Complexity of Two-Variable Logic on Finite Trees

Abstract

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.

Lifted Inference with Tree Axioms

“…However, we do note that languages going "beyond" the expresiveness of first-order logic have been investigated in other contexts. Since two-variable logic over finite trees is equivalent in expressive power to navigational XPath, a popular query language for XML documents, the satisfiability problem on this language has received attention in the logic literature (Benaim et al 2016;Charatonik and Witkowski 2013). Similarly, in the world of probabilistic databases (closely related to asymmetric WFOMC, briefly discussed in the following section), results have been shown for more expressive languages.…”

Finite satisfiability for two‐variable, first‐order logic with one transitive relation is decidable

“…It is known that, over finite trees, navigational XPATH has the same expressive power as two‐variable, first‐order logic with a signature consisting of unary predicates (representing properties of vertices) together with binary ‘navigational’ predicates (representing the modal accessibility relations). The exact complexity of satisfiability for all natural variants of this logic is given in .…”

On the uniform one-dimensional fragment

“…Recent research has mainly concerned decidability and complexity issues in restriction to particular classes of structures and also questions related to different built-in features and operators that increase the expressivity of the base language. Recent articles in the field include for example [3,4,11,25] and several others.…”