sim

: A Concept Similarity Measure under an Agent’s Preferences in

Description Logic ELH

Teeradaj Racharak

1,2

, Boontawee Suntisrivaraporn

and Satoshi Tojo

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

Thammasat University, Pathumthani, Thailand

Graduate School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, Japan

Keywords:

Concept Similarity Measures, Non-standard Reasoning Services, Preference Proﬁle, Description Logics.

Abstract:

In Description Logics (DLs), concept similarity measures (CSMs) aim at identifying a degree of commonality

between two given concepts and are often regarded as a generalization of the classical reasoning problem of

equivalence. That is, any two concepts are equivalent if their similarity degree is one, and vice versa. When

two concepts are not equivalent, the level of similarity varies depending not only on the objective factors

(i.e. the concept descriptions) but also on the subjective factors (i.e. the agent’s preferences). This work

presents the notion of a general preference proﬁle to be used in existing similarity measures and exempliﬁes

its applicability with the similarity measure for the DL ELH , called sim

. We show that our measure is

expressible for all aspects of preference proﬁle and prove that sim

is preference-invariant w.r.t. equivalence,

i.e. similarity between two equivalent concepts is always one regardless of agents’ preferences.

1 INTRODUCTION

Agents’ preferences are used in a variety of related,

but not identical, ways in their daily life: to ex-

press what they like and dislike, to express their de-

sired goals when choosing routes for travelling (Son

et al., 2003), etc. In psychology, preferences may

be conceived of as an agent’s attitude towards a set

of objects when making decisions (Lichtenstein and

Slovic, 2006). Alternatively, preferences can be inter-

preted as a judgment in a sense of liking or disliking

an object (Scherer, 2005).

In Description Logics (DLs), concept similarity

measures (CSMs) aim at identifying a degree of com-

monality between two given concept names and are

often regarded as a generalization of the classical rea-

soning problem of equivalence. That is, any two con-

cepts are equivalent if their similarity degree is one,

and vice versa. To date, many semantic CSMs have

been developed (cf. Section 4). These developments

can induce efﬁcient similarity-oriented DL reasoning

services, i.e., to measure if two concepts are similar,

to check if a given instance is a relaxed instance of

a concept, and to retrieve those instances similar to

a given instance. However, relatively limited efforts

have been placed on addressing real-world similarity

services executed by a user agent, i.e., ﬁnding similar-

ity w.r.t. the needs and preferences of an agent. These

issues can be illustrated with the following example:

Example 1.1. Suppose that Bob, a Ph.D student,

wants to visit a place for active activities, and he feels

like a place where he can enjoy walking. According

to his world, a terminology might have been modeled

in DL as follows:

ActivePlace v Place u ∃canWalk.Trekking

u∃canSail.Kayaking

Mangrove v Place u ∃canWalk.Trekking

Beach v Place u ∃canSail.Kayaking

canWalk v canDo

canSail v canDo

Considering merely the objective aspects of the

world, it is reasonable to conclude that both

Mangrove and Beach are equally similar to the notion

of ActivePlace. Taking into account also Bob’s pref-

erences, however, Mangrove appears more suitable to

his perception of ActivePlace.

The example shows that preferences of an agent

play a decisive role in the choice of alternatives. Thus,

we need to be able to ﬁne-tune the degree of simi-

larity by employing aspects apart from the objective

factors (i.e. the concept descriptions themselves). It

is worth observing that, with a few exceptions like

480

Racharak, T., Suntisrivaraporn, B. and Tojo, S.

sim

: A Concept Similarity Measure under an Agent’s Preferences in Description Logic ELH .

DOI: 10.5220/0005813404800487

In Proceedings of the 8th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2016) - Volume 2, pages 480-487

ISBN: 978-989-758-172-4

sim and simi (cf. Section 4), most CSMs do not al-

low user agents to specify their preferences and use

them to identify a degree of similarity between two

concepts. The responsibility of ﬁnding similar con-

cepts w.r.t. the needs and preferences of an agent rests

solely on that agent.

In this work, we exemplify the applicability of the

so-called preference proﬁle (Racharak et al., 2015),

which is a design guideline for the development of

concept similarity measures under an agent’s pref-

erences, to the similarity measure sim, in symbols

sim

. We also exhibit that sim

is expressible for

all aspects of preference proﬁle and prove that sim

is preference-invariant w.r.t. equivalence, i.e. similar-

ity between two equivalent concepts is always one re-

gardless of agents’ preferences (cf. Section 3).

2 PRELIMINARIES

In Description Logics (DLs), concept descriptions

are inductively deﬁned by the help of a set of con-

structors, a set of concept names CN, and a set of role

names RN. The set of concept descriptions, or simply

concepts, for a speciﬁc DL L is denoted by Con(L).

The set Con(ELH ) of all E LH concepts can be in-

ductively deﬁned by the following grammar,

C, D −→ A | > | C u D | ∃r.C

where > denotes the top concept, C, D ∈ Con(ELH ),

A ∈ CN and r ∈ RN. Conventionally, concept names

are denoted by A and B, concept descriptions are de-

noted by C and D, and role names are denoted by r

and s.

A terminology or TBox O is a ﬁnite set of (possi-

bly primitive) concept deﬁnitions and role hierarchy

axioms, whose syntax is an expression of the form

(A v D) A ≡ D, and r v s, respectively. A TBox is

called unfoldable if it contains at most one concept

deﬁnition for each concept name in CN and does not

contain cyclic dependencies. Concept names occur-

ring on the left-hand side of a concept deﬁnition are

called deﬁned concept names (denoted by CN

def

), all

other concept names are primitive concept names (de-

noted by CN

pri

). A primitive deﬁnition A v D can

easily be transformed into a semantically equivalent

full deﬁnitions A ≡ X u D where X is a fresh con-

cept name. When a TBox O is unfoldable, concept

names can be expanded by exhaustively replacing all

deﬁned concept names by their deﬁnitions until only

primitive concept names remain. Such concept names

are called fully expanded concept names. In what fol-

lows, we assume that concepts are fully expanded,

and as such the TBox can be omitted. Like primi-

tive deﬁnitions, a role hierarchy axiom r v s can be

transformed in to a semantically equivalent role def-

inition r ≡ t u s where t is a fresh role name. Role

names occurring on the left-hand side of a role deﬁni-

tion are called deﬁned role names, denoted by RN

def

All others are primitive role names, collectively de-

noted by RN

pri

. We also denote a set of all r’s super

roles by R

= {s ∈ RN|r = s or r

v r

i+1

∈ O where

1 ≤ i ≤ n, r

= r, r

= s}.

In order to deﬁne a formal semantics for a spe-

ciﬁc DL L, we consider an interpretation I = h∆

, ·

which consists of a nonempty set ∆

as the domain

of the interpretation and an interpretation function ·

which assigns to every concept name A a set A

⊆ ∆

and to every role name r a binary relation r

⊆ ∆

∆

. The interpretation function ·

is inductively ex-

tended to ELH 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 O

(in symbols, I |= O) if it satisﬁes all axioms in O. I

satisﬁes axioms A v, A ≡ C, and r v s, respectively, if

⊆ C

, A

= C

, and r

⊆ s

. One of the main clas-

sical reasoning problems is the subsumption problem.

That is, given two concept descriptions C and D and a

TBox O, C is subsumed by D w.r.t. a TBox O (written

as C v

D) if C

⊆ D

in every model I of O. Fur-

thermore, C and D are equivalent w.r.t. O (written as

C ≡

D) if C v

D and D v

C. When a TBox O is

empty or is clear from the context, we omit to denote

O, i.e. C v D and C ≡ D.

Concept Similarity Measure (CSM). is one of

non-standard DL reasoning services. It determines

how similar two concepts are. Formally, given two

concept descriptions C, D ∈ Con(L) for a speciﬁc DL

L. Then, a concept similarity measure w.r.t. a TBox

O is a function ∼

: Con(L) × Con(L) → [0, 1] such

that C ∼

D = 1 iff C ≡

D (total similarity) and

C ∼

D = 0 indicates total dissimilarity between C

and D. When a TBox O is clear from the context, we

simply write C ∼ D.

Since we present an extension to sim (Suntisri-

varaporn, 2013; Tongphu and Suntisrivaraporn, 2015)

for taking into account an agent’s preferences, the

original deﬁnitions of homomorphism degree and sim

are included here for self-containment. Let C ∈

Con(ELH ) be a fully expanded concept to the form:

u ·· ·uP

u ∃r

u · · · u ∃r

where P

∈ CN

pri

, r

∈ RN, C

∈ Con(ELH ) in the

same format, 1 ≤ i ≤ m, and 1 ≤ j ≤ n. The set

, . . . , P

and the set ∃r

, . . . , ∃r

are denoted

by P

and E

, respectively. An ELH concept de-

sim

: A Concept Similarity Measure under an Agent’s Preferences in Description Logic ELH

481

scription can be structurally transformed into the cor-

responding ELH description tree. The root v

of the

ELH description tree T

has {P

, . . . , P

} as its label

and has n outgoing edges, each labeled with r

to a

vertex v

for 1 ≤ j ≤ n. Then, a subtree with the root

is deﬁned recursively relative to the concept C

Deﬁnition 2.1 (Homomorphism Degree (Tongphu

and Suntisrivaraporn, 2015)). Let T

ELH

be a set of

all ELH description trees and T

, T

∈ T

ELH

cor-

responds to two ELH concept names C and D, re-

spectively. The homomorphism degree function hd :

ELH

× T

ELH

→ [0, 1] is inductively deﬁned as fol-

lows:

hd(T

, T

) = µ · p-hd(P

, P

)

+ (1 − µ) · e-set-hd(E

, E

), (1)

where | · | represents the set cardinality, µ =

∪E

and 0 ≤ µ ≤ 1;

p-hd(P

, P

) =

(

1 if P

∩P

otherwise,

(2)

e-set-hd(E

, E

) =











1 if E

0 if E

0 and E

∗

, E

) otherwise,

(3)

where

∗

, E

) =

∑

∈E

max{e-hd(ε

, ε

) : ε

∈ E

}

(4)

with ε

, ε

existential restrictions; and

e-hd(∃r.X, ∃s.Y ) = γ(ν + (1 − ν) · hd(T

, T

))

(5)

where γ =

∩R

and 0 ≤ ν < 1.

Deﬁnition 2.2 (E LH Similarity Degree (Tongphu

and Suntisrivaraporn, 2015)). Let C and D be ELH

concept names and T

, T

be the corresponding de-

scription trees. Then, the ELH similarity degree be-

tween C and D (in symbols, sim(C, D)) is deﬁned as

follows:

sim(C, D) =

hd(T

, T

) + hd(T

, T

)

(6)

Example 2.1 (Continuation of Example 1.1). Each

primitive deﬁnition can be transformed to a corre-

sponding equivalent full deﬁnition as shown in the

following.

ActivePlace ≡ X uPlace

u∃canWalk.Trekking

u∃canSail.Kayaking

Mangrove ≡ Y u Placeu

u∃canWalk.Trekking

Beach ≡ Z u Place

u∃canSail.Kayaking

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

Furthermore, R

canWalk

= {t, canDo} and R

canSail

{u, canDo} where t and u are fresh primitive role

names. For brevity, let ActivePlace, Mangrove,

Place, Trekking, Kayaking, canWalk, and canSail be

abbreviated as AP, M, P, T, K, cW, and cS, respec-

tively. Using Deﬁnition 2.1, the homomorphism de-

gree from ActivePlace to Mangrove, or hd(T

, T

)

= (

)(

) + (

)

(

max{e-hd(∃cW.T,∃cW.T)}

max{e-hd(∃cS.K,∃cW.T)}

)

= (

)(

) + (

)(

0.5+0.1

) = 0.55

Similarly, hd(T

, T

) = 0.67, hd(T

, T

) = 0.55,

and hd(T

, T

) = 0.67. Thus, sim(M, AP) = 0.61

and sim(B, AP) = 0.61

2.1 Preference Proﬁle

Preference proﬁle is ﬁrst proposed in (Racharak et al.,

2015) as a guideline for developing CSMs under

preferences. It is a quintuple of preference func-

tions which exhibit ﬁve aspects for preference expres-

sions. It can be adopted into the development of ar-

bitrary CSMs and thereby inﬂuencing the calculation

of CSMs. The syntax and semantics of each aspect

are given in term of partial functions since different

agents can have different perspectives of preferences.

Any CSMs that expose those syntactic forms and sat-

isfy their corresponding semantics will infer a similar-

ity value w.r.t. the needs and preferences of an agent.

Each syntax and semantic is presented formally as

follows:

Deﬁnition 2.3 (Primitive Concept Importance). Let

pri

(O) be a set of primitive concept names oc-

curring in O. Then, a primitive concept importance

is a partial function i

: CN → R

≥0

, where CN ⊆

pri

(O).

For any A ∈ CN

pri

(O), i

(A) = 1 captures an ex-

pression of normal importance for A, i

(A) > 1 (and

(A) < 1) indicates that A has higher (and lower, re-

spectively) importance, and i

(A) = 0 indicates that A

is entirely ignored by an agent. For example, suppose

Bob is keenly interested to visit places. Therefore, he

can express as i

(Place) = 2 for his preference proﬁle.

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

482

Deﬁnition 2.4 (Role Importance). Let RN(O) be a

set of role names occurring in O. Then, a role im-

portance is a partial function i

: RN → R

≥0

, where

RN ⊆ RN(O).

For any r ∈ RN(O), i

(r) = 1 captures an ex-

pression of normal importance for r, i

(r) > 1 (and

(r) < 1) indicates that r has higher (and lower, re-

spectively) importance, and i

(r) = 0 indicates that r

is entirely ignored by an agent. For example, Bob is

interested to visit places where he can enjoy walking.

Therefore, he can also express as i

(canWalk) = 2 for

his preference proﬁle.

Deﬁnition 2.5 (Primitive Concepts Similarity). Let

pri

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

ring in O. For A, B ∈ CN

pri

(O), a primitive concepts

similarity is a partial function s

: CN × CN → [0, 1],

where CN ⊆ CN

pri

(O), such that s

(A, B) = s

(B, A)

and s

(A, A) = 1.

For A, B ∈ CN

pri

(O), s

(A, B) = 1 captures an

expression of total similarity between A and B

and s

(A, B) = 0 captures an expression of to-

tal dissimilarity between A and B. For example,

Bob believes that trekking and kayaking are a bit

similar in some sense. Hence, he can express

(Trekking, Kayaking) = 0.1 for his preference pro-

ﬁle.

Deﬁnition 2.6 (Primitive Roles Similarity). Let

pri

(O) be a set of primitive role names occurring in

O. For r, s ∈ RN

pri

(O), a primitive roles similarity is

a partial function s

: RN × RN → [0, 1], where RN ⊆

pri

(O), such that s

(r, s) = s

(s, r) and s

(r, r) = 1.

For r, s ∈ RN(O), s

(r, s) = 1 captures an expres-

sion of total similarity between r and s and s

(r, s) = 0

captures an expression of total dissimilarity between

r and s. For example, Bob believes that walking is

a bit similar to sailing. Hence, he can also express

(t, u) = 0.1 for his preference proﬁle.

Deﬁnition 2.7 (Role Discount Factor). Let RN(O)

be a set of role names occurring in O. Then, a role

discount factor is a partial function d : RN → [0, 1],

where RN ⊆ RN(O).

For any r ∈ RN(O), d(r) = 1 captures an expres-

sion of total importance on a role (over a correspond-

ing nested concept) and d(r) = 0 captures an expres-

sion of total importance on a nested concept (over a

corresponding role). For example, Bob does not con-

cern much if places permit to either walk or to sail.

He would rather consider on actual activities which he

can perform. Thus, he may express d(canWalk) = 0.3

and d(canSail) = 0.3 for his preference proﬁle.

Deﬁnition 2.8 (Preference Proﬁle). (Racharak et al.,

2015) A preference proﬁle, in symbol π, is a quintuple

, i

, s

, di where i

, i

, s

, and d are as deﬁned

above and the default preference proﬁle, in symbol π

is the quintuple hi

, i

, s

, d

i where

(A) = 1 for all A ∈ CN

pri

(O),

(r) = 1 for all r ∈ RN(O),

(A, B) = 0 for all (A, B) ∈ CN

pri

(O) × CN

pri

(O),

(r, s) = 0 for all (r, s) ∈ RN(O) × RN(O), and

(r) = 0.4 for all r ∈ RN(O).

3 CSM UNDER AGENT’S

PREFERENCES

A numerical value obtained by CSMs indicates the

similarity between two concept descriptions. For

instance, sim(ActivePlace, Mangrove) = 0.61 and

sim(ActivePlace, Beach) = 0.61 indicates that the

similarity between ActivePlace and Mangrove, and

that between ActivePlace and Beach are equivalently

61%. This means both Mangrove and Beach match

equally to the general notion of ActivePlace. Un-

fortunately, this is not true because it does not corre-

spond with his needs and his preferences. Indeed, the

similarity degree between ActivePlace and Mangrove

should be greater than the similarity degree between

ActivePlace and Beach in order to consistent with

Bob’s perspectives (as exhibited by Figure 1).

In this section, we adopt those aspects of prefer-

ence proﬁle into our development of concept simi-

larity measure under an agent’s preferences for DL

ELH . In the following, we have presented formal

deﬁnition of concept similarity measures under pref-

erence proﬁle.

Deﬁnition 3.1. Given a CSM ∼, a preference proﬁle

π, and two concepts C, D ∈ Con(L). Then, a con-

cept similarity measure under preference proﬁle π is

a function

∼ : Con(L) × Con(L) → [0, 1]. A CSM ∼

is called preference invariant w.r.t. equivalence if

C ≡ D iff C

∼ D = 1 for any π

By developing a concept similarity measure un-

der preference proﬁle for ELH , we have generalized

the measure sim. To avoid confusion, we write

∼

when referring to an arbitrary CSM in a generic sense,

whereas speciﬁc function symbols, e.g. sim

or hd

are used when talking about speciﬁc CSMs or func-

tions.

In order to consider those aspects of preference

proﬁle, we have presented a total importance func-

tion as

i : CN

pri

∪ RN → R

≥0

based on a concept im-

sim

: A Concept Similarity Measure under an Agent’s Preferences in Description Logic ELH

483

Figure 1: Similarity value with (and without) respect to Bob’s preferences.

portance and a role importance.

i(x) =











(x) if x ∈ CN

pri

and i

is deﬁned on x

(x) if x ∈ RN and i

is deﬁned on x

1 otherwise

(7)

A total similarity function is also presented as

s : (CN

pri

× CN

pri

) ∪ (RN

pri

× RN

pri

) → [0, 1] using a

primitive concept similarity and a primitive role sim-

ilarity.

s(x, y) =











1 if x = y

(x, y) if (x, y) ∈ CN

pri

× CN

pri

and s

is deﬁned on (x, y)

(x, y) if (x, y) ∈ RN

pri

× RN

pri

and s

is deﬁned on (x, y)

0 otherwise

(8)

Similarly, a total role discount factor function is pre-

sented in the following in term of a function

d : RN →

[0, 1] based on a role discount factor.

d(x) =

(

d(x) if d is deﬁned on x

0.4 otherwise

(9)

Let C and D be ELH concept names and r and s

be role names. Let T

, T

, P

, E

, R

, and

are as deﬁned in Deﬁnition 2.1. Let T

ELH

be a set

of all ELH description trees and π = hi

, i

, s

, di

be a preference proﬁle. The homomorphism degree

under preference proﬁle π can be formally deﬁned as

follows:

Deﬁnition 3.2. The homomorphism degree under

preference proﬁle π is a function hd

: T

ELH

→ [0, 1] deﬁned inductively as follows:

, T

) = µ

· p-hd

, P

)

+ (1 − µ

) · e-set-hd

, E

), (10)

where µ

∑

A∈P

i(A)

∑

A∈P

i(A) +

∑

∃r.X ∈E

i(r)

; (11)

p-hd

, P

) =











1 if

∑

A∈P

i(A) = 0

0 if

∑

A∈P

i(A) 6= 0 and

∑

B∈P

i(B) = 0

π∗

, P

) otherwise,

(12)

where

π∗

, P

) =

∑

A∈P

i(A) · max{

s(A, B) : B ∈ P

}

∑

A∈P

i(A)

;

(13)

e-set-hd

, E

) =











1 if

∑

∃r.X ∈E

i(r) = 0

0 if

∑

∃r.X ∈E

i(r) 6= 0

and

∑

∃s.Y ∈E

i(s) = 0

π∗

, E

) otherwise,

(14)

where

π∗

, E

) =

∑

∃r.X ∈E

i(r) · max{e-hd

(∃r.X, ε

) : ε

∈ E

}

∑

∃r.X ∈E

i(r)

(15)

with ε

existential restriction; and

e-hd

(∃r.X, ∃s.Y ) = γ

(

d(r) + (1 −

d(r)) · hd

, T

))

(16)

where γ











1 if

∑

∈R

i(r

) = 0

∑

∈R

i(r

)·max{

s(r

):s

∈R

}

∑

∈R

i(r

)

, otherwise.

(17)

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

484

It is obvious to see that Deﬁnition 3.2 exposes all

elements of preference proﬁle, viz. i

, i

, s

, and d

since it was generalized alongside the use of the func-

tions

s, and

Intuitively, Equation 10 is deﬁned as the weighted

sum of the degree under π of primitive concepts and

the degree under π of matching edges. Equation 11

indicates the weight of primitive concept names w.r.t.

the importance function. Equation 12 calculates the

proportion of best similarity between primitive con-

cept names. Similarly, Equation 14 calculates the pro-

portion of best similarity between existential informa-

tion from Equation 16 and Equation 17. Equation 16

calculates the degree of similarity between matching

edges. Finally, Equation 17 calculates the proportion

of best similarity between role names.

Let C and D be ELH concept names, T

and T

be the corresponding description trees, and π =

, i

, s

, di be a preference proﬁle. The follow-

ing deﬁnition formally describes the ELH similarity

degree under preference proﬁle π.

Deﬁnition 3.3. The ELH similarity degree under

preference proﬁle π between C and D (denoted by

sim

(C, D)) is deﬁned as follows:

sim

(C, D) =

, T

) + hd

, T

)

(18)

Lemma 3.1. For T

, T

∈ T

ELH

, hd

, T

) =

hd(T

, T

Proof

Recall by Deﬁnition 2.8 that the default preference

proﬁle π

is the quintuple hi

, i

, s

, d

i. Also, sup-

pose a concept name D is of the form:

u · · · u P

u ∃r

u · · · u ∃r

where P

∈ CN

pri

, r

∈ CN

pri

, D

∈ Con(E LH ), 1 ≤

i ≤ m, 1 ≤ j ≤ n, P

u ··· u P

is denoted by P

and ∃r

u ··· u ∃r

is denoted by E

. Let d

be the depth of T

. We prove that, for any d ∈ N,

, T

) = hd(T

, T

) with mathematical induc-

tion.

When d = 0, we know that D = P

u · ·· u

. To show that hd

, T

) = hd(T

, T

), we

need to show that µ

= µ and p-hd

, P

) =

p-hd(P

, P

). Let us derive as follows:

∑

A∈P

i(A)

∑

A∈P

i(A) +

∑

∃r.X∈E

i(r)

∑

i=1

∑

i=1

1 + 0

m + 0

= µ.

Furthermore, we only need to show

∑

A∈P

max{

s(A, B) : B ∈ P

} = |P

∩ P

| in or-

der to show p-hd

, P

) = p-hd(P

, P

). We

know that s

maps name identity to 1 and otherwise

to 0. Thus,

∑

A∈P

max{

s(A, B) : B ∈ P

} = |{x : x ∈

and x ∈ P

}| = |P

∩ P

We must now prove that if hd

, T

) =

hd(T

, T

) holds for d = h − 1 where h > 1

and D = P

u · · · u P

u ∃r

u · · · u ∃r

then hd

, T

) = hd(T

, T

) also holds

for d = h. To do that, we have to show

e-set-hd

, E

) = e-set-hd(E

, E

). This

can be done by showing in the similar manner

that γ

= γ and hd

, T

) = hd(T

, T

) from

e-hd

(∃r.X, ∃s.Y ) = e-hd(∃r.X , ∃s.Y ), where

∃r.X ∈ E

and ∃s.Y ∈ E

. Consequently, it

follows by induction that, for T

, T

∈ T

ELH

, T

) = hd(T

, T

Theorem 3.1. For C, D ∈ Con(ELH ),

sim

(C, D) = sim(C, D).

The above theorem follows from Lemma 3.1, Def-

inition 2.2, and Deﬁnition 3.3.

Lemma 3.2. For T

, T

∈ T

ELH

, hd(T

, T

) = 1 iff

, T

) = 1 for any π.

Proof

(Sketch) Let π = hi

, i

, s

, di be an arbitrary pref-

erence proﬁle and π

= hi

, i

, s

, d

i be the de-

fault preference proﬁle. This lemma can be shown

by mathematical induction on the depth of T

(⇒) hd(T

, T

) = 1 implies that there exists a ho-

momorphism mapping from the root of T

to the root

of T

. Consequently, any setting on π does not inﬂu-

ence the calculation on hd

, T

(⇐) hd

, T

) = 1 for any π. In particular,

this holds for the default preference proﬁle π

. By

Lemma 3.1, it is the case that hd(T

, T

) = 1.

Theorem 3.2. sim

is preference invariant w.r.t.

equivalence.

Proof

(Sketch) Given two concepts C and D and an arbi-

trary preference proﬁle π, we have to show C ≡ D

iff sim

(C, D) = 1.

(⇒) By Proposition 7 in (Tongphu and Suntisri-

varaporn, 2015), we can derive that sim(C, D) = 1.

With the usage of Lemma 3.2, Deﬁnition 2.2, and

Deﬁnition 3.3, we can derive that sim

is preference

invariant w.r.t. equivalence.

(⇐) This can be shown similarly as in the forward

direction.

Example 3.1. (Continuation of Example 1.1) Let

enrich the example by assuming Bob’s preference

proﬁle is expressed as follows: (i) i

(Place) = 2;

(ii) i

(canWalk) = 2; (iii) s

(Trekking, Kayaking) =

0.1; (iv) s

(t, u) = 0.1; (v) d(canWalk) = 0.3 and

d(canSail) = 0.3. Let ActivePlace, Mangrove, Place,

sim

: A Concept Similarity Measure under an Agent’s Preferences in Description Logic ELH

485

Trekking, Kayaking, canWalk, and canSail are rewrit-

ten shortly as AP, M, P, T, K, cW, and cS, respec-

tively. Using Deﬁnition 3.2, hd

, T

)

= (

) · p-hd

, P

) + (

) · e-set-hd

, E

)

= (

) · (

i(X)·max{s(X,Y ),s(X ,P)}+i(P)·max{s(P,Y),s(P,P)})

i(X)+i(P)

)

) · e-set-hd

, E

)

= (

)(

1·max{0,0}+2·max{0,1}

1+2

) + (

)·e-set-hd

, E

)

= (

)(

)

i(cW)·max{e-hd

(∃cW.T,∃cW,T)}+1·max{0.2035}

i(cW)+i(cS)

= (

)(

) + (

)

2·max{(1)(0.3+0.7(1))}+1·max{0.2035}

i(cW)+i(cS)

= (

)(

) + (

)

(2)(1)+(1)(0.2035)

2+1

≈ 0.70

Following the same step, we obtain hd

, T

) =

0.75. Hence, sim

(M, AP) ≈ 0.73 by using Deﬁnition

3.3.

Furthermore, using Deﬁnition 3.2, hd

, T

)

= (

) · p-hd

, P

) + (

) · e-set-hd

, E

)

= (

) · (

i(X)·max{s(X,Z),s(X,P)}+i(P)·max{s(P,Z),s(P,P)})

i(X)+i(P)

)

) · e-set-hd

, E

)

= (

)(

1·max{0,0}+2·max{0,1}

1+2

) + (

) · e-set-hd

, E

)

= (

)(

)

2·max{0.2035}+i(cS)·max{e-hd

(∃cS.K,∃cS,K)}

i(cW)+i(cS)

= (

)(

) + (

)

2·max{0.2035}+1·max{(1)(0.3+(0.7)(1))}

i(cW)+i(cS)

= (

)(

) + (

)

(2)(0.2035)+(1)(1)

2+1

≈ 0.57

Following the same step, we obtain hd

, T

) =

0.75. Hence, sim

(B, AP) ≈ 0.66 by using Deﬁnition

3.3.

These results, i.e. sim

(M, AP) > sim

(B, AP),

corresponds with Bob’s needs and preferences.

4 RELATED WORK

Several CSMs abound, but here we investigate those

CSMs that exhibit preferential elements and contrast

them with aspects of our preference proﬁle. Ex-

cept the following two works, most CSMs do not

consider any of preferential elements. Hence, we

omit discussions thereof and merely refer interested

readers to their references for further details, namely

(Janowicz and Wilkes, 2009; Racharak and Suntisri-

varaporn, 2015; D’Amato et al., 2006; Fanizzi and

D’Amato, ) for structural-based similarity measures

and (D’Amato et al., 2009; D’Amato et al., 2008) for

interpretation-based measures.

In an extended work of sim, a range of number

for discount factor (ν) is used in the similarity ap-

plication of SNOMED CT. For instance, when the

roleGroup is found, the value of ν is set to 0. That

approach can be viewed as a speciﬁc application of d

function of preference proﬁle. In simi, pairs of prim-

itive concept names and pairs of role names are per-

mitted to impose the similarity values via the function

pm. For instance, given two primitive concept names

A and B, we can establish the 50% similarity between

A and B by deﬁning pm(A, B) = 0.5. In the similar

manner, given two role names r and s, we can estab-

lish the 50% similarity between r and s by deﬁning

pm(r, s) = 0.5. The former is identical to s

of pref-

erence proﬁle; however, the latter differs from s

preference proﬁle in a sense that pm does not con-

sider primitive role names which contribute to simi-

larity between two arbitrary role names. Furthermore,

each primitive concept name and each existential re-

striction atoms (i.e., those concepts of the form ∃r.C)

is permitted to be weighted w.r.t. a positive real num-

ber via the function g. However, we believe that the

imposition on existential restriction atoms will be im-

practical to use. After all, there can be inﬁnitely many

existential restriction atoms. Thus, our sim

is de-

veloped according to preference proﬁle by allowing

to deﬁne an importance over each role name instead.

Table 1 shows a summary of existing CSMs which

expose elements of preference proﬁle, together with

our proposed sim

, where 4 denotes totally identical

to the speciﬁed function whereas 3 denotes partially

identical to the speciﬁed function.

Table 1: State-of-the-art CSMs embeded preference proﬁle.

CSM i

sim

4 4 4 4 4

sim 4

simi 3 4

5 CONCLUDING REMARKS

In this work, the applicability of the so-called pref-

erence proﬁle is ﬁrst exempliﬁed by generalizing the

mechanism of the similarity measure sim for the DL

ELH , called sim

. Our sim

can nicely utilize pref-

erences of an agent, which are represented in form of

a preference proﬁle, for inﬂuencing the calculation.

We also prove that sim

is backward compatible in the

sense that sim

under the default preference proﬁle

coincides with sim. This ﬁnding together with Propo-

sition 7 in (Tongphu and Suntisrivaraporn, 2015) are

important to show that sim

is preference-invariant

w.r.t. equivalence, i.e. similarity between two equiva-

lent concepts is always one regardless of agents’ pref-

erences. We also investigate existing CSMs and ﬁnd

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

486

that none of them, to the best of our knowledge, en-

tirely comply with the preference proﬁle.

There are some directions for our future work.

Firstly, it appears to be a natural next step to ver-

ify desirable properties in which any CSMs under a

preference proﬁle must have. Secondly, we are go-

ing to investigate deeply on the possibility of other

reasonable aspects to be included in the preference

proﬁle, especially when considering more expressive

DLs. Thirdly, we intend to carry out an implementa-

tion of sim

and perform experiments on realistic on-

tologies. Finally, it is interesting to explore the possi-

bility to extend preference proﬁle beyond other kinds

of similarity-based reasoning services. i.e., relaxed

instance checking and relaxed instance retrieval.

ACKNOWLEDGEMENTS

This research is partially supported by Thammasart

University Research Fund under the TU Research

Scholar, Contract No. TOR POR 1/13/2558; the Cen-

ter of Excellence in Intelligent Informatics, Speech

and Language Technology, and Service Innova-

tion (CILS), Thammasat University; and the JAIST-

NECTEC-SIIT dual doctoral degree program.

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