26,273 research outputs found
An Algebra of Hierarchical Graphs and its Application to Structural Encoding
We define an algebraic theory of hierarchical graphs, whose axioms
characterise graph isomorphism: two terms are equated exactly when
they represent the same graph. Our algebra can be understood as
a high-level language for describing graphs with a node-sharing, embedding
structure, and it is then well suited for defining graphical
representations of software models where nesting and linking are key
aspects. In particular, we propose the use of our graph formalism as a
convenient way to describe configurations in process calculi equipped
with inherently hierarchical features such as sessions, locations, transactions,
membranes or ambients. The graph syntax can be seen as an
intermediate representation language, that facilitates the encodings of
algebraic specifications, since it provides primitives for nesting, name
restriction and parallel composition. In addition, proving soundness
and correctness of an encoding (i.e. proving that structurally equivalent
processes are mapped to isomorphic graphs) becomes easier as it can
be done by induction over the graph syntax
Answering SPARQL queries modulo RDF Schema with paths
SPARQL is the standard query language for RDF graphs. In its strict
instantiation, it only offers querying according to the RDF semantics and would
thus ignore the semantics of data expressed with respect to (RDF) schemas or
(OWL) ontologies. Several extensions to SPARQL have been proposed to query RDF
data modulo RDFS, i.e., interpreting the query with RDFS semantics and/or
considering external ontologies. We introduce a general framework which allows
for expressing query answering modulo a particular semantics in an homogeneous
way. In this paper, we discuss extensions of SPARQL that use regular
expressions to navigate RDF graphs and may be used to answer queries
considering RDFS semantics. We also consider their embedding as extensions of
SPARQL. These SPARQL extensions are interpreted within the proposed framework
and their drawbacks are presented. In particular, we show that the PSPARQL
query language, a strict extension of SPARQL offering transitive closure,
allows for answering SPARQL queries modulo RDFS graphs with the same complexity
as SPARQL through a simple transformation of the queries. We also consider
languages which, in addition to paths, provide constraints. In particular, we
present and compare nSPARQL and our proposal CPSPARQL. We show that CPSPARQL is
expressive enough to answer full SPARQL queries modulo RDFS. Finally, we
compare the expressiveness and complexity of both nSPARQL and the corresponding
fragment of CPSPARQL, that we call cpSPARQL. We show that both languages have
the same complexity through cpSPARQL, being a proper extension of SPARQL graph
patterns, is more expressive than nSPARQL.Comment: RR-8394; alkhateeb2003
Transitivity conditions in infinite graphs
We study transitivity properties of graphs with more than one end. We
completely classify the distance-transitive such graphs and, for all , the -CS-transitive such graphs.Comment: 20 page
Set Unification
The unification problem in algebras capable of describing sets has been
tackled, directly or indirectly, by many researchers and it finds important
applications in various research areas--e.g., deductive databases, theorem
proving, static analysis, rapid software prototyping. The various solutions
proposed are spread across a large literature. In this paper we provide a
uniform presentation of unification of sets, formalizing it at the level of set
theory. We address the problem of deciding existence of solutions at an
abstract level. This provides also the ability to classify different types of
set unification problems. Unification algorithms are uniformly proposed to
solve the unification problem in each of such classes.
The algorithms presented are partly drawn from the literature--and properly
revisited and analyzed--and partly novel proposals. In particular, we present a
new goal-driven algorithm for general ACI1 unification and a new simpler
algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of
Logic Programming (TPLP
Nested Markov Properties for Acyclic Directed Mixed Graphs
Directed acyclic graph (DAG) models may be characterized in at least four
different ways: via a factorization, the d-separation criterion, the
moralization criterion, and the local Markov property. As pointed out by Robins
(1986, 1999), Verma and Pearl (1990), and Tian and Pearl (2002b), marginals of
DAG models also imply equality constraints that are not conditional
independences. The well-known `Verma constraint' is an example. Constraints of
this type were used for testing edges (Shpitser et al., 2009), and an efficient
marginalization scheme via variable elimination (Shpitser et al., 2011).
We show that equality constraints like the `Verma constraint' can be viewed
as conditional independences in kernel objects obtained from joint
distributions via a fixing operation that generalizes conditioning and
marginalization. We use these constraints to define, via Markov properties and
a factorization, a graphical model associated with acyclic directed mixed
graphs (ADMGs). We show that marginal distributions of DAG models lie in this
model, prove that a characterization of these constraints given in (Tian and
Pearl, 2002b) gives an alternative definition of the model, and finally show
that the fixing operation we used to define the model can be used to give a
particularly simple characterization of identifiable causal effects in hidden
variable graphical causal models.Comment: 67 pages (not including appendix and references), 8 figure
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