, FO[↓, ↓+,→,→ ], is ExpSpacecomplete. Further, we consider the restriction of the class of structures to singular trees, i.e., we assume that at every node precisely one unary predicate holds. We observe that FO[↓, ↓+,→,→ ] and even FO2[↓+, ↓] remain ExpSpace-complete over finite singular trees, but the complexity decreases for some weaker logics. Namely, the logic with one binary predicate, ↓+, denoted FO 2[↓+], is NExpTime-complete, and its guarded version, GF2[↓+], is PSpace-complete over finite singular trees, even though both these logics are ExpSpace-complete over arbitrary finite trees.