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

Abstract

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.