Complexity of two-variable logic on finite trees

Saguy Benaim, Michael Benedikt, Witold Charatonik, Emanuel Kieroński, Rastislav Lenhardt, Filip Mazowiecki, James Worrell

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 the exact complexity varies according to the vocabulary used, the presence or absence of a schema, 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 FO 2 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.

Original languageEnglish
Title of host publicationAutomata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Proceedings
Number of pages15
EditionPART 2
StatePublished - 2013
Externally publishedYes
Event40th International Colloquium on Automata, Languages, and Programming, ICALP 2013 - Riga, Latvia
Duration: 8 Jul 201312 Jul 2013

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 2
Volume7966 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference40th International Colloquium on Automata, Languages, and Programming, ICALP 2013


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