DL-Foil:Class Expression Learning Revisited

DL-FOIL: Class Expression Learning Revisited
Nicola Fanizzi, Giuseppe Rizzo, Claudia d’Amato, Floriana Esposito
LACAM - Dipartimento di Informatica, Universit`a degli Studi di Bari Aldo Moro
EKAW 2018, Nancy, France – 15th November 2018

Outline
1 Introduction
2 The problem
3 DL-Foil
4 Evaluation
5 Conclusions, Ongoing & Future Work
Giuseppe Rizzo (LACAM-Dip.Informatica, Bari) DL-Foil EKAW 2018, Nancy, France – 15th November 2018

Introduction
Introduction

Introduction
Motivations
Goal
Eliciting candidate concept descriptions for semi-automatic knowledge base
completion
TBox: candidate (equivalence) axioms
ABox: candidate (class) assertions by classifying individuals
Solutions
(Supervised) Machine learning methods:
E.g. concept learning: symbolic methods for producing a concept
description using a set of pos./neg./unlabeled. examples

Introduction
Motivations:
Previous solutions and current limits
DL-Foil: produces a concept description in a disjunctive form providing a
consistent classiﬁcation of the examples
ternary problem (pos., neg., unlabeled ex.s) – OWA
partial description generated on-the-ﬂy to cover the largest number of pos.
ex.s as possible
selection among a set of candidates generated according to an heuristic
Problems:
generated descriptions not covering positive examples
unlabeled individuals equally contribute to the score for candidate evaluation
Contribution: improving both the specialization procedure and and the
heuristic considering the actual number of unlabeled individuals

The problem
The problem

The problem
The learning problem
Let K = T , A be a DL knowledge base.
Given a concept name C
a training set Tr = (Ps, Ns, Us)
Find a concept description D, such that, letting K = K ∪ {C ≡ D}, A ,
the following entailments hold:
∀a ∈ Ps: K |= C(a)
∀b ∈ Ns: K |= ¬C(b)
i.e. correct w.r.t. the examples and general for predictive purposes

The problem
The learning problem: an example
Let K = { Man Person, Man ¬Woman
Woman Person,
Man (a), Man (b), Man(c), hasChild(a,d), hasChild (b,e), Woman (d), Woman
(f), Artist(e), Dog(z) }
Target concept: Father, i.e. a man with at least a child
Ps={ a, b}
Ns ={ d,f} ( due to Man ¬Woman)
Us= { c, e,z }
induced concept: Father ≡ Man ∃ hasChild

DL-Foil
DL-Foil

DL-Foil
The algorithm
Given Tr and a partial description C in a disjunctive form (initialized C = ⊥):

DL-Foil
The algorithm
C’=

DL-Foil
The algorithm
C’= Reﬁnes C

DL-Foil
The algorithm
C’= Reﬁnes C Find the best Di
{Di |Di C }
neg./unl. example covered

DL-Foil
The algorithm
C’= Reﬁnes C Find the best Di
C = C Di
{Di |Di C }
neg./unl. example covered
no neg.exs coveredRemove pos examples

DL-Foil
DL-Foil: Covering Procedure Example
Input:
Tr = {a, b, d, f, c, e, z}
C = ⊥
Trace of the algorithm
1st ref. step: C = (covered examples: Tr)
D∗
= ¬Woman (covered examples: a, b, c) (c ∈ Us — further specialization
required)
2nd ref. step: C = ¬Woman
D∗
= ¬Woman ∃hasChild.Person (covered examples: a, b –all i ∈ Ps)
C = ⊥ ¬Woman ∃hasChild.Person
Ps = Ps {a, b} = ∅

DL-Foil
Specializing a concept
Generation n concept D C performing a sort of random sampling in the DL
concept space

DL-Foil
concept space
ρ1 D = C A
ρ2 D = C ¬A
ρ3 D = C ∀R. Add a conjunct (randomly selected from the signature)
ρ4 D = C ∃R.

DL-Foil
concept space
Reﬁnes an existing sub-description (randomly selected)
ρ5 D = C1 · · · B · · · Cn
if C = C1 · · · A · · · Cn and B A
ρ6 D = C1 · · · ¬B · · · Cn
if C = C1 · · · ¬A · · · Cn and A B
ρ7 D = C1 · · · ∃R.F · · · Cn
if C = C1 · · · ∃R.E · · · Cn and F ∈ ρ(E)
ρ8 D = C1 · · · ∀R.F · · · Cn
if C = C1 · · · ∀R.E · · · Cn and F ∈ ρ(E)

DL-Foil
concept space
ρ1 D = C A
ρ2 D = C ¬A
ρ3 D = C ∀R.
ρ4 D = C ∃R.
ρ5 D = C1 · · · B · · · Cn
if C = C1 · · · A · · · Cn and B A
ρ6 D = C1 · · · ¬B · · · Cn
if C = C1 · · · ¬A · · · Cn and A B
ρ7 D = C1 · · · ∃R.F · · · Cn
if C = C1 · · · ∃R.E · · · Cn and F ∈ ρ(E)
ρ8 D = C1 · · · ∀R.F · · · Cn
if C = C1 · · · ∀R.E · · · Cn and F ∈ ρ(E)

DL-Foil
Specializing a concept: examples
Let C = Person the concept to be reﬁned
D = Person Man ( using ρ1)
D = Person ¬Woman( using ρ2)
D = Person ∃hasChild. (using ρ3)
D = Person ∀hasChild. (using ρ4)
Let C = Person ∃hasChild.
D = Person ∃hasChild.Man
D = Person ∃hasChild.Dog

DL-Foil
Specializing a concept: examples
Let C = Person the concept to be reﬁned
D = Person Man ( using ρ1)
D = Person ¬Woman( using ρ2)
D = Person ∃hasChild. (using ρ3)
D = Person ∀hasChild. (using ρ4)
Let C = Person ∃hasChild.
D = Person ∃hasChild.Man
D = Person ∃hasChild.Dog ← Satisﬁable w.r.t KB but without pos.exs.!

DL-Foil
Improving concept specialization
Further constraints are used in DL-FOIL for avoiding ”uninformative”
concepts:
the specialization procedure implementing ρ generates concepts that
covers at least a positive example
the score of each specialization exceeds a threshold

DL-Foil
The score
DL-Foil selects the concept maximizing an information-theoretics heuristic:
g(D0, D1) = p1 · log
p1
p1 + n1 + u1
− log
p0
p0 + n0 + u0
D0: the former partial deﬁnition
D1: the specialization
p1, n1, u1: the actual number of pos., neg., unl. exs. covered by D1
p0, n0, u0: the actual number of pos., neg., unl. exs. covered by D0

Evaluation
Evaluation

Evaluation
Preliminary Experiments
Concept membership prediction
5 publicly available ontologies
15 artiﬁcially generated datasets:
random target concept generation
ground truth: individuals labeled according to the membership w.r.t. target
Competitor: CELOE
0.632 bootstrap as the design of the experiment
Indices: membership w.r.t. the induced concept against membership
w.r.t. the target
actual
value
Prediction outcome
pos. neg. unl.
pos. match (M) commission (C) omission (O)
neg. commission match omission
unl. induction (I) induction match

Evaluation
Outcomes
Dataset index DL-FOIL CELOE
BIOPAX M% 95.73 ± 03.74 94.53 ± 01.17
C% 00.13 ± 00.20 03.24 ± 00.85
O% 01.90 ± 03.31 01.62 ± 00.38
I% 02.23 ± 00.40 00.61 ± 00.18
NTN M% 97.78 ± 05.05 97.41 ± 00.15
C% 00.05 ± 00.07 00.00 ± 00.00
O% 02.17 ± 05.00 00.00 ± 00.00
I% 00.01 ± 00.01 02.59 ± 00.15
HDISEASE M% 88.75 ± 01.09 88.08 ± 01.09
C% 00.04 ± 00.10 00.00 ± 00.00
O% 03.64 ± 01.30 07.69 ± 00.90
I% 07.57 ± 01.42 04.23 ± 00.24
FINANCIAL M% 93.52 ± 01.02 87.40 ± 04.74
C% 00.22 ± 00.21 06.33 ± 04.33
O% 00.00 ± 00.00 00.00 ± 00.01
I% 06.26 ± 00.88 06.26 ± 00.52
GEOSKILLS M% 82.60 ± 04.69 50.20 ± 02.31
C% 00.00 ± 00.00 23.66 ± 02.61
O% 13.33 ± 04.43 01.34 ± 00.12
I% 04.07 ± 04.09 24.80 ± 00.89

Conclusions, Ongoing & Future Work

Conclusions & Extensions
Modiﬁed version of DL-Foil with a different specialization procedure and
heuristic
The evaluation shows good results in terms of match rate
Ongoing & Future Work
New evaluations on larger knowledge bases
New specializations procedures
New heuristics
Scalability
Parallel computation
Distributed computation (Spark, Flink...)

Thank You!

DL-Foil:Class Expression Learning Revisited

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