Concept Similarity under the Agent’s Preferences for the Description
Logic FL
0
with Unfoldable TBox
Teeradaj Racharak
1,2
and Satoshi Tojo
2
1
School of Information, Computer and Communication Technology, Sirindhorn International Institute of Technology,
Thammasat University, Pathum Thani, Thailand
2
School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa, Japan
Keywords:
Concept Similarity Measure, Semantic Web Ontology, Preference Profile, Description Logics.
Abstract:
Concept similarity refers to human judgment of a degree to which a pair of concepts is similar. Computational
techniques attempting to imitate such judgment are called concept similarity measures. In Description Logics
(DLs), we could regard them as a generalization of the classical reasoning problem of equivalence. That is,
any two concepts are equivalent if and only if their similarity degree is one. When two concepts are not
equivalent, the level of similarity varies depending not only on the objective factors (e.g. the structure of
concept descriptions) but also on the subjective factors (i.e. the agent’s preferences). The recently introduced
notion called preference profile identified a collection of preferential elements in which any developments
for concept similarity measure should consider. In this paper, we briefly review approaches of identifying
the subsumption degree between FL
0
concept descriptions and exemplify how one can adopt the viewpoint
of preference profile toward the development of concept similarity measure under the agent’s preferences in
FL
0
. Finally, we investigate several properties of the developed measure and discuss future directions.
1 INTRODUCTION
Most Description Logics (DLs) are decidable frag-
ments of first-order logic (FOL) (Baader et al., 2010)
with clearly defined computational properties. DLs
are the logical underpinnings of the DL flavor of
OWL and OWL 2. The advantage of this close con-
nection is that the extensive DLs literature and im-
plementation experiences can be directly exploited by
OWL tools. More specifically, DLs provide unam-
biguous semantics to the modeling constructs avail-
able in the DL flavor of OWL and OWL 2. These
semantics make it possible to formalize and design
algorithms for a number of reasoning services, which
enable the development of ontology applications to
become prominent. For instance, ontology classifi-
cation (or ontology alignment) organizes concepts in
an ontology into a subsumption hierarchy and assists
in detecting potential errors of a modeling ontology.
Though this subsumption hierarchy inevitably bene-
fits ontology modeling, it merely gives two-valued
responses, i.e. inferring a concept is subsumed by
another concept or not. However, certain pairs of
concepts may share commonality even though they
are not subsumed. As a consequence, a considerable
amount of research effort has been devoted to measur-
ing similarity of two given concepts, i.e. a problem of
concept similarity measure in DLs.
Intuitively, concept similarity refers to human
judgment of a degree to which a pair of concepts in
question is similar. Concept similarity measures are
computational techniques attempting to imitate the
human judgments of concept similarity. Indisputably,
they often contribute to various kinds of applications.
For example, they were employed in bio-medical
ontology-based applications to discover functional
similarities of gene such as (Ashburner et al., 2000),
they are often used by ontology alignment algorithms
such as (Euzenat and Valtchev, 2004), they can be em-
ployed in approximate reasoning such as (Raha et al.,
2008; Sessa, 2002) and in analogical reasoning such
as (Racharak et al., 2016c; Racharak et al., 2017b).
Oftentimes, when similarity judgment is performed
by a cognitive agent, the degree of similarity may vary
w.r.t. the need and preferences of the agent. The fol-
lowing example illustrates such a case in which con-
cept similarity measured not only w.r.t. objective fac-
tors (but also w.r.t. subjective factors) can give more
intuitive results.
Example 1.1. An agent A is searching for a hotel
Racharak, T. and Tojo, S.
Concept Similarity under the Agent’s Preferences for the Description Logic FL
0
with Unfoldable TBox.
DOI: 10.5220/0006653402010210
In Proceedings of the 10th International Conference on Agents and Artificial Intelligence (ICAART 2018) - Volume 2, pages 201-210
ISBN: 978-989-758-275-2
Copyright © 2018 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
201
room during his vacation. At that moment, he prefers
to stay in a Japanese-style room or something similar.
In the following, his desired room may be expressed
as the concept DesiredRoom. Suppose RoomA and
RoomB are concepts in a room ontology as follows:
DesiredRoom v Room u floor.Tatami
RoomA v Room u floor.Bamboo
RoomB v Room u floor.Marble
Without considering his preferences, it may be un-
derstood that both RoomA and RoomB are equally
similar to DesiredRoom. However, taking into ac-
count his preferences, RoomA may appear more suit-
able (assuming that tatami and bamboo invoke similar
feeling). In other words, he will not be happy if an in-
telligent system happens to choose RoomB for him.
Other examples can be found in (Tversky, 1977)
where intended behaviors (desirable properties) of
similarity measures were investigated. For example,
people usually speak that “the portrait resembles the
person” rather than “the person resembles the por-
trait”. Also, people usually say that “the son resem-
bles the father” rather than “the father resembles the
son”. These examples clearly point out that cogni-
tive agents make similarity judgment under some sub-
jective factors. Unfortunately, existing measures do
not usually take into account subjective factors dur-
ing computational procedures, though some may con-
sider such as (Lehmann and Turhan, 2012; Tongphu
and Suntisrivaraporn, 2015).
In order to develop similarity measures which
can be performed under subjective factors, (Racharak
et al., 2016b) has introduced a general notion called
concept similarity measure under preference profile
(and later extended in (Racharak et al., 2017a)). In-
stead of implicitly including preferential elements in
the computational representation, (Racharak et al.,
2017a) clearly separated those preferential elements
from the computational procedures. Hence, the gen-
eral notion makes an investigation of concept similar-
ity measure under subjective factors more easily and
provides more natural understanding when concept
similarity measures are used under subjective factors.
It is worth noting that any particular DL L is de-
termined by the concept constructors and the ontolog-
ical constructors it provides. For instance, (Racharak
et al., 2016b; Racharak et al., 2017a) concentrated on
the DL ELH , which offered the constructors conjunc-
tion (u), full existential quantification (r.C), and the
top concept (>); and also, allowed to define role hi-
erarchy axioms in a TBox. In this paper, we concen-
trate on the DL FL
0
, which provides the constructors
conjunction (u), value restriction (r.C), and the top
concept (>) (cf. Section 2). The main contribution of
this paper is to introduce a computational technique
for concept similarity measure under the agent’s pref-
erences for the DL FL
0
(cf. Section 4 - 5). Finally,
we relate the approach to the others (cf. Section 6)
and discuss the future directions (cf. Section 7).
2 PRELIMINARIES
In this section, we review the basics of Description
Logic FL
0
in Subsection 2.1, particularly its syntax,
semantics, and normal form which can be used for
subsumption testing in FL
0
. Then, we review the no-
tion of preference profile in Subsection 2.2.
2.1 Description Logic FL
0
We assume finite sets CN of concept names and RN
of role names that are fixed and disjoint. The set of
concept descriptions, or simply concepts, for a spe-
cific DL L is denoted by Con(L). The set Con(FL
0
)
of all FL
0
concepts can be inductively defined by the
following grammar:
Con(FL
0
) ::= A | > | C u D | r.C
where > denotes the top concept, A CN, r
RN, and C, D Con(FL
0
). Conventionally, concept
names are denoted by A and B, concept descriptions
are denoted by C and D, and role names are denoted
by r and s, all possibly with subscripts.
A terminology or TBox T is a finite set of primi-
tive concept definitions and full concept definitions,
whose syntax is an expression of the form A v D
and A D, respectively. A TBox is called unfold-
able if it contains at most one concept definition for
each concept name in CN and does not contain cyclic
dependencies. Concept names occurring on the left-
hand side of a concept definition are called defined
concept names (denoted by CN
def
), all other con-
cept names are primitive concept names (denoted by
CN
pri
). A primitive definition A v D can easily be
transformed into a semantically equivalent full defi-
nition A X u D where X is a fresh concept name.
When a TBox T is unfoldable, concept names can
be expanded by exhaustively replacing all defined
concept names by their definitions until only primi-
tive concept names remain. Such concept names are
called fully expanded concept names.
1
An interpretation I is a pair I = h
I
, ·
I
i, where
I
is a non-empty set representing the domain of the in-
terpretation and ·
I
is an interpretation function which
1
In this work, we assume that concept names are fully
expanded and the TBox can be omitted.
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
202
assigns to every concept name A a set A
I
I
, and
to every role name r a binary relation r
I
I
×
I
.
The interpretation function ·
I
is inductively extended
to FL
0
concepts in the usual manner:
>
I
=
I
; (C u D)
I
= C
I
D
I
;
(r.C)
I
= {a
I
| b
I
: (a, b) r
I
b C
I
}.
An interpretation I is said to be a model of a TBox
T (in symbols, I |= T ) if it satisfies all axioms in T .
I satisfies axioms A v C and A C, respectively, if
A
I
C
I
and A
I
= C
I
. The main inference problem
in FL
0
is the concept subsumption problem.
Definition 2.1 (Concept Subsumption). Given C, D
Con(FL
0
) and a TBox T , C is subsumed by D w.r.t.
T (denoted by C v
T
D) if C
I
D
I
for every model
I of T . Moreover, C and D are equivalent w.r.t. T
(denoted by C
T
D) if C v
T
D and D v
T
C.
When a Tbox T is clear from the context, we sim-
ply drop T , i.e. C v D or C D.
Using the rewrite rule r.(C u D) r.C u
r.D together with the associativity, the commuta-
tivity, and the idempotence of u, any FL
0
concepts
can be transformed into an equivalent one of the
form r
1
. . . r
n
.A where {r
1
, . . . , r
n
} RN and A
CN. Such concepts can be abbreviated as r
1
. . . r
n
.A
where r
1
. . . r
n
is viewed as a word w over the alphabet
of all role names. We note that when n = 0, i.e. the
empty word ε, ε.A corresponds to A. Furthermore,
a conjunction of the form w
1
.A u . . . w
m
.A can be
abbreviated as L.A where L
:
= {w
1
, . . . , w
m
} is a fi-
nite set of words over the alphabet. We also note that
/
0.A corresponds to >. Using these abbreviations,
any concepts C, D Con(FL
0
) can be rewritten as:
C U
1
.A
1
u ··· u U
k
.A
k
(1)
D V
1
.A
1
u ··· u V
k
.A
k
(2)
where {A
1
, . . . , A
k
} CN and U
i
,V
i
are finite sets of
words over the alphabet of role names. This normal
form provides us the following characterization of
subsumption in FL
0
(Baader and Narendran, 2001):
C v D U
i
V
i
for all i, 1 i k (3)
Theorem 2.1. Concept subsumption and concept
equivalence without TBox (i.e. when the TBox is
empty) in FL
0
can be decided in polynomial time.
Example 2.1. (Continuation of Example 1.1) After
unfolding and transforming into normal forms, each
concept is represented as:
1
DesiredRoom ≡ ∀{ε}.X u
/
0.Y u
/
0.Z u ∀{ε}.R
u ∀{f}.T u
/
0.B u
/
0.M
RoomA ≡ ∀
/
0.X u∀{ε}.Y u
/
0.Z u ∀{ε}.R
u ∀{f}.B u
/
0.T u
/
0.M
RoomB ≡ ∀
/
0.X u
/
0.Y u ∀{ε}.Z u ∀{ε}.R
u ∀{f}.M u
/
0.T u
/
0.B
1
Obvious abbreviations are used for succinctness.
where X,Y, and Z are fresh concept names. Using
Equation 3, it yields DesiredRoom 6v
/
0
RoomA and
DesiredRoom 6v
/
0
RoomB.
2.2 Preference Profile
Preference profile was first introduced in (Racharak
et al., 2016a) as a collection of preferential ele-
ments in which any developments of concept similar-
ity measure should consider (later, it was improved in
(Racharak et al., 2017a)). Its first intuition is to model
different forms of preferences (of an agent) based on
concept names and role names. Measures adopted
this notion are flexible to be tuned by an agent and
can determine the degree of similarity conformable to
that agent’s perception. We give its formal definition
of each preferential aspect in the following definition.
Definition 2.2 (Preference Profile (Racharak et al.,
2017a)). Let CN
pri
(T ), RN
pri
(T ), and RN(T ) be a
set of primitive concept names occurring in T , a set
of primitive role names occurring in T , and a set of
role names occurring in T , respectively. A preference
profile (denoted by π) is a quintuple hi
c
, i
r
, s
c
, s
r
, di
where i
c
, i
r
, s
c
, s
r
, and d are partial functions such
that:
i
c
: CN
pri
(T ) [0, 2] is called a primitive concept
importance;
i
r
: RN(T ) [0, 2] is called a role importance;
s
c
: CN
pri
(T )×CN
pri
(T ) [0, 1] is called a prim-
itive concepts similarity;
s
r
: RN
pri
(T )×RN
pri
(T ) [0, 1] is called a prim-
itive roles similarity; and
d : RN(T ) [0, 1] is called a role discount factor.
We discuss the interpretation of each above func-
tion in order. Firstly, for any A CN
pri
(T ), i
c
(A) = 1
captures an expression of normal importance on A,
i
c
(A) > 1 and i
c
(A) < 1 indicate that A has higher
and lower importance, respectively, and i
c
(A) = 0 in-
dicates that A has no importance to the agent. Sec-
ondly, we define the interpretation of i
r
in the simi-
lar fashion as i
c
for any r RN(T ). Thirdly, for any
A, B CN
pri
(T ), s
c
(A, B) = 1 captures an expression
of total similarity between A and B and s
c
(A, B) = 0
captures an expression of total dissimilarity between
A and B. Fourthly, the interpretation of s
r
is defined
in the similar fashion as s
c
for any r, s RN
pri
(T ).
Lastly, for any r RN(T ), d(r) = 1 captures an ex-
pression of total importance on a role (over a corre-
sponding nested concept) and d(r) = 0 captures an
expression of total importance on a nested concept
(over a corresponding role), e.g. let d(r
1
) = 0.3, then
the degree of similarity under this preference between
Concept Similarity under the Agent’s Preferences for the Description Logic FL
0
with Unfoldable TBox
203
r
1
.A and r
1
.B can be understood as 0.3 degree as
the identical occurrence of r
1
has 0.3 importance.
It is worth noticing that role names appearing in
FL
0
are always primitive. This suggests that both
RN
pri
(T ) and RN(T ) can be considered identically
in Definition 2.2. Furthermore, due to the employed
characterization, d is not used in this paper.
3 FROM CONCEPT
SUBSUMPTION TO
SUBSUMPTION DEGREE
The idea of computing subsumption degree using
the characterization of language inclusion was firstly
proposed in (Racharak and Suntisrivaraporn, 2015).
Two computational techniques on subsumption de-
gree were developed, viz. the skeptical subsump-
tion degree (in symbols,
s
) and the credulous sub-
sumption degree (in symbols,
c
). The names skep-
tical and credulous were motivated by the fact that
the degree obtained from
s
is always less than or
equal to the one obtained from
c
. Basically, the
function
s
checks set inclusions between sets of
words whereas the
c
calculates the proportion be-
tween sets of words. In the following, we rewrite
their original definitions and include them here for
self-containment.
Definition 3.1 (skeptical FL
0
subsumption degree).
Let C, D Con(FL
0
) be in their normal forms and
W(E, A) be a set of words w.r.t. the concept E and the
primitive A. Then, a skeptical FL
0
degree from C to
D (denoted by C
s
D) is defined as follows:
C
s
D =
|{P CN
pri
| W(D, P) W(C, P)}|
|CN
pri
|
,
(4)
where | · | denotes the set cardinality.
Definition 3.2 (credulous FL
0
subsumption degree).
Let C, D Con(FL
0
) be in their normal forms and
W(E, A) be a set of words w.r.t. the concept E and
the primitive A. Then, a credulous FL
0
subsumption
degree from C to D (denoted by C
c
D) is defined as
follows:
1
C
c
D =
PCN
pri
µ(D,C, P)
|CN
pri
|
, (5)
where |· | denotes the set cardinality and µ(D,C, P) =
(
1 if W(D, P) =
/
0
|W(D,P)W(C,P)|
|W(D,P)|
otherwise
(6)
1
We fix a minor error in (Racharak and Suntisrivara-
porn, 2015) here, i.e. the condition W(D, P) =
/
0 is included.
It is worth observing that if
|W(D,P)W(C,P)|
|W(D,P)|
= 1,
then W(D, P) W(C, P) holds (and vice versa). Fol-
lowing this observation, it is not difficult to show that
C
s
D C
c
D for any C, D Con(FL
0
).
Proposition 3.1. For any C, D Con(FL
0
), it follows
that C
s
D C
c
D.
Proof. (Sketch) We only need to show that, for
any P CN
pri
, for any pair of W(D, P) and W(C, P)
which share commonality without being subsumed,
then it follows that C
s
D C
c
D. Fix any P
CN
pri
. Also, let W(D, P) = {r
1
, . . . , r
n
, s
1
, . . . , s
m
} and
W(C, P) = {r
1
, . . . , r
n
,t
1
, . . . ,t
k
}. Then, it is obvious
that C
s
D C
c
D holds. o
Example 3.1. (Continuation of Example 2.1) Let us
abbreviate the concepts DesiredRoom, RoomA, and
RoomB with DR, RA, and RB, respectively. It yields
DR
s
RA =
|{X, Z, R, T, M}|
|{X,Y, Z, R, T, B, M}|
=
5
7
and DR
c
RA
=
1 +
|{ε}∩
/
0|
1
+ 1 +
|{ε}∩{ε}|
1
+ 1 +
|{ f }∩
/
0|
1
+ 1
7
=
5
7
.
Similarly, it yields DR
s
RB = DR
c
RB =
5
7
.
The following theorem shows a very nice property
inherited in both functions, which have not been in-
vestigated in (Racharak and Suntisrivaraporn, 2015).
Theorem 3.1. The functions
s
and
c
can be com-
puted in polynomial time.
Proof. Since the size of normal forms is polyno-
mial in the size of the original concepts, and since the
inclusion checking and the proportion checking can
be also decided in polynomial time, these functions
can be computed in polynomial time. o
The fact that there exists two concept similarity
measures corresponds to an experiment in (Bernstein
et al., 2005). That is, similarity measures might de-
pend on target applications (e.g. target ontologies)
and applicable similarity measures should be person-
alized to the agent’s similarity judgment style. These
observations were also discussed in (Racharak et al.,
2017a). Now, we are ready to exemplify how the
notion of preference profile can be adopted toward
the development of subsumption degree under pref-
erences in FL
0
. We continue this in the next section.
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
204
4 FROM SUBSUMPTION
DEGREE TO SUBSUMPTION
DEGREE UNDER
PREFERENCES
To exemplify a development of subsumption degree
under preferences procedure, we generalize the func-
tion
s
to expose preferential elements of preference
profile. As a result, the new function
π
s is also
driven by the structural subsumption characterization
by means of language inclusion in FL
0
. As aforemen-
tioned in Subsection 2.2, some preferential aspects of
preference profile might be possibly not exposed, e.g.
the role discount factor. This is indeed dependent to
an adopted characterization (e.g. as in
s
).
We start by presenting a relevant aspect of pref-
erence profile in terms of total functions in order to
avoid computing on null values. A total concept simi-
larity function is also presented as
ˆ
s : CN
pri
×CN
pri
[0, 1] as follows:
ˆ
s(x, y) =
1 if x = y
s
c
(x, y) if (x, y) CN
pri
× CN
pri
and s
c
is defined on (x, y)
0 otherwise
(7)
Intuitively, identical concepts are considered totally
similar, i.e. they are set to 1. Otherwise, in case that
they are not defined, different concepts are considered
totally dissimilar by default.
The next step is to generalize the function
s
. We
rewrite the numerator of
s
to:
PCN
pri
max
QCN
pri
{
ˆ
s(P, Q) | W(D, P) W(C, Q)} (8)
Basically, Equation 8 combines value of each maxi-
mal primitive concepts similarity element. Its objec-
tive is to also take into account the value of each sim-
ilar concept pair, if this value is defined.
We may also put the notion of concept impor-
tance into our computational procedure. As suggested
in Subsection 2.2, this results in the flexibility for
weighting on primitive concepts (ranging from having
no importance to having the maximum importance).
To achieve this, we continue with a similar at-
tempt. That is, a total concept importance function
is introduced as
ˆ
i : CN
pri
[0, 2] as follows:
ˆ
i(x) =
(
i
c
(x) if x CN
pri
and i
c
is defined on x
1 otherwise
(9)
Basically, the above equation says that each concept
has normal importance by default, if it is not defined.
To take these matters into account, we should
rewrite both the numerator and the denominator such
that they expose some rooms for tuning with the con-
cept importance. Thus, we rewrite each part, respec-
tively, as follows:
PCN
pri
ˆ
i(P)· max
QCN
pri
{
ˆ
s(P, Q) | W(D, P) W(C, Q)}
(10)
PCN
pri
ˆ
i(P) (11)
Finally, putting each rewritten part together yields
a concrete function for concept subsumption degree
under preference profile. We denote this new function
by
π
s (cf. Definition 4.1) as it presents a generaliza-
tion of
s
w.r.t. preference profile.
Definition 4.1 (skeptical FL
0
subsumption degree un-
der π). Let C, D Con(F L
0
) be in their normal forms
and W(E, A) be a set of words w.r.t. the concept E
and the primitive A. Then, a skeptical FL
0
subsump-
tion degree under π from C to D (denoted by C
π
s D)
is defined as follows:
C
π
s D =
PCN
pri
ˆ
i(P) · max
QCN
pri
{
ˆ
s(P, Q)|W(D, P) W(C, Q)}
PCN
pri
ˆ
i(P)
(12)
Example 4.1. (Continuation of Example 2.1) Sup-
pose that Bamboo is quite similar to Tatami. Then,
ones may express the agent As preferences as:
s
c
(Bamboo, Tatami) = 0.8. Following Definition
4.1, it yields that
DR
π
s RA =
1 + 0 + 1 + 1 + 1 + 0.8 + 1
|X,Y, Z, R, T, B, M|
=
5.8
7
and
RA
π
s DR =
0 + 1 + 1 + 1 + 0.8 +1 + 1
7
=
5.8
7
.
Similarly, it yields DR
π
s RB = RB
π
s DR =
5
7
.
Ones may also observe that this function has a nice
property, i.e. there exists an algorithmic procedure
whose execution time is upper bound by a polynomial
expression. We state this in the following theorem.
Theorem 4.1. The function
π
s can be computed in
polynomial time.
Proof. Since the size of normal forms is polyno-
mial in the size of the original concepts and since the
maximum function and the inclusion checking can be
decided in polynomial time, this function can be com-
puted in polynomial time. o
Concept Similarity under the Agent’s Preferences for the Description Logic FL
0
with Unfoldable TBox
205
It is worth noticing that subsumption degree un-
der preferences for the credulous subsumption degree
can also be developed by incorporating with the no-
tions role importance (i
r
) and primitive roles similar-
ity (s
r
) (cf. Definition 2.2). However, it requires us to
well investigate how those elements should be incor-
porated and we leave this as a future task.
4.1 Backward Compatibility with
s
Under a special setting of preference profile, the func-
tion
π
s can be reduced backward to
s
. This means
that
π
s can be also used for a situation when pref-
erences are not given. Following the convention in-
troduced in (Racharak et al., 2016a; Racharak et al.,
2017a), let us call this special setting the default pref-
erence profile (denoted by π
0
). We give its formal
definition as follows:
Definition 4.2 (Default Preference Profile). Let
CN
pri
(T ) be a set of primitive concept names occur-
ring in T . The default preference profile, in symbol
π
0
, is the pair hi
c
0
, s
c
0
i where
i
c
0
(A) = 1 for all A CN
pri
(T ) and
s
c
0
(A, B) = 0 for all (A, B) CN
pri
(T ) × CN
pri
(T ).
As for its syntactic sugar, let us denote a setting on
π
s by replacing the setting with π. For instance, we
may write the setting with π
0
as
π
0
s. Next, we show
that, under this special setting on
π
s, the computation
produces the same outcome as
s
.
Proposition 4.1. For any C, D Con(FL
0
), C
π
0
s D
= C
s
D.
Proof. (Sketch) We know that, for any P CN
pri
,
if W(D, P) W(C, P), the cardinality is increased by
one. This is indeed equivalent to the value of s
c
w.r.t.
the identical concepts (i.e. s
c
(A, A) = 1 for any A
CN
pri
(T )). This is obvious. o
5 CONCEPT SIMILARITY
UNDER PREFERENCE
PROFILE IN FL
0
A general notion of concept similarity measure un-
der preference profile was originally proposed in
(Racharak et al., 2016b) and was later extended in
(Racharak et al., 2017a). It is worth noting that the
general notion makes a clear distinction between the
core computational procedure and the preference set-
ting, rather than implicitly using or omitting the pref-
erences. Thus, it provides us capabilities to study and
understand intended behaviors on concrete measures
of the concept similarity measure under preference
profile e.g. when they are used under the agent’s pref-
erences. For instance, rather than saying that “the son
resembles the father”, we would say “if certain prefer-
ences or perspectives are fixed, the son and the father
are similar to each other”. This viewpoint is more
natural to use and gives more intuitive computational
understanding.
Definition 5.1 ((Racharak et al., 2017a)). Given a
preference profile π, two concepts C, D Con(L),
and a TBox T , a concept similarity measure un-
der preference profile w.r.t. a TBox T is a function
π
T
: Con(L) × Con(L) [0, 1].
When a TBox T is clear from the context, we sim-
ply write
π
. Furthermore, to avoid confusion on the
symbols,
π
T
is used when referring to arbitrary mea-
sures of the notion.
To develop a concrete measure as an instance of
π
, we make a similar attempt as in (Racharak et al.,
2017a). That is, we take a look on the logical notion
of concept equivalence. Informally, ones may view
the logical notion of concept equivalence as an opera-
tor for comparing two concepts, i.e. it returns true (1)
if both are equivalent concepts; or it returns false (0)
otherwise.
We recall that the logical notion of concept equiv-
alence can be computed from subsumption checking
w.r.t. two corresponding directions (cf. Definition
2.1). Analogously, an instance of concept similarity
measure under preference profile should also be com-
puted from a corresponding subsumption degree un-
der preferences function w.r.t. two corresponding di-
rections. Following this observation, we introduce a
new measure in Definition 5.2. This measure is de-
noted by
π
s as it is motivated from the fact that
π
s is
used for computing subsumption degree under prefer-
ences w.r.t. the two directions.
Definition 5.2. Let C, D Con(FL
0
) be in their nor-
mal forms and π be a preference profile as a prefer-
ence setting from the agent. Then, the skeptical FL
0
similarity measure under preference profile π between
C and D (denoted by C
π
s D) is defined as follows:
C
π
s D =
C
π
s D + D
π
s C
2
(13)
We may also argue to calculate the value by us-
ing alternative binary operators accepting the unit in-
terval, e.g. based on the multiplication (in symbols,
π
×
) on both directions or the root mean square (in
symbols,
π
rms
) on values of both directions. Un-
fortunately, those give unsatisfactory values for the
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
206
extreme cases. For example, A
π
×
> = 0 × 1 = 0
and A
π
rms
> =
q
0
2
+1
2
2
= 0.707, whereas A
π
s > =
0+1
2
= 0.5. Since C
π
×
D C
π
s D C
π
rms
D for
any concepts C and D, we agree with (Suntisrivara-
porn, 2013; Racharak et al., 2017a) that the average-
based definition given above is the most appropriate
method.
1
Based on this form,
π
s basically determines
the degree of concept similarity under preference pro-
file, i.e. behavioral expectation of the measure will
conform to the agent’s perception.
Example 5.1. (Continuation of Example 4.1) It
yields that
DR
π
s RA =
5.8
7
+
5.8
7
2
=
5.8
7
.
Similarly, it yields that DR
π
s RB =
5
7
. Since
DR
π
s RA > DR
π
s RB, it corresponds to the agent
As perception that he may decide to stay in RoomA
when his DesiredRoom is not available.
It is also worth noting that
π
s comes up with a
very nice property as follows:
Theorem 5.1. The measure
π
s can be computed in
polynomial time.
Proof. Since
π
s can be computed in polynomial
time (by Theorem 4.1), this is trivial. o
Finally, it is worth stating that the measure
π
s can
be also used in the case that a preference profile is
not defined by the agent. In such a case, we can tune
the profile setting to π
0
. This property is immediately
followed from Proposition 4.1.
Remark 5.1. Computing
π
0
s yields the degree of con-
cept similarity merely w.r.t. the structure of concept
descriptions in question.
6 RELATED WORK
Concept similarity has been widely studied in vari-
ous fields, e.g. psychological science, computer sci-
ence, artificial intelligence, and linguistic literature.
Roughly, they can be classified into four ways, viz.
path finding, information content, context vector, and
semantic similarity. We review each way as follows.
The path finding approach requires to firstly con-
struct the concept hierarchy. That is, the more gen-
eral concepts they are, the more they are closer to the
1
Though we recommend to use the average, its choice
of operators may be changed and it may produce a different
behavior as discussed.
root of the hierarchy. Also, the more specific they
are, the more they are closer to the leaves of the hi-
erarchy. Once the hierarchy is constructed, the de-
gree of concept similarity is computed from paths be-
tween concepts. Indeed, there are various ways for
determining the degree. For instance, (Rada et al.,
1989) used a path length between concepts according
to successively either more specific concepts or less
specific concepts. A similar approach was introduced
in (Caviedes and Cimino, 2004) where the degree was
computed based on the shortest path between con-
cepts. Ones may also assign different weights to the
role depth as in (Ge and Qiu, 2008). Unfortunately,
this approach fully relies on the concept hierarchy and
ignores constraints defined in the ontology.
The information content approach augments each
concept with a corpus-based statistics. Generally, the
information content of each concept in a hierarchy
is calculated based on the frequency of occurrence
of that concept in a corpus. The more specific con-
cepts they are, the higher information content values
of them will be. For instance, (Resnik, 1995) defined
the degree of similarity between concepts as the in-
formation content of the least common subsumer of
them. Intuitively, this measure was defined to calcu-
late the degree of the shared information between con-
cepts. However, this approach requires a set of world
descriptions such as a text corpus; and also, may be
not sufficient since many concept pairs may share the
same least common subsumer.
On the one hand, the first two approaches may
utilize the concept hierarchy to compute the degree
of concept similarity. On the other hand, the con-
text vector totally relies on the vector representation.
Roughly speaking, each concept is represented by a
context vector and the cosine of the angle between
vectors is used to determine the degree of similarity
between related concepts. Work which employs this
approach includes (Pedersen et al., 2007; Patwardhan,
2006; Sch
¨
utze, 1998).
The semantic similarity approach basically uses
the syntax and semantics of DLs for the develop-
ment of measures. A simple approach was proposed
in (Jaccard, 1901) for the basic DL L
0
(i.e. no use
of roles). Later, the idea was extended in (Lehmann
and Turhan, 2012) for the DL ELH . The extended
work suggested a new framework that satisfied sev-
eral properties for similarity measure. The framework
was defined in general; thus, functions and operators
were parameterized and were left to be specified. A
different approach for the same ELH was proposed
in (Tongphu and Suntisrivaraporn, 2015) in which the
measure was developed based on the structural sub-
sumption characterization of tree homomorphism.
Concept Similarity under the Agent’s Preferences for the Description Logic FL
0
with Unfoldable TBox
207
Indeed, this approach had its root in the study of
similarity measure for the DL EL (Suntisrivaraporn,
2013). Later, it was extended to the DL ALEH in
(Suntisrivaraporn and Tongphu, 2016). Furthermore,
(Racharak and Suntisrivaraporn, 2015) suggested two
measures for the DL FL
0
based on the structural sub-
sumption characterization of language inclusion. It
is worth observing that these measures calculated the
degree of concept similarity according to the structure
of concept descriptions in question.
Instead of using the structure of concept descrip-
tions, ones may try to compute the degree based on
an interpretation of concepts for semantic similar-
ity. These measures often employ the canonical in-
terpretation and the set cardinality such as work in
(D’Amato et al., 2009; D’Amato et al., 2008). Unfor-
tunately, these measures strictly require an ABox.
Another approach for semantic similarity was pro-
posed in (Alsubait et al., 2014). This work introduced
a family of similarity measures in which a classical
subsumption reasoner was used to determine features
for calculating the degree based on the feature model.
While many similarity measures exist, a few of
them utilize the agent’s preferences for calculating the
degree of concept similarity. In addition, existing ap-
proaches may implicitly use the agent’s preferences
in their computational procedures; thus, it is not easy
to investigate intended behaviors of similarity mea-
sures if they are used under the agent’s preferences.
An experiment in (Bernstein et al., 2005) also sug-
gests that measures should be made personalized to
the target application (e.g. the agent). To help such in-
vestigation, a general notion called concept similarity
measure under preference profile was introduced in
(Racharak et al., 2016b) with the developed measure
sim
π
for the DL ELH . This work was continuously
studied in (Racharak et al., 2017a).
7 DISCUSSION AND FUTURE
RESEARCH
This paper introduces a measure for identifying the
degree of concept similarity under preferences in the
DL FL
0
w.r.t. an unfoldable TBox. This introduced
measure is developed based on the calculation of sub-
sumption degree under preferences w.r.t. two corre-
sponding directions. To achieve this desire, we first
review approaches of identifying the subsumption de-
gree between FL
0
concepts and generalize the ap-
proach based on the recently introduced notion called
preference profile. As a result, the proposed measure
can be regarded as an instance of concept similarity
measure under preference profile (cf. Definition 5.1
and Definition 5.2). We have also investigated sev-
eral properties of the measure, viz. its computational
complexity and its backward compatibility. That is,
when the TBox is unfoldable, computing the degree
of concept similarity under preferences can be done
in polynomial time. Furthermore, employing the de-
fault preference profile as its setting yields the degree
w.r.t. the structure of concept descriptions.
In (Racharak et al., 2016b; Racharak et al.,
2017a), the measure sim
π
was introduced as a con-
crete measure of concept similarity measure under
preference profile for the DL ELH
1
. On the one hand,
sim
π
allowed to fully define preferential expressions
over all types of preference profile. On the other
hand, ones might still want to understand how con-
crete measures of
π
for other DLs should be devel-
oped. This work provides an answer to that question.
In this work, we concentrate on another sub-Boolean
DL, i.e. FL
0
. We recall that FL
0
offers the construc-
tors conjunction (u), value restriction (r.C), and the
top concept (>) (cf. Subsection 2.1). The approach
presented in this paper also differs to (Racharak et al.,
2017a) on the adopted characterization, i.e. the lan-
guage inclusion. This work has potential use in de-
veloping knowledge-based systems in which their on-
tologies can be represented in FL
0
, such as the de-
velopment of recommendation systems based on the
agent’s preferences with FL
0
-based knowledge base.
There are several possible directions for its future
work. Firstly, we may try to conduct an experiment
on an appropriate ontology of real-world domains.
Similar experiments as conducted in (Racharak et al.,
2017a) can be carried out. Secondly, it is an obvious
work to investigate its intended behaviors regarding
the properties introduced in (Racharak et al., 2017a).
Thirdly, we are also interested to explore similarity
measures for the more expressive DLs. Lastly, as re-
ported in (Bernstein et al., 2005) about the need of
having multiple measures, it would be interesting to
investigate the possible classes of similarity measures
w.r.t. their potential use cases and applications.
ACKNOWLEDGEMENTS
The authors would like to thank Prachya Boonkwan
from NECTEC and the anonymous reviewers for
comments. This work is part of the JAIST-NECTEC-
SIIT dual doctoral degree program.
1
We recall that ELH offers the constructors conjunction
(u), full existential quantification (r.C), and the top con-
cept (>); also, the TBox can contain (possibly primitive)
concept definitions and role hierarchy axioms.
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
208
REFERENCES
Alsubait, T., Parsia, B., and Sattler, U. (2014). Measur-
ing conceptual similarity in ontologies: how bad is
a cheap measure? In Informal Proceedings of the
27th International Workshop on Description Logics,
Vienna, Austria, July 17-20, 2014., pages 365–377.
Ashburner, M., Ball, C. A., Blake, J. A., Botstein, D.,
Butler, H., Cherry, J. M., Davis, A. P., Dolinski, K.,
Dwight, S. S., Eppig, J. T., Harris, M. A., Hill, D. P.,
Issel-Tarver, L., Kasarskis, A., Lewis, S., Matese,
J. C., Richardson, J. E., Ringwald, M., Rubin, G. M.,
and Sherlock, G. (2000). Gene Ontology: tool for the
unification of biology. Nature Genetics, 25(1):25–29.
Baader, F., Calvanese, D., McGuinness, D. L., Nardi, D.,
and Patel-Schneider, P. F. (2010). The Description
Logic Handbook: Theory, Implementation and Appli-
cations. Cambridge University Press, New York, NY,
USA, 2nd edition.
Baader, F. and Narendran, P. (2001). Unification of con-
cept terms in description logics. Journal of Symbolic
Computation, 31(3):277 – 305.
Bernstein, A., Kaufmann, E., B
¨
urki, C., and Klein, M.
(2005). How Similar Is It? Towards Personalized
Similarity Measures in Ontologies, pages 1347–1366.
Physica-Verlag HD, Heidelberg.
Caviedes, J. E. and Cimino, J. J. (2004). Towards the devel-
opment of a conceptual distance metric for the UMLS.
Journal of Biomedical Informatics, 37(2):77–85.
D’Amato, C., Fanizzi, N., and Esposito, F. (2009). A se-
mantic similarity measure for expressive description
logics. In CoRR, abs/0911.5043.
D’Amato, C., Staab, S., and Fanizzi, N. (2008). On the in-
fluence of description logics ontologies on conceptual
similarity. In Proceedings of Knowledge Engineering:
Practice and Patterns, pages 48–63.
Euzenat, J. and Valtchev, P. (2004). Similarity-based on-
tology alignment in OWL-lite. In de M
´
antaras, R. L.
and Saitta, L., editors, Proceedings of the 16th Euro-
pean Conference on Artificial Intelligence (ECAI-04),
pages 333–337. IOS Press.
Ge, J. and Qiu, Y. (2008). Concept similarity matching
based on semantic distance. In Proceedings of the
4th International Conference on Semantics, Knowl-
edge and Grid, pages 380–383.
Jaccard, P. (1901).
´
Etude comparative de la distribution flo-
rale dans une portion des alpeset des jura. Bulletin de
la Societe Vaudoise des Sciences Naturellese, 37:547–
579.
Lehmann, K. and Turhan, A.-Y. (2012). A framework
for semantic-based similarity measures for ELH -
concepts. In del Cerro, L. F., Herzig, A., and Mengin,
J., editors, JELIA, volume 7519 of Lecture Notes in
Computer Science, pages 307–319. Springer.
Patwardhan, S. (2006). Using wordnet-based context vec-
tors to estimate the semantic relatedness of concepts.
In Proceedings of the EACL 2006 Workshop Making
Sense of Sense-bringing Computational Linguistics
and Psycholinguistics Together, volume 1501, pages
1–8.
Pedersen, T., Pakhomov, S. V., Patwardhan, S., and Chute,
C. G. (2007). Measures of semantic similarity and
relatedness in the biomedical domain. Journal of
Biomedical Informatics, 40(3):288 – 299.
Racharak, T. and Suntisrivaraporn, B. (2015). Similar-
ity measures for FL
0
concept descriptions from an
automata-theoretic point of view. In Proceedings of
the 6th International Conference of Information and
Communication Technology for Embedded Systems
(IC-ICTES), pages 1–6.
Racharak, T., Suntisrivaraporn, B., and Tojo, S. (2016a).
Identifying an Agent’s Preferences Toward Similar-
ity Measures in Description Logics, pages 201–208.
Springer International Publishing, Cham.
Racharak, T., Suntisrivaraporn, B., and Tojo, S. (2016b).
sim
π
: A concept similarity measure under an agent’s
preferences in description logic ELH . In Proceedings
of the 8th International Conference on Agents and Ar-
tificial Intelligence, pages 480–487.
Racharak, T., Suntisrivaraporn, B., and Tojo, S. (2017a).
Personalizing a concept similarity measure in the de-
scription logic ELH with preference profile. Journal
of Computing and Informatics (to appear).
Racharak, T., Tojo, S., Hung, N. D., and Boonkwan, P.
(2016c). Argument-based logic programming for
analogical reasoning. In New Frontiers in Artifi-
cial Intelligence - JSAI-isAI 2016 Workshops, LENLS,
HAT-MASH, AI-Biz, JURISIN and SKL, Kanagawa,
Japan, November 14-16, 2016, Revised Selected Pa-
pers, pages 253–269.
Racharak, T., Tojo, S., Hung, N. D., and Boonkwan, P.
(2017b). Combining answer set programming with
description logics for analogical reasoning under an
agent’s preferences. In Advances in Artificial Intelli-
gence: From Theory to Practice - 30th International
Conference on Industrial Engineering and Other Ap-
plications of Applied Intelligent Systems, IEA/AIE
2017, Arras, France, June 27-30, 2017, Proceedings,
Part II, pages 306–316.
Rada, R., Mili, H., Bicknell, E., and Blettner, M. (1989).
Development and application of a metric on semantic
nets. IEEE Transactions on Systems, Man, and Cyber-
netics, 19(1):17–30.
Raha, S., Hossain, A., and Ghosh, S. (2008). Similarity
based approximate reasoning: fuzzy control. Journal
of Applied Logic, 6(1):47 – 71.
Resnik, P. (1995). Using information content to evaluate se-
mantic similarity in a taxonomy. In Proceedings of the
14th International Joint Conference on Artificial In-
telligence - Volume 1, IJCAI’95, pages 448–453, San
Francisco, CA, USA. Morgan Kaufmann Publishers
Inc.
Sch
¨
utze, H. (1998). Automatic word sense discrimination.
Computational Linguistics, 24(1):97–123.
Sessa, M. I. (2002). Approximate reasoning by similarity-
based SLD resolution. Theoretical Computer Science,
275(12):389 – 426.
Suntisrivaraporn, B. (2013). A similarity measure for the
description logic el with unfoldable terminologies. In
INCoS, pages 408–413.
Concept Similarity under the Agent’s Preferences for the Description Logic FL
0
with Unfoldable TBox
209
Suntisrivaraporn, B. and Tongphu, S. (2016). A structural
subsumption based similarity measure for the descrip-
tion logic ALEH . In Proceedings of the 8th Inter-
national Conference on Agents and Artificial Intelli-
gence, ICAART 2016, pages 204–212. SCITEPRESS
- Science and Technology Publications, Lda.
Tongphu, S. and Suntisrivaraporn, B. (2015). Algorithms
for measuring similarity between ELH concept de-
scriptions: A case study on snomed ct. Journal of
Computing and Informatics (to appear).
Tversky, A. (1977). Features of similarity. Psychological
Review, 84(4):327–352.
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
210