The description logic EL has been used to support ontology design in various domains, and especially in biology and medecine. EL is known for its e cient reasoning and query answering capabilities. By contrast, ontology design and query answering can be supported and guided within an FCA framework. Accordingly, in this paper, we propose a formal transformation of ELI (an extension of EL with inverse roles) ontologies into an FCA framework, i.e. KELI , and we provide a formal characterization of this transformation. Then we show that SPARQL query answering over ELI ontologies can be reduced to lattice query answering over KELI concept lattices. This simpli es the query answering task and shows that some basic semantic web tasks can be improved when considered from an FCA perspective.
with SPARQL. However, including inferred data in query answering requires
either a reasoner to infer all implicit information or query rewriting using property
paths (that enable navigation in a hierarchy) [
Overall, we work towards (i) a formal characterization of the transformation of ontologies into a formal context, (ii) translating the di culty of SPARQL query answering over ontologies into query answering over concept lattices, and nally (iii) providing organization of SPARQL query answers with concept lattices. 2
Transforming E LI Ontologies into Formal Contexts
A formal context represents data using objects and their attributes. Formally, it
is a triple K = (G; M; I) where G is a set of objects, M is a set of attributes,
and I G M is a binary relation. A derivation operator (0) is used to compute
formal concepts of a context. Given a set of objects A, a derivation operator 0
computes the maximal set of attributes shared by objects in A and is denoted
by A0 (this is done dually with set of attributes B). A formal concept is a pair
(A; B) where A0 = B and B0 = A. A set of formal concepts ordered with the set
inclusion relation form a concept lattice [
One di culty of transforming DL ontologies into formal contexts is mainly due to the fact that while DL languages are based on the open world assumption (OWA), FCA relies on the closed world assumption (CWA). The former permits to specify only known data whereas the later demands that all data should be explicitly speci ed. To slightly close the gap between these two worlds, we provide a transformation that maintains a DL semantics into an FCA setting.
To transform an E LI ontology O = hT ; Ai into a formal context K =
(G; M; I), the schema axioms in the TBox become background implications [
E LI O = hT ; Ai FCA formal context KO = (G; M; I) +
background implications L T = fC v Dg fC ! Dg 2 L
A = fC(a); a 2 G, C 2 M , and (a; C) 2 I
R(a; b)g a; b 2 G, 9R:>; 9R :> 2 M , (a; 9R:>) 2 I, and (b; 9R :>) 2 I Art tC Prd
Act
AFNY Example 1. Consider the transformation of the following E LI ontology O = hT ; Ai into a formal context KO and its background implications L. T = fActorsFromNewYork (AFNY) v Actor (Act);
FilmProducer (Prd) v Artist (Art); Actor v Artist; Artist v Person (Per)g A = ftomCruise (tC) 2 ActorsFromNewYorkg
L = f AFNY ! Act; Prd ! Art; KO AFNY Prd Act Art Per tC x
Act ! Art; Art ! Per g
Construction of a concept lattice: In [
This is a simple example but SNOMED-CT and NCIT are much larger than that but not more complex. Let us how a concept lattice based on an E LI ontology can be queried. 3
SPARQL query answering over ontologies vs query
answering over concept lattices
SPARQL query answering over E LI ontologies can be considered from the point
of view of query answering over KELI concept lattices. Querying concept lattices
amounts to fetching the objects given a set of attributes as query constants and
to fetch the attributes given a set of objects as query constants or terms [
SPARQL query answering over ontologies can be done in two main ways: (1) Materialization amounts to computing all implicit data before evaluating the query. This can be done by using a DL reasoner. (2) Query rewriting amounts to converting a query into another one using property paths and schema axioms.
Query answering over E LI ontologies with SPARQL appears to be harder than query answering over KELI concept lattices. SPARQL requires expensive tasks such as materialization and query rewriting but its expressive power is better than lattice querying. By contrast, lattice querying is practically su cient to retrieve instances for E LI ontologies as shown by the following example. Example 2. Let us consider the evaluation of the SPARQL query Q on the ontology O (in Figure 1) and its materialization O0. Q = select all objects, elements who are artists = SELECT ?x WHERE f?x a Artist .g: Under simple entailment evaluation of a SPARQL query, the answer of Q over O is empty. To get non-empty answers for Q, one can evaluate Q over the materialization of O that we call O0, where Q(O0) = ftomCruiseg. Another way is to rewrite Q into Q0 = SELECT ?x WHERE f?x a/rdfs:subClassOf Artist .g. Q0 selects all instances of Artist and that of its subclasses by navigating through the subclass relation. Then, the evaluation of Q0(O) returns ftomCruiseg.
By contrast, Q is converted into a lattice query as q(x) = (x; Artist). The evaluation of this query over a concept lattice KO obtained from O (Figure 1) is Q0(KO) = ftomCruiseg, as it is su cient to return all objects which are instances of Artist or any of its subconcept. 4
It can be convenient to use FCA as a guideline for designing and querying E LI ontologies. In addition, FCA provides visualization and navigation capabilities. The present work does not apply to all ontologies but seems to be well suited to E LI ontologies. We plan to extend and experiment the proposed approach, especially with real-world and large datasets.