APPROXIMATE REASONING BASED ON LINGUISTIC

MODIFIERS IN A LEARNING SYSTEM

Saoussen Bel Hadj Kacem

National School of Computer Sciences, University campus Manouba, 2010, Manouba, Tunisia

Amel Borgi

National Institute of Applied Sciences and Technology, Centre Urbain Nord de Tunis, 1080, Tunis, Tunisia

Moncef Tagina

National School of Computer Sciences, University campus Manouba, 2010, Manouba, Tunisia

Keywords:

Approximate reasoning, Linguistic modiﬁers, Supervised learning, Classiﬁcation rules, Multi-valued logic.

Abstract:

Approximate reasoning, initially introduced in fuzzy logic context, allows reasoning with imperfect knowl-

edge. We have proposed in a previous work an approximate reasoning based on linguistic modiﬁers in a

symbolic context. To apply such reasoning, a base of rules is needed. We propose in this paper to use a super-

vised learning system named SUCRAGE, that automatically generates multi-valued classiﬁcation rules. Our

reasoning is used with this rule base to classify new objects. Experimental tests and comparative study with

two initial reasoning modes of SUCRAGE are presented. This application of approximate reasoning based on

linguistic modiﬁers gives satisfactory results. Besides, it provides a comfortable linguistic interpretation to the

human mind thanks to the use of linguistic modiﬁers.

1 INTRODUCTION

Most information expressed by human beings is un-

certain, vague or imprecise. However, these informa-

tion is necessary for the realization and the use of in-

telligent systems. In the literature, several approaches

have been proposed for the representation of these

types of knowledge, two of which dominate: fuzzy

logic (Zadeh, 1965) and multi-valued logic (Akdag

et al., 1992). To allow systems manipulating and

reasoning with imperfect knowledge, Zadeh (Zadeh,

1975) has introduced approximate reasoning concept

in the fuzzy logic context. This reasoning is based

on a generalization of Modus Ponens to fuzzy data,

known as Generalized Modus Ponens (GMP). It cor-

responds to the following schema:

If X is A then Y is B

X is A

′

Y is B

′

(1)

where X and Y are two variables and A , A

′

, B and

′

are predicates. Approximate reasoning can, not

only infer with an observation equivalent to the rule

premise A , but also with an observation A

′

which is

approximately equivalent to it.

In a previous work (Kacem et al., 2008; Borgi

et al., 2008), we noticed that both fuzzy and multi-

valued GMPs are generally based on the concept

of similarity (Akdag et al., 1992; Khoukhi, 1996;

Bouchon-Meunier et al., 1997). The weakness of this

type of reasoning is that it focuses on the modiﬁca-

tion degree (the degree of similarity between A and

′

) and not to the way A has been modiﬁed to have

′

(weakening, reinforcement, etc.).

We also noted that the concept of linguistic mod-

iﬁers reﬂects a form of similarity which can be used

in the GMP for the evaluation of the changes made on

the premise to lead to the conclusion. The diagram of

approximate reasoning based on linguistic modiﬁers

is as follows:

If X is A then Y is B

X is m(A )

Y is m

′

(B )

(2)

431

Bel Hadj Kacem S., Borgi A. and Tagina M. (2010).

APPROXIMATE REASONING BASED ON LINGUISTIC MODIFIERS IN A LEARNING SYSTEM.

In Proceedings of the 5th International Conference on Software and Data Technologies, pages 431-437

DOI: 10.5220/0002924204310437

 SciTePress

To determine the inference conclusion B

′

= m

′

(B ),

it is enough to determine the modiﬁer m

′

. The lat-

ter is obtained based on the observed modiﬁer m

′

= m(A )) and the correlation intensity between the

premise and the rule’s conclusion.

To apply approximate reasoning, it is necessary to

have a base of rules. Two ways are possible to ob-

tain such a base. First, it can be directly provided

by the expert. Secondly it can be automatically built,

through the use of learning systems. In our work, we

choose to use the second solution, we used a clas-

siﬁcation system for automatic generation of rules,

called SUCRAGE (Borgi, 1999; Borgi and Akdag,

2001). This system generates classiﬁcation multi-

valued rules, the context of our approximate reason-

ing (Kacem et al., 2008). A classiﬁcation rule predicts

the class of a new object. For example, a patient is de-

scribed by a set of attributes such as age, sex, blood

pressure, etc, and the class could be a binary attribute

concluding or not the illness of the patient by a par-

ticular disease.

In this paper, we begin in section 2 by presenting

the symbolic multi-valued logic, the context of our

work. Then, section 3 deals with the concept of lin-

guistic modiﬁers. We present in section 4 the SU-

CRAGE system. Then, in section 5 we explain how

to adapt and apply approximate reasoning based on

linguistic modiﬁers on this system. Finally, before

concluding this work we present in section 6 a com-

parative study of experimental test results.

2 SYMBOLIC MULTI-VALUED

LOGIC

Multi-valued logic is a generalization of Boolean

logic. It provides truth values that are intermediate

between True and False. We denote by M the num-

ber of truth degrees in multi-valued logic. Akdag et

al. (Akdag et al., 1992) have introduced a new gen-

eration of multi-valent logic based on the theory of

multi-sets.

In symbolic multi-valued logic, each linguis-

tic term (such as large) is represented by a multi-

set (Akdag et al., 1992). To express the imprecision of

a predicate, a qualiﬁer v

is associated to each multi-

set (such as rather, little, etc). When a speaker uses

a statement “X is v

A”, v

is the degree to which X

satisﬁes the predicate A

. A truth-degree τ

must cor-

respond to each adverbial expression v

so that:

Denoted mathematically by “X ∈

A”: the object X be-

longs with a degree α to a multi-set A.

X is v

A ⇐⇒ ”X is v

A” is true

⇐⇒ ”X is A” is τ

-true

For example, the statement “John is rather tall”

means that John satisﬁes the predicate tall with the

degree rather.

The set of symbolic truth-degrees forms an or-

dered list L

= {τ

,...,τ

M−1

}

with the total or-

der relation: τ

≤ τ

⇔ i ≤ j, its smallest element is

(false) and the greatest is τ

M−1

(true). In practice,

the number of truth-degrees is often close to 7. The

expert can even propose his own list of truth-degrees;

the only restrictive condition is that they must be or-

dered.

3 GENERALIZED SYMBOLIC

MODIFIERS

A modiﬁer is an operator that builds linguistic terms

from a primary linguistic term. This concept was in-

troduced by Zadeh (Zadeh, 1975) in the fuzzy logic

framework. We distinguish two types of fuzzy mod-

iﬁers. First, reinforcing modiﬁers that reinforce the

notion expressed by the term (as very). Then weaken-

ing modiﬁers, which weaken the notion expressed by

the term (as more or less).

As already mentioned, we have used linguis-

tic modiﬁers in approximate reasoning process in

(Kacem et al., 2008; Borgi et al., 2008). Since our

work falls in multi-valued framework, we use modi-

ﬁers deﬁned in this particular context.

A set of linguistic modiﬁers were proposed in the

multi-valued framework by Akdag and al. in (Akdag

et al., 2001), they were named the Generalized Sym-

bolic Modiﬁers. A Generalized Symbolic Modiﬁer

(GSM) is a semantic triplet of parameters: radius,

nature (i.e dilated, eroded or preserved) and mode

(i.e reinforcing, weakening or central). The radius is

noted ρ with ρ ∈ IN

∗

Deﬁnition 1. Let us consider a symbolic degree τ

with i ∈ IN in a scale L

of a base M ∈ IN

∗

r {1},

and i < M. A GSM m with a radius ρ is denoted m

The modiﬁer m

is a function which applies a linear

transformation to the symbolic degree τ

to obtain a

new degree τ

′

∈ L

′

(where L

′

is the linear trans-

formation of L

) according to a radius ρ such as:

: L

→ L

′

7→ τ

′

A proportion is associated to each symbolic de-

gree within a base denoted Prop(τ

) =

M−1

With M a positive integer not null.

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432

By analogy with fuzzy modiﬁers, the authors pro-

pose a classiﬁcation of symbolic modiﬁers according

to their behavior: weakening and reinforcing mod-

iﬁers, and they add the family of central modiﬁers

(Akdag et al., 2001) that neither reinforce nor weaken

the concept. The deﬁnitions of the reinforcing, weak-

ening and central modiﬁers are given in table 1

We have studied composition of these modiﬁers in

(Kacem et al., 2009).

4 THE LEARNING SYSTEM

SUCRAGE

To apply our approximate reasoning based on linguis-

tic modiﬁers, a rules base is needed. We used a learn-

ing system to obtain this base. SUCRAGE (SUper-

vised Classiﬁcation by Rule Automatic GEneration)

is a supervised learning system which was proposed

by Borgi in (Borgi, 1999; Borgi and Akdag, 2001).

The construction of a classiﬁcation function in

SUCRAGE is done through two phases: the learning

phase and the recognition or classiﬁcation one (Borgi

et al., 2003; Borgi, 2006; Borgi et al., 2007).

4.1 Learning Phase

The rule base is generated thanks to a learning base: a

set of objects already classiﬁed. We denote by B

,...,

the classes deﬁned by the experts, X

,..., X

the

attributes of the objects. The rules generated by SU-

CRAGE have the following form:

If X

is v

A and ... and X

is v

A then the class

is B

with p

where:

is v

A a proposition which means that the

value of X

is in [a,b]

an attribute, X

∈ {X

,...,X

}

[a,b] the sub-interval of index v

in the

ﬁeld of X

A a multi-set for the attributes ﬁelds

the i

class

p a belief degree representing the

uncertainty of the conclusion

The interval [a,b] is a sub-interval of the ﬁeld of

the attribute X

. It is obtained by the a regular dis-

cretization of this ﬁeld.

We have modiﬁed some deﬁnitions in order to respect

the bounds of M and i.

4.1.1 Construction of the Premise

To construct the rule premise, the ﬁrst task consists on

determining what are the attributes to regroup. In SU-

CRAGE, the attributes that appear in a same premise

are the correlated ones. For that, a correlation ma-

trix C is calculated: C = (r

i, j

)

t×t

, with r

i, j

the linear

correlation coefﬁcient between X

and X

Then, one considers that X

and X

are correlated if

the absolute value of r

i, j

is greater than a ﬁxed thresh-

old θ.

One must then discretizes the attributes ﬁelds. In

this work, we retained the regular discretization: it

leads to M sub ranges denoted by v

, v

, ..., v

M−1

Condition parts of rules are then obtained by consid-

ering for each correlated components subset, a sub-

interval (v

) for each component in all possible com-

binations. Figure 1 illustrates such a partition in the

case of two correlated features with a subdivision size

M = 3.

Figure 1: Example of partition of the correlated attributes

space.

4.1.2 Construction of the Rule

Each constructed premise according to the method ex-

posed above conducts to the generation of b rules,

with b the number of classes. The last stage con-

sists in calculating this belief degree p, which can

be represented by the conditional probability to get

the conclusion when the premise is veriﬁed: p =

proba(premise/conclusion). Conditional probabili-

ties are estimated on the training set using a frequen-

tist approach.

4.2 Classiﬁcation Phase

During the classiﬁcation phase, the inference engine

associates a class to a vector representing an object

to classify. Two types of reasoning are used: an ex-

act reasoning and an approximate reasoning. For ex-

act inference, the used method is the classic one. It

consists on the use of the Modus Ponens. The ap-

proximate inference method applies the Generalized

Modus Ponens:

APPROXIMATE REASONING BASED ON LINGUISTIC MODIFIERS IN A LEARNING SYSTEM

433

Table 1: Deﬁnitions of weakening, reinforcing and central modiﬁers.

MODE Weakening Reinforcing Central

NATURE

Erosion

′

= τ

′

= τ

max(0,i−ρ)

′

= L

max(2,i+1,M−ρ)

′

= τ

max(⌊

⌋,1)

′

= L

max(2,M−ρ)

′

= τ

max(0,min(i+ρ,M−ρ−1))

′

= L

max(⌊

⌋+1,2)

′

= L

max(2,M−ρ)

Dilation

′

= τ

′

= L

M+ρ

′

= τ

i+ρ

′

= τ

iρ

′

= τ

max(0,i−ρ)

′

= L

M+ρ

′

= L

Mρ−ρ+1

′

= L

M+ρ

Conservation

′

= τ

max(0,i−ρ)

′

= τ

min(i+ρ,M−1)

′

= τ

′

= L

′

= L

′

= L

* ⌊.⌋ is the ﬂoor function.

If “X

is v

A” and . . . and “X

is v

A” then the class is B

with p

“X

is v

A” and ... and “X

is v

A”

The class is B

with p

′

To be more precise in determining the distance be-

tween the premise and the observation in approximate

reasoning, the attributes values have undergone a dis-

cretization ﬁner than exact reasoning, speciﬁcally M

instead of M.

The approach consists in using a 0+ order infer-

ence engine. The engine has to manage the rules’ un-

certainty and take it into account within the inference

dynamic. More precisely, for a new object O to clas-

sify, the inference engine allows to obtain a ﬁnal be-

lief degree associated to each class. The ﬁnal belief

degree is the result of a triangular co-norm applied on

the probabilities of the ﬁred rules that conclude to this

considered class. Finally, the winner class associated

to the new object is the class where the ﬁnal belief

degree is maximal.

5 INTEGRATION OF

LINGUISTIC MODIFIERS IN

SUCRAGE

The rule of Generalized Modus Ponens with linguis-

tic modiﬁers that we proposed in (Kacem et al., 2008),

and that we intend to use for classifying objects with

SUCRAGE, deals with multi-sets. However, the con-

clusion part of SUCRAGE rules contains a numerical

belief degree. To apply our GMP, the probability mea-

sure of class B

must be symbolic. In this section we

explain the adaptation made on SUCRAGE in order

to use our approximate reasoning based on linguistic

modiﬁers.

5.1 Symbolic Probability

In (Seridi and Akdag, 2001), the authors have deﬁned

a symbolic probability theory. This theory is an alter-

native to the classical theory of probability, in the spe-

cial case where values of probabilities are symbolic

degrees of uncertainty. The authors used this proba-

bility in SUCRAGE in (Seridi et al., 2006), and this

by replacing the probability measure p of the rules

by a symbolic degree of uncertainty. With the spe-

ciﬁc notation of our work, the rules generated by SU-

CRAGE become of the form:

If X

is v

and ... and X

is v

then the

class is v

with B

a class and τ

the symbolic belief degree

associated with this class. Thus, a degrees scale has

been introduced to represent uncertainty L

com-

posed of M

degrees: L

= {τ

,i = 0, . . . , M

− 1}

totally ordered. The boundaries of these sub-intervals

are denoted a

, a

, ..., a

. Therefore, it is associated

with each probability measure p a symbolic degree of

uncertainty τ

The discretization of probability can be regular or

irregular. Seridi et al. (Seridi et al., 2006) chose to

use an irregular discretization to obtain a scale L

7 sub-intervals. The numerical probability is subdi-

vided as follows:

Let’s remind that τ

is the symbolic degree associated

to the linguistic expression v

ICSOFT 2010 - 5th International Conference on Software and Data Technologies

434

p = 0 ⇔ τ

⇔ Impossible

p ∈]0,0.5[ ⇔ τ

⇔ Very little possible

p ∈ [0.5,0.7[ ⇔ τ

⇔ Little possible

p ∈ [0.7,0.8[ ⇔ τ

⇔ Possible

p ∈ [0.8,0.9[ ⇔ τ

⇔ Enough possible

p ∈ [0.9,0.95[ ⇔ τ

⇔ Very possible

p ∈ [0.95,1] ⇔ τ

⇔ Certain

In this work, we use the irregular discretization, as

used by Seridi and al. (Seridi et al., 2006). In (Seridi

et al., 2006) symbolic probabilities are used in SU-

CRAGE with exact inference, unlike our work where

we are interested in approximate reasoning.

5.2 Conclusion Deduction

After building the rules, we have to exploit them. An

inference must be applied in order to classify new ob-

jects.

5.2.1 Simple Premise

Let us start with a simple case where the rule has a

simple premise. The corresponding rule of General-

ized Modus Ponens based on linguistic modiﬁers is as

follows:

Rule : If ”X

is v

A” then ”Y is v

B”

Fact : ”X

is m(v

A)”

Conclusion : ”Y is m

′

B)”

The modiﬁer m

′

to apply to the conclusion is ob-

tained according to the modiﬁer m. The ﬁrst step is to

determine the modiﬁer m. As in the case of the orig-

inal approximate reasoning used in SUCRAGE, the

attributes subdivision cardinal in the rules premises is

M, while the one in the observations is M

. The pro-

posed solution is to ﬁrst ﬁnd the decomposition of the

modiﬁer m with the dilating central DC operator. In-

deed, the observation base is a multiple of the premise

base, so it undergoes a dilatation. Let τ

the member-

ship degree of the observation in the base L

, the

decomposition of the modiﬁer m is given as follows:

m = m

◦ DC

with: ρ

= |γ− αρ

−1

M−1

= M+ 1

(3)

with m

an elementary modiﬁcation operator of

radius ρ

. This operator m

may be eitherCW, CR or

CC, since it acts only on the truth degree

. Its choice

depends on the sign of (γ− α.ρ

). Thus:







, if (γ− α.ρ

) < 0;

, if (γ− α.ρ

) > 0;

CC, else.

(4)

The base is already dilated by the central operator DC.

The operator m

represents the real modiﬁcation

made on the premise. Indeed, the DC operator is cen-

tral, so it acts as a zoom on the base and has no ef-

fect on the proportion of degrees. For this reason, the

operator m

which we denote by m

⋄

is the one that

we consider in determining the modiﬁer m

′

to be de-

duced. The problem in that this operator is compatible

with the observations base L

. It can not be directly

applied to the symbolic probability, given that its base

is different from the base of the symbolic probability.

Thus, it is necessary to convert the modiﬁer m

⋄

to be

compatible with the conclusion base L

. We pro-

pose to keep the same type of operator and change

only the radius ρ

The conversion of the modiﬁer m

⋄

when the sym-

bolic probability is irregular is a complex task. In-

deed, the amplitudes of the probability sub-intervals

are different. For this reason, we propose a solu-

tion that takes into account these amplitudes. We as-

sociate with each symbolic probability degree τ

{α ∈ [0..M − 1]} a value called weight(τ

), which is

equal to the amplitude of the sub-interval number α.

The weight value is given by the following function:

weight : [0..M − 1] → [0,1]

α 7→ a

α+1

− a

with a

the discretization bounds. Then, the new

radius ρ

of formula (3) is obtained by the algorithm

conversion mod above. The principle of this algo-

rithm is to ﬁnd the radius which causes the same mod-

iﬁcation percentage in the probability base L

that is

caused by the radius ρ

in the basis L

Algorithm 1. Begin of algorithm conversion mod.

• Input values:

– The radius ρ

of the modiﬁer to translate;

– The symbolic probability degree to modify;

– The size of the base M

• The values to initialize:

– A real proportion ←

which represents the

proportion of the radius ρ

in the correspond-

ing base L

;

– A real compteurPoids ← 0, a weight counter.

• loop through the symbolic probability degrees

with a decrement, beginning with the degree to

modify. Until compteurPoids ≤ proportion, in-

crease compteurPoids by the weight of the current

degree.

• The new radius is equal to the degree to modify

minus the current degree minus 1.

End of algorithm.

Thus, with this algorithm we determine the radius

′

to use to modify the probability degree, ie: m

′

APPROXIMATE REASONING BASED ON LINGUISTIC MODIFIERS IN A LEARNING SYSTEM

435

5.2.2 Composed Premise

SUCRAGE generates rules whose premise consists

of a conjunction of propositions. Thus, the infer-

ence process can be achieved through the Generalized

Modus Ponens based on generalized linguistic modi-

ﬁers in the case of conjunctive rules. It is as follows:

If ”X

is v

” and ... and ”X

is v

” then ”Y is v

B”

”X

is m

)” and .. . and ”X

is m

)”

”Y is m

′

B)”

The determination of m

is performed by the mod-

iﬁer determination method of simple rule, case that

we have described above. Thus, they must be oper-

ators of type CR, CW or CC. The deduced modiﬁer

′

is determined by aggregating the modiﬁers m

. We

deﬁne for that an operator that aggregate modiﬁers.

In this application framework, the conclusion uncer-

tainty degree weakens when the observation moves

away from the premise. For this reason, we deﬁned

an aggregator which is adaptable to the rules in the

SUCRAGE system:

′

) = CW

so that τ

= S(τ

,τ

) (5)

with:

m and m

′

: modiﬁers of type CR, CW or CC;

, τ

and τ

: symbolic degrees belonging to L

;

S: a T-conorm such as the Lukasiewicz T-conorm.

As in the original SUCRAGE, the rules triggered

are grouped by class. Then, the ﬁnal symbolic belief

degree of each class is calculated, and that by aggre-

gating the belief degrees by the max T-conorm. Fi-

nally, the selected class is whose ﬁnal belief degree is

the greater one.

6 EXPERIMENTAL STUDY

In this section, we ﬁrst describe the extension done

on SUCRAGE, in order to integrate approximate rea-

soning with linguistic modiﬁers. Then, we present

experimental results obtained with this application.

As we noted earlier, to implement our approach,

we need to use symbolic probability as belief degree

of the rules. For this, we integrated into SUCRAGE,

in addition to numerical probability, a new type of be-

lief degrees: irregular symbolic probability.

To perform a comparative study on the classiﬁ-

cation results, we used the learning set Iris. These

data are available on the server of the Irvine Univer-

sity of California

. This database consists of 150 ex-

amples represented by 4 numerical attributes (sepal

//ftp.ics.uci.edu/pub/machine-learning-databases/

lenth, sepal width, petal lenth, petal width). The ex-

amples are divided over 3 classes: Iris setosa, Ver-

sicolor and Virginica. The tests are made by 10-

folds cross-validation. We use different thresholds for

correlation and for discretization cardinals of the at-

tributes. We tested our approximate reasoning based

on linguistic modiﬁers with irregular symbolic prob-

abilities in Table 2.

Table 2: Iris, comparative study of approximate reasoning

with irregular symbolic probability.

Subdivision size 3 5 7

threshold Reasoning

0.95 AR

98 93.33 96

/ER

1 1 1.09

/ER

1 1.01 1.08

/AR

1 1.03 1.08

0.9 AR

98 93.33 96

/ER

1 1 1.09

/ER

1 1.01 1.08

/AR

1 1.03 1.08

0.8 AR

97.33 95.33 98

/ER

1 1.05 1.11

/ER

1.01 1.05 1.13

/AR

1.01 1 1.05

0.5 AR

97.33 95.33 98

/ER

1 1.05 1.11

/ER

1.01 1.05 1.13

/AR

1.01 1 1.05

The ratio rows AR

/ER

gives the division of the

classiﬁcation rate of approximative reasoning AR

whose of exact reasoning ER

with irregular sym-

bolic probability (corresponding to the subscript is).

We note that the approximate reasoning based on

linguistic modiﬁers gives a best results in all cases.

Moreover, it improves the results of exact reasoning

with irregular symbolic probability when the subdivi-

sion cardinal increases. This is because the variation

of this parameter introduces imperfections (Borgi and

Akdag, 2001). Indeed, approximate reasoning helps

to limit borders problems of the discretization: the

imperfections due to a high subdivision size are cor-

rected. The rate reaches at 98% for a subdivision of

7 which represents the best rates obtained by the SU-

CRAGE system.

We also compare our approximate reasoning to

the original exact and approximate reasoning of SU-

CRAGE. The ratio rows AR

/ER

and AR

/AR

present a comparison of our approximate reasoning

with respectively the original exact reasoning ER

and the original approximate reasoning AR

of SU-

CRAGE that is based on numerical probability (cor-

responding to the subscript n). We note through this

table that in all cases, the new version gives better re-

ICSOFT 2010 - 5th International Conference on Software and Data Technologies

436

sults than the original version.

Approximate reasoning with linguistic modiﬁers

gives satisfactory results, moreover it presents a great

advantage with regard to the numerical approach. In-

deed, approximate reasoning with linguistic modi-

ﬁers can reﬁne the interpretation of classiﬁcation re-

sults. The original version of SUCRAGE is a numer-

ical approach, the results of objects assignments to

classes are provided through numerical probabilities.

On the other side, approximate reasoning with lin-

guistic modiﬁers is a linguistic approach which pro-

vides a linguistic interpretation of the results, allow-

ing readability and easy interpretation by the human

mind. Moreover, the use of approximate reasoning

is more advantageous when the data provided by the

experts are imprecise.

7 CONCLUSIONS

In this work, we have presented an application of ap-

proximate reasoning with linguistic modiﬁers that we

have deﬁned in (Kacem et al., 2008; Borgi et al.,

2008). For this purpose, we have used a rule base

generated by a supervised learning system: SU-

CRAGE (Borgi, 1999). Some adaptations have been

made to this system in order to infer with our approx-

imate reasoning. More precisely, we have included

the use of symbolic probability (Seridi and Akdag,

2001) as belief degree of the generated rules. More-

over, we have deﬁned an aggregator of modiﬁers in

order to aggregate the modiﬁers that transform the ob-

servation elements to the premise elements. We have

noticed that classiﬁcation results were improved by

using our approximate reasoning based of linguistic

modiﬁers. This improvement was noticed in compar-

ison with the exact reasoning, as well as with the ap-

proximate reasoning introduced in (Borgi and Akdag,

2001). In addition, our approach provides a linguistic

interpretation through the use of linguistic modiﬁers.

It would be interesting to complete the validation tests

with other data, and more generally to consider an ap-

plication of our approximate reasoning on a base of

rules resulting from expert knowledge acquisition.

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