1. ILP 2008 – Prague
Learning with Kernels
in Description Logics
Nicola Fanizzi
Claudia d’Amato
Floriana Esposito
LACAM - Dipartimento di Informatica
Università degli studi di Bari

2. Motivation
In the Semantic Web context,
uncertainty due to
Incoherence

heterogeneous / distributed knowledge

sources
Inherent incompleteness

Open-World Semantics

need for alternative methods wrt purely logic
(deductive) reasoning
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3. Wine Example
Considering the well-known WINE ontology,
some non logically derivable assertions:
KathrynKennedyLateral

known as a Meritage wine,
but not as a CaliforniaWine and an
AmericanWine
CotturiZinfandel

known as a Zinfandel
it is not a CabernetSauvignon
(a non-disjoint sibling class)
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4. Our Proposal
Inductive Inference
Non-parametric statistical learning methods

applied to standard ontology languages
based on epistemic inference: underlying

semantic similarity between individuals as
elicited from KB
Distance & Kernel functions

A learning framework for based on them

Inductive classifiers for inductive instance

check
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5. Application
Approximate
concept retrieval & query-answering
using inductive classifiers:
Kernel Machines (SVMs)

based on kernel functions

for individuals in KBs
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6. Description Logics
Building blocks:

NC = {C, D, ... } primitive concept names
{C

NR = {R, Q, ... } primitive role names
{R

NI = {a, b, ... } individual names
{a

Complex descriptions built using the language

constructors (next slide)
(∆
set-theoretic Interpretation I = (∆I, ·I)

Int. domain: ∆I of the discourse

Int. mapping: ·I from NI to ∆I

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7. ALCN Logic
CI extension of concept C in the interpretation I

RI extension of role R in the interpretation I

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8. DL Knowledge Bases
KB: K = <T , A >
<T

TBox T : set of axioms regarding the concepts

C≡ D

equivalent to CI ⊆ DI ∧ DI ⊆ CI in the

interpretations satisfying the KB
ABox A : set of assertions on the individuals

C (a )

 R (a , b )
Open-world assumption

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9. Royal Family Example
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10. Normal Form
Negation normal form employed by the tableau-

reasoners [Baader et al., 2003]
Formulae as AND-OR trees

C: └┘ D: └┘
┌┐ { P , P } ┌┐ { }
┌┐ { } { P3 }
1 2
∀R
∃R ∃R ∃R
{ P1, ¬ P2 } { ¬ P1 }
{ P3 } └┘
┌┐ { }
∀R
{ P2 }
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11. Normal Form: a definition
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12. Kernels for ALCN
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13. Kernels for ALCN / 2
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14. Validity
Easily proven

Limited recursion

Closure wrt std operations

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15. Problems with DL KBs
From concepts to indivuals
concepts

Multiple-class classification
Multiple-class

Not necessarily disjoint classes

Open-world semantics

Use MSC approximation

Decompose in multiple learning problems

Training instances may be unlabeled

Possible values for estimator V = {-1, 0, +1}
{-1, +1}

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16. SVMs
Kernels implemented and integrated in LIBSVM

Train 2 classifiers

for the positive classification

for the negative classification

Decision procedure used for instance checking

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17. Experimental Setup
For each KB

classification of all their concepts

 randomly generated query concepts (30)
 composition of primitive or defined
concepts (2 through 8)
SVMs: default parametres values

10-fold Cross Validation test procedure
10-fold

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18. Evaluation Indices
Deductive / Inductive classifier
match rate:
rate:
-1/-1 0/0 +1/+1
omission error rate:
rate:
-1/0 +1/0
commission error rate:
rate:
-1/+1 +1/-1
induction rate:
rate:
0/-1 0/+1
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19. Ontologies
From Protégé's Ontology library

and other benchmarks

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20. Results / 1
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21. Results / 2
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22. Results / 3
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23. Results / 4
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24. Concluding Remarks
Non-parametric What's next:

statistical learning to Better eval. indices

work on DL KBs
New kernels for

Application to

OWL @ ISWC2008
approximate
unsupervised

classification query
learning @
answering
ESWC08
Encouraging

Uncertainty models

experimental results
 Probabilistic m.
 Rough DL
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25. The End
Questions ?
Offline:
Nicola Fanizzi fanizzi@di.uniba.it

Claudia d'Amato claudia.damato@di.uniba.it

Floriana Esposito esposito@di.uniba.it

LACAM - Dipartimento di Informatica
Università degli studi di Bari
Via E. Orabona, 4 - 70125 Bari, Italy
http://lacam.di.uniba.it:8000/
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