Effective computation of maximal sound
approximations of Description Logic ontologies
Marco Console and Jose Mora and Riccardo Rosati and
Valerio Santarelli and Domenico Fabio Savo
Dipartimento di Ingegneria Informatica Automatica e Gestionale Antonio Ruberti
Sapienza Università di Roma, Italia
13th International Semantic Web Conference (ISWC-2014)
Riva del Garda, Italy, October 19-23 2014
Effective computation of maximal sound approximations of Description Logic ontologies
21/10/2014
(1/25)
Ontology approximation
The problem
When using ontologies as a formal description of the domain of
interest, the use of expressive languages (OWL 2) is useful.
When using ontologies for reasoning, high expressivity may be a
problem. In particular, when accessing large quantities of data
(OBDA), computational cost of languages such as OWL 2 is
prohibitive.
Ontology Approximation: Given a ontology O in a language L,
compute an ontology O0 in a target language L0 , in which “as much as
possible” of the semantics of O is preserved.
I can represent a solution for performing costly reasoning services
over ontologies in expressive languages.
Effective computation of maximal sound approximations of Description Logic ontologies
21/10/2014
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Outline
We investigate the problem of approximating ontologies for OBDA
purposes.
1
A new definition of approximation of DL ontologies
2
Approximating OWL 2 ontologies in OWL 2 QL
3
Experimental evaluation
4
Conclusions and future works
Effective computation of maximal sound approximations of Description Logic ontologies
21/10/2014
(3/25)
Outline
1
Preliminaries
2
A new definition of approximation
3
Approximation in OWL 2 QL
4
Experimental evaluation
5
Conclusions and Future Work
Effective computation of maximal sound approximations of Description Logic ontologies
21/10/2014
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Preliminaries
OWL 2 QL is the “data oriented” profile of OWL 2.
Expressions in OWL 2 QL
B
C
Q
R
−→
−→
−→
−→
A
B
P
Q
|
|
|
|
∃Q | δF (U ) | >C | ⊥C
¬B | ∃Q.A
P − | >P | ⊥P
¬Q
E
F
V
W
−→
−→
−→
−→
ρ(U )
>D | T1 | · · · | Tn
U | >A | ⊥ A
V | ¬V
Assertions in OWL 2 QL
BvC
QvR
U vV
EvF
ref (P )
A(a)
P (a, b)
U (a, v)
irref (P )
We say that OWL 2 QL is a closed language: each set of OWL 2 QL
axioms is an OWL 2 QL ontology.
Effective computation of maximal sound approximations of Description Logic ontologies
21/10/2014
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Outline
1
Preliminaries
2
A new definition of approximation
3
Approximation in OWL 2 QL
4
Experimental evaluation
5
Conclusions and Future Work
Effective computation of maximal sound approximations of Description Logic ontologies
21/10/2014
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What we want when we approximate
We deal with semantic approximation:
soundness: only produce correct entailments;
preserve as much as possible of these entailments by means of an
ontology in the target language.
In terms of models of the ontologies:
I Soundness: set of models of the approximation must be a superset
of those of the original ontology
I Minimal change: keep minimal distance between the original
ontology and its approximation
Effective computation of maximal sound approximations of Description Logic ontologies
21/10/2014
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Ontology approximation
We give our notion of approximation in a language LT of an ontology O.
Definition
Let OS be a satisfiable LS -ontology, and let ΣOS be the set of predicate and
constant symbols occurring in OS . An LT -ontology OT over ΣOS is a global
semantic approximation (GSA) in LT of OS if both the following statements
hold:
(i) M od(OS ) ⊆ M od(OT );
(soundness)
(ii) there is no LT -ontology O0 over ΣOS such that
M od(OS ) ⊆ M od(O0 ) ⊂ M od(OT ). (minimal change)
We denote with globalApx (OS , LT ) the set of all the GSAs in LT of OS .
Effective computation of maximal sound approximations of Description Logic ontologies
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Existence and Uniqueness
Lemma (Existence)
Given a language LT and a finite signature Σ, if the set of
non-equivalent axioms in an LT -ontology that one can generate over Σ
is finite, then for any LS -ontology OS globalApx (OS , LT ) 6= ∅.
Lemma (Uniqueness)
Let LT be a closed language, and let OS be an ontology. For each O0
and O00 belonging to globalApx (OS , LT ), we have that O0 and O00 are
logically equivalent.
Effective computation of maximal sound approximations of Description Logic ontologies
21/10/2014
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A constructive notion of approximation
A more constructive definition, based on the notion of Entailment Set
[Pan&Thomas 2007].
Definition
Let O be a satisfiable ontology expressed in a language L over a signature Σ,
and let L’ be a language, not necessarily different from L. The entailment set
of O with respect to L0 , denoted as ES(O, L0 ), is the set which contains all L0
axioms over Σ that are entailed by O.
Theorem
Let OS be a satisfiable LS -ontology and let OT be a satisfiable LT -ontology.
We have that:
(a) M od(OS ) ⊆ M od(OT ) if and only if ES(OT , LT ) ⊆ ES(OS , LT );
(b) there is no LT -ontology O0 such that M od(OS ) ⊆ M od(O0 ) ⊂ M od(OT )
if and only if there is no LT -ontology O00 such that
ES(OT , LT ) ⊂ ES(O00 , LT ) ⊆ ES(OS , LT ).
[Pan&Thomas 2007]: Jeff Z. Pan and Edward Thomas. Approximating OWL-DL ontologies. In Proc. of AAAI, 2007.
Effective computation of maximal sound approximations of Description Logic ontologies
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A new notion of approximation
Computing the entailment set of an ontology is hard: need to
reason over the ontology as a whole!
Idea: instead of reasoning over the whole ontology, we only reason over
portions of it.
k-approximation
I Parametric approximation: only reason over k axioms at a time.
I approximate by computing GSA of each set of k axioms of the
original ontology in isolation.
Effective computation of maximal sound approximations of Description Logic ontologies
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k-approximation
Definition
Let OS be a satisfiable LS -ontology and let ΣOS be the set of predicate and
constant symbols occurring in OS . Let Uk = {Oij | Oij ∈ globalApx (Oi , LT ),
such that Oi ∈ subsetk (OS )}. An LT -ontology OT over ΣOS is a
k-approximation in LT of OS if both the following statements hold:
T
j
(soundness)
O j ∈Uk M od(Oi ) ⊆ M od(OT );
i
there is no LT -ontology O0 over ΣOS such that
T
j
0
(minimal change)
O j ∈Uk M od(Oi ) ⊆ M od(O ) ⊂ M od(OT ).
i
subsetk (O): set of all sets of cardinality k of axioms of O
Effective computation of maximal sound approximations of Description Logic ontologies
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Two notable cases for k-approximation
1
k
| Os|
I for k = |OS |, k-approximation = GSA
I for k = 1, each axiom is treated in isolation, so we consider
ontologies formed by a single axiom
Local Semantic Approximation (LSA)
Effective computation of maximal sound approximations of Description Logic ontologies
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An example of GSA and LSA
Example
Approximation of O in OWL 2
O
=
OGSA = {
{
AvBtC
BuC vF
AvD
AvE
OLSA = {
BvD
CvD
BvD
CvD
BvD
CvD
A v ∃R.D
∃R.D v E
A v ∃R
DvF
}.
A v ∃R.D
}.
A v ∃R
A v ∃R.D
}.
Observation: M od(O) ⊂ M od(OGSA ) ⊂ M od(OLSA )
I OGSA approximates O better than OLSA
Effective computation of maximal sound approximations of Description Logic ontologies
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Outline
1
Preliminaries
2
A new definition of approximation
3
Approximation in OWL 2 QL
4
Experimental evaluation
5
Conclusions and Future Work
Effective computation of maximal sound approximations of Description Logic ontologies
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k-approximation in OWL 2 QL
Theorem
Let OS be a satisfiable OWL 2 ontology. Then the OWL 2 QL ontology
S
Oi ∈subsetk (OS ) ES(Oi , OW L 2 QL) is the k-approximation in OWL 2
QL of OS .
for OWL 2 QL, the set of non-equivalent axioms that can be generated
over a signature is finite → GSA in OWL 2 QL always exists (Existence
Lemma);
OWL 2 QL is closed:
all ontologies in ES(OS , OW L 2 QL) are pairwise logically
equivalent (Uniqueness Lemma)
ES(OS , OW L 2 QL) is an OWL 2 QL ontology for any language of
OS
the union of a set of OWL 2 QL ontologies is still an OWL 2 QL
ontology
Observation: for k = |OS | the k-approximation OT in OWL 2 QL of
OS is unique and coincides with its entailment set in OWL 2 QL.
Effective computation of maximal sound approximations of Description Logic ontologies
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Computing k-approximation in OWL 2 QL
Algorithm: computeKApx(O, k)
Input: a satisfiable OWL 2 ontology O, a positive integer k such that k ≤ |O|
Output: an OWL 2 QL ontology OApx
begin
OApx ← ∅;
foreach ontology Oi ∈ subsetk (OS )
OApx ← OApx ∪ ES(Oi , OW L 2 QL);
return OApx ;
end
Effective computation of maximal sound approximations of Description Logic ontologies
21/10/2014
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Outline
1
Preliminaries
2
A new definition of approximation
3
Approximation in OWL 2 QL
4
Experimental evaluation
5
Conclusions and Future Work
Effective computation of maximal sound approximations of Description Logic ontologies
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Experimental evaluation
1
2
GSA (k = |OS |) vs. LSA (k = 1)
GSA and LSA vs. syntactic sound approximation (baseline)
I Timeout set at 8 hours
I 41 Bioportal ontologies tested
I GSA computable for 26/41 ontologies
I LSA always computable
Ontology
Vertebrate anatomy
Spatial
Translational medicine
Skeletal anatomy
Pato
Lipid
Plant
Mosquito anatomy
Idomal namespace
Cognitive atlas
Fly anatomy
OVERALL AVERAGE
GSA/original
93%
63%
86%
95%
89%
87%
96%
99%
99%
97%
99%
80%
LSA/GSA
97%
86%
99%
92%
100%
97%
81%
44%
98%
100%
67%
87%
Ontology
Protein
Dolce
Galen-A
Fyp
Gene
FMA OBO
OVERALL AVERAGE
SYNT/GSA
56%
42%
30%
57%
78%
89%
81%
44%
59%
26%
67%
72%
SYNT/LSA
67%
52%
64%
99%
99%
97%
100%
100%
100%
30%
100%
90%
LSA/original
47%
78%
70%
85%
99%
99%
72%
LSA time (s)
20
8
26
43
178
113
51
Effective computation of maximal sound approximations of Description Logic ontologies
GSA time (s)
3
9
19
27
99
47
929
214
496
162
25499
1110
LSA time (s)
3
4
7
5
18
10
15
16
16
17
45
41
21/10/2014
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Experimental evaluation
Final considerations:
I GSA provides maximal sound approximation in reasonable time for
majority of tested ontologies (80% average for 26/41);
I LSA provides very fast solution in all cases, and captures on average
significant portion of GSA (87% average);
I LSA provides good approximation even for ontologies for which GSA is
not computable (72% average);
I LSA and GSA both compare favorably against syntactix sound
approximation (respectively 90% and 72% average).
for very large ontologies, 10% difference for LSA/SYNT means
preserving thousands of axioms in very little time
GSA and LSA both useful approaches!
Effective computation of maximal sound approximations of Description Logic ontologies
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Outline
1
Preliminaries
2
A new definition of approximation
3
Approximation in OWL 2 QL
4
Experimental evaluation
5
Conclusions and Future Work
Effective computation of maximal sound approximations of Description Logic ontologies
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Conclusions and Future Work
Conclusions
We have proposed a parameterized semantics for computing sound
approximations of ontologies;
We have provided algorithms for approximations (GSA and LSA) of
OWL 2 ontologies in OWL 2 QL;
Extensive experimental evaluation which demonstrate the validity
of both GSA and LSA.
Future Works
Develop techniques for k-approximations with 1 < k < |OS |;
Integrate ontology module extraction techniques;
When is LSA enough?
Generalizing our approach to OBDA scenario: approximating a
source ontology with both ontology and mappings in the target
OBDA specification.
Effective computation of maximal sound approximations of Description Logic ontologies
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Thank you
Thank you!
Effective computation of maximal sound approximations of Description Logic ontologies
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Effective computation of maximal sound approximations of Description Logic ontologies
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Optimizing the computation of an Entailment Set
Entailment Set: huge number of OWL 2 reasoner invocations for axiom
implications.
Strategy: exploiting acquired knowledge in order to limit the number of
invocation the OWL 2 reasoner (each invocation is 2NExpTime!).
So...
if ∃R has no subsumees, then ∃R.A, ∃R.∃P.A, ∃R.∃P.∃P − , . . .
also have no subsumees.
I save invocations of the OWL 2 reasoner for subsumees of ∃R.A,
∃R.∃P.A, ∃R.∃P.∃P − , . . .
if B1 v B2 and B1 has no disjoint concepts, then also B2 has no
disjoint concepts.
I save invocations of the OWL 2 reasoner for disjoint concepts of
every subsumer of B1 .
Effective computation of maximal sound approximations of Description Logic ontologies
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