6,166 research outputs found
Logics for Unranked Trees: An Overview
Labeled unranked trees are used as a model of XML documents, and logical
languages for them have been studied actively over the past several years. Such
logics have different purposes: some are better suited for extracting data,
some for expressing navigational properties, and some make it easy to relate
complex properties of trees to the existence of tree automata for those
properties. Furthermore, logics differ significantly in their model-checking
properties, their automata models, and their behavior on ordered and unordered
trees. In this paper we present a survey of logics for unranked trees
Deciding the Borel complexity of regular tree languages
We show that it is decidable whether a given a regular tree language belongs
to the class of the Borel hierarchy, or equivalently whether
the Wadge degree of a regular tree language is countable.Comment: 15 pages, 2 figure
An Upper Bound on the Complexity of Recognizable Tree Languages
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular
tree language of infinite trees is in a class
for some natural number , where is the game quantifier. We
first give a detailed exposition of this result. Next, using an embedding of
the Wadge hierarchy of non self-dual Borel subsets of the Cantor space
into the class , and the notions of Wadge degree
and Veblen function, we argue that this upper bound on the topological
complexity of regular tree languages is much better than the usual
Fragments of first-order logic over infinite words
We give topological and algebraic characterizations as well as language
theoretic descriptions of the following subclasses of first-order logic FO[<]
for omega-languages: Sigma_2, FO^2, the intersection of FO^2 and Sigma_2, and
Delta_2 (and by duality Pi_2 and the intersection of FO^2 and Pi_2). These
descriptions extend the respective results for finite words. In particular, we
relate the above fragments to language classes of certain (unambiguous)
polynomials. An immediate consequence is the decidability of the membership
problem of these classes, but this was shown before by Wilke and Bojanczyk and
is therefore not our main focus. The paper is about the interplay of algebraic,
topological, and language theoretic properties.Comment: Conference version presented at 26th International Symposium on
Theoretical Aspects of Computer Science, STACS 200
Languages of Dot-depth One over Infinite Words
Over finite words, languages of dot-depth one are expressively complete for
alternation-free first-order logic. This fragment is also known as the Boolean
closure of existential first-order logic. Here, the atomic formulas comprise
order, successor, minimum, and maximum predicates. Knast (1983) has shown that
it is decidable whether a language has dot-depth one. We extend Knast's result
to infinite words. In particular, we describe the class of languages definable
in alternation-free first-order logic over infinite words, and we give an
effective characterization of this fragment. This characterization has two
components. The first component is identical to Knast's algebraic property for
finite words and the second component is a topological property, namely being a
Boolean combination of Cantor sets.
As an intermediate step we consider finite and infinite words simultaneously.
We then obtain the results for infinite words as well as for finite words as
special cases. In particular, we give a new proof of Knast's Theorem on
languages of dot-depth one over finite words.Comment: Presented at LICS 201
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