Concept Similarity under the Agent’s Preferences for the Description

Logic FL

with Unfoldable TBox

Teeradaj Racharak

1,2

and Satoshi Tojo

School of Information, Computer and Communication Technology, Sirindhorn International Institute of Technology,

Thammasat University, Pathum Thani, Thailand

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

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

. 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

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

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

, 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

(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

in Subsection 2.1, particularly its syntax,

semantics, and normal form which can be used for

subsumption testing in FL

. Then, we review the no-

tion of preference proﬁle in Subsection 2.2.

2.1 Description Logic FL

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

)

of all FL

concepts can be inductively deﬁned by the

following grammar:

Con(FL

) ::= A | > | C u D | ∀r.C

where > denotes the top concept, A ∈ CN, r ∈

RN, and C, D ∈ Con(FL

). 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

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.

An interpretation I is a pair I = h∆

, ·

i, where ∆

is a non-empty set representing the domain of the in-

terpretation and ·

is an interpretation function which

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

⊆ ∆

, and

to every role name r a binary relation r

⊆ ∆

× ∆

The interpretation function ·

is inductively extended

to FL

concepts in the usual manner:

= ∆

; (C u D)

= C

∩ D

;

(∀r.C)

= {a ∈ ∆

| ∀b ∈ ∆

: (a, b) ∈ r

→ b ∈ C

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

⊆ C

and A

= C

. The main inference problem

in FL

is the concept subsumption problem.

Deﬁnition 2.1 (Concept Subsumption). Given C, D ∈

Con(FL

) and a TBox T , C is subsumed by D w.r.t.

T (denoted by C v

D) if C

⊆ D

for every model

I of T . Moreover, C and D are equivalent w.r.t. T

(denoted by C ≡

D) if C v

D and D v

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

concepts

can be transformed into an equivalent one of the

form ∀r

. . . ∀r

.A where {r

, . . . , r

} ⊆ RN and A ∈

CN. Such concepts can be abbreviated as ∀r

. . . r

where r

. . . r

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

.A u . . . ∀w

.A can be

abbreviated as ∀L.A where L

= {w

, . . . , w

} 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

) can be rewritten as:

C ≡ ∀U

u ··· u ∀U

(1)

D ≡ ∀V

u ··· u ∀V

(2)

where {A

, . . . , A

} ⊆ CN and U

are ﬁnite sets of

words over the alphabet of role names. This normal

form provides us the following characterization of

subsumption in FL

(Baader and Narendran, 2001):

C v D ⇔ U

⊇ V

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

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:

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

Obvious abbreviations are used for succinctness.

where X,Y, and Z are fresh concept names. Using

Equation 3, it yields DesiredRoom 6v

RoomA and

DesiredRoom 6v

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

, i

, s

, di

where i

, i

, s

, and d are partial functions such

that:

• i

: CN

pri

(T ) → [0, 2] is called a primitive concept

importance;

• i

: RN(T ) → [0, 2] is called a role importance;

• s

: CN

pri

(T )×CN

pri

(T ) → [0, 1] is called a prim-

itive concepts similarity;

• s

: 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

(A) = 1

captures an expression of normal importance on A,

(A) > 1 and i

(A) < 1 indicate that A has higher

and lower importance, respectively, and i

(A) = 0 in-

dicates that A has no importance to the agent. Sec-

ondly, we deﬁne the interpretation of i

in the simi-

lar fashion as i

for any r ∈ RN(T ). Thirdly, for any

A, B ∈ CN

pri

(T ), s

(A, B) = 1 captures an expression

of total similarity between A and B and s

(A, B) = 0

captures an expression of total dissimilarity between

A and B. Fourthly, the interpretation of s

is deﬁned

in the similar fashion as s

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

) = 0.3, then

the degree of similarity under this preference between

Concept Similarity under the Agent’s Preferences for the Description Logic FL

with Unfoldable TBox

203

∀r

.A and ∀r

.B can be understood as 0.3 degree as

the identical occurrence of r

has 0.3 importance.

It is worth noticing that role names appearing in

are always primitive. This suggests that both

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,

) and the credulous sub-

sumption degree (in symbols,

). The names skep-

tical and credulous were motivated by the fact that

the degree obtained from

is always less than or

equal to the one obtained from

. Basically, the

function

checks set inclusions between sets of

words whereas the

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

subsumption degree).

Let C, D ∈ Con(FL

) 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

degree from C to

D (denoted by C

D) is deﬁned as follows:

D =

|{P ∈ CN

pri

| W(D, P) ⊆ W(C, P)}|

|CN

pri

(4)

where | · | denotes the set cardinality.

Deﬁnition 3.2 (credulous FL

subsumption degree).

Let C, D ∈ Con(FL

) 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

subsumption

degree from C to D (denoted by C

D) is deﬁned as

follows:

D =

∑

P∈CN

pri

µ(D,C, P)

|CN

pri

, (5)

where |· | denotes the set cardinality and µ(D,C, P) =

(

1 if W(D, P) =

|W(D,P)∩W(C,P)|

|W(D,P)|

otherwise

(6)

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

D ≤ C

D for any C, D ∈ Con(FL

Proposition 3.1. For any C, D ∈ Con(FL

), it follows

that C

D ≤ C

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

D ≤ C

D. Fix any P ∈

pri

. Also, let W(D, P) = {r

, . . . , r

, s

, . . . , s

} and

W(C, P) = {r

, . . . , r

, . . . ,t

}. Then, it is obvious

that C

D ≤ 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

RA =

|{X, Z, R, T, M}|

|{X,Y, Z, R, T, B, M}|

and DR

1 +

|{ε}∩

+ 1 +

|{ε}∩{ε}|

+ 1 +

|{ f }∩

+ 1

Similarly, it yields DR

RB = DR

RB =

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

and

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

. 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

to expose preferential elements of preference

proﬁle. As a result, the new function

s is also

driven by the structural subsumption characterization

by means of language inclusion in FL

. 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

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

s : CN

pri

×CN

pri

→

[0, 1] as follows:

s(x, y) =











1 if x = y

(x, y) if (x, y) ∈ CN

pri

× CN

pri

and s

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

. We

rewrite the numerator of

to:

∑

P∈CN

pri

max

Q∈CN

pri

{

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

i : CN

pri

→ [0, 2] as follows:

i(x) =

(

(x) if x ∈ CN

pri

and i

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:

∑

P∈CN

pri

i(P)· max

Q∈CN

pri

{

s(P, Q) | W(D, P) ⊆ W(C, Q)}

(10)

∑

P∈CN

pri

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

s (cf. Deﬁnition 4.1) as it presents a generaliza-

tion of

w.r.t. preference proﬁle.

Deﬁnition 4.1 (skeptical FL

subsumption degree un-

der π). Let C, D ∈ Con(F L

) 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

subsump-

tion degree under π from C to D (denoted by C

s D)

is deﬁned as follows:

s D =

∑

P∈CN

pri

i(P) · max

Q∈CN

pri

{

s(P, Q)|W(D, P) ⊆ W(C, Q)}

∑

P∈CN

pri

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 A’s preferences as:

(Bamboo, Tatami) = 0.8. Following Deﬁnition

4.1, it yields that

s RA =

1 + 0 + 1 + 1 + 1 + 0.8 + 1

|X,Y, Z, R, T, B, M|

5.8

and

s DR =

0 + 1 + 1 + 1 + 0.8 +1 + 1

5.8

Similarly, it yields DR

s RB = RB

s DR =

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

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

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

) and primitive roles similar-

ity (s

) (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

Under a special setting of preference proﬁle, the func-

tion

s can be reduced backward to

. This means

that

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

). We give its formal

deﬁnition as follows:

Deﬁnition 4.2 (Default Preference Proﬁle). Let

pri

(T ) be a set of primitive concept names occur-

ring in T . The default preference proﬁle, in symbol

, is the pair hi

, s

i where

(A) = 1 for all A ∈ CN

pri

(T ) and

(A, B) = 0 for all (A, B) ∈ CN

pri

(T ) × CN

pri

(T ).

As for its syntactic sugar, let us denote a setting on

s by replacing the setting with π. For instance, we

may write the setting with π

s. Next, we show

that, under this special setting on

s, the computation

produces the same outcome as

Proposition 4.1. For any C, D ∈ Con(FL

), C

s D

= C

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

w.r.t.

the identical concepts (i.e. s

(A, A) = 1 for any A ∈

pri

(T )). This is obvious. o

5 CONCEPT SIMILARITY

UNDER PREFERENCE

PROFILE IN FL

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

∼

: Con(L) × Con(L) → [0, 1].

When a TBox T is clear from the context, we sim-

ply write

∼. Furthermore, to avoid confusion on the

symbols,

∼

is used when referring to arbitrary mea-

sures of the notion.

To develop a concrete measure as an instance of

∼, 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

∼s as it is motivated from the fact that

s is

used for computing subsumption degree under prefer-

ences w.r.t. the two directions.

Deﬁnition 5.2. Let C, D ∈ Con(FL

) be in their nor-

mal forms and π be a preference proﬁle as a prefer-

ence setting from the agent. Then, the skeptical FL

similarity measure under preference proﬁle π between

C and D (denoted by C

∼s D) is deﬁned as follows:

∼s D =

s D + D

s C

(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,

∼

) on both directions or the root mean square (in

symbols,

∼

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

∼

> = 0 × 1 = 0

and A

∼

rms

> =

= 0.707, whereas A

∼s > =

0+1

= 0.5. Since C

∼

D ≤ C

∼s D ≤ C

∼

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.

Based on this form,

∼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

∼s RA =

5.8

Similarly, it yields that DR

∼ s RB =

. Since

∼s RA > DR

∼s RB, it corresponds to the agent

A’s perception that he may decide to stay in RoomA

when his DesiredRoom is not available.

It is also worth noting that

∼s comes up with a

very nice property as follows:

Theorem 5.1. The measure

∼s can be computed in

polynomial time.

Proof. Since

s can be computed in polynomial

time (by Theorem 4.1), this is trivial. o

Finally, it is worth stating that the measure

∼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

. This property is immediately

followed from Proposition 4.1.

Remark 5.1. Computing

∼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

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

(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

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

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

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

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

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

was introduced as a con-

crete measure of concept similarity measure under

preference proﬁle for the DL ELH

. On the one hand,

sim

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

∼ 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

. We recall that FL

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

, such as the de-

velopment of recommendation systems based on the

agent’s preferences with FL

-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.

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

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