A hierarchical model of a linguistic variable

Information Sciences 181 (2011) 4394–4408

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A hierarchical model of a linguistic variable Adolfo R. de Soto 1 School of Industrial Engineering and Computer Science, University of León, 24071 LEON, Spain

a r t i c l e

i n f o

Article history: Available online 12 April 2011 Keywords: Interpretable fuzzy systems Computing with words Linguistic variables Multiresolution analysis Hierarchical models Fuzzy sets

a b s t r a c t In this work a theoretical hierarchical model of dichotomous linguistic variables is presented. The model incorporates certain features of the approximate reasoning approach and others of the Fuzzy Control approach to Fuzzy Linguistic Variables. It allows sharing of the same hierarchical structure between the syntactic definition of a linguistic variable and its semantic definition given by fuzzy sets. This fact makes it easier to build symbolic operations between linguistic terms with a better grounded semantic interpretation. Moreover, the family of fuzzy sets which gives the semantics of each linguistic term constitutes a multiresolution system, and thanks to that any fuzzy set can be represented in terms of the set of linguistic terms. The model can also be considered a general framework for building more interpretable fuzzy linguistic variables with a high capacity of accuracy, which could be used to build more interpretable Fuzzy Rule Based Systems (FRBS). Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The concept of Fuzzy Linguistic Variable is a staple of the Fuzzy Set Theory [41,48]. It has the remarkable property of putting together symbols and the meaning of those symbols as proper elements of a computational system [47]. However, calculating with both levels simultaneously is not a trivial problem. Many symbolic operations have a very large set of possible interpretations as fuzzy operations. Besides that, many fuzzy operations on fuzzy sets do not have a clear correspondence with symbolic operations or do not correspond to inner operations in the set of symbolic terms. For these reasons, both aspects, the symbolic and the aspect relative to the meaning, have been developed differently in Fuzzy Set Theory and its applications. Calculus with fuzzy sets has had a higher prominence than calculus with symbols or words. The main hypothesis of this work is that a stronger structure within the set of linguistic terms of a linguistic variable is needed, but with the condition of a semantic model being associated with the same structure. In this way, inner symbolic operations would also have as a result a fuzzy set which gives their meaning. Essentially, the linguistic variable concept introduces two levels for manipulating ‘‘words’’, the syntactic or symbolic level, where the names of the words are given and certain operations can be defined working on those symbols to generate new symbols, and the semantic or meaning level, where Fuzzy Sets are introduced to give the meaning of each symbolic word. Both levels are expressed explicitly in the initial definition of a Linguistic Variable given by Zadeh [43–45]. In the original definition, no structure is considered for the set of linguistic terms, in spite of the fact that both linguistic connectives such as and, or, not and linguistic modifiers such as very, little, at most, were considered from the beginning for building derived terms from the basic terms. Zadeh’s definition does not fix either the semantic mapping which gives the semantic value of each linguistic term. Traditionally two different ways have been considered. The first consists of defining a different fuzzy set for each linguistic term, regardless of whether the term is simple or includes various operators or modifiers. This approach has usually been used in Fuzzy Rule Based Systems. In contrast, the second approach gives a meaning to the basic 1 This work has been partially funded by the Spanish Ministry of Science and Innovation through Project TIN2008-06890-C02-01 and the Human Resources National Mobility Program through grant PR2008-0143. E-mail address: [email protected]

0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.04.006

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terms by means of fuzzy sets directly but calculates the meaning of the derived linguistic terms. Using operations on the class of fuzzy sets, a meaning for the connectives and for the linguistic modifiers is set by means of certain fuzzy operations and then those operations are used to carry out the calculation. As is well known, t-norms, t-conorms [4] and negation functions [36] are used to represent connectives, while several types of fuzzy operators are employed to represent linguistic modifiers [6,31,32]. In this work the first approach described above is considered and a hierarchical interpretation of a dichotomous Fuzzy Linguistic Variable is given. This kind of variable is characterized by the use of two antonymous linguistic terms as basic terms. The use of this type of linguistic variable is very common in natural language and human beings process a great amount of information using linguistic terms derived from these basic antonymous linguistic terms. The use of this kind of variable is perhaps an embodied property of the human mind [26], and its representation as a hierarchical classification system is able to show the two basic properties of hierarchical classifications [34]: cognitive economy and correspondence with the structure of the world, this latter being understood as the pairing of world and observer. Furthermore, in this type of linguistic variable the middle term is very important. In the model presented here, the middle term comes from one assumption in the model. The linguistic variables used in Fuzzy Rules Based Systems customarily have a finite and totally ordered number of linguistic terms. It is evident that with a finite number of linguistic terms the capacity for accuracy is restricted. In fact, when more accuracy is needed in a fuzzy rule based system the usual procedure is to increase the number of fuzzy labels. Working with a finite number of labels is easier than using an infinite number. For that reason, many expert systems use a finite number of symbolic linguistic terms, without a fuzzy interpretation, to carry out reasoning, following the model of finite multivalued logics. The problem with this approach is that a serious restriction is imposed on the accuracy. On the contrary, if a natural language is considered, an infinite number of linguistic terms seems more plausible. At least potentially, natural languages have the possibility to produce a non-bounded number of linguistic terms associated with a linguistic variable through the use of connectives or linguistic modifiers. As far as approximate reasoning is concerned, this would appear to be true. Nevertheless, it is also true that human beings make use of only a few linguistic terms, and employ linguistic modifiers to build new linguistic expressions through the intensification or weakening of the meaning of a linguistic term. In this way, it is possible to attain greater accuracy. However, the number of different linguistic expressions that people use in their vocabulary is quite limited, in spite of their potential capacity to produce an infinite number of them. The model in this work is also based on a multiresolution scheme which permits working simultaneously with several levels of resolution, and in this way tries to solve the problem of finite resolution versus potentially infinite resolution. In many problems that human beings solve easily, resolution is set at the level which is sufficient to solve the problem in question. For example, to find out if a piece of furniture will fit against a wall, people use an accuracy of centimetres, which is good enough. On the contrary, for a carpenter to make that piece of furniture probably requires an accuracy of millimetres. The objective is to be able to work in Fuzzy Rule Based Systems with an adequate resolution for the problem. Multiresolution Analysis Theory, which was introduced by Mallat in [27], is usually used in Wavelet Theory [9] as the main method for building wavelet functions. Another important issue in this work is the matter of interpretability of Fuzzy Rule Based Systems [7]. At the beginning of their development, human interpretability of FRBS was perceived to be an essential aspect of them. However, in many works on the topic of FRBS accuracy is the main concern. This work considers interpretability as a fundamental property of FRBS and interpretability is reached through a simple structure in the linguistic label set and through having as few rules as possible. Nevertheless, the problem is how to achieve a high degree of accuracy with few rules and a simple linguistic term set. A hierarchical model can help to solve this problem. Hierarchical Fuzzy Rule Based Systems have frequently been used for modeling systems [35,39] but fewer examples exist that use Hierarchical Linguistic Variables [14,19]. The approach to interpretability of FRBS adopted here is mainly focused on getting a good structure in the set of linguistic terms of a Linguistic Variable used in FRBS. Following the taxonomy given in [18], this paper is only concerned with the semantic interpretability of a Linguistic Variable. But a relevant difference with usual approaches to FRBS interpretability (see, for example, [2,3,7,22,23]) has to be remarked. The final goal here is not to obtain a high evaluation in the interpretability of one FRBS in an isolated way, but to build interpretable Linguistic Variables to be used in several related FRBS which incorporate a common variable with different levels of precision. For example, in the field of home automation, several FRBS can use the temperature variable with different precision in different systems. Using the same set of linguistic terms for the temperature, in spite of the different precision required in each FRBS, can improve the interpretability of the whole system. Several other works have studied the problem of the interpretability in Fuzzy Rule Based Systems through defining an adequate structure in the set of linguistic terms of a Linguistic Variable. In their works, Cat Ho and collaborators [8] built an abstract algebraical structure to model linguistic modifiers as operations on the set of linguistic terms. This was solely a symbolic approach and the meaning of the terms was not considered. Work reference [33] modeled linguistic expressions of natural language which involved trichotomous linguistic variables, i.e. linguistic variables with three basic linguistic terms, such as small–medium–large using a formal system of higher-order Fuzzy Logic. Again, being a symbolic approach, this model was independent of the meaning given to the basic linguistic terms and to the linguistic modifiers. Finally, work reference [5] introduced a definition of interpretability in Linguistic Variables by means of defining certain relationships within the set of linguistic terms which are preserved in the semantic interpretation counterpart (given by fuzzy sets). None of these works considers the approach adopted here, where the meaning of every linguistic term is defined directly without using operators to derive the meaning of a composite linguistic term. Similarly, a multiresolution perspective has not been

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considered anywhere. Finally, some aspects of the approach adopted in this work have been partially outlined in previous works by the author in conferences [14,15], but never in the complete form given here. The rest of this paper is structured as follows. The first section introduces the model of a Hierarchical Linguistic Variable in accordance with the Zadeh’s definition. Two main interpretations of the model can be considered, both of which are explained in the section. Special importance is given to the semantic aspects of the model through multiresolution systems of functions. Various basic issues relating to ordering in the set of hierarchical linguistic terms are given with the aim of showing the possibility of defining symbolic operations on the set of linguistic terms. Thereafter, a Hierarchical Linguistic Variable allows the definition of a multiresolution descriptor for a fuzzy set and it is shown how in this model the basic linguistic labels can be considered as a basis of the space of fuzzy sets. Finally, a basic learning method grounded in multiresolution analysis is presented. With this method, it is possible to fix the meaning of the set of linguistic terms in a Hierarchical Linguistic Variable, adapting the meaning to the context without losing the hierarchical structure of the Linguistic Variable. 2. Hierarchical Linguistic Variables Following Zadeh’s definition, a linguistic variable is given by a quintuplet LV ¼ hV; T ; U; G; Mi where V is the name of the variable, T is the set of linguistic terms, U is the universe of discourse, G is a set of syntactic rules which generate the set of linguistic terms, and M is a set of semantic rules which assign a meaning, given by a fuzzy set, to each linguistic term in T. In the hierarchical model which is proposed here the main elements of a linguistic variable are given as follows:  T: only a unique basic term is considered: e0, and the rest is generated by the grammar G,  U: the unit interval [0, 1] will be the basic universe but any real closed interval can be considered,  G: there is only one rule in the set of syntactic rules which generate the linguistic terms:

S ) e0 jlðSÞjrðSÞ; where l, r are two operators named left and right operators respectively.  M: The semantic rule can be given by means of a biparametric family of fuzzy sets which are shifting and scaling versions of a mother fuzzy set in a similar way to the definition of a family of scaling functions in a multiresolution system for building a Wavelet System [9,27]. In fact that case is a particular case of using basis B-spline to set the semantics of a Hierarchical Linguistic Variable. As is shown below, it is possible to assign different semantics to these labels, depending on the basic properties of the linguistic term set. The hierarchical model tries to cope with the very frequent linguistic variables which have a pair of basic linguistic terms with a relationship of antinomy between them such as short–tall, small–large, etc. These terms are l(e0) and r(e0). In the model, each linguistic term e belongs to a hierarchical level given by the number n of times that the operators l, r are applied to generate e. So e0 has level 0, l(e), r(e) have level 1, ll(e), lr(e), rl(e), rr(e) have level 2 and so forth. Let Tn denote the set of linguistic terms of level n and T ¼ [n2N T n the set of all linguistic terms. In Fig. 1 a graphic representation of T is shown. The operators l, r can have different interpretations. For example, it is possible to consider that these operators allow the building of new linguistic terms as linguistic modifiers do. So, on each level there are different linguistic terms achieving a greater capacity for accuracy. The basic terms maintain their meaning and when new levels are considered, new linguistic terms are introduced. This interpretation will be named the approximate reasoning interpretation. On the other hand, it is possible to adopt a Fuzzy Control point of view of a Hierarchical Linguistic Variable and to consider that each level gives a whole set of linguistic labels. Then, when a change of level occurs, a new set of linguistic labels is obtained with a higher capacity for accuracy. In this interpretation, the meaning of a linguistic label changes in a way dependent upon the level under consideration. It would be as if the presence of new labels restrict the meaning of the old ones. For example, the concept of young can change if only two concepts, young and old, are considered when classifying all people, in contrast with the situation when further concepts, such as child, young, middle-aged and old, are considered. This interpretation is here designed the Fuzzy Control interpretation. Under this interpretation, the operators l, r can be viewed as mappings from a coarser level of resolution to a finer one. Lastly, the operators l, r can be interpreted as the linguistic expressions less than and more than, respectively, similarly to what was done in Ref. [5]. They can be used to build a partition at a given interval. With this interpretation, each label e will be partitioned into two parts given by l(e) and r(e). This interpretation is here termed the partition interpretation.

Fig. 1. A hierarchical linguistic variable.

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2.1. Two hierarchical models Hierarchies consist of different levels which may or may not be nested. In terms of the theory of categorization, a hierarchical structure where the categories of one level are always included in the level above it is called a taxonomy. In the model of a Hierarchical Linguistic Variable it is possible to assign different semantics to the linguistic terms depending on whether a taxonomy between the linguistic labels is assumed or not. To discuss different ways to define the semantic rule M for a Hierarchical Linguistic Variable, account is taken of the family of triangular fuzzy sets given by

  m  m; n 2 N; n P m; /mn ðxÞ ¼ max 0; 1  nx   n where /00(x) = 1. Other parametric families can be considered using higher-order basis B-spline functions [11], as is shown in the next section. They are introduced in the setting of multiresolution analysis. Triangular fuzzy sets are basis B-splines of degree 1 according the notation used later in this paper. Basically, two possible models can be posited; they depend on whether or not the operators l and r commute. Under the Approximate Reasoning interpretation, it is possible to interpret the different levels as new levels of precision in the set of linguistic terms in the same way as linguistic modifiers do. The use of a linguistic modifier would in that case involve the step of going from a lower level to a higher level of precision. For example, taking the linguistic modifier very and the linguistic modifier slightly, it is possible to question whether very slightly old is equivalent to slightly very old. If these are seen as equivalent terms, it is being assumed that the operators l and r commute and lr(S) = rl(S). If this is the case, the set of linguistic terms Tn of the same level can be written as

T n ¼ femn : n 2 N; m 2 f0; . . . ; ngg and it is possible then to use the following notation:

e00 ¼ e0 e01 ¼ lðe00 Þ e11 ¼ rðe00 Þ  emnþ1 ¼ lðemn Þ emþ1nþ1 ¼ rðemn Þ 

Fig. 2. The model when l, r commute.

Fig. 3. A semantic interpretation when l, r commute.

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The equality emn = lnmrm(e00) gives the syntactic production used to generate the label emn. Fig. 2 gives a graphical representation of this model. With fuzzy triangular numbers, the semantic rule M would be direct (see Fig. 3)

  m  Mðemn Þ ¼ /mn ðxÞ ¼ max 0; 1  nx   m; n 2 N; n P m: n If, on the contrary, it is assumed that l and r are interpreted under the partition interpretation, the Hierarchical Linguistic Variable defines a taxonomy and each label contains two labels of a finer level. This model can be obtained merely just positing a non-commutative interaction between the operators l, r. In this case the linguistic term set can be defined as

T n ¼ femn : m; n 2 N; 1 6 m 6 2n g and the different levels are written as

e10 ¼ e0 e11 ¼ lðe00 Þ e21 ¼ rðe00 Þ  e2m1nþ1 ¼ lðemn Þ e2mnþ1 ¼ rðemn Þ  So on each level n there are 2n linguistic terms and the subindex n gives the level, while the subindex m gives the position of the label within the level n (see Fig. 4). This model was used in [2] to define a method to learn a FRBS using different levels of resolution in the set of linguistic terms. In this case, recovering the syntactic expression of a linguistic term emn from its position within the level is a little more complicated. Given a linguistic term e, let es1 ;...;sn be its expression in terms of the operators l, r, where si 2 {l, r}. The expression es1 ;...;sn is interpreted as es1 ;...;sn ¼ s1 . . . sn ðe0 Þ. Then, for each emn it will be the case that emn ¼ es1 ;...;sn when the indexes si are obtained as

si ¼

8 > :

if m > 2n1 þ

nP 1

ð1Þsjþ1 2j1 ;

j¼i

l

otherwise;

Fig. 4. A non-commutative model.

Fig. 5. A semantic interpretation when l, r do not commute.

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where

ð1Þsi ¼



4399

if si ¼ r;

1

1 if si ¼ l:

The semantic rule in this case will be (see Fig. 5)

    m  1  Mðemn Þ ¼ /m12n 1 ðxÞ ¼ max 0; 1  ð2n  1Þx  n 2  1 taking into account that the particular case of M(e10) = 1. 2.2. The semantic mapping of a Hierarchical Linguistic Variable To establish the meaning of a linguistic term in a linguistic variable, two approaches are possible. One of them involves fixing the meaning of the basic linguistic terms and calculating the meaning of the composite linguistic terms through the fuzzy interpretation of the connectives and linguistic modifiers. The other consists of direct assignment of a fuzzy set to each linguistic term independently of whether it is considered a basic term or a composite term. This is the approach adopted here. The whole set of linguistic terms is seen as a multiresolution system with the capacity to represent information with different levels of resolution. Multiresolution analysis is based on a nested sequence of successive approximation functional spaces

    V 2  V 1  V 0  V 1  V 2    

ð1Þ

[ V j ¼ L2 ðRÞ

ð2Þ

with

j2Z

and

\

V j ¼ f0g;

ð3Þ

j2Z

where L2 ðRÞ is the space of all square Lebesgue integrable functions in the real space R. The spaces Vi are used to approximate general functions and since the union of all of them is dense in L2 ðRÞ, any given function can be approximated arbitrarily closely by means of its projections Pjf on Vj. Multiresolution analysis is mainly used in Wavelet Theory [9] to build wavelet families. In the previous section, the semantics of the linguistic terms was expressed by a family of triangular fuzzy sets. That example is simply a particular case of using basis B-splines functions to model fuzzy sets. In fact, B-splines can be used to build multiresolution analysis and so the set of linguistic terms of a Hierarchical Linguistic Variable can be represented as a multiresolution system. 2.2.1. B-splines B-spline functions, of degree k, are piecewise polynomial functions which can be generated as a linear combination of particular B-splines named basis B-splines. The i-th basis B-spline Bi,k(jt) of degree2 k for the non-decreasing real sequence t = (ti)i=1,. . .,m of knots can be defined by the Cox–de Boor recursive formula:



Bi;0 ðxjtÞ ¼

Bi;k ðxjtÞ ¼

1 if t i 6 x < t iþ1 ; 0

otherwise;

i ¼ 1; . . . ; m  k  1;

x  ti t iþkþ1  x Bi;k1 ðxjtÞ þ Biþ1;k1 ðxjtÞ i ¼ 1; . . . ; m  k  1: t iþkþ1  t iþ1 t iþk  t i

Hence, there are m  k  1 basis B-splines of degree k from a sequence of m knots. As example, basis B-splines of degree 0 are indicator functions:

 Bi;0 ðxÞ ¼ 1½ti ;tiþ1  ¼

1 if t i 6 x < tiþ1 ; 0 otherwise;

while basis B-splines of degree 1 are linear functions and they are triangular fuzzy sets:

Bi;1 ðxÞ ¼

2

8 xt i > > < tiþ1 ti

t iþ2 x > tiþ2 tiþ1

> :

0

if t i 6 x < t iþ1 ; if t iþ1 6 x < t iþ2 ; otherwise:

k is the degree of the polynomial constituents of Bi,k,t.

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Basis B-splines of degree 2 are quadratic functions, basis B-splines of degree 3 are cubic functions and so on. All these functions can be used to represent fuzzy sets. In fact several properties of basis B-splines are important when they are used as membership functions: 1. They are positive: Bi,k(xjt) P 0 for all x, 2. They have local support: Bi,k(xjt) = 0 for all x R [ti, ti+1] 3. The basis B-splines of degree k are a partition of the unity: mk1 X

Bi;k ðxjtÞ ¼ 1:

i¼0

Hence they have the usual basic properties needed to have a good interpretability as membership functions used to represent linguistic terms of a linguistic variable. In [28], a set of basic properties of an interpretable fuzzy set and an interpretable set of linguistic terms of a linguistic variable was given. The properties were formulated as a set of constraints for any automatic method to obtain information granules from data. In the case of the model given here, any basis B-spline is always a strictly convex, unimodal, continuous fuzzy set and it is normalize in the case of degree 1. Moreover, each level i is a set of fuzzy sets which are properly ordered, distinguishable, complete and constitute a complementary uniform granulation. Another important case is obtained when B0 is taken as the characteristic function of the interval ½ 12 ; 12 and Bk is calculated as the convolution product

the constant Haar function equal to 1;

b0 ¼ X ½0;1Þ ;

br ¼ br1  b0 ; where ⁄ is the convolution operator

ðf  gÞðxÞ ¼

Z

f ðx  tÞgðtÞdt:

R

These functions are named centred or cardinal B-splines. A B-spline S of degree k is given by a linear combination of basis B-splines Bi,k of degree k

SðxjtÞ ¼

mk1 X

pi Bi;k ðxjtÞ;

x 2 ½tkþ1 ; t mk1 ;

pi 2 R:

i¼1

When the knot vector is uniformly spaced, the B-splines are named uniform B-splines. In this case, the basis B-splines are just shifted copies of each other, i.e. Bi,k = Bk(x  ti) and the space of B-splines forms a multiresolution system and it is possible to obtain a Wavelet System from the family of basis B-splines [30]. When non-uniform knot sequences are considered, it is also possible to obtain a multiresolution system if the set of knot vectors form a nested family of vectors. A family L of non-decreasing sequences {tl}l2L is a nested family of sequences if tl  tl+1, taking the vectors as sets of points. A level of linguistic terms Tn (under the approximate reasoning interpolation described in the previous section) has n + 1 labels. It is possible to map those labels to the set of n + 1 basis B-splines of degree k. It must be taken into account that to obtain n + 1 basis B-splines functions it is necessary to define a sequence of knots of at least m = n + k + 2 points. Hence if a set of nested sequences {tl}l=1,. . .,1 is defined with length(tl) = n + k + 2 a meaning to T ¼ [n2N T n is obtained. That system constitutes a multiresolution system and thanks to the properties of the basis B-splines of degree k it is possible to choose the smoothness of the functions. 3. Symbolic aspects of a Hierarchical Linguistic Variable Having a fixed structure on the linguistic term set allows the definition of certain symbolic operations on the linguistic terms, from which the meaning of the result can be obtained, thanks to the mapping between the set of linguistic terms and the set of fuzzy sets which give their meaning. In fact, as will be shown in a later section, it is possible to obtain the meaning of each linguistic term of a Hierarchical Linguistic Variable with automatic procedures, so the meaning of each linguistic term can be different but the structural relationship between linguistic terms will be the same. In fact, it may be possible to define various common-sense relationships for this kind of linguistic variable, valid for any Hierarchical Linguistic Variable. 3.1. Orderings on the set of hierarchical linguistic terms Hierarchical Systems are characterized by a partial ordering between their elements [1]. In the model presented here, several orderings can be defined for the set of linguistic terms. At this point, only the case where the operators l, r commute is taken into account, so the set of linguistic terms to consider is T ¼ [n2N T n where

T n ¼ femn : n 2 N; m 2 f0; . . . ; ngg:

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Fig. 6. The basic ordering relations for any admissible ordering.

Fig. 7. The lexicographic ordering.

Of particular interest are orderings which maintain the usual ordering within a level, and which also take into account the hierarchical structure between levels in such a way that labels at lower levels are smaller than labels at higher levels. Definition 3.1. An ordering relation 6s is said to be admissible for T if the next two conditions are satisfied: 1. 8n 2 N emn 6s em0 n when m 6 m0 2. 8n 2 N e0n 6s e0nþ1 and enn 6s enþ1nþ1 This definition imposes a minimal set of ordering relations between the elements of Tn, as can be seen in Fig. 6. Definition 3.1 can be satisfied with total ordering and partial orderings. Proposition 3.2. The ordering relation 62 defined as:

emn 62 em0 n0 () ðn < n0 Þ _ ðn ¼ n0 ^ m 6 m0 Þ is a total admissible ordering named lexicographic order. Fig. 7 illustrates this ordering. Consideration will now be given to the relationship defined by

emn ~2 em0 n0

if



n 6 n0

and mn0 6 m0 n; or

n > n0

and 0 – m 6 m0 :

This relation is reflexive and anti-symmetrical, but it is a non-transitive relationship because with n0 > n > m > 0 it is the case that e0n0 ~2 e1n0 ; e1n0 ~2 e1n ; e1n ~2 emn but e0n0  ~ 2 emn . Taking 2 (see Fig. 8) as the transitive closure of ~2 the following result ensues. Proposition 3.3. The relation 2 is an admissible partial ordering relation for T. The following result shows an interesting property of the ordering given above. Theorem 3.4. Under the ordering 2, two labels on the same level always have an intermediate label.3 This theorem is not necessarily valid for any given pair of labels, but in a certain sense it states that it is possible to refine any information given by two labels in a Hierarchical Linguistic Variable. 3

The proof of this theorem is simply a proof by cases and because of his length it is omitted.

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Fig. 8. The ordering relation 2 (some relationships are not shown).

Another method to define an admissible ordering on T is by the extension principle. The operations:

MINðl1 ; l2 ÞðzÞ ¼ MAXðl1 ; l2 ÞðzÞ ¼

sup z¼minðx;yÞ

minðl1 ðxÞ; l2 ðyÞÞ;

sup z¼maxðx;yÞ

minðl1 ðxÞ; l2 ðyÞÞ

define a distribution lattice for the class of fuzzy sets with the order given by

l1 6 l2 () MINðl1 ; l2 Þ ¼ l1 : If the meaning of the labels emn is given by the fuzzy sets defined in the universe [0, 1] by triangular membership

  o m  emn ðxÞ ¼ max 0; 1  nx  ; m; n 2 N; n P m n the order defined from the extension principle for the set T is an admissible ordering, provided that the label e00 is not considered. As a symbolic ordering it can be defined by the expression

emn 3 em0 n0

8 > < m ¼ n ¼ 0 or () m ¼ n; m0 ¼ n0 ; n 6 n0 or > : jn  n0 j 6 m0 n  mn0 :

4. A multiresolution descriptor of a fuzzy set Zadeh introduced in his paper [46] the concept of a descriptor for the elements of a universe of discourse U given by a set of linguistic terms T = {t1, . . . , tn}. For each element u 2 U a fuzzy set D(u) in the set F ðTÞ of fuzzy sets over T is defined. The value D(u)(ti) = ti(u) 2 [0, 1] expresses the compatibility level between the element u and the linguistic term ti. Following these ideas, Willaeys and Malvache [40] defined an approximate descriptor for any fuzzy set in a universe U by a weighted set of labels belonging to a basic vocabulary covering the universe of discourse. At a later point, Dubois et al. [16] defined several other approximate descriptors of a fuzzy set. That definition is followed here. Let l 2 F ðUÞ be a fuzzy set over the universe U and T ¼ ft 1 ; . . . ; t n g a set of linguistic labels. An approximate descriptor D is a mapping from F ðUÞ to the class of fuzzy sets F ðT Þ over T . D expresses the fuzzy set l in terms of the linguistic labels of T . Some examples of approximate descriptors are: 1. Dþ ðlÞðti Þ ¼ supu2U minðlðuÞ; ti ðuÞÞ. This descriptor can be interpreted as the degree of possibility that an element u of U compatible with l is among the elements of U covered by the term ti. 2. D ðlÞðti Þ ¼ inf u2U maxð1  lðuÞ; t i ðuÞÞ; which can be interpreted as the degree of inclusion of l into ti. 3. D ðlÞðt i Þ ¼ inf u2U maxð1  ti ðuÞ; lðuÞÞ; which corresponds to the fuzzy set of terms which certainly entail l.

In the reference mentioned above, several properties of these descriptors were shown. In particular, the pair (D⁄(l), D+(l)) can be viewed as the fuzzy extension of a rough set where D⁄(l) is a lower approximation and D+(l) is an upper approximation. Several other definitions are possible. For example, if the linguistic labels are normalized, one can have

Dc ðlÞðt i Þ ¼ lðcti Þ; where cti is a point such that ti ðcti Þ ¼ 1. Then, it is proved that

D ðlÞðt i Þ 6 Dc ðlÞðt i Þ 6 Dþ ðlÞðt i Þ for all t i 2 T .

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Fig. 9. Hierarchical linguistic terms as vectors.

In [15] a multiresolution descriptor of a fuzzy set was introduced. That descriptor is based on multiresolution schemata and can be defined from a Hierarchical Linguistic Variable under the taxonomic or non-commutative interpretation of the operators l, r. 4.1. The multiresolution descriptor A multiresolution analysis can be built by fixing conditions (see [9]) with regard to a scaling function /. From here onwards, the function /(x) = max(0, 1  jxj), which is a fuzzy triangular number centred at 0 and a second-order B-spline, will be considered. However, other scaling functions can be taken into consideration, in particular other B-spline functions [11] of higher order which are also accepted as fuzzy sets. From the scaling function / a multiresolution analysis M/ is generated by defining functions4 /ij(x) = /(2ix  j) = max(0, 1  j2ix  jj) and considering the linear span functional vector spaces V i ¼ spanf/ij : j 2 Zg. The functions /ij are triangular fuzzy sets constructed with integer shifts and dilations by powers of 2, with support 2i[j  1, j + 1] and symmetric with respect to 21j. R Let the space of all measurable and square integrable fuzzy sets, that is, those l such that R ðlðxÞÞ2 dx < 1 over a compact 5 universe U now be considered. Given l 2 F ðUÞ and the multiresolution system M/, the projection of l in Vi can be calculated as:

lij ¼

Z

lðxÞ/ij ðxÞdx

U

and by defining

li ðxÞ ¼

X

lij /ij ðxÞ

j2Z

an approximation is obtained for the fuzzy set l in Vi. The sequence lij is finite because U is a compact set, and it represents l in Vi. Therefore, an approximate descriptor of l at resolution i can be defined as

Di ðlÞð/ij Þ ¼ lij :

ð4Þ

By fixing the resolution i, it is possible to represent each fuzzy set l with the sequence u = (lij). Since every fuzzy set l R R satisfies l(x) 6 1, it holds that U /ij lðxÞ 6 U /ij . The last integral is equal to 1 for /0 and it is less than 1 for all /ij with i > 0. Only resolutions with i P 0 are here taken into account, hence the sequence u belongs to the product space ½0; 1ni where the number of factors ni depends on the chosen resolution i. When there is no ambiguity, the multiresolution factor may be denoted by n. The space [0, 1]n = {u = (u1, . . . , un)} may be taken to be the fuzzy sets space at a fixed resolution given by the multiresolution analysis M/. The fuzzy sets /ij will be the base term set of linguistic labels. Again, if no problems of ambiguity arise, they will be denoted by /j, where the index j takes values between 1 and n. It is of interest to note various points with regard to this representation: 1. Singletons ux0 all go to the sequence 0 because of the equation

Z

U

ux0 ðxÞ/ij ðxÞdx ¼ 0 8i; j 2 Z:

So, in this model, the calculus on fuzzy sets is not a generalization of classical set theory because the singletons are lost. 2. Constant fuzzy sets l(x) = k go to constants sequences because of the equation

Z U

k/ij ðxÞdx ¼ k

Z

/ij ðxÞdx

U

and for a fixed i, the integral of /ij is always constant. 4

Note that the interpretation of the indexes in this formula is different of the notation given in the Section 2.1. The problems with the borders in the setting of Wavelet Theory are not taking in consideration here because it would does unnecessarily complicate the exposition. 5

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3. The basic linguistic labels are represented with unit vectors. That is, /j is associated with the vector (0, . . . , 1, . . . 0), with the 1 in position j. Fig. 9 illustrates the idea of a set of linguistic terms in a Hierarchical Linguistic variable as a set of basis vectors, where each basis vector is associated with a linguistic term. P 4. Given a vector u = (u1, . . . , un), the approximate fuzzy set l ¼ uj /j can be calculated by linear interpolation of the values u = (u1, . . . , un). 5. Basic Fuzzy Calculus in F i ðUÞ ¼ ½0; 1n A vector u ¼ ðu1 ; . . . ; un Þ 2 F i ðUÞ ¼ ½0; 1n represents a fuzzy set at a fixed resolution i. The usual operations in Fuzzy Set Theory can be extended in a natural way to the product space [0, 1]n. 5.1. t-Norms and t-Conorms T-norms and t-conorms are habitually used as functions to model intersection and union in Fuzzy Set Theory [4]. A t-norm T defines an operation T : F i ðUÞ F i ðUÞ ! F i ðUÞ in a pointwise way:

Tðu; v Þ ¼ ðTðu1 ; v 1 Þ; . . . ; Tðun ; v n ÞÞ: A similar operation is defined pointwise for a t-conorm. These operations on F i ðUÞ inherit the usual properties of t-norms and t-conorms, so the outcome is that t-norms and t-conorms for the product space [0, 1]n are obtained. The unit element for t-norms is the unit vector 1 = (1, . . . , 1) and the absorbent element is the zero vector 0 = (0, . . . , 0). The order 6 defined pointwise allows the proving of monotonic properties. It would be of interest to find the difference between first projecting two fuzzy sets and then computing the pointwise t-norm on the one hand, and first calculating the t-norm of both fuzzy sets and then obtaining their projection. Let li, gi be two projections of two fuzzy sets at resolution i and let lij, gij be their components in that resolution. The following properties are satisfied: 1. T(l, g)ij 6 min(lij, gij), 2. S(l, g)ij P max(lij, gij). These formulae state that the components of the projection of T(l, g) (S(l, g)) are always less than (more than) or equal to the minimum (the maximum) of the projection components of each fuzzy set. The proof in the case of a t-norm, (it is analogous for a t-conorm), is evident. From T(l, g) 6 l, it is deduced that T(l, g)/ij 6 l/ij because /ij is always positive, and by integrating both sides one has

Tðl; gÞij ¼

Z

Tðl; gÞðxÞ/ij ðxÞdx 6

Z

U

U

lðxÞ/ij dx ¼ lij :

The same can be done for gij, so the result follows. On the other hand, it is satisfied that

minðli ; gi Þ ¼ min

X

lij ;

X



gij P

X

minðlij ; gij Þ

P P P because of the truth of the relation minð ai bi ; ai ci Þ P ai minðbi ; ci Þ. It is of interest to note that hF ðcalUÞ; min; max; 0; 1i is a distributed lattice and if the basic term set of linguistic labels is considered, the outcome is a finite Boolean algebra. The lattice order is pointwise order and it may be noted that two projections li, gi verify li 6 gi if and only if each of their components satisfy (lij 6 gij). 5.2. Negation Classical negation in Fuzzy Set Theory is defined by the involution 1  x [36]. A pointwise negation can be defined for F i ðUÞ as

notðuÞ ¼ ð1  u1 ; . . . ; 1  un Þ and it is easy to prove that this is an involutive negation. Moreover, it holds that:

notðlÞij ¼

Z U

ð1  lðxÞÞ/ij ðxÞdx ¼

Z

/ij ðxÞdx  U

Z U

lðxÞ/ij ðxÞdx ¼ ki  lij

and

X X X /ij  lij /ij ¼ 1  li ; ð1  lij Þ/ij ¼ whenever

P

/ij ¼ 1, which is always the case in the multiresolution analysis M/ under consideration here.

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Note that although the negation (1, . . . , 1, 0, 1, . . . , 1) of a basic vector (0, . . . , 1, . . . , 0) is not itself a basic vector, it can still be interpreted as the sum of the rest of the basic vectors. Usually, when a finite set of symbolic labels is considered, negation is defined as the antonymous label, without the possibility of the simultaneous existence of both a real negation and the concept of an antonymous label. In the model presented here, the antonym6 of a label can be defined as:

antðuÞ ¼ ðun ; un1 ; . . . ; u1 Þ: With any admissible order I, the basic linguistic labels are in increasing order:

/1 I /2 I    I /n : Taking the antonym of each label inverts this order, and so it holds that

antð/1 ÞI antð/2 ÞI    I antð/n Þ: Pointwise negation inverts the usual pointwise order between fuzzy set projections and the lexicographic order between basic linguistic labels:

notð/1 Þ  notð/2 Þ      notð/n Þ A usual property of the antonymous label [12] is that ant (l)  not (l). If standard pointwise inclusion, that is, l  g () l 6 g is considered, this property will be satisfied when

uniþ1 6 1  ui () uniþ1 þ ui 6 1 and that relation will be true for all basic linguistic labels except for the central one, which exists when n is an odd number and it is its own antonym. 5.3. Linguistic modifiers Zadeh proposed in his earlier work [42] a representation for several linguistic modifiers. This representation was based on operations, such as square or square root, upon membership functions. For example, if ‘‘A’’ is a fuzzy set and its membership function is lA, then the membership function of ‘‘very A’’ can be calculated as lv eryA ¼ l2A . This idea can be directly taken into

multiresolution descriptors using the same operator in a pointwise way such as u2 ¼ u2i and from a semantic point of view its behaviour will be correct. However, a relationship cannot be assumed between the projections of l and the projections of l2 because this depends on the chosen multiresolution system. It is possible to assert that

ðl2 Þij ¼

Z

l2 /ij 6

Z

l/ij ¼ lij ;

R

R but in general it is not possible to compare l2 /ij directly with l/ij 2 . Nevertheless, several works [20,25,38] have shown that the behaviour of hedges is not exactly given by Zadeh’s vision of linguistic modifiers, so other models have been proposed in fuzzy theory [6,31,32]. The model based on translations within the universe of discourse is especially interesting for the present model. In this case the membership function of ‘‘very A’’ is calculated as lveryA(x) = lA(x  t). If the projections in the multiresolution system are calculated, the outcome is

ðlv eryA Þij ¼ ¼

Z

Z

lv eryA ðxÞ/ij ðxÞdx ¼

Z

lcA ðn  tÞ /cij ðnÞdn ¼

Z

c ðnÞdn ¼ cA ðnÞ / e2pitn l ij

Z

lcA ðnÞ /cij ðn  tÞdn

lA ðxÞ/ij ðx  tÞdx;

where the Parseval Identity and Fourier Calculus have been used. This relation indicates that if the appropriate value is taken for t, such as t = 2i in the case of a multiresolution system based on triangular fuzzy sets, it is possible to achieve the result that (lvery A)ij = (lA)ij1 and therefore, a unitary left or right shift of the vector (uij) can be interpreted as a linguistic modifier when this is modeled by an inner translation in the universe of discourse. It is relevant to note that the operators r, l of a Hierarchical Linguistic Variable can be interpreted as linguistic modifiers. When the operators l, r commute with each other, it holds that

lðemn Þ ¼ emnþ1 ; rðemn Þ ¼ emþ1nþ1 ; so they operate through increasing the level of resolution. In terms of the elements of the basis the result is that:

lðv Þ ¼ ðv ; 0Þ; rðv Þ ¼ ð0; v Þ;

6

This is the classic definition of the antonym label in Fuzzy Set Theory, to see another definitions the reader is referred to Ref. [37].

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where v represents any linguistic term. So these can be seen as linguistic modifiers which permit refinement of the meaning of a fuzzy label. 6. Changing the resolution level The multiresolution system M/ allows change of the resolution level with an efficient algorithm thanks to the fast wavelet transform. Moreover, it is possible to analyze the errors introduced when a lower resolution level is chosen. In particular, thanks to the multiresolution properties of the model, a FRBS which uses a Hierarchical Linguistic Variable can be changed in a local way, introducing or deleting labels in a neighborhood where the resolution is not high or low enough. In [29] can be seen an example of that with triangular fuzzy sets, which are splines of degree 1. Briefly, since Vi  Vi+1, there exists a complementary space Wi of Vi in Vi+1 such that Vi+1 = Vi Wi. In accordance with wavelet theory, there is a functional base wij in Wi such that for all li+1 2 Vi+1

liþ1 ¼

X

muiþ1j /iþ1j ¼

X

lij /ij þ

X

l~ ij wij ;

R

~ ij ¼ U lwij are the so called detail coefficients. The fast wavelet transform is an efficient method to compute the coefwhere l ~ ij Þ from li+1j and, reciprocally, these can be recovered from ðlij ; l ~ ij Þ. Consequently, from a given projection of a ficients ðlij ; l fuzzy set l in Vi+1 it is possible easily to calculate its projection in Vi and to change the error level when a lower resolution is used. For example, consider the biorthogonal (2, 2) wavelet transform [9]. Biorthogonal wavelets are characterized by the exis that generate a dual multiresolution analysis. The primal and dual tence of a dual scaling function / and a dual wavelet w systems are used for analysis and synthesis respectively. The biorthogonal (2, 2) system uses as one of its scaling functions the second-order B-spline /ðxÞ ¼ b2 ðxÞ ¼ maxð0; 1  jxjÞ. Without giving details at this point, it is the case that a wavelet transform can be performed by means of the lifting scheme [10]. This method allows a faster and more economical computation of the scaling and detail coefficients when a resolution change is needed. Let l be a fuzzy set and let (li+1,j)j=1,. . .,n be the projections of l in the subspace Vi+1. The coefficients li+1,j can be extended with zero values when j < 1 or j > n. Thanks to the lifting scheme it is possible to calculate the biorthogonal (2, 2) wavelet transform easily. The detail and scaling coefficients in a lower resolution are calculated as:

1 2

l~ i;j ¼ liþ1;2jþ1  ðliþ1;2j þ liþ1;2jþ2 Þ; 1 4

li;j ¼ liþ1;2j þ ðl~ i;j1 þ l~ i;j Þ: Again, the wavelet inverse transform is easily calculated from the detail and scaling coefficients of a lower resolution:

1 4

liþ1;2j ¼ li;j  ðl~ i;j1 þ l~ i;j Þ; 1 2

liþ1;2jþ1 ¼ l~ i;jþ1 þ ðliþ1;2j þ li;2jþ2 Þ: 7. Changing the semantics of a Hierarchical Linguistic Variable A Hierarchical Linguistic Variable can be seen as a multiresolution system to approximate any square integrable function. In fact, as was shown in [13,14] any function can be approximated to any degree of accuracy with a Takagi–Sugeno Fuzzy Rule Based System based on a multiresolution system. That means it is possible to express any data to any degree of accuracy in terms of the labels of a Hierarchical Linguistic Variable. Two approaches are admissible to achieve this expressiveness. First, without changing the meaning of linguistic terms, use can be made of the multiresolution analysis given above. This approach is based on fixing an admissible error or threshold of approximation and using any linguistic terms of any level to obtain an error below that threshold. In this approach, the meaning of the linguistic labels is fixed, and a high degree of approximation is obtained by using labels of a finer level where they are necessary and labels of a coarser level where a rough approximation is enough. Another approach to obtain a high accuracy without changing the meaning of the fuzzy sets is to use different weights for each rule in the set of rules of a FRBS [22,23]. However, another approach is also possible. It consists of changing the meaning of the linguistic terms as a function of the data. In fact, this is the approach taken in [13,14]. With this method, after fixing the greatest error level to be admitted, using the Fast Wavelet Transform algorithm, it is possible to build a fuzzy partition of a certain number of linguistic labels which can be considered the semantic definition of the linguistic terms of the appropriate level in a Hierarchical Linguistic Variable. The remaining levels can be built by adding details to obtain more linguistic terms or eliminating details to obtain fewer linguistic terms. In this case, the meaning of each linguistic term changes as a function of the data, and this method can be used as a supervised learning method. Using the Fast Wavelet Transform Algorithm is not the only possible way of fixing the meaning of each linguistic term in a Hierarchical Linguistic Variable. In a more general learning setting, methods more similar to the usual human learning

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methods are needed. In these cases, semi-supervised or unsupervised approaches would be more suitable, and different clustering methods can be used. Generally speaking, any clustering method which produces a hierarchical structure of clusters can be adapted to find the meaning of the linguistic terms of a Hierarchical Linguistic Variable. In [17] a fuzzy hierarchical clustering was studied, and it can easily be applied to get a Hierarchical Linguistic Variable which follows the partition model. For this kind of model, it is also possible to apply recursively to any linguistic term a fuzzy version of a linear Support Vector Machine [21] and so obtain a taxonomy of fuzzy bipartitioned classes. If an approximate reasoning model is considered, it is possible to use other Hierarchical Clustering Algorithms [24]. However, more careful research is needed into this subject because there is an important difference between a general fuzzy multi-class classification problem and the problem of obtaining a hierarchical family of fuzzy sets which can model a Hierarchical Linguistic Variable in the sense defined here. In the present instance, a fixed structural relationship between the clusters is given, so more information is available. Perhaps this information can be used to obtain general methods of classification similar to those used by humans. 8. Conclusion The concept of Linguistic Variable has played a fundamental role in applying the Fuzzy Set Theory to the building of Fuzzy Rule Based Systems as well as to the development of techniques for Approximate Reasoning. Moreover, Linguistic Variables must be the foundation for advancing along the Computing With Words path. In this work, a model of a Hierarchical Linguistic Variable has been introduced. It has been shown how (in accordance with the classical definition of a Linguistic Variable) a very simple hierarchical classification of the linguistic terms can be built in the specific case of dichotomous linguistic variables. The model uses just one basic term and two operators. The semantics of the linguistic terms is given by a mapping which directly assigns a fuzzy set to each linguistic term as is usually done in Fuzzy Rule Based Systems. However, in this model an infinite set of linguistic terms is generated. As opposed to other models of Linguistic Variable which consider an infinite number of linguistic terms, in this model it is not necessary to define the meaning of the operators which generate new linguistic terms as operators on the set of fuzzy sets. For example, this way of proceeding is common when linguistic modifiers are used to generate new linguistic terms, and some fuzzy interpretation is needed to set the meaning of a composite linguistic term. The set of fuzzy sets which fix the meaning of a Hierarchical Linguistic Variable forms a multiresolution system of functions. Several good examples of these families are given in this work by using the family of basis B-splines functions. Every family can be characterized by a set of nested intervals defined by points where each higher level is a refinement of the level below. Using B-splines of different smoothness, linear and non-linear examples can be built. Different nested sequences of knots allow to the designer create non-symmetrical or non-uniform partitions. Under this model, the set of linguistic terms and the set of fuzzy sets which give their meaning have the same structure. That arrangement has several advantages. It allows an adequate interpretation to be given of the capacity of the set of linguistic terms of a linguistic variable to reach an arbitrary degree of accuracy. It is possible to define operators on the set of linguistic terms as symbolic operations without losing an acceptable interpretation of the operation result as a fuzzy set. In other words, it is possible to define inner operations on the set of words which always have an adequate meaning given by a fuzzy set. Another important issue is that thanks to the properties of multiresolution systems, any fuzzy set can be described in terms of the basic linguistic terms, and operations between those descriptions can be defined. Several automatic methods for obtaining the meaning of each linguistic term have been commented upon. Using Bsplines to obtain Fuzzy Rule Based Systems is a well-known method in the field but how to exploit the multiresolution capacity of the model to improve the construction of accurate and interpretable Fuzzy Rule Based Systems presented here is a subject that needs further research. Finally, another interesting line of research is to study how having the same structure in different Linguistic Variables could allow the building of common-sense rules valid for any linguistic variable which follows that structure and thus allow the creation of rules which belong to a higher abstraction level. Acknowledgements The author would like to acknowledge the anonymous reviewers for their very careful reading and their useful comments and suggestions. References [1] V. Ahl, T.F.H. Allen, Hierarchy Theory: A Vision, Vocabulary, and Epistemology, Columbia University Press, 1996. [2] R. Alcalá, J.R. Cano, O. Cordón, F. Herrera, Pedro Villar, Igor Zwir, Linguistic modeling with hierarchical systems of weighted linguistic rules, International Journal of Approximate Reasoning 32 (2-3) (2003) 187–215. [3] J.M. Alonso, L. Magdalena, G. Serge, HILK: a new methodology for designing highly interpretable linguistic knowledge bases using the fuzzy logic formalism, International Journal of Intelligence Systems 23 (2008) 761–794. [4] C. Alsina, E. Trillas, L. Valverde, On some logical connectives for fuzzy sets theory, Journal of Mathematical Analysis and Applications 93 (1) (1983) 15– 26.

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A.R. de Soto / Information Sciences 181 (2011) 4394–4408

[5] U. Bodenhofer, P. Bauer, Interpretability of linguistic variables: a formal account, Kybernetika 41 (2) (2005) 227–248. [6] B. Bouchon-Meunier, Y. Yia, Linguistic modifiers and imprecise categories, International Journal of Intelligent Systems 7 (1992) 25–36. [7] J. Casillas, O. Cordón, F. Herrera, L. Magdalena (Eds.), Interpretability Issues in Fuzzy Modeling, Studies in Fuzziness and Soft Computing, vol. 12, Springer, 2003. [8] N. Cat Ho, Huynh V. Nam, An algebraic approach to linguistic hedges in zadeh’s fuzzy logic, Fuzzy Sets and Systems 129 (2002) 229–254. [9] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1992. [10] I. Daubechies, W. Sweldens, Factoring Wavelet Transforms into Lifting Steps, Journal of Fourier Analysis and Applications 4 (3) (1998) 245–267. [11] C. de Boor, A Practical Guide to Splines, Springer, Verlag, 1978. [12] A.R. de Soto, E. Trillas, On antonym and negate in fuzzy logic, International Journal of Intelligent Systems 14 (1999) 295–303. [13] A.R. de Soto, Building a hierarchical set of fuzzy rules using wavelet theory, in: International Conference on Information Processing and Management of Uncertainty Knowledge-Based Systems, vol. 3, Madrid, July 2000, pp. 1764–1769. [14] A.R. de Soto, Uniform fuzzy partitions with cardinal splines and wavelets: getting interpretable linguistic fuzzy models, in: J. Casillas, O. Cordon, L. Herrera, L. Magdalena (Eds.), Interpretability Issues in Fuzzy Modeling, Studies in Fuzziness and Soft Computing, vol. 128, Physica-Verlag, 2003, pp. 477–498. [15] A.R. de Soto, A multiresolution approximate descriptor of a fuzzy set, in: Proceedings of 2007 FUZZ-IEEE Conference on Fuzzy Systems, London, 2007, pp. 1186–1191. [16] D. Dubois, L. Foulloy, S. Galichet, H. Prade, Performing approximate reasoning with words?, in: L.A. Zadeh, J. Kacprzyk (Eds.), Computing With Words in Information/Intelligent Systems, Physica-Verlag, Berlin,Germany, 1999, pp. 24–49. [17] D. Dumitrescu, B. Lazzerini, L.C. Jain, Fuzzy Sets and their Application to Clustering and Training, CRC Press, Inc., Boca Raton, FL,USA, 2000. [18] M.J. Gacto, R. Alcalá, F. Herrera, Interpretability of linguistic fuzzy rule-based systems: an overview of interpretability measures, Information Sciences 181 (20) (2011) 4340–4360. [19] F. Herrera, L. Martinez, A model based on linguistic 2-tuples for dealing with multigranularity hierarchical linguistic contexts in multiexpert decisionmaking, IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics 31 (2001) 227–234. [20] H.M. Hersh, A. Caramazza, A fuzzy set approach to modifiers and vagueness in natural language, Journal of Experimental Psychology General 105 (3) (1976) 254–276. [21] T. Inoue, S. Abe, Fuzzy support vector machines for pattern classification, Proceedings of International Joint Conference on Neural Networks, IJCNN’01, vol. 2, 2001, pp. 2:1449–1454. [22] H. Ishibuchi, T. Murata, I.B. Türksen, Single-objective and two-objective genetic algorithms for selecting linguistic rules for pattern classification problems, Fuzzy Sets and Systems 89 (1997) 135–150. [23] H. Ishibuchi, T. Yamamoto, Rule weight specification in fuzzy rule-based classification systems, IEEE Transactions on Fuzzy Systems 13 (4) (2005) 428– 435. [24] A.K. Jain, M.N. Murty, P.J. Flynn, Data clustering: a review, ACM Computing Surveys 31 (3) (1999) 264–323. [25] G. Lakoff, Hedges: A study of meaning criteria and the logic of fuzzy concepts, Journal of Philosophical Logic 2 (1973) 458–508. [26] G. Lakoff, M. Johnson, Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought, Basic Books, 1999. [27] S.G. Mallat, Multiresolution approximations and wavelet orthogonal bases of L2 ðR), Transactions of the American Mathematical Society 315 (1989) 69– 87. [28] C. Mencar, A.M. Fanelli, Interpretability constraints for fuzzy information granulation, Information Sciences 178 (24) (2008) 4585–4618. [29] C. Moraga, A Metasemantics to refine fuzzy if–then rules, in: Proceedings of the 34th International Symposium on Multiple-Valued Logic, May 2004, pp. 148 – 153. [30] F. Müller, P. Brigger, K. Illgner, M. Unser, Multiresolution approximation using shifted splines, IEEE Transactions on Signal Processing 46 (9) (1998) 2555–2558. [31] A.M. Norwich, I.B. Turksen, A model for the measurement of membership and the consequences of its empirical implementation, Fuzzy Sets and Systems 12 (1984) 1–25. [32] V. Novák, Fuzzy Sets and their Applications, Adam Hilger, Bristol-Philadelphia, 1989. [33] V. Novák, A comprehensive theory of trichotomous evaluative linguistic expressions, Fuzzy Sets and Systems 159 (22) (2008) 2939–2969. [34] E. Rosch, Principles of Categorization, in: Eleanor Rosch, Barbara B. Lloyd (Eds.), Cognition and Categorization, John Wiley & Sons Inc., 1978, pp. 27–48. [35] V. Torra, Special issue on hierarchical fuzzy systems, International Journal of Intelligent Systems 17 (447) (2002). [36] E. Trillas, Sobre funciones de negación en la teorı´a de conjuntos difusos, Stochastica III (1) (1979) 47–60 (In Spanish. Reprinted (English version) in Advances of Fuzzy Logic, S. Barro et al. (Eds.), Press of the Universidad de Santiago de Compostela, 1998, pp. 31–43). [37] E. Trillas, C. Moraga, S. Guadarrama, S. Cubillo, E. Castiñeira, Computing with Antonyms, in: M. Nikravesh, J. Kacprzyk, L. Zadeh (Eds.), Forging New Frontiers: Fuzzy Pioneers I, vol. 217, Springer, Berlin/ Heidelberg, 2007, pp. 133–153. [38] P.J. Mac Vicar-Whelan, Fuzzy sets, the concept of height, and the hedge ‘‘Very’’, IEEE Transactions on Systems, Man and Cybernetics 8 (1978) 507–511. [39] D. Wang, X.-J. Zeng, J.A. Keane, A survey of hierarchical fuzzy systems, International Journal of Computational Cognition 4 (1) (2006). [40] D. Willaeys, N. Malvache, The use of fuzzy sets for the treatment of fuzzy information by computer, Fuzzy Sets and Systems 5 (3) (1981) 323–327. [41] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. [42] L.A. Zadeh, A fuzzy-set-theoretic interpretation of linguistic hedges, Journal of Cybernetics 2 (1972) 4–34. [43] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. Part I, Information Sciences 8 (1975) 199–249. [44] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. Part II, Information Sciences 8 (1975) 301–357. [45] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. Part III, Information Sciences 9 (1976) 43–80. [46] L.A. Zadeh, Quantitative fuzzy semantics, in: A. Lotfi Zadeh (Ed.), Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers, World Scientific Publishing Co.,, Inc., River Edge, NJ, USA, 1996, pp. 105–122. [47] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – An outline, Information Sciences 172 (1–2) (2005) 1–40. [48] L.A. Zadeh, Is there a need for fuzzy logic?, Information Sciences 13 (2008) 2751–2779