Complexity of two-variable logic on finite trees

Verification of properties expressed in the two-variable fragment of first-order logic FO² has been investigated in a number of contexts. The satisfiability problem for FO² over arbitrary structures is known to be NEXPTIME-complete, with satisfiable formulas having exponential-sized models. Over words, where FO² is known to have the same expressiveness as unary temporal logic, satisfiability is again NEXPTIME-complete. Over finite labelled ordered trees, FO² has the same expressiveness as navigational XPath, a popular query language for XML documents. Prior work on XPath and FO² gives a 2EXPTIME bound for satisfiability of FO² over trees. This work contains a comprehensive analysis of the complexity of FO² 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 FO², its guarded version, GF². Our results depend on an analysis of types in models of FO² formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to satisfiability for FO² sentences over finite trees.