Tuning Agent’s Proﬁle for Similarity Measure in Description Logic ELH

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:

In Description Logics (DLs), concept similarity measure aims at identifying a degree of commonality of two

given concepts and is often regarded 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). Realistic ontologies

are generally complex. Methodologies for tuning a measure to conform with the agent’s preferences should

be practical, i.e. it is doable in practice. In this work, we investigate and formalize the task of tuning the

preference functions based on the information deﬁned in a TBox and an ABox. We also show how the proposed

approaches can be reconciled with the measure sim

, i.e. a concept similarity measure under preference proﬁle

for DL ELH . Finally, the paper relates the approach to others and discusses future direction.

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 on mea-

suring similarity of two given concepts, i.e. a problem

of concept similarity measure in DLs.

Basically, a concept similarity measure (CSM) is

a function mapping from a concept pair to a unit in-

terval (i.e. 0 ≤ x ≤ 1 for any real number x). The

higher the value is mapped to, the more likely sim-

ilarity of them may hold. Intuitively, the value 0

can be interpreted as total dissimilarity whereas the

value 1 can be interpreted as total similarity or equiv-

alence. Hence, one may regard concept similarity

measure as a generalization of the classical reason-

ing problem of equivalence. Its idealistic objective

is to imitate similarity identiﬁcation performed by a

human expert. It plays a major role in the discovery

of similar concepts in an ontology. For example, it is

employed in bio-medical ontology-based applications

to discover functional similarities of gene (Ashburner

et al., 2000), it is often used by ontology alignment

algorithms (Euzenat and Valtchev, 2004). There is

currently a signiﬁcant number of measures in DLs.

Prominent examples are (Janowicz and Wilkes, 2009;

Racharak and Suntisrivaraporn, 2015; D’Amato et al.,

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

2009; D’Amato et al., 2008; Racharak et al., 2016b).

However, these measures are devised based on objec-

tive factors (a notable exception is (Racharak et al.,

2016b) where a concept similarity measure under an

agent’s preferences is discussed.). For example, they

use the structure (or interpretations) of concept de-

scriptions in question to measure. When these mea-

Racharak T. and Tojo S.

Tuning Agent’s Proﬁle for Similarity Measure in Description Logic ELH.

DOI: 10.5220/0006249602870298

In Proceedings of the 9th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2017), pages 287-298

ISBN: 978-989-758-220-2

287

sures are employed to characterize similar concepts

in an ontology, they may lead to unintuitive results.

It may even be contradictory to an intuitive under-

standing of an application domain. The following ex-

ample illustrates that using objective-based measures

may not sufﬁce to answer an agent’s request.

Example 1.1 An agent A wants to visit a place for do-

ing some active activities. At that moment, he would

like to enjoy walking. Suppose that a place ontology

has been modeled 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 canMoveWithLegs

canSail v canTravelWithSails

Suppose that a measure used by that Agent A

considers merely the objective aspects, it is reason-

able to conclude that both Mangrove and Beach are

equally similar to the concept ActivePlace. However,

by taking into account also the agent’s preferences,

Mangrove appears more suitable to his perception of

ActivePlace at that moment. In other words, he will

not be happy if an intelligent system happens to rec-

ommend him to go for a Beach.

The example shows that preferences of an agent

play a decisive role in the choice of alternatives. In

essence, when the choices of an answer are not to-

tally similar to a concept in question, a measure may

need to be tuned by subjective factors, e.g. an agent’s

preferences. A set of preferential aspects is identi-

ﬁed in (Racharak et al., 2016a) called preference pro-

ﬁle and is extended toward (Racharak et al., 2016b)

called sim

. However, realistic ontologies are com-

plex – consisting in plenty of concept names and role

names. Tuning a measure to conform with the agent’s

preferences should be practical, i.e. it is doable in

practice. Hence, our primary motivation of this work

is a deeper understanding of how preference proﬁle is

conﬁgured. In this work, we investigate and formal-

ize the task of conﬁguring the preference functions

based on the information deﬁned in a TBox and an

ABox (cf. Section 3). This work is an extension of

(Racharak et al., 2016b): with respect to its prede-

cessor, it shows how the proposed approaches can be

reconciled with the measure sim

, i.e. a concept simi-

larity measure under preference proﬁle for DL E LH

(cf. Section 4). Preliminaries is brieﬂy reviewed in

Section 2. Finally, the paper relates the approach to

others (cf. Section 5 and Section 6) and discusses fu-

ture direction (cf. Section 7).

2 PRELIMINARIES

In this section, we review the basics of Description

Logic ELH (cf. Subsection 2.1), which provides the

logical underpinning for OWL 2 EL (Group, 2012;

Grau et al., 2008) and our developed measure sim

(originally introduced in (Racharak et al., 2016b)) is

based. After that, we brieﬂy explain the notion of

preference proﬁle in Subsection 2.2.

2.1 Description Logic ELH

We assume countably inﬁnite sets CN of concept

names, RN of role names, and IN of individual names

that are ﬁxed and disjoint. The set of concept descrip-

tions, or simply concepts, for a speciﬁc DL L is de-

noted by Con(L). The set Con(ELH ) of all ELH

concepts can be inductively deﬁned by the following

grammar:

Con(ELH ) ::= A | > | C u D | ∃r.C

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

and C, D ∈ Con(E LH ). 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 (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 T 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. Like prim-

itive 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

∈ T where

1 ≤ i ≤ n, r

= r, r

= s}.

An assertion or ABox A is a ﬁnite set of concept

assertions and role assertions whose syntax is an ex-

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

288

pression of the form C(a) and r(a, b) where a, b ∈ IN,

respectively. An ontology O consists of a TBox T

and an ABox A, i.e. O = hT , Ai. However, some

existing ontologies may omit an ABox A in practice.

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

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 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 T

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

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

if A

⊆ C

, A

= C

, and r

⊆ s

. Also, an interpre-

tation I is said to be a model of an ABox A (in sym-

bols, I |= A) if it satisﬁes all axioms in A. I satisﬁes

axioms C(a) and r(a, b) if a

∈ C

and (a,b) ∈ r

, re-

spectively. Furthermore, an interpretation I is said to

be a model of an ontology O if it satisﬁes all axioms in

T and A. An interpretation I

is called the canonical

interpretation if:

1. ∆

of I

consists of all individual names in A;

2. ∀A ∈ CN, we deﬁne A

= {x | A(x) ∈ A}; and

3. ∀r ∈ RN, we deﬁne r

= {(x, y) | r(x, y) ∈ A}.

That is, the canonical interpretation I

is the interpre-

tation which takes the set of ABox as the interpreta-

tion domain.

The main inference problem for ELH is the sub-

sumption problem. That is, given C, D ∈ Con(ELH )

and an ontology O, C is subsumed by D w.r.t. O (in

symbols, C v

D) if C

⊆ D

for every model I of

O. Furthermore, C and D are equivalent w.r.t. O (in

symbols, C ≡

D) if C v

D and D v

C. A much

more interesting inference problem, which is based on

concept subsumption, is the concept hierarchy. That

is, let CN(O) be the concept names occurring in O,

the concept hierarchy of O is the most compact repre-

sentation of the partial ordering (CN(O), v

) induced

by the subsumption relation w.r.t O. When an ontol-

ogy O is empty or is clear from the context, we omit

to denote O, i.e. C v D or C ≡ D. Furthermore, a

more generalization of the concept equivalence is a

concept similarity measure under preference proﬁle

(cf. Deﬁnition 2.3), which is originally introduced in

(Racharak et al., 2016b).

2.2 Preference Proﬁle

We ﬁrst introduced preference proﬁle (denoted by π)

in (Racharak et al., 2016a) as a collection of pref-

erential elements in which the development of con-

cept similarity measure should consider. Its ﬁrst intu-

ition is to model different forms of preferences (of an

agent) based on concept names and role names. Mea-

sures adopted this notion are ﬂexible to be tuned by an

agent and can determine the similarity conformable to

that agent’s perception. We give a formal deﬁnition of

each preferential aspect in the following deﬁnition.

Deﬁnition 2.1 (Preference Proﬁle (Racharak et al.,

2016a)). 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, in symbol π, is a quintuple hi

, i

, s

, di

where

• i

: CN → [0, 2] where CN ⊆ CN

pri

(T ) is called

primitive concept importance;

• i

: RN → [0, 2] where RN ⊆ RN(T ) is called role

importance;

• s

: CN × CN → [0, 1] where CN ⊆ CN

pri

(T ) is

called primitive concepts similarity;

• s

: RN × RN → [0, 1] where RN ⊆ RN

pri

(T ) is

called primitive roles similarity; and

• d : RN → [0, 1] where RN ⊆ RN(T ) is called 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 (i

(A) < 1) indicates that A has higher (and

lower, respectively) importance, and i

(A) = 0 indi-

cates 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 ex-

pression of total importance on a nested concept (over

a corresponding role).

Deﬁnition 2.2 (Default Preference Proﬁle

(Racharak et al., 2016a)). The default preference

proﬁle, in symbol π

, is the quintuple hi

, i

, s

, d

In the original deﬁnition of preference proﬁle

(Racharak et al., 2016a; Racharak et al., 2016b), both i

and i

are mapped to R

≥0

, which is a minor error.

Tuning Agent’s Proﬁle for Similarity Measure in Description Logic ELH

289

where

(A) = 1 for all A ∈ CN

pri

(T ),

(r) = 1 for all r ∈ RN(T ),

(A, B) = 0 for all (A, B) ∈ CN

pri

(T ) × CN

pri

(T ),

(r, s) = 0 for all (r, s) ∈ RN

pri

(T ) × RN

pri

(T ),

(r) = 0.4 for all r ∈ RN(T ).

Let us also note that the value of d

determines

how important the existential information should be

considered by a measure in the default manner (see

the interpretation of d). This information is indeed

dependent on an application domain and might be re-

deﬁned. For instance, if d

is deﬁned as 0.3, 0.4, 0.5,

then ∃part.Heart

∼

part.Colon yields 0.3, 0.4, 0.5,

respectively.

In the following, we give the formal

deﬁnition of concept similarity measure under prefer-

ence proﬁle.

Deﬁnition 2.3 ((Racharak et al., 2016b)). 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 func-

tion

∼

: Con(L) × Con(L) → [0, 1]. A function

∼

is called preference invariance w.r.t equivalence

if C ≡ D ⇔ (C

∼

D = 1 for any π).

Taking π = π

for a concept similarity measure

under preference proﬁle, i.e.

∼

, obtains an objective

similarity degree. We prove this in Theorem 3.1 of

(Racharak et al., 2016b).

3 STRATEGIES OF TUNING π

3.1 Tuning i

This subsection exhibits a strategy for tuning primi-

tive concept importance i

in practice. Realistic on-

tologies are generally complex – consisting in plenty

of concept names. Hence, having some strategies of

tuning is useful since it helps to pave the way for a

more convenient use of preference proﬁle.

As a starting point, we seek to observe characteris-

tics of realistic ontologies whose TBox is unfoldable

e.g. a popular medical ontology SNOMED, denoted by

med

, (Stearns et al., 2001). Figure 1 gives an exam-

ple of concept deﬁnitions in O

med

and Figure 2 shows

the concept hierarchy w.r.t. O

med

These numbers is obtained from an application of sim

. Its

precise deﬁnition is given in Section 4.

According to this investigation, we assume an ontology

O has an unfoldable TBox T in this work.

Pericardium v Tissue u ∃part.Heart

Endocardium v Tissue u ∃part.Heart

Appendicitis ≡ Inﬂammation

u∃loc.Appendix

Pericarditis ≡ Inﬂammation

u∃loc.Pericardium

Endocarditis ≡ Inﬂammation

u∃loc.Endocardium

Inﬂammation v Disease

HeartDisease ≡ Disease u ∃loc.∃part.Heart

Figure 1: Example of concept deﬁnitions in O

med

Tissue

Endocardium

Pericardium

Heart

Disease

HeartDisease

Endocarditis

Pericarditis

Inﬂammation

Appendicitis

Figure 2: The concept hierarchy of O

med

According to the above ﬁgures, it is intuitive to ex-

press primitive concept importance through the con-

cept hierarchy. For instance, an agent may say my

concept importance goes through Disease; or my

concept importance goes through HeartDisease. In-

formally investigating, let CN

pri

med

) be a set of

primitive concept names occurring in O

med

. Since

Disease ∈ CN

pri

med

), the former case is simple,

e.g. an agent may mean i

(Disease) = 1.2. The lat-

ter case is a bit complicated since HeartDisease 6∈

pri

med

). However, the agent’s intention may

mean i

(Disease) = 1.2 and i

(Heart) = 1.2. This

informal investigation shapes the development as fol-

lows:

Deﬁnition 3.1. Let CN(T ) (CN

pri

(T ) and CN

def

(T ))

be a set of concept names (primitive concept names

and deﬁned concept names, respectively) occurring

in T . Then, a propagation for primitive concept im-

portance is a partial function I

: CN

→ [0, 2], where

⊆ CN(T ) ∪ {>}, such that a mapping n of I

X (i.e. I

(X) = n) is deﬁned inductively as follows:

1. X ∈ CN

pri

(T ) ⇒ i

(X) = n;

2. X

= > ⇒ ∀x ∈ CN

pri

(T ) : i

(x) = n; and

3. X ∈ CN

def

(T ) ⇒ ∀x ∈ RHS(X) : I

(x) = n.

where RHS(X ) is a set of concept names appearing

on the right-hand side of X.

Its interpretation is deﬁned in a usual way. That is,

for any A ∈ CN(T ), I

(A) = 1 captures an expression

of normal importance on A, I

(A) > 1 (and I

(A) <

Later, we discuss some restrictions the readers should

take into account when the notion I

is employed.

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

290

1) indicates that A has higher (and lower, respectively)

importance, I

(A) = 0 indicates that A has no impor-

tance to the agent. For the special case, I

(>) = n

indicates that every primitive concept name occurring

on T is of equal importance at n.

Example 3.1. From Figure 1, suppose that an agent

A is using a concept similarity measure under pref-

erence proﬁle for querying some names that expose

the similar characteristics to HeartDisease. Thus, the

agent can express a preference I

(HeartDisease) =

1.2 instead of individually specifying i

(Disease) =

1.2 and i

(Heart) = 1.2.

There are a few concerns that we should take into

account, i.e. (1) inconsistent preferences of concepts

occurring on the same branch of the concept hierar-

chy; (2) inconsistent preferences of deﬁned concepts

occurring on different branches of the concept hierar-

chy; and (3) expressing preferences when a TBox T

contains equivalent deﬁned concepts.

Let us take a look on our ﬁrst concern through the

following preference expression: my concept impor-

tance goes through Disease, especially HeartDisease.

In this example, we may take I

(Disease) = 1.2 and

(HeartDisease) = 1.3. Here, Disease is redeﬁned

on i

twice. This kind of scenarios is possible to

happen because we are extending the aspect of primi-

tive concept importance toward both types of concept

names. There are many ways to handle this with the

use of operators ⊕ : [0, 2]

→ [0, 2].

Deﬁnition 3.2. Let A ∈ CN

pri

(T ) be a set of primi-

tive concept names occurring in T and x

, x

∈ [0, 2].

Also, let i

(A) = x

be the previous mapping on A.

We compute a new mapping i

(A) = x

as follows:

(A) =

(

if i

is not deﬁned on A

⊕ x

otherwise

(1)

The notion of the operator remains abstract here as

its concrete operators may vary on the context of use.

In the following, we establish some of the abstract

notion ⊕, i.e. ⊕

max

, ⊕

ﬁrst

, and ⊕

last

. Let two real

numbers x

, x

∈ [0, 2]. Then,

⊕

max

= max{x

, x

} (2)

⊕

ﬁrst

= x

(3)

⊕

last

= x

(4)

Example 3.2. From Figure 1, an agent might say

My interest is Disease except HeartDisease. That is,

we may take I

(Disease) = 1.2 (i.e. i

(Disease) =

1.2) and I

(HeartDisease) = 0 (i.e. i

(Disease) =

1.2 ⊕ 0 and i

(Heart) = 0). Taking ⊕ as ⊕

max

yields

(Disease) = 1.2 and i

(Heart) = 0. It also yields the

same results by taking ⊕ as ⊕

ﬁrst

Example 3.3. From Figure 1, an agent might say

My concern is nothing except HeartDisease. That is,

we may take I

(>) = 0 (i.e. i

(Disease) = 0,

(Tissue) = 0, and i

(Heart) = 0) and

(HeartDisease) = 1 (i.e. i

(Disease) = 0 ⊕ 1

and i

(Heart) = 0 ⊕ 1). Taking ⊕ as ⊕

max

yields

(Disease) = 1, i

(Tissue) = 0, and i

(Heart) = 1. It

also yields the same results by taking ⊕ as ⊕

last

Now, we discuss our second concern on the ap-

plication of I

. Let us consider the following pref-

erence expression of an agent: I

(Pericarditis) = 1.2

and I

(Endocarditis) = 1.6. One may notice that both

use the primitive concept name Disease in common.

Now, a natural question to ask is how a concept im-

portance value should be propagated since a propaga-

tion may cause an inconsistency of preference values

for a primitive concept name, such as Disease in this

example. This requires further work to study. How-

ever, one simple way for handling this problem is to

prevent a mapping leading to this situation. Instead,

an agent has to tune primitive concept names via the

primitive concept importance i

individually.

Lastly, we contemplate our third concern. That

is, what happens if a deﬁned concept name C

deﬁned on I

and there exists another deﬁned con-

cept name C

such that C

≡

? A natural way

for handling this problem is to treat C

in the same

way as C

(because they are equivalent). In particular,

∀C

∈ CN

def

(T ) : C

≡

⇒ I

) = I

Nevertheless, this also requires further work to ex-

plore other possibilities for coping with this prob-

lem and investigate desired properties the notion

∼

should hold when it is used with I

3.2 Tuning i

Let us remind that i

is a function which maps a role

name r ∈ RN to a value x ∈ [0, 2]. Its primary motiva-

tion is to deﬁne a user-identiﬁed importance value for

an individual role name. A distinguished characteris-

tic of i

to i

is that, not only restricted to primitive

ones, ones may also deﬁne an importance on deﬁned

role names. We bear this understanding on the devel-

opment of a strategy to tune i

as follows.

Our primary motivation of providing a strategy to

help tuning i

is similar to that one of i

. That is,

realistic ontologies are complex – consisting in plenty

Tuning Agent’s Proﬁle for Similarity Measure in Description Logic ELH

291

of role names. An intuitive way to simplify the task of

tuning i

is to proceed on a more general role name.

We note that, suppose r v s ∈ T , a role s is said to be

more general than a role r. This intuition shapes our

development as follows:

Deﬁnition 3.3. Let RN(T ) be a set of role names

occurring in T . Then, a propagation for role impor-

tance is a partial function I

: RN

→ [0, 2], where

⊆ RN(T ), such that a mapping n of I

on X (i.e.

(X) = n) is inductively deﬁned as follows:

1. ∀X

∈ RN(T ): (X

v X ∈ T ⇒ i

(X) = n and

) = n); and

2. ∀X

∈ RN(T ): (X

v X 6∈ T ⇒ i

(X) = n).

There are a few concerns that we should take into

account, i.e. (1) inconsistent preferences of roles oc-

curring on the same branch of the role hierarchy; and

(2) expressing preferences when a TBox T contains

equivalent role names. We discuss these in order.

Let us take a look on our ﬁrst concern through

the following preference expression (according to

an ontology given in Example 1.1): my role im-

portance goes through canMoveWithLegs, especially

canWalk. Suppose we take I

(canMoveWithLegs) =

1.2 and I

(canWalk) = 1.3. Here, canWalk is rede-

ﬁned on i

twice. Similar to I

, we handle this prob-

lem with the use of operators ⊕ : [0, 2]

→ [0, 2].

Deﬁnition 3.4. Let r ∈ RN(T ) be a set of role names

occurring in T and x

, x

∈ [0, 2]. Also, let i

(r) = x

be the previous mapping on r. We compute a new

mapping i

(r) = x

as follows:

(r) =

(

if i

is not deﬁned on r

⊕ x

otherwise

(5)

The operator remains abstract here as its concrete

operators may vary on the context of use and may be

deﬁned in the same sense as I

(e.g. Equation 2 to

4). For example, an agent may prefer to take the last

mapping when that agent says exceptional cases (e.g.

r except s where r ∈ R

) in order to suppress the pre-

viously propagated value. Also, an agent may prefer

to take the last mapping when that agent would like

to emphasize some special circumstances (e.g. r es-

pecially s where r ∈ R

) in order to suppress the pre-

viously propagated value.

Example 3.4. From Example 1.1, an agent might say

My interest is canMoveWithLegs except canWalk.

That is, we may take I

(canMoveWithLegs) = 1.2

(i.e. i

(canMoveWithLegs) = 1.2 and i

(canWalk) =

1.2) and I

(canWalk) = 0 (i.e. i

(canWalk) = 1.2 ⊕

Later, we discuss some restrictions the readers should

take into account when the notion I

is employed.

2). Using ⊕

ﬁrst

for ⊕ yields i

(canMoveWithLegs) =

1.2 and i

(canWalk) = 0.

Example 3.5. From Example 1.1, an agent might

say My interest is canMoveWithLegs, especially

canWalk. Let us take I

(canMoveWithLegs) = 1.2

(i.e. i

(canMoveWithLegs) = 1.2 and

(canWalk) = 1.2) and I

(canWalk) = 1.3 (i.e.

(canWalk) = 1.2 ⊕ 1.3). Using ⊕

last

for ⊕ yields

(canMoveWithLegs) = 1.2 and i

(canWalk) = 1.3.

Lastly, we discuss the second concern. That is,

what happens if a deﬁned role name r

is deﬁned on

and there exists another deﬁned role name r

such

that r

and r

? Similar to our basic

handling of this case in I

, we recommend to treat

in the same way as r

(because they are equiva-

lent). Nevertheless, this also requires further work to

explore other possibilities for coping with this prob-

lem and investigate desired properties the notion

∼

should hold when it is used with I

3.3 Tuning s

In this subsection, we present a strategy for tuning

primitive concepts similarity. If an ABox A is pre-

sented, then we can induce the canonical interpreta-

tion I

from A to calculate primitive concepts simi-

larity for all possible primitive concept pairs. Suppose

that I

is constructed and let A, B ∈ CN

pri

(T ), we es-

tablish the following calculation for the function s

(A, B) =

(

1 if A

= B

∩B

∪B

otherwise

(6)

where | · | represents the set cardinality.

Intuitively, Equation 6 computes the commonality

of both primitive concept names. Since O

med

does not

contain an ABox A, let us use a handcraft ontology to

exemplify the calculation.

Example 3.6. Let a family ontology O = hT , Ai in

which T is deﬁned as follows:

Grandfather ≡ Man u ∃child.Parent

Parent ≡ Person u ∃child.Person

Man ≡ Male u Person

Let an ABox A is deﬁned as follows:

child(john, elise) child(emma, watson)

Person(john) Person(elise)

Person(emma) Person(watson)

Male(john) Male(watson)

Thus, ∆

= {elise, john, emma, watson},

Person

= {john, elise, emma, watson}, Male

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

292

= {john, watson}, and s

(Person, Male) =

|john,watson|

|john,elise,emma,watson|

= 0.5.

3.4 Tuning s

This subsection presents a strategy for tuning primi-

tive roles similarity. Indeed, we attempt in the similar

fashion as what we do for s

. That is, we use the

canonical interpretation I

to obtain primitive roles

similarity. Let r, s ∈ RN

pri

(T ) and deﬁne operators ·

and ·

for any primitive role r as r

= {x | (x, y) ∈ r

}

and r

= {y | (x, y) ∈ r

}, respectively, then:

(r, s) =











1 if r

= s

λ ·

∩s

∪s

+(1 − λ) ·

∩s

∪s

otherwise

(7)

where 0 < λ < 1 and | · | represents the set cardinality.

Intuitively, Equation 7 is deﬁned as the weighted

sum of the commonality on the ﬁrst arguments of

roles and the commonality on the second arguments

of roles. It is recommended to set the weight λ =

∪s

|+|r

∪s

, i.e. the proportion of all individuals ap-

pearing on the ﬁrst arguments to all individuals ap-

pearing on both arguments. The following example

exempliﬁes the calculation.

Example 3.7. Let a family ontology O = hT , Ai in

which T is deﬁned as follows:

Parent ≡ Person u ∃child.Person

BrotherSister ≡ Person u ∃sibling.Person

Let an ABox A is deﬁned as follows:

sibling(john, max) sibling(yok, watson)

child(emma, yok) child(john, elise)

child(emma, watson) Person(john)

Person(elise) Person(emma)

Person(watson) Person(max)

Person(yok)

Thus, ∆

= {elise, john, emma, watson,

max, yok}, child

= {emma, john}, sibling

{yok, john}, child

= {yok, watson, elise}, sibling

{watson, max}, and s

(child, sibling) =

≈ 0.48.

3.5 Tuning d

The primary motivation of this aspect is to capture an

expression of total expression on a role beyond a cor-

responding nested concept (Racharak et al., 2016a).

Hence, tuning this aspect may requires skilled do-

main expertise. For example, SNOMED ontology

engineers realize that roleGroup is used to nestedly

group existential restrictions; hence, it can uninten-

tionally increase the degree of similarity due to role

commonality. Considering this fact, they may set

d(roleGroup) = 0. This shows that role discount fac-

tor of different role names may be independent. How-

ever, the same strategy as I

can be employed to com-

fort on conﬁguring this aspect, i.e. a propagation for

role discount factor via a more general role name.

4 APPLYING PROPOSED

STRATEGIES TO MEASURE

ELH CONCEPTS

In this section, we show an applicability of the pro-

posed strategies for tuning preference proﬁle π to be

used with the measure sim

(Racharak et al., 2016b).

We note that sim

is an instance of

∼

(Deﬁnition

2.3) for DL ELH . Using sim

requires that concept

deﬁnitions in a TBox T must be fully expanded, i.e.

for each deﬁned concept name A ∈ CN

def

(T ), such

that A ≡ D, we simply replace A with D wherever it

occurs in C and continue to recursively expand D. If

A is of the form A v D, then we replace A with X uD

such that X is a fresh concept wherever A occurs in C

and recursively expand D. We note that X represents

the primitiveness of A, i.e. the unspeciﬁed character-

istics that differentiate it from D.

For the purpose of self-containment, we include

the original deﬁnition of homomorphism degree un-

der preference proﬁle hd

and the similarity degree

under preference proﬁle sim

here.

In order to consider all aspects of preference pro-

ﬁle, we have presented a total importance function as

i : CN

pri

∪ RN → [0, 2]

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

(8)

A total similarity function is also presented as

s : (CN

pri

× CN

pri

) ∪ (RN

pri

× RN

pri

) → [0, 1] using

primitive concepts similarity and primitive roles sim-

ilarity.

Tuning Agent’s Proﬁle for Similarity Measure in Description Logic ELH

293

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

(9)

Similarly, a total role discount factor function is pre-

sented in the following in term of a function

d : RN →

[0, 1] based on role discount factor.

d(x) =

(

d(x) if d is deﬁned on x

0.4 otherwise

(10)

Let us note that the default value of Equation 8 - 10

is set according to the default preference proﬁle π

(Deﬁnition 2.2).

Let C ∈ Con(E LH ) be a fully expanded concept

to the form:

u · · · u P

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 E LH concept de-

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 la-

bel and has n outgoing edges, each labeled with r

a vertex v

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

root v

is deﬁned recursively relative to the concept

. Let π = hi

, i

, s

, di be a preference proﬁle.

The homomorphism degree under preference proﬁle

π can be formally deﬁned as follows:

Deﬁnition 4.1 ((Racharak et al., 2016b)) . Let

ELH

be a set of all ELH description trees and

, T

∈ T

ELH

corresponds to two ELH concept

names C and D, respectively. The homomorphism de-

gree under preference proﬁle π is a function hd

ELH

× T

ELH

→ [0, 1] deﬁned inductively as fol-

lows:

, T

) = µ

· p-hd

, P

)

+ (1 − µ

) · e-set-hd

, E

), (11)

where











1 if

∑

A∈P

i(A)

and

∑

∃r.X ∈E

i(r) = 0

∑

A∈P

i(A)

∑

A∈P

i(A)+

∑

∃r.X∈E

i(r)

otherwise;

(12)

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,

(13)

where

π∗

, P

) =

∑

A∈P

i(A) · max{

s(A, B) : B ∈ P

}

∑

A∈P

i(A)

;

(14)

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,

(15)

where

π∗

, E

) =

∑

∃r.X ∈E

i(r) · max{e-hd

(∃r.X, ε

) : ε

∈ E

}

∑

∃r.X ∈E

i(r)

(16)

with ε

existential restriction; and

e-hd

(∃r.X, ∃s.Y ) = γ

(

d(r)+(1−

d(r))·hd

, T

))

(17)

where γ











1 if

∑

∈R

i(r

) = 0

∑

∈R

i(r

)·max{

s(r

):s

∈R

}

∑

∈R

i(r

)

, otherwise.

(18)

Intuitively, Equation 11 is deﬁned as the weighted

sum of the degree under π of primitive concepts and

the degree under π of matching edges. Equation 12

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

the importance function. Equation 13 calculates the

proportion of best similarity between primitive con-

cept names. Similarly, Equation 15 calculates the pro-

portion of best similarity between existential informa-

tion from Equation 17 and Equation 18. Equation 17

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

294

calculates the degree of similarity between matching

edges. Finally, Equation 18 calculates the proportion

of best similarity between role names.

Let C and D be fully expanded ELH concept

names, T

and T

be the corresponding description

trees, and π = hi

, i

, s

, di be a preference proﬁle.

The following deﬁnition formally describes the E LH

similarity degree under preference proﬁle π.

Deﬁnition 4.2 ((Racharak et al., 2016b)) . The

ELH similarity degree under preference proﬁle π be-

tween C and D (denoted by sim

(C, D)) is deﬁned as

follows:

sim

(C, D) =

, T

) + hd

, T

)

(19)

Intuitively, the degree of similarity under prefer-

ence proﬁle of two concepts is the average of the de-

gree of having homomorphisms under preference pro-

ﬁle in both directions. We note that sim

(C, D) = 1

indicates that C and D are total similarity under a par-

ticular π and sim

(C, D) = 0 indicates total dissimi-

larity between C and D under a particular π.

4.1 A Simple Methodology

Now, we are ready to discuss a simple methodol-

ogy for reconciling the previously proposed strate-

gies with the measure sim

(Deﬁnition 4.2). Section 3

shows that each aspect of preference proﬁle may need

different strategies for tuning. For example, i

may be

tuned via a deﬁned concept name using a propagation

for primitive concept importance I

whereas i

may

be tuned via a more general role name using a prop-

agation for role importance I

. Also, both s

and s

may employ the canonical interpretation I

to initial-

ize values. In the following, we explain procedural

steps that the readers may follow to tune preference

proﬁle for their use. These steps also hint a system

ﬂow of similarity-based under the agent’s proﬁle ap-

plications, such as the best matching concept under

the agent’s proﬁle application.

1. An agent may start with tuning each aspect of

preference proﬁle individually;

2. To help tuning i

, a system may present the con-

cept hierarchy w.r.t. an ontology. Then, an agent

indirectly specify primitive concept names via a

deﬁned concept name depicted on the hierarchy

with the notion I

(cf. Deﬁnition 3.1 and Deﬁni-

tion 3.2). Some patterns of an agent’s utterance

may be associated with certain operators, e.g.

(a) We may associate A especially B with ⊕

ﬁrst

where depth(A) < depth(B) and depth(X ) is the

depth of X on the concept hierarchy;

(b) We may associate A except B with ⊕

ﬁrst

, where

depth(A) < depth(B) and C

6= >;

last

; and

(d) Otherwise, the agent-deﬁned default concrete

operator is used;

3. To help tuning i

, a system may present the role

hierarchy w.r.t. an ontology. Then, an agent indi-

rectly specify role names via a more general role

name name depicted on the hierarchy with the

noion I

(cf. Deﬁnition 3.3 and Deﬁnition 3.4).

Similarly, some patterns of an agent’s utterance

may be associated with certain operators, e.g.

(a) We may associate r especially s with ⊕

last

where r ∈ R

;

(b) We may associate r except s with ⊕

last

, where

r ∈ R

; and

operator is used;

4. To help tuning s

and s

, a system may con-

struct the canonical interpretation, which is in-

duced from A. Then, each initial value for all

possible primitive concept pairs and primitive role

pairs is calculated according to Equation 6 and

Equation 7, respectively;

5. An agent may reﬁne the agent’s preference proﬁle

if that agent wishes.

We exemplify the methodology in its applicable

use cases, such as trip planning (Example 4.1).

Example 4.1. (Continuation from Example 1.1) We

expand each deﬁnition in T as follows:

ActivePlace ≡ X uPlace u ∃canWalk.Trekking

u∃canSail.Kayaking

Mangrove ≡ Y u Place u ∃canWalk.Trekking

Beach ≡ Z u Place u ∃canSail.Kayaking

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

Furthermore, R

canWalk

= {t, cMWL}

and R

canSail

{t, cTWS} where t and u are also fresh primitive role

names.

Let an ABox A be deﬁned as follows:

cMWL(p

) cTWS(p

, k

)

canWalk(p

) canSail(p

, k

)

Trekking(t

) Kayaking(k

)

Place(p

) Place(p

)

Place(p

) Mangrove(p

)

Beach(p

)

To query for a desired place, an agent needs to

express his preferences. Suppose the agent says

Obvious abbreviations are used here for the sake of suc-

cinctness.

Tuning Agent’s Proﬁle for Similarity Measure in Description Logic ELH

295

My interest is a place where I can travel with

by feet, especially walking, i.e. i

(Place) = 1.5,

(canMoveWithLegs) = 1.5, and I

(canWalk) =

1.8. Also, it yields i

(canMoveWithLegs) = 1.5 and

(canWalk) = 1.5 ⊕

last

1.8 = 1.8.

Constructing the canonical interpreta-

tion from A, we obtain ∆

= {p

, p

, t

, k

}, s

(Trekking, Kayaking) = 0, and

(canMoveWithLegs, canTravelWithSails) = 0.

Let ActivePlace, Mangrove, Place, Trekking,

Kayaking, canWalk, and canSail are rewritten shortly

as AP, M, P, T, K, cW, and cS, respectively. Using

Deﬁnition 4.1, hd

, T

)

= (

2.5

5.3

) · p-hd

, P

) + (

2.8

5.3

) · e-set-hd

, E

)

= (

2.5

5.3

) · (

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

i(X)+i(P)

)

2.8

5.3

) · e-set-hd

, E

)

= (

2.5

5.3

)(

1·max{0,0}+1.5·max{0,1}

1+1.5

)

2.8

5.3

) · e-set-hd

, E

)

= (

2.5

5.3

)(

1.5

2.5

) + (

2.8

5.3

)

i(cW)·max{e-hd

(∃cW.T,∃cW,T)}+1·0

i(cW)+i(cS)

= (

2.5

5.3

)(

1.5

5.3

) + (

2.8

5.3

)



1.8·1+1·0

1+1.8



≈ 0.623

Following the same step, we obtain hd

, T

) ≈

0.767. Hence, sim

(M, AP) ≈ 0.695 by using Deﬁni-

tion 4.2. Also, we obtain hd

, T

) ≈ 0.472 and

, T

) ≈ 0.714. Hence, sim

(B, AP) ≈ 0.593.

The fact that sim

(M, AP) > sim

(B, AP) corre-

sponds to the agent’s perception.

5 RELATIONSHIP TO

LEARNING-BASED APPROACH

Our proposed development uses the canonical inter-

pretation I

to compute numerical values for map-

pings on s

(cf. Subsection 3.3) and s

(cf. Subsection

3.4). Its drawback is that an existence of the canon-

ical interpretation I

is required. This section rather

discusses an alternative approach to obtain values for

mappings on s

and s

In addition to our proposed logic-based approach,

another natural way to conﬁgure both s

and s

to employ existing machine learning techniques on a

large corpus. For example, one may use Word2vec

(Mikolov et al., 2013) with a large corpus of text to

produce a vector space. Each word in the corpus will

be assigned by a corresponding vector in the space.

Word vectors are positioned in the vector space such

that words sharing common features in the corpus are

located in close proximity to one another in the space.

This characterization can later be converted into ele-

ments of the mapping s

and s

. Reconciling an on-

tology with machine learning techniques to improve

an application of

∼

is interesting but is outside the

scope of this paper. We leave this as a future task.

6 RELATED WORK

While there has been substantial work on concept

similarity measures in the context of DLs, the topic

of concept similarity measure under an agent’s prefer-

ences remains relatively unaddressed. Notable excep-

tions include (Tongphu and Suntisrivaraporn, 2015;

Lehmann and Turhan, 2012); however, measures pre-

sented in these papers may not include preferential

aspects evidently in the formal deﬁnition. We discuss

the differences of ours to others in the following.

As similarity may be subjective, the techniques in-

volved in concept similarity measure can be classiﬁed

into two main classes: ones which address the prob-

lem under an agent’s preferences, e.g. sim

(orig-

inally introduced in (Racharak et al., 2016b)), and

ones which do not, e.g. (Janowicz and Wilkes, 2009;

Racharak and Suntisrivaraporn, 2015; D’Amato et al.,

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

2009; D’Amato et al., 2008). The measure sim

gen-

eralizes the notion of homomorphism structural sub-

sumption with an aim to develop a similarity mea-

sure under preference proﬁle for DL ELH . As pre-

viously mentioned, (Tongphu and Suntisrivaraporn,

2015; Lehmann and Turhan, 2012) may not include

preferential aspects evidently in the formal deﬁnition;

however, their approaches share some viewpoints in

common to preference proﬁle. For instance, (Tong-

phu and Suntisrivaraporn, 2015) provides some facili-

ties similar to i

and d whereas (Lehmann and Turhan,

2012) provides some facilities similar to i

, s

, and

. We refer the readers to (Racharak et al., 2016a;

Racharak et al., 2016b) for the detailed discussion.

Speaking out in the context of DLs, we may

classify techniques in the other way round, i.e.

structure-based measure and interpretation-based

measure. Structure-based measure, e.g. (Janowicz

and Wilkes, 2009; Racharak and Suntisrivaraporn,

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

2006; Tongphu and Suntisrivaraporn, 2015; Lehmann

and Turhan, 2012; Racharak et al., 2016b), is

deﬁned using the syntax of concepts being mea-

sured. On the other hand, interpretation-based mea-

sure, e.g. (D’Amato et al., 2009; D’Amato et al.,

2008), is deﬁned using interpretations and cardinal-

ity. Some measures also include elements of both,

e.g. (D’Amato et al., 2006; Fanizzi and D’Amato,

2006) use structure to measure concepts but use the

canonical interpretation I

to measure similarity of

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

296

Figure 3: A mock-up of the best-matching concepts under the agent’s proﬁle application.

primitive concept names. The measure sim

per se

is categorized as a structure-based measure; however,

reconciling sim

with our strategies to help tuning

and s

may be considered as the last category as

it uses the canonical interpretation I

for measuring

similarity of primitive concept names and similarity

of primitive role names.

This work is an extension of the original sim

(Racharak et al., 2016b) in which we investigate and

formalize the task of conﬁguring the preference func-

tions based on the information deﬁned in a TBox and

an ABox (cf. Section 3). We also show that the pro-

posed strategies can be reconciled with the original

development, i.e. sim

(cf. Section 4).

7 DISCUSSION AND FUTURE

RESEARCH

As realistic ontologies are generally complex – con-

sisting in plenty of concept names and role names,

having some strategies to tune a measure helps ontol-

ogy engineers, researchers, and application users to

use a measure for similarity-based under the agent’s

proﬁle applications. That is, instead of specifying

each aspect of preference proﬁle individually and

manually, an agent may automatically assign an im-

portance of each primitive concept name through a

deﬁned concept name with I

. Similarly, an agent

may automatically assign an importance of each role

name through a more general role name with I

However, these notions have some restrictions and we

discuss these in Subsection 3.1 and Subsection 3.2,

respectively. If an ABox is presented, the canonical

interpretation can be induced and be used to further

compute s

and s

for each primitive concept pair and

primitive role pair (cf. Subsection 3.3 and Subsection

3.4). In any cases, these strategies are recommended

to use for the initial preference tuning and these val-

ues may be reﬁned wherever the agent wishes.

We also present a simple guideline to implement

similarity-based under the agent’s proﬁle applications

(cf. Subsection 4.1). As exhibited by the guideline, a

system may depict the concept hierarchy and the role

hierarchy w.r.t. an ontology to let an agent ﬁne tune

via I

and I

, respectively. Figure 3 shows a mock-up

of the best-matching concept under the agent’s pro-

ﬁle application. In this mock-up, we permit an agent

to identify his preferences through the notion I

highlighting concept names occurring on the hierar-

chy. We note that this mock-up does not show all the

strategies we have discussed in this work. Also, it

uses ⊕

max

to handle conﬂicting values on i

Currently, we are under an implementation of the

best-matching application under the agent’s proﬁle

using our extended measure sim

. There are several

possible directions for the theoretical future research.

Tuning Agent’s Proﬁle for Similarity Measure in Description Logic ELH

297

Firstly, our current strategies cannot be used to ex-

press complex preferences, such as multi-dimensional

preferences. Hence, it appears to be a natural step

to develop a high-level language for the speciﬁcation

of an agent’s preferences in the context of similarity-

based problems. Secondly, we are interested to ex-

tend the notion of preference proﬁle to support a more

expressive DL family, e.g. universal restriction, con-

cept negation, and also, to support an ABox. Thirdly,

we intend to devise a concept similarity measure un-

der preference proﬁle which can handle more expres-

sive DLs. Finally, we intend to employ our devel-

oped notion of concept similarity measure under pref-

erence proﬁle toward a system of analogical reason-

ing. As we have developed an argument-based logic

programming for analogical reasoning in (Racharak

et al., 2016c), it would be interesting to connect these

two research studies.

ACKNOWLEDGEMENTS

The authors would like to thank Prachya Boonkwan

for his proofreading and useful comments. This re-

search is part of the JAIST-NECTEC-SIIT dual doc-

toral degree program.

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