33 research outputs found
Piecewise testable tree languages
This paper presents a decidable characterization of tree languages that can
be defined by a boolean combination of Sigma_1 sentences. This is a tree
extension of the Simon theorem, which says that a string language can be
defined by a boolean combination of Sigma_1 sentences if and only if its
syntactic monoid is J-trivial
On shuffle ideals of general algebras
We extend a word language concept called shuffle ideal to general algebras. For this purpose, we introduce the relation SH and show that there exists a natural connection between this relation and the homeomorphic embedding order on trees. We establish connections between shuffle ideals, monotonically ordered algebras and automata, and piecewise testable tree languages
EF+EX Forest Algebras
We examine languages of unranked forests definable using the temporal
operators EF and EX. We characterize the languages definable in this logic, and
various fragments thereof, using the syntactic forest algebras introduced by
Bojanczyk and Walukiewicz. Our algebraic characterizations yield efficient
algorithms for deciding when a given language of forests is definable in this
logic. The proofs are based on understanding the wreath product closures of a
few small algebras, for which we introduce a general ideal theory for forest
algebras. This combines ideas from the work of Bojanczyk and Walukiewicz for
the analogous logics on binary trees and from early work of Stiffler on wreath
product of finite semigroups
Tree Languages Defined in First-Order Logic with One Quantifier Alternation
We study tree languages that can be defined in \Delta_2 . These are tree
languages definable by a first-order formula whose quantifier prefix is forall
exists, and simultaneously by a first-order formula whose quantifier prefix is
. For the quantifier free part we consider two signatures, either the
descendant relation alone or together with the lexicographical order relation
on nodes. We provide an effective characterization of tree and forest languages
definable in \Delta_2 . This characterization is in terms of algebraic
equations. Over words, the class of word languages definable in \Delta_2 forms
a robust class, which was given an effective algebraic characterization by Pin
and Weil
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