100,545 research outputs found
Two-Way Unary Temporal Logic over Trees
We consider a temporal logic EF+F^-1 for unranked, unordered finite trees.
The logic has two operators: EF\phi, which says "in some proper descendant \phi
holds", and F^-1\phi, which says "in some proper ancestor \phi holds". We
present an algorithm for deciding if a regular language of unranked finite
trees can be expressed in EF+F^-1. The algorithm uses a characterization
expressed in terms of forest algebras.Comment: 29 pages. Journal version of a LICS 07 pape
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
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Wreath Products of Forest Algebras, with Applications to Tree Logics
We use the recently developed theory of forest algebras to find algebraic
characterizations of the languages of unranked trees and forests definable in
various logics. These include the temporal logics CTL and EF, and first-order
logic over the ancestor relation. While the characterizations are in general
non-effective, we are able to use them to formulate necessary conditions for
definability and provide new proofs that a number of languages are not
definable in these logics
Robustness: a New Form of Heredity Motivated by Dynamic Networks
We investigate a special case of hereditary property in graphs, referred to
as {\em robustness}. A property (or structure) is called robust in a graph
if it is inherited by all the connected spanning subgraphs of . We motivate
this definition using two different settings of dynamic networks. The first
corresponds to networks of low dynamicity, where some links may be permanently
removed so long as the network remains connected. The second corresponds to
highly-dynamic networks, where communication links appear and disappear
arbitrarily often, subject only to the requirement that the entities are
temporally connected in a recurrent fashion ({\it i.e.} they can always reach
each other through temporal paths). Each context induces a different
interpretation of the notion of robustness.
We start by motivating the definition and discussing the two interpretations,
after what we consider the notion independently from its interpretation, taking
as our focus the robustness of {\em maximal independent sets} (MIS). A graph
may or may not admit a robust MIS. We characterize the set of graphs \forallMIS
in which {\em all} MISs are robust. Then, we turn our attention to the graphs
that {\em admit} a robust MIS (\existsMIS). This class has a more complex
structure; we give a partial characterization in terms of elementary graph
properties, then a complete characterization by means of a (polynomial time)
decision algorithm that accepts if and only if a robust MIS exists. This
algorithm can be adapted to construct such a solution if one exists
Two-variable logics with some betweenness relations: Expressiveness, satisfiability and membership
We study two extensions of FO2[<], first-order logic interpreted in finite
words, in which formulas are restricted to use only two variables. We adjoin to
this language two-variable atomic formulas that say, "the letter appears
between positions and " and "the factor appears between positions
and ". These are, in a sense, the simplest properties that are not
expressible using only two variables.
We present several logics, both first-order and temporal, that have the same
expressive power, and find matching lower and upper bounds for the complexity
of satisfiability for each of these formulations. We give effective conditions,
in terms of the syntactic monoid of a regular language, for a property to be
expressible in these logics. This algebraic analysis allows us to prove, among
other things, that our new logics have strictly less expressive power than full
first-order logic FO[<]. Our proofs required the development of novel
techniques concerning factorizations of words
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
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
- …