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Representation model of information in linguistics terms
Fuzzy Sets and Systems 107 (1999) 277–287 www.elsevier.com/locate/fss
Representation model of information in linguistics terms1 A. Blanco ∗ , M. Delgado, W. Fajardo Departamento Ciencias de la ComputaciÃon e Inteligencia Artiÿcial, Universidad de Granada, E.T.S.I. InformÃatica, 18071 Granada, Spain Received August 1996; received in revised form October 1997
Abstract We present a codi cation model that allows to implement a system of fuzzy relations on any discrete decision model. The paper begins with an introduction about the representation of fuzzy information, to present afterwards the codi cation c 1999 Elsevier Science B.V. method and nally enumerates the advantages and disadvantages of the presented methods. All rights reserved.
1. Introduction Automatic systems for information management have usually been designed in such a way that they need a specialized protocol for the input and output information’s trac. Human beings cannot be seen as a precision mechanism. They usually express the knowledge about the world using natural language full of vague and imprecise concepts. Thus, he is obliged to use arti cial devices (such as sensors) to obtain the pieces of information requested by the mentioned protocols. It seems reasonable to design systems which are able to process the information as the user gives it, with fuzzy and imprecise concepts. There are two solutions to this question: • To design new systems to work directly with fuzzy information, forgetting all the previous systems (with all the advantages and disadvantages). 1
Paper nanced by the DGICYT’s project PB 92-0945. Corresponding author. Tel.: +58 243 100; e-mail: aragorn@ robinson.ugr.es. ∗
• To design a method that, like a lter, admits the information given by the user in order to convert it into pieces of information needed by the speci c system’s protocol. This solution allows us to use the old systems. In this paper we look for methods to extend Arti cial Intelligence’s classic models (nonfuzzy models) to fuzzy models. In other words, we research for a knowledge representation that allows a classic model (crisp model) to be transformed into a model that works with fuzzy information. A computer is a discrete machine that works with discrete models in a natural form. It seems reasonable to think that the best form to research a computational model is to use a discrete representation of the linguistic information in models designed as crisp models. For this reason, it is obviously desirable to work with the fuzzy information with a discrete representation instead of the usual continuous representation. According to what has been said before and considering that the system must adapt to the user, we are going to develop a method of imprecise data
c 1999 Elsevier Science B.V. All rights reserved. 0165-0114/99/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 3 6 5 - 5
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Fig. 1. Input=output interface as a functional element of the system.
codiÿcation with linguistics labels. In this form, we can use crisp models of tested ecacy to obtain models that can work directly with linguistic information. For this, we will work with linguistic labels and we will introduce a codi cation method that permits us to work with systems that were not designed for this proposal. When we apply the codiÿcation method to a pattern set, we obtain a representation that can be processed by every crisp system. In this form, we obtain a procedure to extend non fuzzy models, particularly crisp models, to fuzzy models. These fuzzy models can work with information expressed by linguistic labels. Therefore, we make use of the advantages of the crisp systems, because we can use these crisp systems to work with continuous fuzzy information. The continuous fuzzy information will codify in function of the requirements of the particular crisp model. We call these systems linguistic systems because they work with linguistic terms. We wish to emphasize that we can use discrete models of arti cial intelligence as a host to research the fuzzy system. This is possible because the codi cation method that we propose is easily transportable and it has been researched to work as an input=output interface (as we can see in Figure 1). If we know the necessary requirements of the host to process the discrete information we can codify fuzzy information to be processed by this discrete host system. We divide this study into four parts: • A rst part (paragraphs 2 and 3) where we show how to codify and decodify in a discrete form a piece of information that is given in a linguistic form using the incremental discretization method.
• A second part (paragraph 4) where we describe how to codify IF–THEN linguistic rules using incremental discretization method. • And a third part (paragraph 5) where we introduce an extension of the incremental discretization method that permits us to represent uncertainty using a simultaneous representation of multiple uncertainty metrics. • Finally, we conclude with a conclusion set (paragraph 6) obtained from this paper and the bibliography (paragraph 7) used.
2. Codifying linguistic variables by incremental discretization method We performed our study on the basis of fuzzy information representation’s method highly accepted and very extended in the reasoning with fuzzy information (see [3,7–9]), where any element u from V is represented with terms over T (H ) as u ≡ 1 t1 ; : : : ; n tn ;
i ∈ [0; 1];
ti ∈ T (H );
(1)
where i notes the compatibility of u and ti (it is calculated using an ad hoc function, particularly the possibility, necessity or any other uncertainty measure). Founded on this representation we propose to codify the values’ variable as follows. Supposing that T (H ) is composed by n elements. We give to T (H ) an arbitrary order and every term from T (H ) is associated to a vector of dimension m (supposing that we want a global precision of order 1=m, although we can really determine a di erent precision for each term). A vector of m×n dimension is associated to the totality of T (H ). A pattern can be presented in a discrete form by di erent protocols as a function of the particular goal pretended by the correspondent codi cation. There are many possibilities of discrete codi cations, for example, the use by Blanco [2], Ishibuchi [5], etc. Their characteristics depend on the speci c problem to solve and the mode particularly for which they have been designed. One of these methods codi es the fuzzy information using a discretization where A represents a fuzzy set (and its ownership function), whose support is the interval [u0 ; u1 ]. A k natural is elected and made
A. Blanco et al. / Fuzzy Sets and Systems 107 (1999) 277–287
We have obtained a codi cation system that can express any element of the discourse universe V with terms from the set T (H ).
a domain partition, obtaining the elements: si = u0 + (i − 1)
u1 − u0 ; k −1
279
i = {1; 2; : : : ; K}
Scheme 1.
starting from these they associated the vector: a = (a1 ; a2 ; : : : ; ak ) = (A(s1 ); A(s2 ); : : : ; A(sk )); ai = A(si ) ∈ [0; 1] to the vector s = (s1 ; s2 ; : : : ; sk ): When a fuzzy system described by rules with fuzzy antecedents and consequents, is discretized in such a form that if the rules are composed by variables r and t in the antecedent and consequent, respectively, each rule can be considered as a function from [0, 1]n1 +···+nr to [0,1]m1 +···+mt , where ni and mj are the corresponding sizes of the discretizations referenced by i =1; : : : ; n and j=1; : : : ; m. In this form, and to every fuzzy system, the input can be modeled by n-dimensional vector (x1 ; x2 ; : : : ; xn ) and the output by the m-dimensional vector (y1 ; y2 ; : : : ; ym ) with xi ; yj ∈[0; 1], i = 1; : : : ; n; j = 1; : : : ; m. This codi cation solves the fuzzy sets’ representation problem, but it does not use the label form. For this reason, when the codi cation is restricted to the linguistic approach of the linguistic problems it can represent only information expressed in linguistic terms. We are going to propose another codi cation method that permits to represent any linguistic variable’s value purely expressed or not in linguistic terms. Then, we codify u with a binary (or bipolar) vector of dimension m×n: C(u) = (C11 ; : : : ; C1m ; : : : ; Cn1 ; : : : ; Cnm ); where i = 0 → Cij = 0; i = 1; : : : ; n; j = 1; : : : ; m; j j+1 ∃j t :q : m 6 i ¡ m ; i 6= 0 → Ci1 = 1 if 16j; 0 if 1¿j
(2)
(3)
in another form, C(u) is a vector whose components Cij are calculated with the algorithm summarized in the scheme 1.
With this idea as a basis we can extend the codi cation to the general case when the variable value is given by a fuzzy subset A over V . For this reason, we again consider i as the compatibility grade of A with ti adequately measured. In particular, if A = tk then it is codi ed with 1 if i = k; ∀j; (4) Cij = 0 in other case: Here, when it is used only as linguistic label, it is redundant to use a high number of bits for the label, for this, we can economize using only one bit (m = 1) per term ti ∈ T (H ). As mentioned above, in this case and in the precedent cases, when we xed the bits’ number to use in the codi cation, we are determining the maximum precision that we will permit in the problem. From the beginning until the end, if we use 1 or 10 bits per label we determine that the most ne appreciations developed along the problem are, respectively, unary or decimal. For this reason, we must anticipate at the beginning the maximum precision to work. Let us see an example that clari es the de nition. Example Let us suppose that we pretend to codify the linguistic variable high, which is sited where the terms set T (H ) is equal to (short, medium, high). The discourse universe V is equal to [0.50, 2.50] meters and the semantic representation of the terms are shown in Fig. 2. Then we can nd three di erent situations: Case 1: The discourse universe’s values V are expressed by experts in terms of elements over the set T (H ) such as shown in (1), more particularly the
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Fig. 2. Semantic representation of the linguistic variable high.
presented information is the next: 0.2 short, 0.8 medium. Then as we have seen in (2) and (3) and 1 supposing that we want a global precision of order 10 we will have C(u) = (0:8 short; 0:2 med; 0:0 high) = (1111111100 1100000000 00000000); which will be the de nitive codi cation of the presented information. Case 2: The values of the discourse universe V are obtained by measures with devices that give crisp information. In this case we are obliged to calculate the i that determine the compatibility of the observations u and ti . We can determine this compatibility by the semantic representation of the linguistic variable thanks to the ownership’s functions of the associated fuzzy sets (see Fig. 2). Therefore if the appreciation of the sampling device gets us a value equal to 1.79, we’ll obtain (see Fig. 3): – short = short (1:79) = 0:0, – medium = medium (1:79) = 0:1, – high = high (1:79) = 0:9: With this, we pretend to codify the value 0.1 medium, 0.9 high of the linguistic variable high. We will proceed as in the previous case. Thus, supposing 1 such as we have we like a precision level equal to 10 seen in (2) and (3) we will have C(u) = (0:0 short; 0:1 medium; 0:9 high) = (0000000000 1000000000 1111111110):
This is going to be the codi cate result of the information given by the sensor. Case 3: Finally, it can happen that the information will be given by a fuzzy subset over V . As we saw, it is enough to consider i as the compatibility grade of u over ti measured adequately. Particularly, if u = tk = medium and we use 10 bits for each term then it is codi ed according to (4), as C(u) = (0:0 short; 1:0 medium; 0:0 high) = (0000000000 1111111111 0000000000) however, as we commented, here we must use a bit per term, according to (4) we have C(u) = (0:0 short; 1:0 medium; 0:0 high) = (0 1 0):
3. A note about decodiÿcation of results At this point, we must comment on the decodi cation process. Our process has two steps that can be described in the next form: • Step 1: A → 1 t1 ; : : : ; n tn . • Step 2: 1 t1 ; : : : ; n tn → C(A) ≡ (C11 ; : : : ; Cm1 ; : : : ; C1n ; : : : ; Cmn ). If we have C(A), we obtain the representation 1 t1 ; : : : ; n tn immediately, because we only must make the inverse process to the codi cation (step 2), according to the scheme 2.
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Fig. 3. Ownership grade of the element 1.79 to the linguistic labels short, medium and high.
The correspondence is biunique. There is a unique codi cation to a representation and vice versa. Now, we must obtain A. The diculty is that the correspondence between the information representation and the codi cation is biunique but the correspondence between the information and its representation expressed in terms of compatibility with t from T (H ) is not. That is, to each piece of information corresponds a unique representation, but to each representation does not correspond a unique piece of information. We can see an example in Fig. (4). If we are restoring a piece of information that is the solution of a control system, we can use a standard decodi cation method, for example the centroid method. Scheme 2.
4. Codiÿcation of IF–THEN linguistic rules by incremental discretization method A fuzzy system may be identi ed by a rule based system ([1,4]). A rule is a cause–e ect association and can be codi ed in a discrete form. We look for a
method to represent fuzzy information by a discrete codi cation. A rule can be seen as a pattern association with a complexity level directly proportional to itself. Particularly, the complexity level is given by the number of the components of antecedents and consequents and it is logic complexity. According to what has been said before about the codi cation of linguistics variable values and the different situations that can be present (pieces of information expressed in compatibility terms, in crisp terms or in purely linguistics terms), we must have no problem to codify rules expressed in linguistics terms. As in the linguistics variables case, we can nd three di erent situations with respect to the values of the antecedent and consequent. In this form, a rule (A; B) with a fuzzy antecedent and consequent, where A and B are given by the n and p labels where their respective linguistic variables can be valued, is expressed by vectors of characteristics. The components of these vectors are the n and p linguistics labels where A and B can be valued, respectively. In this form, each vector component (linguistic label) is codi ed by a sequence of bits. This sequence has a variable length in function of the precision of the description used. If the precision is j for all linguistics labels of the antecedent A and k to all linguistics labels of the consequent B, we can express the fuzzy rule (A; B) with a pair of vectors with n × j and p × k elements, respectively. If we look for a system to memorize pairs of patterns (A; B), where A is described by n labels and B is described by p labels and if the precision to the labels of A is j and k to the labels of B, then, (A; B) is expressed by a pair of vectors with dimension equal to n × j and p × k.
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Fig. 4. The representation “0.4 short, 1 medium, 0.2 high” is the same to the fuzzy sets I and I0 . Table 1 1. 2. 3. 4. 5. 6. 7.
IF IF IF IF IF IF IF
position position position position position position position
positive medium and velocity zero THEN force negative medium. positive short and velocity positive short THEN force negative short. positive short and velocity negative short THEN force zero. negative medium and velocity zero THEN force positive medium. negative short and velocity negative short THEN force positive short. negative short and velocity positive short THEN force zero. zero and velocity zero THEN force zero.
In another form, we describe (A; B) at the beginning as Ai = (ai1 ; ai2 ; : : : ; ain ); Bi = (bi1 ; bi2 ; : : : ; bip ): And now, how we codify it is shown in the following Table A
B
a11 ; : : : ; a1j : : : an1 ; : : : ; anj b11 ; : : : ; b1k : : : bp1 ; : : : ; bpk In paragraph 2 of this function we explained clearly how to codify a linguistic variable. According to that method we show an example to obtain the binary (or bipolar) vectors corresponding to the representation of rules as cause–e ect associations expressed in linguistics terms. Example We suppose that we pretend to make a brake system model. This brake system pretends to stop a mobile at a determined point. The model is characterized by the rules set that is showed in the Table 1. Furthermore, we pretend to integrate the decision model (expressed
in linguistic terms) over a system that process crisp information only. We can codifying the position, velocity and force according to the linguistics labels showed in Figs. 5–7. We are interested in codifying the information with a decimal precision level. Consequently, we codify the labels using 10 bits and according to its -cuts (obtained from the membership function to the respective fuzzy sets) to the di erent cases that can be as we showed earlier. We only show the case of information expressed in terms of compatibility level over each label. The compatibility level can be a value in [0; 1]. The result codi ed to the component Y1 from the set of learning patterns, that correspond to “a negative medium force with membership grade equal to 1” will be: 1111111111 0000000000 0000000000 0000000000 0000000000 its discretization appearance is shown in Fig. 8. In this form, we codify the m learning pattern pars corresponding to the rules as associations pars (X; Y ) where X corresponds to Position-Velocity, and Y to Force.
A. Blanco et al. / Fuzzy Sets and Systems 107 (1999) 277–287
Fig. 5. Semantic representation of the linguistic variable position.
Fig. 6. Semantic representation of the linguistic variable velocity.
Fig. 7. Linguistic representation of the linguistic variable force.
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Fig. 8. Discretization of the force linguistics labels.
Table 2
Table 3
Force NM P O S I T I O N
NS
N NS
Values PS
PM
PM PS
ZE PS PM
Velocity ZE
ZE
NM NS ZE PS PM
Negative medium Negative short Zero Positive short Positive medium
ZE ZE
NS NM
Using this representation method, it seems a logical consequence that if we take into account the limitations of codi cation and the capacity from the host system, we can implement fuzzy systems using crisp systems. We can do it using the techniques of di erent authors to process pieces of information guaranteing the recovery of the learning pattern. Therefore, we will have the fuzzy information represented in a discrete form and we will be able to use it according to the particular protocol from the elected host model to the fuzzy system. When we have codi ed the information from Tables 2 and 3, we represent the information in the form of membership grades to the antecedent linguistic label. In this form if the membership grade from a determinate element to a determinate label is 0.3, the corresponding sub-vector will have its three rst bits equal to 1 and the rest equal to 0 (we suppose a decimal precision grade). Particularly and in function
of the example, if we wanted to know the force necessary to a velocity 0.2 negative short, 0.8 zero and a position 0.7 negative medium and 0.3 negative short, we give to the system the next vector: 2 1100000000 1111111100 0000000000 1111111000 1110000000 0000000000 0000000000 0000000000 and the system will give us the corresponding solution (according to the system’s characteristics). This solution will be interpreted as we described in the paragraph 3.
5. Extension from the codiÿcation of linguistic labels by incremental discretization method to represent uncertainty If we come back to the codi cation of linguistic labels by the incremental discretization de nition that 2 We present it discomposed in two sub-strings corresponding to position and velocity because of the evident space problems.
A. Blanco et al. / Fuzzy Sets and Systems 107 (1999) 277–287
we introduced earlier and seeing the Pedrycz’s uncertainty representation [9]. We are now interested in extending the information presented to the system codifying it using a simultaneous representation of possibility and necessity or with the help of an other two uncertainty metrics. In this form, we will have the option of analyzing the consistency for the information represented. To do it, we will represent u ∈ V in terms of T (H ) as u ≡ 1 t1 ; ÿ1 t1 ; : : : ; n tn ; ÿn tn ; i ; ÿi ∈ [0; 1]; ti ∈T (H ); (5) where i is the possibility and ÿi is the necessity from u in respect to ti . We are going to extend the variable codi cation as we will explain now. As we have seen, T (H ) is composed of n elements. We use an arbitrary order to T (H ), to each term in T (H ) is associated two vectors which are mdimensional (if we like a global 3 precision of order 1=m), one to the possibility and the other to the necessity. To all H is associated a vector with dimension 2m × n. In this form, u is codi ed with a binary or bipolar vector of dimension 2m × n:
And in an analogous form ÿi = 0 → CijN = 0; i = 1; : : : ; n; j = 1; : : : ; m; j j+1 ∃j t : q : m 6ÿi ¡ m ; (8) ÿi 6= 0 → 1 if 16j; Ci1N = 0 if 1¿j that is C(u) is a vector with components CijP ; CijN calculated according to the algorithm that we can see in the schemes 3 and 4. We have extended our linguistics variables codi cation system that can codify every element from the discourse universe V in terms of elements from set T (H ) to a system that allows codifying each of these elements as a function of any two measures of uncertainty. We can extend the system easily to use simultaneously for as many uncertainty representation functions as we like. Scheme 3.
P P N N ; : : : ; C1m ; C11 ; : : : ; C1m ;:::; (C11 P P N N ; : : : ; Cnm ; Cn1 ; : : : ; Cnm ) Cn1
(6)
where, as we saw, if i = 0 → CijP = 0; i = 1; : : : ; n; j = 1; : : : ; m; j j+1 ∃j t :q : m 6 i ¡ m ; (7) i 6= 0 → 1 if 16j; Ci1P = 0 if 1¿j:
3 We talk about global precision to make the exposition understanding easier. Really, we can use a distinct precision to each term.
285
Scheme 4.
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In this form we can obtain an information that codi es imprecise information, and furthermore we can make a posterior analysis of the con ict or ignorance grade from the knowledge given to the system. To do it we must obtain the datum according to two dual uncertainty measures, to quantify uncertainty resident within the input datum in a posterior analysis. To this objective we can use the possibility and necessity as uncertainty measures. For this, B represents one of the elements of the frame of cognition while X constitutes an input datum. X and B are de ned in the same universe of discourse. The possibility measure is given by (X |B) = sup [min(X (z); B(z))] z∈ X
(9)
and the necessity by Nec(X |B) = inf [max((1 − X (z)); B(z))]: z∈ X
(10)
When X is a precise numerical information, these two measures coincide. If X becomes a numerical interval, the di erences between the possibility and necessity measures could be di erent from zero. In fact, the following monotonicity property holds: I (X1 |B) − Nec(X1 |B)6(X2 |B) − Nec(X2 |B); X1 ⊂ X
(11)
Pedrycz [9] suggests that we consider the two measures collectively and treat them as a convenient vehicle to quantify uncertainty resident within the input datum. To do it, it is de ned as = ;
= 1 − Nec (X |B);
(12)
three cases can be distinguished: • + = 1: no uncertainty. • + ¿1: con ict. X is a con icting piece of evidence as it invokes both B and its complement. • ++1: ignorance. X indicates a lack of a sucient support making it dicult to make any decision that is either in favor of B or against it. 6. Conclusions Now, we must show its advantages and disadvantages
– We have made a method to represent information expressed in terms of a linguistic system with a binary or bipolar code. – With this codi cation we can work on linguistic problems using procedures developed for crisp problems. – Considering that it is a codi cation method to be implanted over a host model, the resulting system obtains the characteristic typical of the host model. – The most important advantage of this codi cation method is the simplicity. The models obtained as a result are easy to implement because the method is easily intelligible and afterwards is implanted in a very studied host model. – Also, we must show up its operation facility. The method is not only easy to manipulate but it does not suppose either a big computational cost to increment the pattern’s number to codify. – Finally, we must talk about the system’s versatility. In previous systems it was necessary to design a speci c model for every problem, and particularly for every discourse universe in case that the actuation interval had changed. – The versatility is not only limited to work with any system described by using linguistics labels, furthermore the resulting system after the codi cation can work with mixed systems. It can work with systems that use antecedents (or consequents) crisp and consequents (or antecedents) expressed using linguistics labels. – The application of this representation to mechanisms of measure of uncertainty, fuzzy and nonfuzzy, shows the utility of the numeric approximation presented. The idea of combining fuzzy and and nonfuzzy information can be useful when the number of nonfuzzy pieces of information is limited by input problems (problems with the size of the measures and=or interferences). In these situations, the subjective linguistics appreciations given by an expert are very useful in the construction of fuzzy machines. – We can use representations that use simultaneously more than one mechanism of uncertainty representation. In this way, we can represent the information and analyze the consistency level of the source that gives us this information. This is important, because an expert can be an information source with an unestimable value, particularly when we cannot
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monitor the sytsem to emulate, but always when we work with this or another information source we must verify its reliability level. To these advantages we must add all the speci cally inherent advantages of the host system. References [1] A. Blanco, Identi cacion de Sistemas Mediante Redes Neuronales, Tesis Doctoral, Universidad de Granada, 1993. [2] A. Blanco, A learning procedure to identify weighted rules by neural networks, Fuzzy Sets and Systems 69 (1994) 29–36. [3] M. Delgado, J.L. Verdegay, M.A. Vila, A linguistic version of the compositional rule of inference, preprints Sicica’92, Symposium of Intelligent Components and Instruments for Control Applications, 1992, pp. 141–148. [4] W. Fajardo, Recuperacion de informacion con memorias asociativas difusas, Tesis Doctoral, Universidad de Granada, 1995. [5] H. Ishibuchi, H. Tanaka, H. Okada, Interpolation of fuzzy ifthen rules by neural networks, Internat. J. Approx. Reasoning, 10 (1994) 3–27. [6] M. Mizumoto, H.J. Zimmermann, Comparison of fuzzy reasoning methods, Fuzzy Sets and Systems 8 (1982) 253–283.
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[7] W. Pedrycz, Numerical and applicational aspects of fuzzy relational equations, Fuzzy Sets and Systems 11 (1983) 1–18. [8] W. Pedrycz, Applications of fuzzy relational equations for method of reasoning in presence of fuzzy data, Fuzzy Sets and Systems 16 (1985) 163–175. [9] W. Pedrycz, Fuzzy Sets Engineering, CRC Press, Boca Raton. [10] T. Yamakawa, A simple fuzzy controller hardware system employing MIN & MAX operations, a challenge to 6th generation computer, preprints of Second IFSA Congress, Tokyo, 20 –25 July, 1987, 827– 830. [11] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems Man Cybernet. SSMC-3, 1973, 82 – 84. [12] L.A. Zadeh, The concept of a linguistic variable and its applications to approximate reasoning, Part I., Inform. Sci. 8 (1975) 199–249, Part II. Inform. Sci. 8 (1975) 301–357, Part III. Inform. Sci. 9 (1975) 43 – 80. [13] L.A. Zadeh, A theory of approximate reasoning, in: J.E. Hayes, D. Michie, L.I. Mikulich (Eds.), Machine Intelligence, vol. 9, Elsevier, Amsterdam, 1979, pp. 149 –194. [14] L.A. Zadeh, Fuzzy sets and information granularity, in: M.M. Gupta et al. (Eds.), Advances in Fuzzy Sets Theory and Applications, North-Holland New York, 1979, pp. 3 –19.
Representation model of information in linguistics terms1 A. Blanco ∗ , M. Delgado, W. Fajardo Departamento Ciencias de la ComputaciÃon e Inteligencia Artiÿcial, Universidad de Granada, E.T.S.I. InformÃatica, 18071 Granada, Spain Received August 1996; received in revised form October 1997
Abstract We present a codi cation model that allows to implement a system of fuzzy relations on any discrete decision model. The paper begins with an introduction about the representation of fuzzy information, to present afterwards the codi cation c 1999 Elsevier Science B.V. method and nally enumerates the advantages and disadvantages of the presented methods. All rights reserved.
1. Introduction Automatic systems for information management have usually been designed in such a way that they need a specialized protocol for the input and output information’s trac. Human beings cannot be seen as a precision mechanism. They usually express the knowledge about the world using natural language full of vague and imprecise concepts. Thus, he is obliged to use arti cial devices (such as sensors) to obtain the pieces of information requested by the mentioned protocols. It seems reasonable to design systems which are able to process the information as the user gives it, with fuzzy and imprecise concepts. There are two solutions to this question: • To design new systems to work directly with fuzzy information, forgetting all the previous systems (with all the advantages and disadvantages). 1
Paper nanced by the DGICYT’s project PB 92-0945. Corresponding author. Tel.: +58 243 100; e-mail: aragorn@ robinson.ugr.es. ∗
• To design a method that, like a lter, admits the information given by the user in order to convert it into pieces of information needed by the speci c system’s protocol. This solution allows us to use the old systems. In this paper we look for methods to extend Arti cial Intelligence’s classic models (nonfuzzy models) to fuzzy models. In other words, we research for a knowledge representation that allows a classic model (crisp model) to be transformed into a model that works with fuzzy information. A computer is a discrete machine that works with discrete models in a natural form. It seems reasonable to think that the best form to research a computational model is to use a discrete representation of the linguistic information in models designed as crisp models. For this reason, it is obviously desirable to work with the fuzzy information with a discrete representation instead of the usual continuous representation. According to what has been said before and considering that the system must adapt to the user, we are going to develop a method of imprecise data
c 1999 Elsevier Science B.V. All rights reserved. 0165-0114/99/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 3 6 5 - 5
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Fig. 1. Input=output interface as a functional element of the system.
codiÿcation with linguistics labels. In this form, we can use crisp models of tested ecacy to obtain models that can work directly with linguistic information. For this, we will work with linguistic labels and we will introduce a codi cation method that permits us to work with systems that were not designed for this proposal. When we apply the codiÿcation method to a pattern set, we obtain a representation that can be processed by every crisp system. In this form, we obtain a procedure to extend non fuzzy models, particularly crisp models, to fuzzy models. These fuzzy models can work with information expressed by linguistic labels. Therefore, we make use of the advantages of the crisp systems, because we can use these crisp systems to work with continuous fuzzy information. The continuous fuzzy information will codify in function of the requirements of the particular crisp model. We call these systems linguistic systems because they work with linguistic terms. We wish to emphasize that we can use discrete models of arti cial intelligence as a host to research the fuzzy system. This is possible because the codi cation method that we propose is easily transportable and it has been researched to work as an input=output interface (as we can see in Figure 1). If we know the necessary requirements of the host to process the discrete information we can codify fuzzy information to be processed by this discrete host system. We divide this study into four parts: • A rst part (paragraphs 2 and 3) where we show how to codify and decodify in a discrete form a piece of information that is given in a linguistic form using the incremental discretization method.
• A second part (paragraph 4) where we describe how to codify IF–THEN linguistic rules using incremental discretization method. • And a third part (paragraph 5) where we introduce an extension of the incremental discretization method that permits us to represent uncertainty using a simultaneous representation of multiple uncertainty metrics. • Finally, we conclude with a conclusion set (paragraph 6) obtained from this paper and the bibliography (paragraph 7) used.
2. Codifying linguistic variables by incremental discretization method We performed our study on the basis of fuzzy information representation’s method highly accepted and very extended in the reasoning with fuzzy information (see [3,7–9]), where any element u from V is represented with terms over T (H ) as u ≡ 1 t1 ; : : : ; n tn ;
i ∈ [0; 1];
ti ∈ T (H );
(1)
where i notes the compatibility of u and ti (it is calculated using an ad hoc function, particularly the possibility, necessity or any other uncertainty measure). Founded on this representation we propose to codify the values’ variable as follows. Supposing that T (H ) is composed by n elements. We give to T (H ) an arbitrary order and every term from T (H ) is associated to a vector of dimension m (supposing that we want a global precision of order 1=m, although we can really determine a di erent precision for each term). A vector of m×n dimension is associated to the totality of T (H ). A pattern can be presented in a discrete form by di erent protocols as a function of the particular goal pretended by the correspondent codi cation. There are many possibilities of discrete codi cations, for example, the use by Blanco [2], Ishibuchi [5], etc. Their characteristics depend on the speci c problem to solve and the mode particularly for which they have been designed. One of these methods codi es the fuzzy information using a discretization where A represents a fuzzy set (and its ownership function), whose support is the interval [u0 ; u1 ]. A k natural is elected and made
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We have obtained a codi cation system that can express any element of the discourse universe V with terms from the set T (H ).
a domain partition, obtaining the elements: si = u0 + (i − 1)
u1 − u0 ; k −1
279
i = {1; 2; : : : ; K}
Scheme 1.
starting from these they associated the vector: a = (a1 ; a2 ; : : : ; ak ) = (A(s1 ); A(s2 ); : : : ; A(sk )); ai = A(si ) ∈ [0; 1] to the vector s = (s1 ; s2 ; : : : ; sk ): When a fuzzy system described by rules with fuzzy antecedents and consequents, is discretized in such a form that if the rules are composed by variables r and t in the antecedent and consequent, respectively, each rule can be considered as a function from [0, 1]n1 +···+nr to [0,1]m1 +···+mt , where ni and mj are the corresponding sizes of the discretizations referenced by i =1; : : : ; n and j=1; : : : ; m. In this form, and to every fuzzy system, the input can be modeled by n-dimensional vector (x1 ; x2 ; : : : ; xn ) and the output by the m-dimensional vector (y1 ; y2 ; : : : ; ym ) with xi ; yj ∈[0; 1], i = 1; : : : ; n; j = 1; : : : ; m. This codi cation solves the fuzzy sets’ representation problem, but it does not use the label form. For this reason, when the codi cation is restricted to the linguistic approach of the linguistic problems it can represent only information expressed in linguistic terms. We are going to propose another codi cation method that permits to represent any linguistic variable’s value purely expressed or not in linguistic terms. Then, we codify u with a binary (or bipolar) vector of dimension m×n: C(u) = (C11 ; : : : ; C1m ; : : : ; Cn1 ; : : : ; Cnm ); where i = 0 → Cij = 0; i = 1; : : : ; n; j = 1; : : : ; m; j j+1 ∃j t :q : m 6 i ¡ m ; i 6= 0 → Ci1 = 1 if 16j; 0 if 1¿j
(2)
(3)
in another form, C(u) is a vector whose components Cij are calculated with the algorithm summarized in the scheme 1.
With this idea as a basis we can extend the codi cation to the general case when the variable value is given by a fuzzy subset A over V . For this reason, we again consider i as the compatibility grade of A with ti adequately measured. In particular, if A = tk then it is codi ed with 1 if i = k; ∀j; (4) Cij = 0 in other case: Here, when it is used only as linguistic label, it is redundant to use a high number of bits for the label, for this, we can economize using only one bit (m = 1) per term ti ∈ T (H ). As mentioned above, in this case and in the precedent cases, when we xed the bits’ number to use in the codi cation, we are determining the maximum precision that we will permit in the problem. From the beginning until the end, if we use 1 or 10 bits per label we determine that the most ne appreciations developed along the problem are, respectively, unary or decimal. For this reason, we must anticipate at the beginning the maximum precision to work. Let us see an example that clari es the de nition. Example Let us suppose that we pretend to codify the linguistic variable high, which is sited where the terms set T (H ) is equal to (short, medium, high). The discourse universe V is equal to [0.50, 2.50] meters and the semantic representation of the terms are shown in Fig. 2. Then we can nd three di erent situations: Case 1: The discourse universe’s values V are expressed by experts in terms of elements over the set T (H ) such as shown in (1), more particularly the
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Fig. 2. Semantic representation of the linguistic variable high.
presented information is the next: 0.2 short, 0.8 medium. Then as we have seen in (2) and (3) and 1 supposing that we want a global precision of order 10 we will have C(u) = (0:8 short; 0:2 med; 0:0 high) = (1111111100 1100000000 00000000); which will be the de nitive codi cation of the presented information. Case 2: The values of the discourse universe V are obtained by measures with devices that give crisp information. In this case we are obliged to calculate the i that determine the compatibility of the observations u and ti . We can determine this compatibility by the semantic representation of the linguistic variable thanks to the ownership’s functions of the associated fuzzy sets (see Fig. 2). Therefore if the appreciation of the sampling device gets us a value equal to 1.79, we’ll obtain (see Fig. 3): – short = short (1:79) = 0:0, – medium = medium (1:79) = 0:1, – high = high (1:79) = 0:9: With this, we pretend to codify the value 0.1 medium, 0.9 high of the linguistic variable high. We will proceed as in the previous case. Thus, supposing 1 such as we have we like a precision level equal to 10 seen in (2) and (3) we will have C(u) = (0:0 short; 0:1 medium; 0:9 high) = (0000000000 1000000000 1111111110):
This is going to be the codi cate result of the information given by the sensor. Case 3: Finally, it can happen that the information will be given by a fuzzy subset over V . As we saw, it is enough to consider i as the compatibility grade of u over ti measured adequately. Particularly, if u = tk = medium and we use 10 bits for each term then it is codi ed according to (4), as C(u) = (0:0 short; 1:0 medium; 0:0 high) = (0000000000 1111111111 0000000000) however, as we commented, here we must use a bit per term, according to (4) we have C(u) = (0:0 short; 1:0 medium; 0:0 high) = (0 1 0):
3. A note about decodiÿcation of results At this point, we must comment on the decodi cation process. Our process has two steps that can be described in the next form: • Step 1: A → 1 t1 ; : : : ; n tn . • Step 2: 1 t1 ; : : : ; n tn → C(A) ≡ (C11 ; : : : ; Cm1 ; : : : ; C1n ; : : : ; Cmn ). If we have C(A), we obtain the representation 1 t1 ; : : : ; n tn immediately, because we only must make the inverse process to the codi cation (step 2), according to the scheme 2.
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Fig. 3. Ownership grade of the element 1.79 to the linguistic labels short, medium and high.
The correspondence is biunique. There is a unique codi cation to a representation and vice versa. Now, we must obtain A. The diculty is that the correspondence between the information representation and the codi cation is biunique but the correspondence between the information and its representation expressed in terms of compatibility with t from T (H ) is not. That is, to each piece of information corresponds a unique representation, but to each representation does not correspond a unique piece of information. We can see an example in Fig. (4). If we are restoring a piece of information that is the solution of a control system, we can use a standard decodi cation method, for example the centroid method. Scheme 2.
4. Codiÿcation of IF–THEN linguistic rules by incremental discretization method A fuzzy system may be identi ed by a rule based system ([1,4]). A rule is a cause–e ect association and can be codi ed in a discrete form. We look for a
method to represent fuzzy information by a discrete codi cation. A rule can be seen as a pattern association with a complexity level directly proportional to itself. Particularly, the complexity level is given by the number of the components of antecedents and consequents and it is logic complexity. According to what has been said before about the codi cation of linguistics variable values and the different situations that can be present (pieces of information expressed in compatibility terms, in crisp terms or in purely linguistics terms), we must have no problem to codify rules expressed in linguistics terms. As in the linguistics variables case, we can nd three di erent situations with respect to the values of the antecedent and consequent. In this form, a rule (A; B) with a fuzzy antecedent and consequent, where A and B are given by the n and p labels where their respective linguistic variables can be valued, is expressed by vectors of characteristics. The components of these vectors are the n and p linguistics labels where A and B can be valued, respectively. In this form, each vector component (linguistic label) is codi ed by a sequence of bits. This sequence has a variable length in function of the precision of the description used. If the precision is j for all linguistics labels of the antecedent A and k to all linguistics labels of the consequent B, we can express the fuzzy rule (A; B) with a pair of vectors with n × j and p × k elements, respectively. If we look for a system to memorize pairs of patterns (A; B), where A is described by n labels and B is described by p labels and if the precision to the labels of A is j and k to the labels of B, then, (A; B) is expressed by a pair of vectors with dimension equal to n × j and p × k.
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Fig. 4. The representation “0.4 short, 1 medium, 0.2 high” is the same to the fuzzy sets I and I0 . Table 1 1. 2. 3. 4. 5. 6. 7.
IF IF IF IF IF IF IF
position position position position position position position
positive medium and velocity zero THEN force negative medium. positive short and velocity positive short THEN force negative short. positive short and velocity negative short THEN force zero. negative medium and velocity zero THEN force positive medium. negative short and velocity negative short THEN force positive short. negative short and velocity positive short THEN force zero. zero and velocity zero THEN force zero.
In another form, we describe (A; B) at the beginning as Ai = (ai1 ; ai2 ; : : : ; ain ); Bi = (bi1 ; bi2 ; : : : ; bip ): And now, how we codify it is shown in the following Table A
B
a11 ; : : : ; a1j : : : an1 ; : : : ; anj b11 ; : : : ; b1k : : : bp1 ; : : : ; bpk In paragraph 2 of this function we explained clearly how to codify a linguistic variable. According to that method we show an example to obtain the binary (or bipolar) vectors corresponding to the representation of rules as cause–e ect associations expressed in linguistics terms. Example We suppose that we pretend to make a brake system model. This brake system pretends to stop a mobile at a determined point. The model is characterized by the rules set that is showed in the Table 1. Furthermore, we pretend to integrate the decision model (expressed
in linguistic terms) over a system that process crisp information only. We can codifying the position, velocity and force according to the linguistics labels showed in Figs. 5–7. We are interested in codifying the information with a decimal precision level. Consequently, we codify the labels using 10 bits and according to its -cuts (obtained from the membership function to the respective fuzzy sets) to the di erent cases that can be as we showed earlier. We only show the case of information expressed in terms of compatibility level over each label. The compatibility level can be a value in [0; 1]. The result codi ed to the component Y1 from the set of learning patterns, that correspond to “a negative medium force with membership grade equal to 1” will be: 1111111111 0000000000 0000000000 0000000000 0000000000 its discretization appearance is shown in Fig. 8. In this form, we codify the m learning pattern pars corresponding to the rules as associations pars (X; Y ) where X corresponds to Position-Velocity, and Y to Force.
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Fig. 5. Semantic representation of the linguistic variable position.
Fig. 6. Semantic representation of the linguistic variable velocity.
Fig. 7. Linguistic representation of the linguistic variable force.
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Fig. 8. Discretization of the force linguistics labels.
Table 2
Table 3
Force NM P O S I T I O N
NS
N NS
Values PS
PM
PM PS
ZE PS PM
Velocity ZE
ZE
NM NS ZE PS PM
Negative medium Negative short Zero Positive short Positive medium
ZE ZE
NS NM
Using this representation method, it seems a logical consequence that if we take into account the limitations of codi cation and the capacity from the host system, we can implement fuzzy systems using crisp systems. We can do it using the techniques of di erent authors to process pieces of information guaranteing the recovery of the learning pattern. Therefore, we will have the fuzzy information represented in a discrete form and we will be able to use it according to the particular protocol from the elected host model to the fuzzy system. When we have codi ed the information from Tables 2 and 3, we represent the information in the form of membership grades to the antecedent linguistic label. In this form if the membership grade from a determinate element to a determinate label is 0.3, the corresponding sub-vector will have its three rst bits equal to 1 and the rest equal to 0 (we suppose a decimal precision grade). Particularly and in function
of the example, if we wanted to know the force necessary to a velocity 0.2 negative short, 0.8 zero and a position 0.7 negative medium and 0.3 negative short, we give to the system the next vector: 2 1100000000 1111111100 0000000000 1111111000 1110000000 0000000000 0000000000 0000000000 and the system will give us the corresponding solution (according to the system’s characteristics). This solution will be interpreted as we described in the paragraph 3.
5. Extension from the codiÿcation of linguistic labels by incremental discretization method to represent uncertainty If we come back to the codi cation of linguistic labels by the incremental discretization de nition that 2 We present it discomposed in two sub-strings corresponding to position and velocity because of the evident space problems.
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we introduced earlier and seeing the Pedrycz’s uncertainty representation [9]. We are now interested in extending the information presented to the system codifying it using a simultaneous representation of possibility and necessity or with the help of an other two uncertainty metrics. In this form, we will have the option of analyzing the consistency for the information represented. To do it, we will represent u ∈ V in terms of T (H ) as u ≡ 1 t1 ; ÿ1 t1 ; : : : ; n tn ; ÿn tn ; i ; ÿi ∈ [0; 1]; ti ∈T (H ); (5) where i is the possibility and ÿi is the necessity from u in respect to ti . We are going to extend the variable codi cation as we will explain now. As we have seen, T (H ) is composed of n elements. We use an arbitrary order to T (H ), to each term in T (H ) is associated two vectors which are mdimensional (if we like a global 3 precision of order 1=m), one to the possibility and the other to the necessity. To all H is associated a vector with dimension 2m × n. In this form, u is codi ed with a binary or bipolar vector of dimension 2m × n:
And in an analogous form ÿi = 0 → CijN = 0; i = 1; : : : ; n; j = 1; : : : ; m; j j+1 ∃j t : q : m 6ÿi ¡ m ; (8) ÿi 6= 0 → 1 if 16j; Ci1N = 0 if 1¿j that is C(u) is a vector with components CijP ; CijN calculated according to the algorithm that we can see in the schemes 3 and 4. We have extended our linguistics variables codi cation system that can codify every element from the discourse universe V in terms of elements from set T (H ) to a system that allows codifying each of these elements as a function of any two measures of uncertainty. We can extend the system easily to use simultaneously for as many uncertainty representation functions as we like. Scheme 3.
P P N N ; : : : ; C1m ; C11 ; : : : ; C1m ;:::; (C11 P P N N ; : : : ; Cnm ; Cn1 ; : : : ; Cnm ) Cn1
(6)
where, as we saw, if i = 0 → CijP = 0; i = 1; : : : ; n; j = 1; : : : ; m; j j+1 ∃j t :q : m 6 i ¡ m ; (7) i 6= 0 → 1 if 16j; Ci1P = 0 if 1¿j:
3 We talk about global precision to make the exposition understanding easier. Really, we can use a distinct precision to each term.
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Scheme 4.
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In this form we can obtain an information that codi es imprecise information, and furthermore we can make a posterior analysis of the con ict or ignorance grade from the knowledge given to the system. To do it we must obtain the datum according to two dual uncertainty measures, to quantify uncertainty resident within the input datum in a posterior analysis. To this objective we can use the possibility and necessity as uncertainty measures. For this, B represents one of the elements of the frame of cognition while X constitutes an input datum. X and B are de ned in the same universe of discourse. The possibility measure is given by (X |B) = sup [min(X (z); B(z))] z∈ X
(9)
and the necessity by Nec(X |B) = inf [max((1 − X (z)); B(z))]: z∈ X
(10)
When X is a precise numerical information, these two measures coincide. If X becomes a numerical interval, the di erences between the possibility and necessity measures could be di erent from zero. In fact, the following monotonicity property holds: I (X1 |B) − Nec(X1 |B)6(X2 |B) − Nec(X2 |B); X1 ⊂ X
(11)
Pedrycz [9] suggests that we consider the two measures collectively and treat them as a convenient vehicle to quantify uncertainty resident within the input datum. To do it, it is de ned as = ;
= 1 − Nec (X |B);
(12)
three cases can be distinguished: • + = 1: no uncertainty. • + ¿1: con ict. X is a con icting piece of evidence as it invokes both B and its complement. • ++1: ignorance. X indicates a lack of a sucient support making it dicult to make any decision that is either in favor of B or against it. 6. Conclusions Now, we must show its advantages and disadvantages
– We have made a method to represent information expressed in terms of a linguistic system with a binary or bipolar code. – With this codi cation we can work on linguistic problems using procedures developed for crisp problems. – Considering that it is a codi cation method to be implanted over a host model, the resulting system obtains the characteristic typical of the host model. – The most important advantage of this codi cation method is the simplicity. The models obtained as a result are easy to implement because the method is easily intelligible and afterwards is implanted in a very studied host model. – Also, we must show up its operation facility. The method is not only easy to manipulate but it does not suppose either a big computational cost to increment the pattern’s number to codify. – Finally, we must talk about the system’s versatility. In previous systems it was necessary to design a speci c model for every problem, and particularly for every discourse universe in case that the actuation interval had changed. – The versatility is not only limited to work with any system described by using linguistics labels, furthermore the resulting system after the codi cation can work with mixed systems. It can work with systems that use antecedents (or consequents) crisp and consequents (or antecedents) expressed using linguistics labels. – The application of this representation to mechanisms of measure of uncertainty, fuzzy and nonfuzzy, shows the utility of the numeric approximation presented. The idea of combining fuzzy and and nonfuzzy information can be useful when the number of nonfuzzy pieces of information is limited by input problems (problems with the size of the measures and=or interferences). In these situations, the subjective linguistics appreciations given by an expert are very useful in the construction of fuzzy machines. – We can use representations that use simultaneously more than one mechanism of uncertainty representation. In this way, we can represent the information and analyze the consistency level of the source that gives us this information. This is important, because an expert can be an information source with an unestimable value, particularly when we cannot
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monitor the sytsem to emulate, but always when we work with this or another information source we must verify its reliability level. To these advantages we must add all the speci cally inherent advantages of the host system. References [1] A. Blanco, Identi cacion de Sistemas Mediante Redes Neuronales, Tesis Doctoral, Universidad de Granada, 1993. [2] A. Blanco, A learning procedure to identify weighted rules by neural networks, Fuzzy Sets and Systems 69 (1994) 29–36. [3] M. Delgado, J.L. Verdegay, M.A. Vila, A linguistic version of the compositional rule of inference, preprints Sicica’92, Symposium of Intelligent Components and Instruments for Control Applications, 1992, pp. 141–148. [4] W. Fajardo, Recuperacion de informacion con memorias asociativas difusas, Tesis Doctoral, Universidad de Granada, 1995. [5] H. Ishibuchi, H. Tanaka, H. Okada, Interpolation of fuzzy ifthen rules by neural networks, Internat. J. Approx. Reasoning, 10 (1994) 3–27. [6] M. Mizumoto, H.J. Zimmermann, Comparison of fuzzy reasoning methods, Fuzzy Sets and Systems 8 (1982) 253–283.
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