Verification of properties expressed in the two-variable fragment of first-order logic FO 2 has been investigated in a number of contexts. The satisfiability problem for FO 2 over arbitrary structures is known to be NEXPTIME-complete, with satisfiable formulas having exponential-sized models. Over words, where FO 2 is known to have the same expressiveness as unary temporal logic, satisfiability is again NEXPTIME-complete. Over finite labelled ordered trees FO 2 has the same expressiveness as navigational XPath, a popular query language for XML documents. Prior work on XPath and FO 2 gives a 2EXPTIME bound for satisfiability of FO 2 over trees. This work contains a comprehensive analysis of the complexity of FO 2 on trees, and on the size and depth of models. We show that different techniques are required depending on the vocabulary used, whether the trees are ranked or unranked, and the encoding of labels on trees. We also look at a natural restriction of FO 2 , its guarded version, GF 2 . Our results depend on an analysis of types in models of FO 2 formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to satisfiability for FO 2 sentences over finite trees.

We consider the satisfiability and finite satisfiability problems for extensions of the two-variable fragment of firstorder logic in which an equivalence closure operator can be applied to a fixed number of binary predicates. We show that the satisfiability problem for two-variable, first-order logic with equivalence closure applied to two binary predicates is in 2NEXPTIME, and we obtain a matching lower bound by showing that the satisfiability problem for two-variable first-order logic in the presence of two equivalence relations is 2NEXPTIME-hard. The logics in question lack the finite model property; however, we show that the same complexity bounds hold for the corresponding finite satisfiability problems. We further show that the satisfiability (=finite satisfiability) problem for the twovariable fragment of first-order logic with equivalence closure applied to a single binary predicate is NEXPTIME-complete.

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Uniform one-dimensional fragment UF = 1 is a formalism obtained from first-order logic by limiting quantification to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulas with two or more variables. The fragment can be seen as a canonical generalization of two-variable logic, defined in order to be able to deal with relations of arbitrary arities. The fragment was introduced recently, and it was shown that the satisfiability problem of the equality-free fragment UF 1 of UF = 1 is decidable. In this article we establish that the satisfiability and finite satisfiability problems of UF = 1 are NEXPTIME-complete. We also show that the corresponding problems for the extension of UF = 1 with counting quantifiers are undecidable. In addition to decidability questions, we compare the expressivities of UF = 1 and two-variable logic with counting quantifiers FOC 2 . We show that while the logics are incomparable in general, UF = 1 is strictly contained in FOC 2 when attention is restricted to vocabularies with the arity bound two.