Samantha Bail The Logical Diversity of Explanations in OWL Ontologies
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Most frequent templates
31
tailment (σ = 9.5, m = 2), which is a reduction by 33.7%
compared to the full corpus.
On average, a template covers 1.5 justifications (standard
deviation σ = 2.3, median m = 1), with some ontologies con-
taining entailments with large numbers of isomorphic justi-
fications. One such example is the Orphanet Ontology of
Rare Diseases, whose dominating templates are of the type
Θ1
= {C1 C2, C2 ∃p1.C4, domain(p1, C3)} |= C1 C3
with atomic subsumption chains of arbitrary size in place
of the first subsumption axiom, and some variations that
include subproperty axioms. Two of the templates of this
type cover the majority (110 and 105 justifications, respec-
tively) of the 220 justifications each for several entailments
in the ontology. From personal contact with the Orphanet
developers we have learned that this OWL ontology is in fact
8https://sites.google.com/site/isocikm2013/
both strict and subexpression-isomorphism.
A total of 1,492 entailments (7.8% of the total corpus)
rom 43 ontologies are affected by lemma-isomorphism, with
an average reduction of 30.3% compared to strict isomor-
phism for those entailments. The strongest effects can be
seen in the Fission Yeast Phenotype ontology, where the jus-
tifications for several entailments only differ in the length of
their atomic subsumption chains and thus are each reduced
to a single template of the type
Θ2
= {C1 . . . Cn, Cn
≡ C2 . . .} |= C1 C2.
5.3.2 Isomorphism within ontologies
Across the justifications for all entailments of an ontol-
ogy, the reductions caused by the three equivalence rela-
tions are clearly more visible than for individual entailments.
However, the effects of the relations differ strongly across
the 78 ontologies, with strict isomorphism generally having
the strongest impact, and subexpression-isomorphism hav-
duction by 92.2% and a 1% difference compared
somorphism. The number of justifications covered
e template is slightly increased with an average of
fications (σ = 59, m = 2) per template.
ost frequent template (by numbers of justifications)
Θ2, which covers 2,128 (1.5% of the total set) justifi-
26 ontologies. Across the ontologies in the corpus,
frequent template occurs in 28 of the 78 ontologies.
plate is a single equivalence axiom which we have
een in the Lipid ontology:
Θ3 = {C1 ≡ C2 x} |= C1 C2
rfluous part x matches a number of operands such
classes and existential restrictions. Interestingly,
template occurs in the highest number of ontolo-
nly covers 573 justifications across the corpus.
Note tha
Θ7 corre
length. T
justificat
the corp
plate occ
both of w
corpus.
isomorph
in multip
out of 5,
5.3.4
As we
isomorph
only due
do not c
for lemma-isomorphism.
On average, a template covers 25.8 justifications (σ =
208.5, m =3); however, the large standard deviation shows
that the distribution of justifications per template has shifted
towards a few very frequent templates, whereas there is still
a ‘long tail’ of 1,878 templates that match only a single jus-
tification. If we consider the distribution of justifications
per template over the quartiles of the corpus, 25% of the
justifications in Ss can be covered by the 8 most frequent
templates, 50% by the 44 most frequent templates, and 75%
by the 277 (out of 5,487) most frequent templates. The
most frequent templates are all subtle variations of a tem-
plate containing only two or three axioms:
Θ4 ={C1 C2, C3 ≡ C2 C4, C3 C5} |= C1 C5
Θ5 ={C1 C2, C5 ≡ C2 C3} |= C1 C5
Θ6 ={C1 C2, C3 ≡ C2 C4 C6, C3 C5} |= C1 C6
Θ ={C1 C2, C5 ≡ C2 C3 C4} |= C1 C5
•Atomic subsumption chains - but then also:
some superfluous part