Controlling the Depth, Size, and Number of Subtrees for Two-variable Logic on Trees

Verification of properties of first order logic with two variables FO 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 is known to have the same expressiveness as unary temporal logic, the same properties hold. Over finite labelled ordered trees FO 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 gives a 2EXPTIME bound for satisfiability of FO. In this work we give the first in-depth look at the complexity of FO 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 formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to finite satisfiability for FO sentences over trees.