![](bg4.png)
Definition 7. Two individuals a and b are called One
Step Node-similar for an ABox A, if we have OSN
a
A
=
OSN
b
A
. We denote the One Step Node-similarity rela-
tion for an ABox A with R
OSNSim
.
And last but not least, we define the notion of an OSN-
based ABox-Summarization.
Definition 8. An ABox-OSN-Summarization (for
an ABox A and an One Step Node-similarity rela-
tion R
OSNSim
) is an ABox-Summarization AOSNS
A
=
hφ
sum
, ω
sum
i, s.t.
• N
sum
=[R
OSNSim
]
• We define φ
sum
by the equivalence relation
R
OSNSim
• For each osn ∈ [R
OSNSim
]
ω
sum
(osn) ={C(osn
∗
)|C ∈ osn.rootconset}∪
[
ans∈osn.ansset
{R(a, ans
∗
)|R ∈ ans.rs}∪
[
ans∈osn.ansset
{C(ans
∗
)|C ∈ ans.cs}
, where x
∗
denotes a fresh and unique individual name
for each OSN-/ANS-object x.
Example 7. One ABox-OSN-Summarization AS
EX 3
for A
EX
and R
φ
sum
EX 3
= {( joe, lara, alice, luis),
(cl, ai, ai2), (tl), (ana), (amanda), ( f rank), ( jim), (in f )} is
N
sum
EX 3
={s1, s2, c1, c2, p1, p2, p3, d1}
φ
sum
( joe) = s1
φ
sum
(lara) = s1
..
φ
sum
(cl) = c1
..
φ
sum
(in f ) = d1
ω
sum
EX 3
(s1) = {Student(s1), takes(s1, cnew1),
Course(cnew1)}
ω
sum
EX 3
(s2) = {GradStudent(s2), takes(s2, cnew2),
Course(cnew2)}
..
ω
sum
EX 3
(p2) = {Pro f (p2), teaches(p2, cnew5),
Course(cnew5), headO f (p2, newd1),
Department(newd1)}
..
5 HOW TO SOLVE DECISION
PROBLEMS?
The main theorem of our work is presented in Theo-
rem 5.1.
Theorem 5.1. Given an Ontology hT , R , Ai, every
ABox-OSN-Summarization AOSNS
A
for A is sound.
This is, informally, clear, since the ABox of each
summarization individual is a subset of the original
ABox A (modulo renaming).
Proof. We have to show that hT , R , ω
sum
(φ
sum
(a))i
C(φ
sum
(a)) =⇒ hT , R , Ai C(a).
We show the proof by contraposition and then
derive a contradiction: Given hT , R , Ai 2 C(a), we
have that there exists a model I
1
for hT , R , A ∪
{¬C(a)}i. Now, I
1
has to satisfy all the asser-
tions in A ∪ {¬C(a)}, and since ω
sum
(φ
sum
(a)) ∪
{¬C(φ
sum
(a))} is structurally equivalent (by con-
struction in Definitions 5, 6 and 8) to a subset of A ∪
{¬C(a)}, we have that a rewriting of I
1
has to satisfy
all the assertions in ω
sum
(φ
sum
(a)) ∪ {¬C(φ
sum
(a))}.
Thus, there exists a model for hT , R , ω
sum
(φ
sum
(a))∪
{¬C(φ
sum
(a))}i. Contradiction, since we assumed
hT , R , ω
sum
(φ
sum
(a))i 2 C(φ
sum
(a)) (by Contraposi-
tion).
Given Theorem 5.1 above, we can speedup query an-
swering over description logics in the following ways.
Given an individual a and an atomic concept de-
scription C, Instance Checking is the problem to de-
termine, whether an Ontology hT , R , Ai C(a). In
common description logic systems, this is done by
checking, whether hT , R , A ∪ {¬C(a)}i is inconsis-
tent. Once the ABox is reasonably big, the underlying
exponential behavior of tableau algorithms shows up
easily, albeit all known optimizations techniques.
Given an ABox-OSN-Summarization AOSNS
A
for A, we can perform some kind of sanity check with
existing tableau algorithms, but dramatically reduced
ABoxes.
1. Check, whether hT , R , ω
sum
(φ
sum
(a))i 2
C(φ
sum
(a)). If this is true, then we already
know that hT , R , Ai C(a).
2. Check, whether hT , R , ω
sum
(φ
sum
(a))i 2
¬C(φ
sum
(a)). If this is true, than we already
know that hT , R , Ai ¬C(a), and it is safe to
assume that hT , R , Ai 2 C(a) (if the underlying
ontology is consistent).
Please note that we deal with a dramatically reduced
assertional part in both cases. Only if both sanity
checks fail, then we still have to apply the full reason-
ing machine, i.e. we have to check hT , R , Ai C(a).
The main contribution of this work is the speedup
of Instance Retrieval. Given an atomic concept de-
scription C, Grounded Instance Retrieval is the prob-
lem to determine all individuals a ∈ A, s.t. we have
hT , R , Ai C(a).
SOUND SUMMARIZATIONS FOR ALCHI ONTOLOGIES - How to Speedup Instance Checking and Instance Retrieval
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