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.

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.

Verification of properties of first order logic with two variables FO 2 has been investigated in a number of contexts. Over arbitrary structures it is known to be decidable with NEXPTIME complexity, with finitely satisfiable formulas having exponential-sized models. Over word structures, where FO 2 is known to have the same expressiveness as unary temporal logic, the same properties hold. Over finite labelled ordered trees FO 2 is also of interest: it is known to have the same expressiveness as navigational XPath, a common query language for XML documents. Prior work on XPath and FO 2 gives a 2EXPTIME bound for satisfiability of FO 2 . In this work we give the first in-depth look at the complexity of FO 2 on trees, and on the size and depth of models. We show that the doubly-exponential bound is not tight, and neither do the NEXPTIMEcompleteness results from the word case carry over: the exact complexity varies depending on the vocabulary used, the presence or absence of a schema, and the encoding used for labels. Our results depend on an analysis of subformula 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 finite satisfiability for FO 2 sentences over trees.