Linguistic Modiﬁers with Unbalanced Term Sets in Multi-valued Logic

Nouha Chaoued

1,2

, Amel Borgi

and Anne Laurent

Université de Tunis El Manar, Faculté des Sciences de Tunis,

LR11ES14 Informatique en Programmation, Algorithmique et Heuristique, 2092 Tunis, Tunisie

Univ. Montpellier, LIRMM, UMR 5506, F-34395, Montpellier Cedex 5, France

Keywords:

Imperfect Knowledge, Multi-valued Logic, Unbalanced Terms, Linguistic Modiﬁers.

Abstract:

Modeling human knowledge by machines should be as faithful as possible to reality. Therefore, it is imper-

ative to take account of inaccuracies and uncertainties in this knowledge. This problem has been dealt with

through different approaches. The most common approaches are fuzzy logic and multi-valued logic. These

two logics propose a linguistic term modeling. Generally, problems modeling qualitative aspect use linguistic

variables assessed in linguistic terms that are uniformly distributed on the scale. However, in many cases,

linguistic information needs to be deﬁned by unbalanced term sets whose terms are not uniformly and/or not

symmetrically distributed. In the literature, it is shown that many researchers have dealt with these term sets

in the context of fuzzy logic. Thereby, in our work, we introduce a new approach to represent and treat such

term sets in the context of multi-valued logic. First, we propose an algorithm that allows representing terms

within an unbalanced set. Then, we describe a second algorithm that permits the use of linguistic modiﬁers

within unbalanced multi-sets.

1 INTRODUCTION

Knowledge handled by humans is often imperfect.

These imperfections may be due to ambiguity, incom-

pleteness, imprecision, uncertainty, inconsistency,

etc. Several approaches were suggested in the lit-

erature for such knowledge representation and treat-

ment. The most known are fuzzy logic (Zadeh, 1965)

and multi-valued logic (De Glas, 1989; Akdag et al.,

1992).

We notice that humans are able to perform reason-

ing without any exact measurements. They mostly

use abstract terms of natural language (young, old,

mature, etc.) and symbolic data rather than numer-

ical values or qualitative ones. Terms can also be

composed by using adverbs, such as little, more or

less and slightly. Both fuzzy logic and multi-valued

logic propose a linguistic term modeling to allow us-

ing words in reasoning process. They use linguistic

variables that take values in a set of linguistic terms

(Zadeh, 1975). These latter express the various nu-

ances of the processed information using words.

Generally in Knowledge-Based Systems, experts

use linguistic terms that are uniformly and symmetri-

cally distributed on a scale. However, in some cases,

we need to assess qualitative aspects by means of vari-

ables using linguistic term sets which are not uni-

formly distributed. For example, in the evaluation

process, we often consider a single negative term, e.g.

Fail, and many positive terms such as Medium, Good,

Excellent, etc. The gap between these terms is un-

equal.

In this paper, we focus on multi-valued logic. It al-

lows to symbolically represent imprecise knowledge

using ordered adverbial expressions of natural lan-

guage (De Glas, 1989). We have noticed that in the

context of this logic, few studies have treated unbal-

anced linguistic term sets (Abchir, 2013; Chaoued

and Borgi, 2015). The aim of this paper is to establish

a methodology to represent and manage this kind of

data. It is based on our previous work (Chaoued and

Borgi, 2015) that expresses unbalanced terms using a

uniform multi-set. In the present work, we apply our

proposal to an Information Retrieval System (IRS).

The latter aims to retrieve a set of documents that sat-

isﬁes a user query. This system is composed of three

units (Herrera-Viedma and López-Herrera, 2007):

1. A documentary archive or a database including a

set of documents. They are represented by means

of index terms describing their subject content.

2. A query subsystem presenting user needs by

means of weighted queries. It indicates the top-

ics that he/she is asking for.

Chaoued, N., Borgi, A. and Laurent, A..

Linguistic Modiﬁers with Unbalanced Term Sets in Multi-valued Logic.

In Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2015) - Volume 2: KEOD, pages 50-60

ISBN: 978-989-758-158-8

3. An evaluation subsystem allowing the evaluation

of documents according to their relevance com-

pared to the user query.

Usually, users are interested in documents whose

contents are the most relevant to their queries. This

implies the use of more precise labels in the left

(positive) interval than in the right (negative) one

(Herrera-Viedma and López-Herrera, 2007). Thus,

in IRS the use of non-uniformly distributed term

sets is recommended. Indeed, it is more appropriate

to use such a kind of multi-sets to represent the

relevance degrees of the documents or to express the

weights of index terms in the queries. To achieve

this, we will use the unbalanced set S

= {None,

Low, Medium, High, Quite High, Very High, Total}

= {N, L, M, H, QH,V H, T } (Herrera-Viedma and

López-Herrera, 2007) (Fig. 1).

Figure 1: Unbalanced Linguistic Term Set.

Two aspects are discussed in the present work.

The ﬁrst concerns the representation of terms within

an unbalanced multi-set. The second deals with the

management of such a kind of knowledge. Fig. 2 il-

lustrates the different steps of this process. In the ﬁrst

one, we apply the single scale algorithm (Chaoued

and Borgi, 2015) to express a term (L in the ﬁgure)

using a uniform multi-set (L is represented by τ

). Af-

terwards, an existing tool, as aggregation operator or

modiﬁers, is applied to the obtained term (τ

is modi-

ﬁed into τ

). The last step aims to express the result of

the computational phase with a term from the initial

multi-set (Approximately L in the ﬁgure).

Figure 2: Management of Unbalanced Linguistic Term Set.

This article is organized as follows. We start, in

section 2, with an explanation of what is unbalanced

term sets. In section 3, we introduce the basic con-

cepts of multi-valued logic and of linguistic modiﬁers.

Existing works that concern the representation of un-

balanced multi-sets are presented in section 4. Then,

section 5 introduces our approach to express terms

within an unbalanced set (Fig. 2 Step 3). Finally, in

section 6, we propose a new way to use Generalized

Symbolic Modiﬁers (GSM) with unbalanced linguis-

tic terms (Fig. 2 Step 2).

2 PRELIMINARIES

Most Knowledge-Based Systems use linguistic sets

with terms that are uniformly and symmetrically dis-

tributed. However, there are other cases that need to

assess qualitative aspect by means of variables using

linguistic terms which are not uniformly and/or sym-

metrically distributed, named unbalanced linguistic

sets. In many real-life situations, these latter are used

as in project investment, negotiation process, evalua-

tion process, etc. Asymmetric linguistic information

can be a consequence of the nature of the linguistic

variables involved in the problem such as personal

examination or evaluation system (Fig. 3) (Martínez

and Herrera, 2012). Terms are not equidistant, e.g.

the distance between Poor and Average is greater than

between Average and Good. This difference indicates

the expert’s interest in having a more precise deﬁni-

tion of a part of the domain, that leads to the use of

more labels in this interval.

Figure 3: Set of 5 Linguistic Terms not Uniformly Dis-

tributed.

Moreover, in many problems, several decision

makers are involved. In fact, it is more reliable to

obtain a decision based on the opinion of several ex-

perts than on a single one. In a multi-experts decision

making process, each expert may assess his knowl-

edge with a particular scale having a speciﬁc granu-

larity (Fig. 4) (Martínez and Herrera, 2012). It indi-

cates their different knowledge backgrounds or judg-

ing abilities. In each used linguistic set, terms are uni-

formly distributed.

Herrera et al. (Francisco Herrera and Martínez,

2008) considered that an unbalanced fuzzy set is a set

with a minimum term, a maximum term and a central

term s

called also midpoint. The remaining terms

are neither uniformly distributed nor asymmetrically

on both side of central term :

S = S

∪ S

(1)

With :

• S

: the subset including terms on the left of the

central term s

and #(S

) its cardinality;

Linguistic Modiﬁers with Unbalanced Term Sets in Multi-valued Logic

Figure 4: Fuzzy Multi-Granular Scales Used by Experts.

• S

: the singleton containing the term s

;

• S

: the subset including terms on the right of the

central term s

and #(S

) its cardinality.

Hence, S is described by:

S = {(#(S

), density

), 1, (#(S

), density

)} (2)

where 1 is the midpoint. The density corresponds to

a high granularity around the central term or the end

of fuzzy partition. Therefore, the density value can be

middle or extreme.

For example, the set represented by Fig. 3 will

be described by S = {(1, extreme), 1, (3, extreme)}. In

fact, the central term is {Average}. The left subset in-

cludes only the term {Poor} which is the minimum

term. While the right subset includes {Good, Very

Good, Excellent}. These terms are closer to the max-

imum term Excellent. So the density corresponds to

extreme for the two subset.

Xu (Xu, 2009) proposed the unbalanced set as a

set of (2t-1) labels, with t a positive integer, deﬁned

by:

(t)

= {s

(t)

|β = (1 −t),

(2 − t),

(3 − t), ..., 0,

...,

(t − 3),

(t − 2), (t − 1)} (3)

Thus, the central term has index 0 and other terms

have positive or negative indices. The main idea of

this deﬁnition is that the absolute value of the devia-

tion between the indices of two successive terms in-

crease regularly proceeding from the center term to

the end of the scale. Fig. 5 (Xu, 2009) illustrates an

unbalanced set with t = 4. We notice that the terms

are symmetrically distributed around the central term.

In this work, we propose to deﬁne an unbalanced

linguistic term set S

as a set of degrees that includes

a minimum degree, False, a maximum degree, True,

and the remaining ones are not necessarily uniformly

Figure 5: A Set of Seven Linguistic Labels S

(4)

distributed on the scale: the gaps between adjacent

terms may be unequal. Set granularity, i.e. its cardi-

nality, can be odd or even. Each term is deﬁned by its

position on the scale. They are supplied directly or by

specifying the gap between each successive terms.

To illustrate our proposal, let us use the set repre-

sented in Fig. 3. This set will be described using the

distance between terms :

• Distance Poor - Average: d

• Distance Average - Good: d

• Distance Good - Very Good: d

• Distance Very Good - Excellent: d

There are two important subjects to treat when

dealing with imperfect knowledge: (1) their represen-

tation and (2) their management. Many researches

have dealt with these term sets in the fuzzy logic

context (Herrera-Viedma and López-Herrera, 2007;

Martínez and Herrera, 2012; Herrera and Martínez,

2001; Wang et al., 2015; Xu, 2009; Marin et al.,

2014; Bartczuk et al., 2012; Jiang et al., 2015). Re-

searchers have proposed approaches for their repre-

sentation (Martínez and Herrera, 2012; Herrera and

Martínez, 2001; Bartczuk et al., 2012) and their man-

agement, such as operators for unbalanced aggrega-

tion (Marin et al., 2014; Jiang et al., 2015). In the

context of multi-valued logic, only we and Abchir, in

(Chaoued and Borgi, 2015) and (Abchir, 2013) re-

spectively, proposed algorithms to represent unbal-

anced term sets within a uniform set. Our work, de-

veloped in the multi-valued context, has two targets:

the ﬁrst one is to propose a new way to represent terms

within unbalanced multi-sets; the second intends to

deﬁne an approach to use linguistic modiﬁers with

such knowledge.

3 MULTI-VALUED LOGIC AND

GENERALIZED SYMBOLIC

MODIFIERS

Multi-valued logic is based on De Glas’s multi-set

theory (De Glas, 1989). In this approach, each lin-

guistic term is represented by a multi-set. Knowledge

is expressed thanks to an ordered and ﬁnite scale of

M symbols denoted by (De Glas, 1989; Akdag and

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

Pacholczyk, 1989):

= {τ

, τ

, ..., τ

(M−1)

};M ≥ 2 (4)

is the membership degree to the multi-set (i ∈ [0,

M-1]).

It should be noted that symbolic degrees are con-

nected only by the total order relation ≤, deﬁned by

(Adkag, 1992)

≤ τ

⇐⇒ α ≤ β;∀α and β ∈ [0, M − 1] (5)

In most existing works on multi-valued logic,

these degrees are assumed to be uniformly distributed

on the scale. The membership relation in multi-valued

logic is partial:

x ∈

A ⇐⇒ x belongs to A at a degree α (6)

To express the imprecision of a predicate, a quali-

ﬁer ϑ

is associated with each degree:

x is ϑ

A ⇐⇒ (x is ϑ

A) is true

⇐⇒ ”x is A” is τ

true

The work of Zadeh (Zadeh, 1972), in fuzzy logic, and

more recently those of Akdag et al. (Akdag et al.,

2000; Akdag et al., 2001) and Truck (Truck, 2002),

in multi-valued logic context, consider that any fuzzy

subset or multi-valued symbol can be considered as a

modiﬁer of another fuzzy subset or multi-valued sym-

bol respectively. A modiﬁer allows modeling knowl-

edge and gradual reasoning to build new terms from

the initial ones, or to compare two values by ﬁnding

the modiﬁer that allows the transformation from one

to another.

In the multi-valued logic, a membership in a

multi-set is characterized by a symbolic membership

degree τ

deﬁned on a scale of ordered degrees L

Thus, the data modiﬁcation is a transformation of a

degree and/or the scale of the multi-set. Indeed, some

linguistic modiﬁers preserve the same multi-set, but

modify the membership degree. Others transform a

multi-set towards another one. Hence, it leads to an

expansion or erosion of the original scale.

Symbolic linguistic modiﬁers have been proposed

by Akdag et al. (Akdag et al., 2000; Akdag et al.,

2001) and they were generalized and formalized by

Truck (Truck, 2002; Truck et al., 2002). They were

named Generalized Symbolic Modiﬁers (GSM). Ac-

cording to Truck (Truck, 2002), a GSM (Deﬁnition 1)

is a triplet of parameters: radius, nature (i.e. di-

lated, eroded or preserved) and mode (i.e. reinforc-

ing, weakening or central). The radius is denoted by

ρ with ρ ∈ N

∗

. The higher ρ, the more powerful the

modiﬁer.

Deﬁnition 1. (Truck, 2002) Let τ

be a symbolic de-

gree, such that i ∈ N, in a scale L

of M terms

(M ∈ N

∗

\{1}) with i < M. Let m be a GSM with ra-

dius ρ denoted m

. The modiﬁer m

is a function that

performs a linear transformation of τ

to a new degree

∈ L

(where L

is the linear transformation of

) according to a radius ρ : m

(i) = i

; m

(M) = M

For each linguistic degree, Akdag et al. (Akdag

et al., 2000) associate a numerical rate, i.e. an in-

tensity level. In fact, an item from the multi-set can

be considered as a precision degree of a proposition.

This latter is the quotient prop(τ

) =

p(τ

)

M−1

associated

with each τ

such that p(τ

) is its position in the scale.

Thus, Prop(τ

) is the weight of the degree τ

relatively

to the linguistic set granularity, i.e. its intensity com-

pared to the truth degree τ

(M−1)

(True).

Symbolic modiﬁers are classiﬁed as: weaken-

ing when the proportion decreases, i.e. prop(τ

) <

prop(τ

) (the four weakening modiﬁers deﬁned in

(Truck, 2002) are EW

, DW

and CW

); rein-

forcing when it increases, i.e. prop(τ

) > prop(τ

)

(the four reinforcing modiﬁers proposed in (Truck,

2002) are ER

, DR

and CR

) or central if

prop(τ

) does not change, such modiﬁers may act as

a zoom on the base (the four central modiﬁers pre-

sented in (Truck, 2002) are EC

, EC

, DC

and DC

Some examples of GSM are presented in Table 1.

4 REPRESENTING

UNBALANCED MULTI-SETS

In the context of multi-valued logic, few works

have treated the non-uniformly distributed knowl-

edge. Abchir proposed in the discussion of his the-

sis (Abchir, 2013) a ﬁrst approach based on the use

of Generalized Symbolic Modiﬁers (GSM). We also

proposed in a previous work (Chaoued and Borgi,

2015) a modiﬁed version of the Abchir’s algorithm

to represent unbalanced degrees on a single uniform

scale .

4.1 Abchir’s Algorithm

This approach aims to represent numerical input val-

ues v

, v

, ..., representing the position of terms on

a scale, as symbolic degrees of a uniform multi-set

. The algorithm starts by calculating a γ coefﬁ-

cient multiplier to transform input values (v

) to in-

teger ones, denoted val

, if they are not. Then, each

obtained integer value is transformed into a couple (τ

b.c is the ﬂour function.

Linguistic Modiﬁers with Unbalanced Term Sets in Multi-valued Logic

Table 1: Some Examples of Generalized Symbolic Modiﬁers (GSM) (Truck, 2002).

Mode Nature Modiﬁer Effect

Weakening

Erosion EW

m(i) = max (0, i - ρ )

m(M) = max (2, M - ρ )

Dilatation

m(i) = i

m(M) = M + ρ

m(i) = max (0, i - ρ )

m(M) = M + ρ

Conservation CW

m(i) = max (0, i - ρ )

m(M) = M

Reinforcing

Erosion

m(i) = i

m(M) = max (i + 1, M - ρ )

m(i) = min (i + ρ , M - ρ - 1)

m(M) = max (1, M - ρ )

Dilatation DR

m(i) = i + ρ

m(M) = M - ρ

Conservation CR

m(i) = min (i + ρ , M - 1)

m(M) = M

Central

Erosion EC

m(i) = max ( b

c , 1)

m(M) = max (b

c + 1, 2)

Dilatation DC

m(i) = i ρ

m(M) = M ρ - ρ + 1

) according to the following rule:

M = max(2, val

i+1

);τ

= τ

val

(7)

The procedure is iterative. It stops when all in-

put values have been treated. The author denotes the

ﬁrst M by M

. If the next value to treat cannot be

integrated in this set, i.e., val

≥ M

, a new value M,

denoted by M

, is calculated such that: M

= val

+1.

Thus, the couple (val

, M

) leads to (τ

val

, L

When M changes, previously calculated couples must

be represented within the new scale L

. The weak-

ening expanding modiﬁer DW(M

− M

) (Table 1) is

used to do this.

The process continues for each value, ensuring

that the multiplier coefﬁcient γ always generates in-

teger values. If this is not the case, the multiplier γ

should be changed in a new γ that allows transform-

ing all considered values into integers, and its old

value will be denoted by γ

old

. Then, all previous val-

ues must be recalculated using the central expanding

modiﬁer DC(c) (Table 1). The multiplier c allows to

transform γ

old

into γ. It is obtained as follows:

γ = c ∗ γ

old

(8)

For this algorithm, neither considered values to

treat nor their number is known in advance. Partition-

ing will be done as the value is speciﬁed. The ﬁnal

result remains the same if a change is made in input

terms order. The main criticism made to this approach

is its iterative part for recalculating pairs representing

terms already treated. This is done each time that the

used multi-set or the value of the multiplier γ is mod-

iﬁed.

4.2 Single Scale Algorithm

The difference between Abchir’s algorithm (Abchir,

2013) and the single scale approach (Chaoued and

Borgi, 2015) is in the input data as well as in their

treatment. In this latter, input data can be the po-

sitions of the values to partition or the distances be-

tween them. This last case, not allowed with Abchir’s

approach, offers more ﬂexibility to the users.

The gap between each pair of successive terms re-

ﬂects a difference in their meaning. In this proposal,

inputs are expressed by numerical values which are

supplied directly (d

) (Fig. 6-a) or speciﬁed by the po-

sition of each term on the scale (v

) (Fig. 6-b). In the

latter case, the distance between the terms (more pre-

cisely their positions) v

i+1

and v

are calculated as fol-

lows:

= v

i+1

− v

(9)

(a) Distances between terms

(b) Terms position

Figure 6: Algorithm Inputs.

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

This proposal is based on Espinilla et al.’s work

(Espinilla et al., 2011). The authors proposed a new

deﬁnition of linguistic hierarchies: Extended Linguis-

tic Hierarchies (ELH), within fuzzy logic. To build an

ELH, they use a ﬁnite number of levels l(t, n(t)) with:

• t: the number indicating the hierarchy level with

t=1,..,m; such m is the number of experts;

• n(t): the granularity of the linguistic term sets cor-

responding to the level t. Each linguistic term has

a triangular membership function, uniformly and

symmetrically distributed on [0,1]. The granular-

ity of each level is always odd.

To express his knowledge, each expert can use a

speciﬁc level, i.e. a particular granularity (Espinilla

et al., 2011). Espinilla et al. proposed to add a new

level l(t

∗

, n(t

∗

)) (Deﬁnition 2) to the hierarchy with

∗

= m + 1. This level retains all the modal points

of all the previous levels. The modal points have a

membership degree that is equal to one.

Deﬁnition 2. (Espinilla et al., 2011) Let

n(1)

, ..., S

n(m)

} be the set of linguistic scales

with any odd value of granularity. A new level,

l(t

∗

, n(t

∗

)) with t

∗

= m + 1, that keeps the former

modal points of the previous m levels can have the

following granularity:

n(t

∗

) = 1 + LCM[(n(1) − 1), (n(2) − 1), ...,

(n(m + 1) − 1)]

With m the number of experts

Fig. 7 (Espinilla et al., 2011) shows an ELH with

three levels of 3, 5 and 7 terms. The fourth level (t

∗

)

has as granularity the LCM of the previous ones:

1 + LCM(3 − 1, 5 − 1, 7 − 1) = 13.

Figure 7: Extended Linguistic Hierarchy (ELH) with 3, 5,

7 and 13 Terms.

In a similar way, we proposed to express unbal-

anced terms within a new uniform multi-set (Chaoued

and Borgi, 2015). In fact, the granularity of the uni-

form multi-set equals the LCM of the sets represent-

ing initial terms granularity. The input data are unbal-

anced linguistic terms set S

, M the number of

terms) and the gap between terms. These are provided

directly or by specifying the position of each term.

First, the granularity M

of each uniformly dis-

tributed linguistic set L

used to represent each input

term v

is calculated. In this set, the distance between

each two successive terms is denoted by d

(1 + d

)

;k = 1, ..., (M − 1) (10)

Once the sets L

including all the terms of the

initial set are determined, the granularity of the uni-

form set L

is deduced as:

= 1 + LCM

M−1

k=1

− 1) (11)

Each value will be expressed on the new uniform

scale L

as a couple (τ

, L

) according to the fol-

lowing rule:

= γ

∗ d

;k = 1, ..., (M − 1) (12)

Where d

is the distance between any pair of succes-

sive degrees in L

. For a uniform linguistic set, this

distance is:

− 1)

(13)

To express the distances d

according to the dis-

tance d

, the multiplier γ

is calculated using the rule

(12).

The minimum term is represented by (τ

, L

) and

denoted by s

. The process, presented in Algorithm

1, continues for other values. The obtained couples

(τ

, L

) are denoted by s

. To ensure the succession

of terms, the calculated multiplier γ

is added to the

one calculated in the previous iteration named γ

old

Thus, the comparison is always done regarding the

ﬁrst term of the set L

, i.e., τ

This algorithm allows to bring us back to the uni-

form case which will give us the capacity to use differ-

ent existing tools as linguistic modiﬁers (Kacem et al.,

2015), aggregation operators, etc.

Considering the complexity, this proposal is less

complex (O(p)) than Abchir’s (O(p

)), with p the

number of input terms (Chaoued and Borgi, 2015).

Indeed, in this approach the treatment is done after

introducing all data. Hence, the couples representing

terms are calculated once.

To illustrate this algorithm, let us consider an In-

formation Retrieval System (IRS) and the linguis-

tic term set S

= L

= {N, L, M, H, QH, VH, T}

Linguistic Modiﬁers with Unbalanced Term Sets in Multi-valued Logic

(Herrera-Viedma and López-Herrera, 2007) (Fig. 1).

This latter is used to express the relevance of docu-

ments in the retrieval process. We aim to represent

each term of L

using a term from a uniform linguistic

set. As input, we consider the normalized distances in

as:

• Distance N - L: d

= 0.25

• Distance L - M: d

= 0.25

• Distance M - H: d

= 0.125

• Distance H - QH: d

= 0.125

• Distance QH - VH: d

= 0.125

• Distance VH - T: d

= 0.125

First, we calculate the values M

1+d

= 5 = M

; M

1+d

= 9 = M

= M

Algorithm 1: Single Scale algorithm.

Input:

M unbalanced linguistic terms

Normalized distances d

with k=1, ..., (M-1)

begin

Calculate granularities M

; M

∈ N

∗

\{1}

with k=1, ..., (M-1)

Calculate granularity:

= 1 + LCM

M−1

k=1

− 1)

Deduce distance d

−1)

← (τ

, L

)

Calculate γ

with k=1, ..., (M-1)

← (τ

, L

)

Output: S

= {(τ

, L

),(τ

, L

),...,

(τ

p−1

, L

)}

Then, we determine the granularity M

of the uni-

form set and the distance between any of its succes-

sive terms d

: M

= 1 + LCM(5 − 1, 9 − 1) = 9;

−1

= 0.125

We associate the couple (τ

, L

) to the term N, and

we treat the other values:

• For the term L (k = 1)

old

← 0; γ

← γ

old

= 2; L ← (τ

, L

)

• For the term M (k = 2)

old

← 2; γ

← γ

old

= 4; M ← (τ

, L

)

• For the term H (k = 3)

old

← 4; γ

← γ

old

= 5; H ← (τ

, L

)

• For the term QH (k = 4)

old

← 5; γ

← γ

old

= 6; QH ← (τ

, L

)

• For the term VH (k = 5)

old

← 6; γ

← γ

old

= 7;V H ← (τ

, L

)

• For the term T (k = 6)

old

← 7; γ

← γ

old

= 8; T ← (τ

, L

)

The algorithm output is illustrated in Fig. 8.

Figure 8: Result of the Single Scale Algorithm.

5 REPRESENTATION OF TERMS

WITHIN AN UNBALANCED

SET

In the previous section, we have presented our pre-

vious work (Chaoued and Borgi, 2015) to represent

an unbalanced term set S

within a uniform term set

. Thereby, existing linguistic modiﬁers or aggre-

gation operators can be applied to its terms. The ob-

tained values are also expressed using the uniform set

. However, it is more understandable for the user

to represent these results using terms from the initial

set S

In this section, we focus on the representation of a

term τ

from the uniform set L

with a term from S

denoted by L

. We suppose that the used distances on

both sets are normalized. For more clarity, we denote

the degrees of the uniform set L

by τ

, ..., τ

... and

those of S

by τ

, ..., τ

... (knowing that τ

and τ

cor-

respond to the same position 0 and τ

and τ

to 1).

We propose a way to determine the nearest position,

in S

, to an initial term τ

, represented by its position

in L

. For that purpose, ﬁrst, we need to deﬁne the

proportion of an unbalanced degree from S

Deﬁnition 3. Let τ

be a symbolic degree of an unbal-

anced multi-set S

). Its proportion is deﬁned as:

Prop(τ

) =

∑

j=1

With d

the normalized distance between each pair of

successive degrees in the unbalanced multi-set S

We aim to identify the term τ

pos

in S

which dis-

tance between it and the ﬁrst term (τ

) of L

is the

closest to v

, the position of τ

on L

. Thus, we will

compare v

with the sum l

of the distances separating

successive terms in L

: l

∑

j=1

The process will stop when the value of l

higher or equal to v

. If the values v

and l

are equal,

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

the position of τ

in L

is that of τ

k−1

. If v

< l

, we

check the closest term to v

between τ

k−1

and τ

. In

this case, a proportion error, denoted by α, exists and

its value is the difference between the position of a

term τ

and that of the closest term, i.e. τ

k−1

or τ

. If

these terms are equidistant to τ

, we choose the term

with the smallest index. This approach is described in

Algorithm 2.

The inputs for this function are the position v

the term τ

, the M terms of the unbalanced multi-

set S

and the normalized distances d

between its

successive terms. As output, we obtain the couple

(τ

pos

, α).

To illustrate our approach, let us continue with the

IRS already described. Let us suppose that a docu-

ment D is described in the database with a set of 5

index t

(i ∈ {1, ..., 5}):

D = 0.7/t

+ 0.5/t

+ 0.9/t

+ 0.6/t

+ 0.4/t

In fact, the system indicates for each index term

a numeric value corresponding to its importance in

describing the subject discussed in the document. If

the index term t

is not linked to the document subject

content, its value is equal to 0. However, the index

value that is equal to 1 means that t

is too important

in the document subject. We aim to represent each

index value, i.e. v

, by means of a couple (τ

pos

, α);

such τ

pos

is a membership degree from L

The counter k is initialized to 1 and the distance l

to 0. For the ﬁrst index value v

• k = 1: l

= 0.25 < 0.7 = v

• k = 2: l

= 0.25 + 0.25 = 0.5 < 0.7

• k = 3: l

= 0.5+ 0.125 = 0.625 < 0.7

• k = 4: l

=0.625 + 0.125 = 0.75 > 0.7

In this case v

is lower than l

(= 0.75). We can

say that the index value is between τ

, High (H), and

, Quite High (QH). So, we check the closest term

between τ

and τ

by comparing values of (l

- v

)

and (v

- (l

- d

)): (0.75 - 0.7) < (0.7 - (0.75 - 0.125)).

Hence, τ

is closer than τ

and the proportion error

is α = -0.05. Thus, v

is represented by (QH, -0.05)

(Fig. 9). We can say that the degree τ

is weakened

of 0.05. We proceed using the same approach for the

other index values. Thus, we obtain:

D = (QH, -0.05)/t

+ (M, 0)/t

+ (VH, 0.025)/t

(H, -0.025)/t

+ (M, -0.1)/t

Figure 9: Representation of the Index v

within an Unbal-

anced Set.

Algorithm 2: Representation of terms within unbal-

anced multi-set.

Input:

The position v

of the term τ

, τ

∈ L

M linguistic terms of L

)

Normalized distances d

in L

; k=1, ..., (M-1)

begin

k = 1

= 0

while l

< v

∑

j=1

k ← k + 1

if l

= v

then

pos ← k-1

α ← 0

else

if l

- v

< v

- (l

- d

) then

pos ← k

α ← - (l

- v

)

else

pos ← k-1

α ← v

- (l

- d

)

Output: The couple (τ

pos

, α)

6 LINGUISTIC MODIFIERS

WITH UNBALANCED

MULTI-SETS

In multi-valued logic, any symbol can be considered

as a modiﬁer of another one. Thereby, modiﬁers allow

to represent small variations of imprecise characteri-

zations of a linguistic variable. Symbolic modiﬁers

may transform simultaneously the membership de-

grees and the scale of the multi-set. We propose in this

section to illustrate how to use linguistic modiﬁers

with unbalanced imperfect knowledge. We present

an approach to apply Generalized Symbolic Modi-

ﬁers (Truck, 2002) (Table 1), initially designed for

balanced sets, to unbalanced term sets.

To perform this, ﬁrst, we express the unbalanced

term to modify with a term from a balanced multi-set.

Afterwards, we apply the GSM modiﬁer on the ob-

tained term within a balanced set. Then, we should

ﬁnd the closest matching term back in the original

unbalanced multi-set. Our proposal has as input M

linguistic terms of the unbalanced multi-set S

(rep-

resented on a scale L

), a term τ

from this set, the

normalized distances d

between its successive terms

and the GSM m to apply. The treatment will be as

follows (Algorithm 3):

Linguistic Modiﬁers with Unbalanced Term Sets in Multi-valued Logic

Algorithm 3: Using GSM with unbalanced multi-set.

Input:

M unbalanced linguistic terms of L

)

The term τ

∈ L

Normalized distances d

of L

; k=1, ..., (M-1)

The GSM m

begin

← Single scale approach (M

unbalanced terms, d

) (Algo.1)

Identify the couple (τ

, L

) corresponding

to (τ

, L

) in L

(τ

, M

) ← m (τ

, M

)

←

−1)

Calculate the closest term τ

pos

to v

in S

and the proportion error α (Algo.2)

Output: The couple (τ

pos

, α)

1. Express the term τ

within a uniform multi-set L

using the single uniform scale approach (Subsec-

tion 4.2). The new term is denoted by τ

2. Apply the GSM m to the term τ

. Thus, we get

a new uniform linguistic set L

, such that M

m(M

), and a new term τ

= m(τ

3. Represent the term τ

within the initial unbal-

anced linguistic set L

) using the approxi-

mation function presented in Algorithm 2.

Thus, for this last step, we have to determine the

nearest position to τ

in S

). First, we calculate

the position v

, i.e. the distance between the term τ

and τ

. Afterwards, we determine the term τ

pos

comparing v

to the sum l

of the distances between

each successive terms in L

To illustrate our approach, we use the same ex-

ample of IRS previously presented. We consider the

unbalanced set L

of relevance degrees (Fig. 10). We

apply the reinforcing modiﬁer CR(1) to the relevance

degree Medium, i.e. τ

. The process is represented in

Fig. 11.

Figure 10: Unbalanced Set of Relevance Degrees.

As mentioned before, the distances d

are:

= d

= 0.25; d

= d

= 0.125.

The proportion of the term τ

in the unbalanced

term set L

(Deﬁnition 3) is:

Prop (τ

) = d

+ d

= 0.25 + 0.25 = 0.5

The granularity of the uniform set, previously cal-

culated in Subsection 4.2, is 9 and the term τ

from

is represented by τ

in L

. We notice that the pro-

portion is preserved Prop(τ

) =

= 0.5.

For this example, we apply the GSM CR (1) to the

couple (τ

, L

) (Table 1):

= m(4) = min(4 + 1, 9 − 1) = min(5, 8) = 5;

= m(9) = 9

The obtained couple is (τ

, L

). We remind that

the modiﬁer CR is a reinforcing one, i.e. the propor-

tion is increased: Prop (τ

) =

> Prop(τ

) =

Finally, we must express the term τ

within the

initial set L

. We ﬁrst calculate the distance: v

−1)

= 0.625.

The counter k is initialized to 1 and the distance l

to 0.

• For k = 1; l

← 0.25 < v

• For k = 2 ; l

← 0.25 + 0.25 = 0.5 < v

• For k = 3 ; l

← 0.5 + 0.125 = 0.625 = v

In this case, the value of l

is equal to v

. Thus,

is the correspondent term in the unbalanced set L

The proportion of this term is Prop (τ

) = 0.625. It

corresponds to the proportion of the term τ

in the

balanced multi-set L

We can say that the result of applying CR (1) to

the relevance degree Medium corresponds to the rele-

vance degree High.

Figure 11: The Process of Applying CR (1) to Relevance

Degree M.

7 CONCLUSIONS

This work deals with imperfect knowledge manage-

ment in the multi-valued logic context. Speciﬁcally,

it deals with the representation and management of

unbalanced multi-sets. In fact, the available literature

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

that involves unbalanced term sets are only concerned

with the fuzzy logic. They also do not propose a for-

mal deﬁnition of unbalanced term sets, which could

be an important improvement in imperfect knowledge

management.

This paper proposes an algorithm that allows to

represent uniformly distributed terms within an unbal-

anced term set. This is performed by an approxima-

tion function that provides the closest term in the un-

balanced multi-set to the desired value. In some cases,

a proportion error may exist. Besides that, based on

our previous work (Chaoued and Borgi, 2015) that al-

lows the representation of terms on a single uniform

scale, we introduce a second algorithm that applies

linguistic modiﬁers, initially designed for balanced

terms, to unbalanced ones.

As future work, we notice that, in both Abchir’s

and our algorithms, the input terms for the represen-

tation are numerical. It would be interesting to pro-

pose a way to treat symbolic cases where the dis-

tances between terms are deﬁned with words instead

of numbers. It would also be more understandable

by humans to express the proportion error using ad-

verbs like little, more or less, slightly,... Another

aspect in the management of imperfect knowledge

that should be treated is the approximate reasoning

(Zadeh, 1975). It is based on the Generalized Modus

Ponens. Its principle is to deduce a fact similar to the

conclusion rule from an observation approximately

equal to the premise rule. It would be important to

propose a new Generalized Modus Ponens rules deal-

ing with unbalanced multi-sets.

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KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development