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) and the function associated to the corresponding symbol. For simplicity, we have considered in the program usual functions, like Zadeh's logic (dmin,dmax,comp) or Lukaszewicz's logic (luka.comp), but each user can introduce alternative functions. In both cases the function comp is defined as comp(x) = 1-x . Lists in Haskell are written as sequences of elements between square brackets separated with commas (e.g., [ 1 , 2 , 3 , 4 ] is the list of naturales formed by such numbers). - univ is universe of discourse of the logic. The universe is introduced as a list of characters not overlapping any other name neither variable nor predicate symbol. Such characters are written in Haskell between quotation marks. As we can only deal with finite lists, we restrict our universe of discourse to be a finite domain. - intlist is the list of pairs of predicates and associated interpretation
23 eval :: [Char] -> [(Char,Float->Float->Float)] -> [Char] -> [(Char,Float->Float)] -> [(Char,Char,Float)] -> [(Char,Char)] -> [(Char,Float->Float)] -> Float eval xs fs us rs cs es ms = evalR xs fs 0 us rs cs es ms evalR :: [Char] -> [(Char,Float->Float->Float)] -> Float -> [Char] -> [(Char,Float -> Float)] -> [(Char,Char,Float)] -> [(Char,Char)] -> [(Char,Float->Float)] -> Float evalR [] _ v _ _ _ _ _ = v evalR C(':xs) fs v us rs cs es ms= let (ys,zs)=formulaS('(':xs) in evalR zs fs (evalR ys fs v us rs cs es ms) us rs cs es ms evalR C&':xs) fs v us rs cs es ms = let (ys,zs)=formulaS(xs) in evalR zs fs ((funcion '&' fs) v (evalR ys fs v us rs cs es ms)) us rs cs es ms evalR (M':xs) fs v us rs cs es ms = let (ys,zs)=formulaS(xs) in evalR zs fs (dor v (evalR ys fs v us rs cs es ms) (funcion '&' fs) (funcion '"' fs)) us rs cs es ms evalR C>':xs) fs v us rs cs es ms = let (ys,zs)=formulaS(xs) in evalR zs fs (funcionl (funcion '>' fs) (funcion '"' fs) (funcion '&' fs) v (evalR ys fs v us rs cs es ms)) us rs cs es ms evalR C"':xs) fs v us rs cs es ms = let (ys,zs)=formulaS(xs) in evalR zs fs ((funcion '*' fs) (evalR ys fs v us rs cs es ms) 0) us rs cs es ms evalR CA':xs) fs v us rs cs es ms = let (ys,zs,var)=formulaSC(xs) in evalR zs fs (valoraA ys var fs v us rs cs es ms) us rs cs es ms evalR CE':xs) fs v us rs cs es ms = let (ys,zs,var)=formulaSC(xs) in evalR zs fs (valoraE ys var fs v us rs cs es ms) us rs cs es ms evalR (xixs) fs v us rs cs es ms = let (y,m,zs) = formulaRel(xs) in evalR zs fs (valorRel x (head y) m us rs cs es ms) us rs cs es ms Figure 1. The eval function implemented
24 functions. This collection of predicates is inserted as a list of pairs formed by the predicate symbol and the interpretation function. - numcharlist is the list of numeric characteristic values. The numeric characteristic for each universe symbol is introduced as a list of triples (S,P,V), where S is an element from the universe of discourse, P is a predicate symbol, and V is the numeric characteristic associated to S for the predicate symbol P. We use this numeric characteristic to obtain the interpretation of the element for the corresponding predicate. - cont is the environment used to evaluate free variables. It is represented as a list of pairs formed by the variable and the value within a bounded universe. In almost every execution this environment is empty, i.e., there are no free variables. - modlist is the list of modifiers of the logic. The list consists of pairs formed by the modifier symbol and its interpretation function. The modifier symbols are not standard for the program, therefore the user should give them with the associated interpretation function. 4. E x a m p l e Let us consider three individuals for this case named John, Michael, Ann, to be observed under the predicates tall, short and old, young. Lets for example assume that their respective heights are 1.7, 1.9, 1.65 meters and that their respective ages are 18, 20 and 35 years. The interpretation function for these values can be seen in the table below.
John Michael Ann
HEIGHT 1.7 1.9 1.65
tall 0.4 1 0.3
short 0.6 0 0.7
AGE 18 22 35
young 1 0.9 0.6
old 0 0.1 0.4
Lets then evaluate two specific formulae with our tool: Example 1: - Michael is neither very old nor very young: this property is specified as -(PT(M)AQT(M)) where in this case M represents Michael, P and Q are the predicates old and young and the | represents the syntactic modifier very. After performing the syntactic translation we obtain the following formula valid in the system: ~(P+(M)&Q+(M))
25 where + represents the modifier | . To evaluate the formula in our tool we have to give an element in the list of modifiers and its associated function. The call to the system in this case is the following: evaluate "~(P+(M))&(Q+(M))" [('k> ,luka),('~',comp)] ['M'] [ ( ' P ' . y o u n g ) , C Q ' , o l d ) ] [ ( ' M \ ' P ' , 2 2 ) , ( ' M \ ' Q ' , 2 2 ) ] [] [ ( ' + ' , s q r ) ] where sqr is the predefined function to calculate the square of a given number. The formula is evaluated in the system using the Lukaszewicz's logic together with the square function as interpretation of the modifier symbol for predicates. Example 2: - Everybody is very tall and very young: this property is formalized as follows. Vx.(PT0=)AQT0=)) where P and Q represents the predicates tall and young. And after the translation process we obtain the expression: Ax.(P+(x)&Q+(x)) To ask the tool for the value of the expression, we need to give the universe of discourse as the list of elements A, B, C. The environment is again empty, and the expression introduced to the system will be evaluate "Ax.((P+(x))&(Q+(x)))" [('&',dmin),('"',comp)] ['J'.'M'.'A'] [ ( ' P \ tall),('Q',young)] [(,J',,P',1.7),('J',,Q',18),,('M','P',1.9),('M','Q',22), C A ' , , P , , 1 . 6 5 ) , ( , A ' , ' Q ' , 3 5 ) ] [] [ ( ' + ' , s q r ) ] The logic used to evaluate this expression is the Zadeh's one and for the universal quantification we can use the aggregation of the conjunction of every instance of the formula, substituting the variable with a universal symbol (see table below with the values for expressions in this example). Example 1 2
Zadeh 0.19 0.09
Product 0.1881 0.0042
Lukaszewicz 0.18 0
Although results are no dramatically different, it is easy to understand that a fuzzy logic selection and properties settings reach different output values and those values can be used by the expert to make strong decisions.
26 5. Conclusions and future work We have presented in this paper a functional tool for the evaluation of fuzzy first order formulae. This tool is useful to assist an expert to decide about the convenience of different logics. The functional programming language we chose, Haskell, allows to consider functions as parameters, producing a general but efficient tool. Nevertheless, it is still needed that the user writes down the whole expression to be evaluated. In order to make this tool more friendly to users, we plan to improve it by embedding the system into another language with graphic interface capabilities. We also plan to extend the above tool to carry out the progress of the program, in terms of the involved logic formulae, so the user can obtain at any time the evaluation of the formula transformed by the program instruction. Acknowledgments This Research has been partially supported by grants MTM2005-08982 and TIN2005-09207 from the Government of Spain. References 1. B. Berard, M. Bidoit, A. Finkel, F. Laroussinie, A. Petit, L. Petrucci and Ph. Shnobelen. Systems and Software verification: model-checking techniques and tools, Springer, 2001. 2. R. Bird. Introduction to functional programming using Haskell, Prentice Hall, 1998. 3. Y. Bertot and P. Casteran. Interactive Theorem Proving and Program Development Coq'Art: The Calculus of Inductive Constructions, Springer, 2004. 4. J.M. Cleva, J. Leach and F.J.Lopez-Praguas. A logic programming approach to the verification of functional-logic programs, Proc. Principles and Practice of Declarative Programming (PPDP'04), ACM Press, 2004, pp. 9-19. 5. P. Cousot and R. Cousot. Refining model checking by abstract interpretation, Automated Software Engineering Journal 6:69-95, 1999. 6. H.B. Enderton. A Mathematical Introduction to Logic, Academic Press, 2001. 7. M.J.C. Gordon and T.F. Melham. Introduction to HOL, Cambridge Univ. Press, 1993. 8. C.A.R. Hoare. An axiomatic basis for computer programming, Comm. ACM 12:89-100, 1969. 9. V. Lopez, J. Montero y L. Garmendia. Fuzzy specification of algorithms, Technical Report (www.mat.ucm.es/ fuzzycs/fsa.pdf). 10. G.J. Myers. The Art of Software Testing, John Wiley, 1979.
FIELD THEORY AND COMPUTING WITH WORDS GERMANO RESCONI ( , ) and MASOUD NIKRAVESH
(2)
(1) Catholic Univeristy.Brescia , Italy, Email resconi(a),numerica,it BISC Program, EECS Department, University of California, Berkeley, CA 94720, US, Email: Nikravesh(d).eecs. berkelev. edu
In this paper, we assume that computing with words is equivalent to computation by field. A field is generated by a word or a sentence as sources. Computation by field means the research of the intensity of the sources of fields (holography) and construction by the intensity of the sources of the fields (fields modelling process). A field is a map between points in the reference space and values. For example, in a library, the reference space would be where die documents are located. At any given word, we define the field as a map of the position of the documents in the library and the number of the occurrences (values) of the word in the document. The word or source is located in one point of the reference space (query) but the field (answer) can be located in any part of the reference. Complex strings of words (structured query) generate a complex field or complex answer by which structure is obtained by the superposition of the fields of the words as sources with different intensity. Any field is a vector in the space of the documents. A set of basic fields is a vector space and form a concept. We break the traditional idea that a concept is one word in the conceptual map. Internal structure (entanglement) of the concept is the relation of dependence among the basic fields. Geometric image of the concept X and the field R in Zadeh's rule "X isr R" is given. Fields can be fuzzy sets where values are the membership values of the fuzzy set. The ambiguous word is the source ( query) of the fuzzy set ( field or answer).
1. Introduction We know that a key assumption in computing with words is that the information which is conveyed by a proposition expressed in a natural language or word may be represented as a generalized constraint of the form "X isr R", where X is a constrained variable; R is a constraining relation; and r is an indexing variable whose value defines the way in which R constrains X. Thus, if p is a proposition expressed in a natural language, then "X isr R" representing the meaning of p, equivalently, the information conveyed by p. Therefore, the generalised constraint model can be represented by field theory in this way. The meaning of any natural proposition p is given by the space X of the fields that form a concept in the reference space or objective space, and by a field R in the same reference.. We note that a concept is not only a word, but is a domain or context X where the propositions p represented by the field R are locate. The word in the new image is not a passive entity but is an active entity. In fact the word is the source of the field. We can also use the idea that the word as an abstract entity is a query and the field as set of instances of the query is the answer. 27
28
In the agent image, where only one word (query) as a source is used for any agent. The field generated by the word (answer) is a Boolean field (values true/false). We can thus compose the words by logic operations to create complex Boolean expression or complex Boolean query. This query generates a Boolean field for any agent. The set of agents create a set of elementary complex fields whose superposition is the fuzzy set represented by field with fuzzy values. The field is the answer to the ambiguous structured query whose source is the complex expression p. The fields with fuzzy values for complex logic expression is coherent with traditional fuzzy logic with a more conceptual transparency because is founded on agents and Boolean logic structure. As points out [Masoud Nikravesh] Web is a large unstructured and in many cases conflicting set of data. So in the Web, fuzzy logic and fuzzy sets are essential to put query and find appropriate searches to obtain the answer. For the agent interpretation of the fuzzy set, the net of the Web is structured as a set of conflicting and in many case irrational agents whose task is to create any concept. Agents produce actions to create answers for ambiguous words in the Web. A structured query in RDF can be represented as a graph of three elementary concepts as subject, predicate and complement in a conceptual map. Every word and relationship in the conceptual map are variables whose values are fields which superposition gives the answer to the query. Because we are more interested in the meaning of the query than how we write the query itself, we are more interested in the field than how we produce the field by the query. In fact, different linguistic representations of the query can give the same field or answer. In the construction of the query we use words as sources of fields with different intensity. With the superposition we obtain the answer for our structured query. We structure the text or query to build the described field or meaning. It is also possible to use the answer, as a field, to generate the intensity of the words as sources inside a structured query. The first process is denoted READ process by which we can read the answer (meaning) of the structured query. The second process is the WRITE process by which we give the intensity or rank of the words in a query when we know the answer. In analogy with the holography, the WRITE process is the construction of the hologram when we know the light field of the object. The READ is the construction of the light field image by the hologram. In the holography the READ process uses a beam of coherent light as a laser to obtain the image. Now in our structured query the words inside of the text are activated in the same time. The words as sources are coherent in the construction by superposition of the desired answer or field.
29
2. Representation of the space of the fields inside a reference space Given the n dimensional reference space Xi , X 2 ,.. .,Xn , a scalar field is a function G that associate at one point of the space one value . So the field is G = G( Xi , X 2 ,.. .,X n ). A linear model of the field G is obtained by the weighted superposition of basic fields F! , F 2 ,.. .,FP. in this way G = S,F 1 (X 1 ,...,X n ) + S 2 F 2 (X,,...,X n ) +
+ S D F n (X ] ,...,X I 1 )
(1)
In the equation (1), Fi , F 2 ,..., F n are the basic fields and Si , S 2 ,..., Sn are the weights or source values of the basic fields. We assume that any basic field is generated by an entity as source. So the intensity of the fields is proportional to the intensity of the sources that generate the field itself.
2.1 Example of the basic field and sources In Figure 1, we show an example of two different basic fields in a two dimensional reference space (x,y). The general equation of the fields is
F(x>y) = S[e
U
U
]
(2)
the parameters of the field Fi are S=l h =2 and x0 = -0.5 and y0 = -0.5, the parameters of the field F2 are S=l h =2 and x0 = 0.5 and y0 = 0.5
fo^..,
-1 -1
-0.5
0_5
1
-0.5
Figure 1. Two different basic fields in the two dimensional reference space (x,y).
30 For the sources Si = 1 and S2 = 1 the superposition field F that we show in figure 2 is F = F] + F2. For the sources Si = 1 and S2 = 2 the superposition field F that we show again in figure 2 is F = Fi + 2 F 2 . F = F, + 2F2.
F = F, + F2
Figure 2. Example of superposition of elementary fields Fj , F2
2.2 Computation of the sources To compute the sources Sk we put the values of the elementary fields Fk in the different points Ph of the reference space in the colons of the table 1 . The field G is put in the last colon of the table. Table 1. Fields values for M points in the reference space
Pi
p2 PM
F, F,., F2,
F2 F,.2 F2.2
F2.N
Field G G, G2
F
FM,2
FM,N
GM
M,1
•
FN
>•
Fl.N
The values in the table 1 is represented by this matrices K 1,1 2,1
X F
M,\
U L
K
IN
F
2,2
F
M,2
G,
2,N
r
M,N.
GM.
31 The matrix X is the relation among the elementary fields Fk and the points Ph. Now we are interested in the computation of the sources S by which we give the best linear model of G by the elementary field values. So we have the superposition expression F
'
F
and G* = S,
' F12 '
l,l"
2,l
_FM,\_
+s2
^2,2
' \ n ' + ... + S„ \ n
= xs
(3)
_FM,n_
_FM,2_
Now we compute the best sources S in a way the difference IC-Gl is the minimum distance for any possible choice of the set of sources. It is easy to show that the best sources is obtained by the expression T S = (XX)
-1
T X G
(4)
when X is a square matrix we have G* = G and -1 S = X G and G* = G = XS (5) In a geometric representation, any column of the table 1 is a vector in the orthogonal multidimensional space of the points P of the reference space. The set of the vectors for any elementary field is a general reference that forms the space of the fields. The contro-variant coordinates of the field G in the space of the field are the best sources S. In figure 3 we show an example of the space of the fields
•
P.
Pi Figure 3 The fields F] and F 2 are the space of the fields. The coordinates of the vectors Fj and F2 are the values of the fields in the three points P] , P 2 , P3.
32 When the vector G is out of the space of the fields as we show in figure 4
•
P.
P2 Figure 4 The vector G is out of the space of the fields Fi and F2. The variable QG = X S = G* is the projection of G on the space of the fields The value of the variable D (metric of the space of the field) computed by the expression (6) D2 = ( X S )T ( X S ) = ST XT X S = ST A S = [ ( X T X ) " ' X T G ]TXTX [ ( X T X ) ' X T G ] = ( Q G ) T Q G
(6)
Is the intensity of the field QG where QG = X [ (X T X )•' XT G ] = X S
(7)
is the projection of G on the space of the fields X. All the fields G with the same projection on the space of the fields have the same D. When X is a square matrix D is always invariant because Q G = G = G*. We remark that A = XT X is a quadratic matrix that gives the metric tensor of the space of the fields. When A is a diagonal matrix all the elementary field are independent one from the other. But when A has non diagonal elements, in this case the elementary fields are dependent on from the other. Among the elementary field there is a correlation or a relationship. We note that with A it is possible to represent a complex network of relationships between fields.
33
3. Field theory, concepts and Web search To search in the web, we use the term-document matrix to obtain the information retrieved. In table 2, we show the data useful in obtaining the desired information in the Web. Table 2. Term ( word), document and complex text G
Document] Document2 DocumentM
Wordi K,.,
Word2 K,.2
K2.1
K2.2
K-2.N
Concept G G, G,
KM,I
KM,2
KM,N
GM
•
• •
Wordn Kl.N
Where Ky is the value of the wordj in the document. The word in table 2 is a source of a field which values are the values in the position space of the documents. Any document is one of the possible positions of the word. In a Web search, it is useful to denote the words and complex text in a symbolic form as queries, the answers are the fields generated by the words or text as sources. Any ontology map is a conceptual graph in RDF language where we structure the query as a structured variable. The conceptual map in figure 5 is the source G of a complex field obtained by the superposition of the individual words in the map.
Figure 5. Conceptual map as structured query. The map is a structured variable whose answer or meaning is the field G in the documents space located in the Web
34 Differential equation of the fields can give the dynamic of the field G in the space of the documents or reference space. Given the word "warm" and three agents that are named Carlo , Anna and Antonio, we assume that any agent is a source for a Boolean fields in figure 6.
CARLO
ANNA
ANTONIO
Figure 6 Three Boolean fields Fi(X,Y), F 2 (X,Y), F 3 (X,Y) for three agents. The superposition of the thee Boolean field in figure 6 is the fuzzy set fi(X, Y) = Sj Fj (X,Y) + S2 F2 (X,Y) + S3 F3 (X,Y)
(8)
At the word "warm" that is the query we associate the answer ft(X, Y) that is a field. Different words generate Boolean expressions (query) as G = IF [ ( X AND Y ) ORZ ] THENH
(9)
any agent generate the Boolean fields for the words X , Y , Z and H. After any agent by the (9) generate a new field for G by the composition of the fields for the words X,Y,Z,H. The superposition of the three elementary Boolean fields Gi , G2 , G3 one for any agent, gives the fuzzy set field or answer to the query in (9). We remark that the elementary Boolean fields generate the space of the fields and inside the have the fuzzy set for (9). Relationship among the agents is obtained by A = XT X where X is the space of the fields.
Reference 1. L. A. Zadeh and M. Nikravesh, Perception-Based Intelligent Decision Systems; Office of Naval Research, Summer 2002 Program Review, Covel Commons, University of California, Los Angeles, July 30th-August 1st, 2002. 2. L. A. Zadeh, and J. Kacprzyk, (eds.), Computing With Words in Information/Intelligent Systems 1: Foundations, Physica-Verlag, Germany, 1999a. 3. L. A. Zadeh, L. and J. Kacprzyk (eds.), Computing With Words in Information/Intelligent Systems 2: Applications, Physica-Verlag, Germany, 1999b. 4. G. Resconi and L. C.Jain, Intelligent Agents , Springer, 2004
NEW OPERATORS FOR CONTEXT ADAPTATION OF MAMDANI FUZZY SYSTEMS ALESSIO BOTTA 1MT Lucca Institute for Advanced Studies, Via San Micheletto 3', 55100Lucca, alessio. botta@imtlucca. it
Italy
BEATRICE LAZZERINI, FRANCESCO MARCELLONI Dipartimento di Ingegneria dell 'Informazione: Elettronica, Informatica, Telecomunicazioni, University of Pisa, Via Diotisalvi 2, 56122 Pisa, Italy {b.lazzerini,fmarcelloni}@iet.unipi.it In this paper we introduce a set of tuning operators that allow us to implement context adaptation of fuzzy rule-based systems while keeping semantics and interpretability. The idea is to achieve context adaptation by starting from a (possibly generic) fuzzy system and adjusting one or more its components, such as membership function shape, fuzzy set support, distribution of membership functions, etc. We make use of a genetic optimization process to appropriately choose the operator parameters. Finally, we show the application of the proposed operators to Mamdani fuzzy systems.
1. Introduction The issue of context in fuzzy rule-based systems (FRBS) has been addressed in previous papers [1,2,3,4,5], where context adaptation is mainly regarded as scaling fuzzy sets from one universe of discourse to another. This approach, which has been used, e.g., in FRBS identification problems [1], is, however, unsatisfactory since it does not take into account such issues as the modification of the shape of the membership functions (MFs), of the center of mass and/or the radius of fuzzy sets, and of the extent to which each rule contributes to the final result. On the other hand, context adaptation can be reasonably regarded as the tuning process of an initial normalized FRBS, which can just be built based on a normalized universe of discourse with uniform distribution of the linguistic terms associated with the variables of interest. Of course, an appropriate set of context adaptation operators is required to perform the tuning process. From this perspective, context adaptation also allows reusability of FRBSs: a new FRBS can be derived through context adaptation from a previously available FRBS, provided by a domain expert or obtained by means of an identification technique. A very intuitive example of context adaptation is related, e.g., to 35
36 tuning the sample curves, which represent the general, similar structure of the wages of workers, in order to create a specific curve instance pertinent to a specific context (i.e., educational level, type of work, sex, etc.). In this paper we propose a set of tuning operators that can be used to perform context adaptation with several degrees of freedom. For the sake of simplicity, we will consider triangular MFs described by the triplet (/, m, u), where / and u represent the lower and upper bounds of the support and m the modal value, and will refer to Mamdani FRBSs because of their very immediate interpretability. In the following, we first introduce the operators used to adapt the initial normalized FRBS to the context and then we discuss the application of a genetic algorithm to optimize the choice of the operator parameters. Finally, we show an application of our approach to sample data. 2. Non-Linear Scaling Function for Fuzzy Domains Since we start from a normalized FRBS, the first context adaptation we have to perform is to adjust the universes so as to cover all possible input and output values and possibly to make granularity finer in some parts of the universe and coarser in the other parts. This is carried out by non-linear scaling functions. The scaling function proposed in [1] and [5] can concentrate/dilate the MFs around a single point (a sort of center of gravity), that can be either the center of the universe or one of the two extremes. In [4], the authors propose a more general scaling function, that is a linear composition of sigmoidal-like functions. These two approaches have some limits: the former allows concentrating/dilating the universes around only three predefined points, while the latter requires several parameters, whose identification can be a long and difficult process in the case of strong non-linear behavior of the scaling function. In this paper, we introduce a scaling function which, extending the approach proposed in [5], can both compress/dilate the universe of discourse (from a normalized interval [0,1] to any context-specific interval [a,b]) and nonuniformly distribute the MFs, allowing to select any point in [a,b] as center of gravity. The scaling function is defined as: [a + (b-a)[l-(\-Ay
''(1-*)']
ifx>X,
where ks > 0 is the degree of compression/dilation and X e [0,1] is the relative position within interval [a,b] of the center of gravity. Figure 1 shows three different applications of the function with different values for ks and A. The upper part of the figure displays function s(x) and the lower part shows the scaled MFs (continuous lines) obtained by transforming the original MFs (dotted lines) through s(x).
37
(a) (b) ' (C) ' ' Figure 1. Examples of application of the non-linear scaling function with different values of X and ks: (a) X = 0.5, ks = 0.5; (b) X = 0.25, ks = 0.3; (c) X = 0.8,fe= 1.2.
3. Fuzzy Modifiers As discussed in the Section 1, the effects of the context should be modeled so as to modify the FRBS coherently. Thus, when performing the tuning of the FRBS, we should avoid modifying each single parameter of each MF separately, but rather we should perform the same action on all MFs at once. The scaling function shown in (1) is an example of this approach, but it allows only very restricted modifications of the MFs. To the aim of providing a more flexible and customizable tuning, we use a set of appropriately defined fuzzy modifiers. As stated in [6], a fuzzy modifier is any mapping from a fuzzy set to another fuzzy set. Usually, a fuzzy modifier is associated with a linguistic meaning (modifier semantic), thus composing a linguistic hedge. A modifier is inclusive, if it generates either a superset (expansive modifier) or a subset (restrictive modifier) of the original fuzzy set. In the following, we describe four different modifiers, which allow adapting each fuzzy set of a partition to a specific context, without affecting the meaning of the fuzzy set. The modifiers are concurrently applied to all fuzzy sets of a partition by using the same parameters. 3.1. Coverage-Level Modifier The level of coverage e of a fuzzy partition P = {Ah...^iN} is defined as [7]: Vxe[a,b],3i\le.
on the universe [a,b] (2)
To determine the level of coverage can be very expensive, because it involves computation of the membership values for each point of the universe. Fortunately, some fuzzy partitions show nice properties that simplify the
38 calculation of the level of coverage, restricting the search to a small, finite set of points. For instance, in the case of triangular fuzzy sets, if the following conditions hold: (a) Vi | 2< i < N,mM - cut) • kE ]
ifAr£ 0,
where w, =(CM / -C/ / )/[(C/ / -J/,) + (5M J -CM J )]. Figure 3.c shows the application of the generalized enough modifier with kE = 0.3. The generalized enough is an inclusive modifier. Some interpretability problems can arise when kE —»• 1, because adjacent MFs tend to overlap very much.
Figure 3. Examples of application of the generalized positively (a-b) and the generalized-enough modifiers (c).
4. The Genetic Algorithm The problem of modeling the effects of a context on a normalized FRBS reduces to searching for an optimal configuration of the nine parameters used in the tuning operators. To find the optimal parameters, we adopt a genetic algorithm (GA). In the following, we briefly describe some specific features of the GA. Coding scheme. We use a real-coded chromosome consisting of all the parameters needed to configure the tuning operators for each linguistic variable. Assuming that V is the number of the linguistic variables, the length of the chromosome is equal to 9 V. Fitness function. Let {(xj,yj)} be a set of M experimental data, where x, is a vector of V-\ input data, and yj is the corresponding output of the real system that we are trying to model. We use as fitness function the mean square error (MSE) estimator, defined as MSE = (2A/)"1 ^T" (F{x.)-yj)\
where Fix]) is the
output of the contextualized FRBS computed on input Xj. Mutation and crossover operators. As mutation operator, we use the random uniform mutation described in [9], As regards the crossover operator,
41 we use the BLX-a crossover operator [9], with a = 0.25 for the genes corresponding to parameters av and bv of the non-linear scaling function, and a = 0 for the other genes. The choice of using a = 0.25 for parameters av and bv allows tuning the extremes of the universe to the experimental data. Phenotype generation. The generation of a phenotype depends not only on the values of the parameters, but also on the order of application of the scaling function and the MF modifiers. Indeed, their composition is not a commutative operation. For the sake of simplicity, in the experiments we chose the following pre-fixed order: non-linear scaling, coverage-level modifier, core-position modifier, generalized enough modifier and generalized positively modifier. We observe that the two modifiers (generalized enough and generalized positively), which change the shape of the MFs, are applied at the end of the sequence: this allows to apply the non-linear scaling and the first modifiers to triangular MFs. 5. Example: Structure of Wages We applied the context adaptation technique to the structure of wages shown in Figure 4.a. The structure of wages [10] is studied in economics by means of a set of curves that show how the hourly wage changes with the amount of years of experience. We have different curves depending on the different educational levels of the people: college graduates, college drop-outs, high-school graduates and high-school drop-outs. Hence, we can consider the educational level as the context. The input and the output linguistic variables are uniformly partitioned with 6 and 5 fuzzy sets, respectively. Thus, the initial Single Input Single Output (SISO) normalized FRBS is composed of 6 IF-THEN rules. For each of the four curves, we executed ten runs of the GA, with a population of 20 individuals, a stopping condition over the total number of generations set to 5000 and a training set of 100 uniformly distributed points. Table 1 shows the averaged MSE in the form of mean ± standard deviation (o) and the best MSE achieved in the runs. Figures 4.a-d display, respectively, the original curves, the output of the initial FRBS, the results of the best fitting models (solid) compared to the original curves (dotted) and an example of a contextualized output linguistic variable. We observe that, although we used an extremely simple FRBS, we obtained a very low MSE on all models.
Figure 4. Example of context adaptation performed on the wage curves inspired by [10].
42 Table 1. Main results of 40 runs of the context adaptation GA on the structure of wages example. Curve
Mean MSE ± a
Best MSE
College graduates
0.0148 ±0.0142
0.0066
College drop-outs High-school graduates
0.0103 ±0.0118 0.0072 ± 0.0029
0.0047 0.0037
High-school drop-outs
0.0034 ± 0.0030
0.0012
6. Conclusion We have shown how context adaptation of a fuzzy rule-based system can be obtained as a tuning process of a given FRBS which aims both to adjust the components of the FRBS to the specific domain and to keep the interpretabihty degree of the original FRBS. We have introduced a set of powerful context adaptation operators, we have described the genetic tuning of their parameters, and we have presented their application to Mamdani fuzzy systems. References 1. L. Magdalena, Adapting the gain of an FLC with genetic algorithms, Int. J. Approximate Reasoning 17 (4) (1997). 2. R. R. Gudwin, F.A.C. Gomide, Context adaptation in fuzzy processing, Proceedings of the Brazil-Japan Joint Symposium on Fuzzy Systems (1994). 3. W. Pedrycz, R.R. Gudwin, F.A.C. Gomide, Nonlinear context adaptation in the calibration of fuzzy sets, Fuzzy Sets and Systems 88 (1997). 4. R. Gudwin, F. Gomide, W. Pedrycz, Context adaptation in fuzzy processing and genetic algorithms, Int. J. Intell Systems 13 (10/11) (1998). 5. O. Cordon, F. Herrera, L. Magdalena, P. Villar, A genetic learning process for the scaling factors, granularity and contexts of the fuzzy rule-based system data base, Inform. Sci. 136 (1-4) (2001). 6. M. De Cock, E.E. Kerre, A new class of fuzzy modifiers, Proceedings of IEEE-ISMVL (2000). 7. J. Valente de Oliveira, Semantic constraints for membership function optimization, IEEE Trans. Systems, Man, Cybern. 29 (1999). 8. H. Shi, R. Ward, N. Kharma, Expanding the definitions of linguistic hedges, IFSA World Congress and 20th NAFIPS Int. Conf. (2001). 9. F. Herrera, M. Lozano, J.L. Verdegay, Tackling real-coded genetic algorithms: Operators and tools for behavioral analysis, Artificial Intell. Rev. 12(1998). 10. K.M. Murphy, F. Welch, The structure of wages, Quart. J. Econ. 107:1 (1992).
USING PARAMETRIC FUNCTIONS TO SOLVE SYSTEMS OF LINEAR FUZZY EQUATIONS-AN IMPROVED ALGORITHM
ANNELIES V R O M A N , GLAD D E S C H R I J V E R , E T I E N N E E. K E R R E Department
of Applied Mathematics and Computer Science, Ghent University Fuzziness and Uncertainty Modelling Research Unit Krijgslaan 281 (S9), B-9000 Gent, Belgium. {Annelies. Vroman\ Glad.Deschrijver]Etienne.Kerre} @UGent. be Homepage: http ://www. fuzzy. UGent. be
Buckley and Qu proposed a method to solve systems of linear fuzzy equations. Basically, in their method the solutions of all systems of linear crisp equations formed by the a-levels are calculated. We proposed a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system. By observing the monotonicity of the parametric functions in each variable, i.e. each fuzzy coefficient in the system, we improve the algorithm by calculating less parametric functions and less evaluations of these parametric functions. We show that our algorithm is much more efficient than the method of Buckley and Qu.
1. Introduction In this paper we search for a solution of the matrix equation: Ax = b for x = [£fc]nxi where A = [a,ij}nXn is a matrix with fuzzy numbers as entries and b = [bfc]nxi is a vector of fuzzy numbers. Such equations are hard to solve exactly and often the exact solution does not exist or is a vector of fuzzy sets which are not fuzzy numbers. Therefore the search for an alternative solution has a solid ground. Buckley and Qu * have already proposed a solution. We follow their line of reasoning, although the solution can be adjusted a little bit. A practical algorithm to find this solution, which is an improved version of the basic method described in a previous paper5, is proposed here. 2. Preliminaries First we recall some definitions concerning fuzzy numbers4. Let A be a fuzzy set on R. Then A is called convex if A(\x\ + (1 — A)x2) > 43
44 min(j4(xi) )J 4(a;2))), for all xit x2 £ E and A £ [0,1]. If for x £ K it holds that -A(a:) = 1, then we call x a modal value of A. The support of A is defined as supp/1 = {x \ x € R and J4(X) > 0}. A mapping / : R —> R, or in particular / : R —> [0,1], is called upper-semicontinuous when / is right-continuous where / is increasing, and left-continuous where / is decreasing. Definition 1. 4 A fuzzy number is defined as a convex uppersemicontinuous fuzzy set on R with a unique modal value and bounded support. From now on fuzzy numbers will be denoted by a lowercase letter with a tilde, e.g. a, and a vector of fuzzy numbers will be denoted as b . Sometimes we will denote the i-th component of b by (b)(. Crisp numbers will be represented by a lowercase letter, e.g. a, and vectors of crisp numbers will be denoted as b = (61,62, ••• , 6 n ) T . The notions of support and a-level are extended componentwise for vectors or matrices of fuzzy numbers. The arithmetic of fuzzy numbers is based on Zadeh's extension principle and is equivalent to interval arithmetic applied to the a-levels. 3. Solving s y s t e m s of linear fuzzy e q u a t i o n s First of all, we require that the matrix A of fuzzy numbers is regular in the sense that the matrix A~l exists for all a$j £ supp(ay). Buckley and Qu 1 proposed to construct a set of all crisp solutions corresponding to the crisp systems formed by the elements in a certain a-level. They define the solution by, for all a £ ]0,1], fi(a) = { x | x £ R " and (3A = [aij}nxn
£ E n x n ) ( 3 b = [6 f c ] n x l £ R n )
((V(«,j,fc) £ { l , 2 , . . . , n } 3 ) ( o i j £ {aij)a and bk £ (bk)a)
and Ax = b)}
and for all x e R " , x B ( x ) = sup{a | a £ ]0,1] and x £ fi(a)}. We see that XB is defined as a fuzzy set on R n and not as a vector of fuzzy numbers. The solution xjg(x) expresses to what extent the crisp vector x is a solution of the system of linear fuzzy equations Ax = b. We prefer to define a vector as solution of fuzzy numbers to avoid information loss. Therefore we give a membership degree to every component of the solution vector and then (XB)J(X) expresses the degree to which x belongs to the fuzzy set (xg)j, independent of ( X B ) J , for all j =f i. We thus define for all x £ R and for a l H £ { 1 , 2 , . . . , n} (xfi)i(x) = sup{a I a £ [0,1] and (3x £ fi(a))(x = xi)},
(1)
45 where X; denotes the i-th component of x. This method is purely theoretical: in fact all crisp systems are solved. When all these systems have to be solved, the computation time will be large. In this paper we propose a practical algorithm to compute the solution. Instead of solving all these crisp systems, we determine parametric functions of these solutions and we only calculate the necessary solutions. 3.1. Systems
with one fuzzy
coefficient
We first consider the case that we have to solve a system of linear fuzzy equations in which exactly one of the coefficients is a fuzzy number and the other coefficients are crisp. Without loss of generality we may assume that fin is a fuzzy number. In order to obtain the solution xs of Ax. = b, we have to solve the crisp systems A(an)x. = b, where fin is replaced by an in the matrix A for all a n e [ a u , a n ] = supp(fin). We can solve all of these systems through Cramer's rule thanks to the non-singularity of the crisp matrix A(an), for all a n € supp(fin). So we can write the solution for every component as a quotient of two determinants. The determinant of a matrix A is denoted as \A\. By expanding the determinants in the numerator and the denominator along the first row, we can write each component of the solution using parameters cy, C2j, C3 and c\: *j = fj(an)
=
„
„
, .
•
(2)
Due to this result, every solution can be written using parametric functions with variable a n . Note that c\j and c-ij are dependent of j due to the fact that the j-th column in the numerator contains the components of b. On the other hand, the denominator is the same for all j € { 1 , . . . , n } , so C3 and C4 are independent of j . Thus we propose the following method to solve .. It is easy to see that for large n, K and m the method described above needs less computation time than the method of Buckley and Qu. We have implemented our algorithm and the improved algorithm in Matlab 6.5 and we have compared them with the method of Buckley and Qu, which consists of solving all crisp systems directly without using parametric functions. In Figure 2(a) the computation time a for the three algorithms is plotted in function of the dimension of the system with four fuzzy coefficients and four a-levels. b In Figure 2(b) the difference in computation time between the basic method with parametric functions and the improved version is shown in function of the dimension of the system with four fuzzy coefficients and four a-levels considered.
4. C o n c l u s i o n In this paper we have proposed an improved version of our method to solve nxn systems in which some (or all) coefficients are fuzzy. While in the method of Buckley and Qu for every element in the support of each fuzzy number the corresponding crisp nxn system is solved, in our method only the crisp nxn systems corresponding to the bounds of each support must be solved, and only the solution for the combinations of the lower and the upper limits of the considered a-level are obtained using parametric functions. Our method performs much better when a lot of a-levels are considered and when n is large.
a
Actually the average of the computation time of 5 different n x n systems for each considered n. b These results are obtained on a computer with an Intel Pentium 4 CPU 2.40 GHz processor (and 512 MB RAM).
50
dimanwxicfUidvriam(•.» »-. SIXMOO)
(a) Comparison of the method of Buckley and Qu, the basic method and the improved method with parametric functions
dmndoi d Of >y«ani(*{|. 5^S00*5C»t
(b) Comparison of the basic method and the improved method with parametric functions
Figure 2. Computation time in function of the dimension of the system. The system contains 4 fuzzy numbers and considers 4 a-levels.
Acknowledgment Annelies Vroman would like to thank the Fund for Scientific ResearchFlanders for funding the research project G.0476.04 elaborated on in this paper. References 1. J. J. Buckley and Y. Qu,"Solving systems of linear fuzzy equation", Fuzzy Sets and Systems, vol. 43, pp. 33-43, 1991. 2. M. Eisenberg, "Axiomatic Theory of Sets and Classes", Holt, Rinehart and Winston, Inc., New York, 1971. 3. R. Moore, "Interval Arithmetic", Prentice-Hall, Englewood Cliffs, NJ, USA, 1966. 4. E. E. Kerre, "Fuzzy Sets and Approximate Reasoning", Xian Jiaotong University Press, Xian, People's Republic of China, 1999. 5. A. Vroman, G. Deschrijver and E. E. Kerre, "Solving systems of linear fuzzy equations by parametric functions", IEEE Transactions on Fuzzy Systems, in press.
NUMERICAL IMPLEMENTATION STRATEGIES OF THE FUZZY FINITE ELEMENT METHOD FOR APPLICATION IN STRUCTURAL DYNAMICS
D. MOENSr D. V A N D E P I T T E K. U.Leuven,
dept. Mechanical Engineering, div. Kasteelpark Arenberg 41 B - 3001, Heverlee, Belgium E-mail: [email protected]
PMA
This paper gives an overview of numerical strategies for the implementation of the fuzzy finite element method for structural dynamic analysis. It is shown how the deterministic finite element procedure is translated to a fuzzy counterpart. Some general solution strategies for the underlying interval problem are reviewed. The specific requirements for the implementation of the fuzzy method for structural dynamic analysis are then discussed based on two typical dynamic analysis procedures, i.e., the eigenvalue analysis and the frequency response analysis.
1. Introduction The exponential growth of computational capabilities of modern computers clearly has an impact on the use of the finite element (FE) method for engineering purposes. This evolution has paved the way for a number of computationally intensive analysis techniques derived from the classical FE technique, as e.g. non-deterministic FE analyses. In this context, recently a number of non-probabilistic approaches are emerging as alternative for the more common probabilistic FE analysis. The interval FE (IFE) analysis is based on the interval concept for the description of uncertain model properties, and so far has been studied only on an academic level li2. The fuzzy FE (FFE) analysis is basically an extension of the IFE analysis, and has been studied in a number of specific research domains as e.g. structural static and dynamic analysis 3>4'5. Section 2 of this paper summarises the methodology behind the implementation of the FFE method. Different generally applicable solution * Postdoctoral Fellow of the Research Foundation - Flanders
51
52 strategies for the underlying interval problem are reviewed in section 3. Section 4 then focuses on the specific interval solution schemes for dynamic analysis, focusing respectively on eigenvalue analysis and frequency response analysis.
2. T h e Fuzzy F i n i t e E l e m e n t M e t h o d Whereas a classical set clearly distinguishes between members and nonmembers of a set, the fuzzy set introduces a degree of membership, represented by the membership function. This function describes the grade of membership to the fuzzy set for each element in the domain. A fuzzy set x is defined as fix (x): x=[(x,
fix (x)) \{xeX)
{m (x) € [0,1])}
(1)
for all x that belong to the domain X. The support of a fuzzy set equals: supp(x) = {x £ X,fiXi(xi)
> 0}
(2)
The difference with the classical (or also called crisp) set is that the fuzzy concept allows for membership values different from zero and one. This enables the representation of a value that is only to a certain degree member of the set. The goal of the F F E analysis is to obtain a fuzzy description of some output quantities of a F E analysis in which the non-deterministic input is modelled using the fuzzy set model. It consequently aims at the derivation of the membership function of the output quantities y = / ( x ) given the membership functions of all input quantities in x. Therefore, the F F E analysis requires an arithmetic which handles the numerical evaluation of functions of fuzzy sets. A general concept follows directly from ZADEH'S extension principle 6 . The strategy consists of searching in the output domain for sets which have an equal degree of membership. This is achieved by analysing the input domain on a specific level of membership a. At this level, the a-cuts of the components of the input vector x are defined as: Xi'a = {xi e X{,p,Xi(xi)
> a}
(3)
This means that an a-cut is the interval resulting from intersecting the membership function at jxXi (xi) = a. After deriving the a-cuts of all input quantities at a specific level, a general interval analysis is performed: Via = {Vi I (Vj € { 1 , . . . ,n}){Xj
e Xj'a) and y = / ( x ) }
(4)
53 It can be proved (see e.g. MOENS et al. 7 ) that the obtained output interval is an intersection of the output membership function at the a-level, and consequently represents an a-cut of the output. This means that a discretized approximation of the output membership function can be obtained from repeating the a-level procedure at a number of levels. Note that in order to obtain the support of the resulting output membership function, the interval analysis has to be performed on the supports of the input membership functions. Also, in case of a multi-dimensional output vector y, this procedure has to be repeated on each individual component y% of the output vector. Based on this a-cut strategy, a number of F F E applications have been published in specific research domains. Since, through the a-cut strategy, the IFE analysis forms the numerical core of the F F E method, this paper now focuses on the implementation of the IFE analysis. 3. General i m p l e m e n t a t i o n s c h e m e s for I F E analysis 3.1. The interval
arithmetic
approach
The interval arithmetic approach consists of translating the complete deterministic numerical F E procedure to an equivalent interval procedure using the interval arithmetic operations for addition, subtraction, multiplication and division of interval scalars. This approach has an important drawback, due to the fact that an arithmetic interval operation overestimates the interval result if it neglects correlation between the operands. This will be further referred to as conservatism. In the interval translation of the F E analysis, this phenomenon cannot be avoided as it is impossible to keep track of the relationships between all intermediate results of the algorithm. Since numerous operations are required in a general F E solution procedure, the amount of conservatism due to this phenomenon can be substantial. Therefore, this approach is not applied as stand-alone principle in F F E implementations. However, it can be usefully applied in a hybrid implementation scheme, as discussed in section 4. 3.2. The global optimisation
approach
Basically, calculating the smallest hypercube around the solution set expressed in Eq. (4) is equivalent to minimising and maximising each output component of the deterministic analysis result y. This can be implemented using a global optimisation procedure in which the interval vector containing the uncertain parameters defines the constraints for the variables. The
54 solution set is an interval vector y 1 describing the hypercube around the exact solution: yl = {y{vi--yIm}
(5)
with: 1i
=
min r /i (x) > * = * • •• ™ X6X 1
(6)
y{ = max fc (x), i = 1 . . . m (7) xex 1 An efficient and robust optimisation algorithm is primordial for this solution strategy. The input interval vector defines the number of constraints and, therefore, strongly influences the performance of the procedure. Also, as an execution of the deterministic FE analysis is required in each goal function evaluation, this approach is numerically expensive, and most appropriate for rather small FE models with a limited number of input uncertainties. 3.3. The vertex
analysis
The vertex method 8 approximates the result of an interval analysis by introducing all possible combinations of boundary values of the input intervals into the analysis. For n input intervals, there are 2™ vertices for which the analysis has to be performed. These vertices are denoted by Cyj = 1,... 2". Each of these represents one unique combination of lower and upper bounds on the n input intervals. The approximate analysis range is deduced from the extreme values of the set of results for these vertices:
Vii
mm{/( c j)}i> m a x {/( c j)}i 3
(8)
3
Despite its simplicity, this method has some disadvantages. Most importantly, this approach cannot identify local optima of the analysis function which are not on the vertex of the input space. It only results in the exact output interval if the analysis function is monotonic over the considered input range. This is a strong condition that is difficult to verify for FE analysis because of the often complex relation between analysis output and physical input uncertainties. The approximation obtained when monotonicity is not guaranteed is not necessarily conservative. This reduces the validity of this method for design validation purposes. Secondly, it is clear that the computational cost of this method increases exponentially with the number of input intervals, which limits its applicability to systems with very few interval uncertainties.
55 4. IFE implementation strategies for structural dynamic analysis 4.1. Eigenvalue
analysis
The deterministic procedure of the FE eigenvalue analysis consists of the assembly of the system stiffness and mass matrices K and M, after which the deterministic eigenvalue Aj satisfies the equation: K # , = XM&i
(9)
with 3?j the corresponding eigenvector. In the interval eigenvalue procedure, the aim is to calculate the bounds on specific eigenvalues, given that the uncertain parameters are within their defined ranges x 1 . This comes down to the calculation of the solution set: {A* I (K e K 1 ) ( M € M 1 ) (K*{ = AiM*i) |
(10)
with K and M incorporating implicitly the dependency of the system matrices on the input parameters. Note that in this equation, # t is different for each A, in the solution set. It can be shown that, assuming independent interval system matrices, the exact bounds of this solution set are achieved for vertex matrix combinations 9 . This means that, based on the assumption that all interval entries appearing in the system matrices are independent, the exact solution of the interval eigenvalue problem can be found. Some algorithms have been developed which efficiently calculate this exact vertex solution of the interval eigenvalue problem. CHEN et al. 9 introduced a non-iterative procedure based on the RAYLEIGH quotient, which states that the lower and upper bound on the ith eigenvalue follow directly from two deterministic eigenvalue problems: (K + S i K S i ) ¥ i = A 7 ( M - S i M S i ) ¥ i 1
1
(K - S'KS ) * ; = Xi (M + S'MS ) * ,
(11) (12)
with S1 = diag (sgn($j,i),... sgn(<J?j,n)) and 4>, the ith eigenvector from the deterministic analysis at the midpoints K and M of the interval matrices. K and M represent the radius of the interval matrices. This method requires all the components of the eigenvector to have a constant sign over the considered domain and does not allow the occurrence of an eigenfrequency cross-over in the input parameter space. An enhanced methodology was developed by EL-GEBEILY et al. 10. It provides a solution for the original problem with an extra restriction of symmetry on the considered system
56 matrices: (\i
| (Ks e K1) (Ms e M1) (Ks*i = AiM.#i) |
(13)
with Ks and Ms symmetric. The most important effect of this extra restriction is that it intrinsically removes the conservatism resulting from allowing artificial non-symmetric system matrices. Currently, a number of research activities aiming at approximating the solution of the problem stated in Eq. (10) are still ongoing n(') + ^ 2 ( 0 ) 1, p=l, -1 < p < 0 and 0 < p < 1, respectively. Genetic algorithm is one of the effective methods to solve economic dispatch problem [7]. General genetic algorithm, contrary to conventional optimization method, has strong global search capability and weak local search ability. It can increase search performance to combine genetic algorithm with conventional optimization method. Based on this idea, a new optimization method, hybrid genetic algorithm with quasi-simplex techniques (GAQST) is proposed in this study to solve general single nonlinear optimization problems. The iteration steps of the GAQST are described as follows:
64 Step 1: Initialize a population of size \x = K (w+1); Step 2: Evaluate the fitness values for each individual Xt of the population based on the objective function/(x.) ; Step 3: Subdivide the population into K subpopulations; Step 4: For each subpopulation, create an offspring by genetic operations and quasi-simplex techniques in parallel. In order to increase subpopulation varieties, select respectively the best point as offspring from the points obtained by the formulas (10) and (12). The rest offspring of subpopulation are created by reproduction, crossover and mutation operations; Step 5: Unite all offspring created in Step 4 to form a new generation population; Step 6: Stop the iteration if the termination criterion is satisfied, otherwise, go back to Step 2. 4. Experiment Results By combining the proposed WIPM, GAQST and fuzzy number ranking methods, a new approach is developed to solve the bi-objective constrained fuzzy non-linear programming problem described by formula (1). Firstly, convert the FDEELD into the single objective optimization problem by using WIPM. Secondly, use the Lagrange relaxation method to form a Lagrange function. Finally, use the GAQST to optimize the Lagrange function. In the process of the iteration, fuzzy number ranking method proposed in the literature [8] is used to compare fuzzy function values of different points for the single objective function. 4.1. Objective Function Using the Lagrange relaxation method, penalty function h can be written as h(x) = w, (1—1.—f +
wJ^-^—)2
+ Mmax{0,PJhw-PJ(t)}
+ Mmax{Q,Pj(t)-Pjhigh} 4.2 Test Data Table 1 to Table 4 show respectively the test data of the units output, cost function, emission function, and load demand.
65 Table 1 Limits of unit output and ramp rate Unit No.
P«,(MW)
1 2 3 4 5 6 7
20 20 35 35 130 120 125
f _ W
R
*>J
125 150 225 210 325 310 315
J
40 40 50 50 60 60 60
30 30 40 40 50 50 50
Table 2 Fuzzy coefficients of the cost functions Unit no 1 2 3 4 5 6 7
a
a2
825.72578 645.32513 1135.89710 1198.86520 1586.73960 1295.65920 1496.65170
846.36892 661.45826 1158.61504 1222.84250 1610.54069 1315.09409 1519.10148
\
«0 800.95401 625.96538 1107.49967 1168.89357 1555.00481 1269.74602 1466.71867
bo 37.46062 41.32673 38.83637 36.90654 36.58126 38.29901 36.52011
b,
h
38.53973 42.51721 39.83217 37.85286 37.32782 39.08062 37.26542
39.46468 43.53762 40.62881 38.60992 37.92507 39.70591 37.86167
c
o
c
C
i
0.16218 0.12359 0.02705 0.03472 0.02521 0.01682 0.02013
0.15813 0.12050 0.02651 0.03403 0.02478 0.01653 0.01979
2
0.16559 0.12619 0.02754 0.03534 0.02559 0.01707 0.02043
Tat)le 3 Fuzz)f coefficients of the emission functions Unit No. 1 2 3 4 5 6 7
«o
a,
a2
A
A
A
r0
h
Yi
15.18178 15.18178 34.69310 34.69310 42.03762 40.92147 40.92147
15.65132 15.65132 35.58267 35.58267 42.89553 41.75660 41.75660
16.04260 16.04260 36.29432 36.29432 43.53896 42.38295 42.38295
0.28456 0.28456 -0.54136 -0.54136 -0.52138 -0.53245 -0.53245
0.29276 0.29276 -0.52816 -0.52816 -0.51116 -0.52201 -0.52201
0.29979 0.29979 -0.51760 -0.51760 -0.50298 -0.51366 -0.51366
0.00382 0.00382 0.00698 0.00698 0.00453 0.00464 0.00464
0.00392 0.00392 0.00712 0.00712 0.00461 0.00472 0.00472
0.00400 0.00400 0.00725 0.00725 0.00468 0.00479 0.00479
Table 4 Load demand T D T D
1 690
2 670
3 670
4 680
5 730
6 800
7 870
8 840
9 890
10 920
It 950
13 890
14 890
15 930
16 970
17 930
18 950
19 1070
20 1040
21 950
22 850
23 760
12 91 0 24 73 0
T-time segment; D—correspondence load demand
4.3 Test Results Penalty function h is a high-dimension nonlinear function, and therefore it is hard to know where the global minimum point is. In order to demonstrate the effectiveness of the proposed algorithm, the mean and standard deviation of fuzzy fuel cost, fuzzy emission and fuzzy total cost corresponding with the optimal outputs are tested. In addition, in order to compare the magnitude of effect of the set of weights to fuzzy fuel cost and fuzzy emission, we calculate 3 group weights. Table 5 lists the means and standard deviations of fuzzy fuel cost, fuzzy emission and fuzzy total cost by the proposed algorithm through running independently 10 times, where MFC, MEC, and MTC present the means of the fuel cost, the emission, and the total cost, respectively; STDEV-FC, STDEV-EC and STDEVTC present corresponding standard deviations.
66 Table S The comparison of the results obtained for different weights
0,,w>) (0.3,0.7)
(0.5, 0.5)
(0.7, 0.3)
MFC
STDEV-FC
MEC
STDEV-EC
MTC
STDEV-TC
1067359 1092154 1112300 1061711 1086218 1106213 1053936 1078110 1097695
291.4 303.8 312.4 472.8 377.1 615.9 57 58.3 60
11423.23 11993.81 12539.55 11466.7 12041.25 12596.67 11600.89 12184.12 12744.69
2.2 2.5 2.7 4.2 5.3 9.4 1.3 1.4 1.5
1078780 1104148 1124838 1073181 1098320 1118805 1065S37 1090295 1110440
290.7 300.7 310.4 468.5 540.6 607.9 55 58.2 60.9
As the standard deviations of every result are all significantly small, the results are believable. It can be seen that the fuel cost decreases and the emission increase when the weight of the fuel cost is assigned higher. 5. Conclusions This study proposes a new environmental economic load dispatch model with a consideration of uncertainty in the coefficients of fuel cost and emission functions. The weighting ideal point method, hybrid genetic algorithm with quasi-simplex techniques and fuzzy number ranking method are used to solve the optimization problem described in the model. The proposed method has two main advantages: (1) it characterizes more precisely the fuel cost and emission; and (2) it provides more information than real number-based methods. References [1] Abido, M. A. (2003), "Environmental/economic power dispatch using multi-objective evolutionary algorithms", IEEE Transactions on Power Systems, Vol. 18, No. 4, pp. 1529-1537. [2] Venkatesh, P., Gnanadass, R, Padhy, N.P.(2003), "Comparison and application of evolutionary programming techniques to combined economic emission dispatch with line flow constraints", IEEE Transactions on Power Systems, Vol. 18, No. 2, pp. 688-697. [3] Rughooputh, H.C.S., Ah King, R.T.F. (2003), "Environmental/economic dispatch of thermal units using an elitist multi-objective evolutionary algorithm", Industrial Technology, Vol. I, pp. 48-53. [4] Huang, C. M., Yang, H. T. and Huang, C. L. (1997), "Bi-objective power dispatch using fuzzy satisfaction-maximizing decision approach", IEEE Transactions on Power Systems, Vol. 12, No. 4, pp.1715-1721. [5] Watts, D., Atienza, P. and Rudnick, H. (2002), "Application of the Power Exchange-Independent System Operator Model in Chile", Power Engineering Society Summer Meeting, 2002 IEEE, Vol. 3, pp. 1392-1396. [6] Nelder. J. A., Mead. R. (1965), "A simplex method for function minimization", the computer Journal, 5. [7] Damousis, I. G., Bakirtzis. A. G., Dokopoulos. P. S. (2003), "Network-constrained economic dispatch using real-coded genetic algorithm", IEEE Transactions on Power Systems, Vol. 18, No. l,pp. 198-205. [8] Lee, E. S., Li. R. L. (1988), "Comparison of fuzzy numbers based on the probability measure of fuzzy events", Comput. Math. Appl. 15, pp. 887-896.
MINIMIZING THE NUMBER OF AFFECTED CONCEPTS IN HANDLING INCONSISTENT KNOWLEDGE* ERIC GREGOIRE + CRIL CNRS & IRCICA Lens, F-62307, France
In this paper, we introduce a new family of approaches to fuse inconsistent logic-based knowledge sources. They accommodate two preference criteria to arbitrate between conflicting information: namely, the minimisation of the number of contradicted formulas and the minimisation of the number of the different terms that are involved in those formulas.
1. Introduction Logic-based approaches to knowledge fusion can be classified according to several criteria (see e.g. [5] for a survey). Among them, the syntax dependence/independence criterion remains a controversial one [7]. Many authors claim that any acceptable logic-based approach to knowledge fusion should be syntax-irrelevant in the sense that the way according to which knowledge is expressed within formulas should not influence the resulted fused knowledge [5,6]. Such a claim is widely accepted when fusing the beliefs of rational agents is modelled. Indeed, it does not seem acceptable to consider that the beliefs of rational agents should depend on their syntactical representation. In this paper, we are not concerned with fusing the beliefs of rational agents. We are rather concerned with the information that should be fused from mere computerized knowledge sources, simply. In this context, syntax can be important and must sometimes be taken into account, at least to some extent. For example, in earliest expert systems, inference engines a la OPS implemented a different treatment for mere facts and longer formulas, based on the assumption that the longer a formula is, the more specific it should be. More generally, the way according to which knowledge is expressed may embody important implicit information. Such an implicit information can be the result of the -conscious or * This work is supported in part by an EC FEDER grant and by the Region Nord/Pas-de-Calais [email protected]
67
68 unconscious- will of the user, or e.g. simply the result of previous fusion steps. For example, a user might want to insist on the true character of a formula by asserting it several times. Else, the multiple occurrences of a formula can be the result of several converging information sources, contributing to its acceptability. Also, the intended meaning of the formula a and b and c in a knowledge base can be different from the meaning of the presence of the three formulas a, b and c. Indeed, to some extent, the unique formula a and b and c can be a way to assert that the three terms are correlated true, whereas the three mere individual facts can sometimes be independent ones. From a technical point of view, a purely semantic-based approach to knowledge fusion interprets the initial knowledge sources as sets of models, namely truth assignments satisfying all the involved formulas. Sources exhibiting the same models are thus indistinguishable in this respect. Accordingly, it can be claimed that such a semantic-based approach is too coarse-grained with respect to some artificial intelligence applications. On the other hand, a purely syntaxbased approach can be interpreted as too fine-grained, since it can make a difference between formulas a and b and b and a. In this paper, a trade-off between syntax and semantic-based approaches is proposed. It is syntax-based in the sense that it might be influenced according to the way knowledge is split into formulas. However, it will make no difference between e.g. formulas a and b and b and a. It is semantic-oriented in the sense that it will define the resulted fused knowledge by means of its set of models. 2. Semantic-oriented approaches For the simplicity of the presentation, we use standard propositional logic, although all results in the paper can be easily extended to the first-order finite Herbrand case. We follow the notations from [1], Let L be a propositional language of formulas over a finite alphabet P of Boolean variables, also called atoms. The A , v, -i and => symbols represent the standard conjunctive, disjunctive, negation and material implication connectives, respectively. A literal is an atom or a negated atom. A term designs a literal without its sign. Q. denotes the set of all interpretations of L, which are functions assigning either true ox false to every atom. A model of a knowledge base KB is an interpretation that satisfies every formula of KB. An interpretation or a model will be represented by the set of literals that it satisfies. The set of models of KB will be denoted [[.O]]. KB is consistent when [[KB]] is not empty. KB |= x expresses that the literal x can be deduced from KB, i.e. that it belongs to all models of KB.
69 Let us consider a multi-set of n (« > 1) consistent prepositional knowledge bases E = {KB\,...JCBn} to be fused. Fusion operators will be defined as functions A that associate a knowledge base, denoted A(E), to each information set E. From a syntactical point of view, a knowledge base KB is thus a set of formulas of L. Many syntactical approaches to knowledge fusion amount to taking preferred maximal (with respect to cardinality) consistent subset(s) of formulas of the set-theoretic union of the bases to be fused (see e.g. [3]). From a purely model-theoretical point of view, a KB is a set of models. Most modeltheoretic approaches rank order the set of interpretations w.r.t. all KBi, using the Hamming distance (also called Dalai's distance [4]) from a model of KBt. A(E) is then characterized by the set of interpretations that are minimal in some sense with respect to this rank-ordering. In the following, we shall adopt a trade-off between these two points of view. But let us first describe the model-theoretic approach in more details. The Hamming distance between an interpretation ro and a prepositional knowledge base KBt is defined as the smallest number of atoms about which this interpretation differs from some model of KB\. Definition 1. d(aJCBi) = min m^ [[oi]] dist(co,co') where dist(co,co') is the number of atoms whose evaluation differs in the two interpretations. In the following we shall investigate several definitions for rank-ordering interpretations from Q. Accordingly, assume for the moment that an overall distance noted dA(co,E) between an interpretation and the multi-set E has been defined, already. Definition 2.
co •
Definition3. [[A(E)]] = min(Q,P2(co,A2?j)
72 Definition 4'. (Majority operator for dPi>P2(co,Aj5j)) d2Ti>P2 (P2 (co',E) p2
Now, how can the Max-based egalitarist operator be adapted? Definition 6'.
dmaxPi>P2 (co,E) = (Max(j 6[1..n]) min,, Max(i e[1..B]) min2)
Let us note that the two Max operators are uncoupled in the sense that the maximal number of involved literals in falsified formulas does not necessarily concern the maximal number of falsified formulas among the knowledge sources where the number of falsified formulas are minimized.
73 Definition 10. roe [[A(E)]] ij^there is no ro' s.t. dmaxPl>?2 (ro'.E) ?2 (co,E). 4.
Example
Let us illustrate these new operators on a simple example. Assume that we need to fuse the following knowledge sources. KB] = {a, - i c } , KB2 = {a => (b A C), 6} and . 0 3 = {-.a A C}. Clearly, KB\ u £ B 2 ^ ^ ? 3 is inconsistent. Let us now compute the above operators. In the following table, we indicate for each interpretation col, the resulting distances and couples of distances. The interpretations corresponding to distances and couples that are underlined are models of the fused knowledge source. Distances and couples that are bared represent solutions that are withdrawn by the new operators with respect to the corresponding initial ones. From this example, we clearly see that the d£pi>P2 and the dmax p1>P2 operators narrow the range of solutions that could be obtained by the dx and d , ^ operators. We can see in the example that the interpretation {a, b, —\c} is not a model anymore of the fused knowledge sources when the d£Pi>P2 operator is considered. Indeed, it requires dropping or weakening formulas that globally involve three terms, whereas it is sufficient to regain consistency to drop a similar number of formulas involving less terms. Obviously enough, similar results can be obtained for the weighted sum operators as well. Table 1. Example to 0)1
a>2 »3 004 0)5 0)6 0)7 0)8
5.
a false false true false true true true false
b false true false true true false true false
c true false false true false true true false
dj(o),E)
dzPi>P2(o),E)
dmax(0),E)
3 2 3 2 2 4 2 3
(3,3)
2
(2,2)
02)
I
(L2)
(3,3) (2,2)
2 2 1 2
(2,3) (2,2)
I I
£L2)
(4,3)
02J (3,3)
dmax P1>P2(0),E)
(2,3)
02}
Conclusions
In this paper, a new family of syntax-based approaches to knowledge fusion has been introduced. Clearly, it provides a fair trade-off between purely syntax-based and semantic-oriented ones. Its main original feature is that it allows several possible solutions to be discriminated based on the number of different terms that it requires dropping.
74 References 1. S. Benferhat, D. Dubois, S. Kaci, and H. Prade, "Encoding classical fusion in possibilistic logic: a general framework for rational syntactic merging", Proc. ofECAI'2000, pp. 3-7 (2000). 2. B. Bessant, E. Gregoire, P. Marquis and L. Sais, "Iterated Syntax-Based Revision in a Nonmonotonic Setting", Frontiers in Belief Revision, Kluwer, Applied Logic Series, 22, pp. 369-391 (2001). 3. C. Cayrol, M.-C. Lagasquie-Schiex and T. Schiex, "Nonmonotonic reasoning: from complexity to algorithms", Annals of Mathematics and Artificial Intelligence, 22, pp. 207-236 (1998). 4. M. Dalai, "Investigations into a theory a knowledge base revision: preliminary report", Proc. ofAAAI'88, pp. 475-479 (1988). 5. E. Gregoire and S. Konieczny, "Logic-based approaches to information fusion", Information Fusion, 7, pp. 4-18 (2006). 6. S. Konieczny, and R. Pino Perez, "Merging with integrity constraints", Proc. ofEcsqaru'99, pp. 233-244, LNCS 1638, Springer (1999). 7. E. Gregoire, " Syntax and semantics in knowledge fusion: a mixed approach", Proc. of the 6' Int. Conf on Sensor Fusion: Architectures, Algorithms and Applications, pp. 60-64, Orlando, (2002). 8. E. Gregoire, "Extension of a distance-based fusion framework, Proc. of the 8th Int. Conf. on Sensor Fusion", pp. 282-286, Orlando (2004). 9. J. Lin, "Integration of weighted knowledge bases", Artificial Intelligence, 83, pp. 363-378(1996). 10. J. Lin, and A.O. Mendelson, "Merging databases under constraints", Int. Journ. of Cooperative Information Systems, 7(1), pp. 55-76 (1998). 11. N. Rescher, and R. Manor, "On inference from inconsistent premises", Theory and Decision, 1, pp. 179-219 (1970). 12. P.Z. Revesz, "On the semantics of theory change: arbitration between old and new information", Proc. of the 12th ACM SIGACT-SIGMOD-SIGART Symp. on Principles of Databases, pp. 71-92 (1993).
A KNOWLEDGE MANAGEMENT BASED FUZZY MODEL FOR INTELLIGENT INFORMATION DISPOSAL * XIAOBEI LIANG 1 Shanghai Business School, Shanghai, 200235, China School of Management, Fudan University, Shanghai, 200433, China ZIJING ZHANG Glorious Sun School of Business and Management, Dong Hua University, Shanghai, 200051, China DAOLIZHU School of Management, Fudan University, Shanghai, 200433, China BINGYONG TANG Glorious Sun School of Business and Management, Dong Hua University, Shanghai, 200051, China Based on analysis of a knowledge management fuzzy model, the judgment thinking process is discussed in this paper. Then, fuzzy models of three thinking forms based on intelligent information disposal are established for abstract thinking, fuzzy pattern recognition model of imagination thinking, and fuzzy state equation model of intuitive thinking. Finally, a fuzzy integrated judgment model can be established on the basis of three thinking forms and different fuzzy models by fuzzy large-scale system modeling techniques.
1. Introduction Knowledge management is a systemization, a process that drives management and uses group knowledge, and includes knowledge foundation, knowledge issuance and sharing together1"2. Generally, a knowledge management system is composed of knowledge gaining, generating system, knowledge issuance, and sharing system. Both knowledge gaining and generating system mainly use the knowledge gaining relevant knowledge and experiences or supplies with a ' This work is supported by grant 05ZR14091 of the Shanghai Natural Science Foundation. Corresponding author. Tel.; 0086-21-62373937; Fax: 0086-21-62708696. E-mail address: [email protected] (X.B. Liang)
1
75
76 definite knowledge environment to create relevant knowledge. Moreover they document it and save it for the later use3"4. Knowledge issuance system mainly uses computer network technology, data storage technology management and issues relevant knowledge. Knowledge sharing system mainly uses group apparatus and workflow system to reach knowledge sharing. In this paper, based on an analysis of knowledge management fuzzy model5, the judgment thinking process is discussed. Then, fuzzy models of three thinking forms based on intelligent information processing are set up. 2. Human's thinking activity processes Most knowledge management processes are human's thinking activity processes, in fact they are typical intelligent information processing. The related information in this process mostly is fuzzy intelligent information except little precise information. Consequently, in the model base system of knowledge management-based enterprise strategic decision support systems we should start with the human brain thinking process and establish corresponding models. The typical thinking activity process could be regarded as an intelligent information processing, besides, it could be divided into four specific process: (1) information processing via recognition process; (2) information transmission via communication process; (3) information storage via memory process; and (4) information usage via decision process. Such process could regarded as some input information, some output information by processing. That is, employing great system control modeling idea, looking on the human brain as a black box, we emphasize to study the relation of thinking process input and output, represent the external behavior, function and result. Then we should establish the index system of input and output information, study the fuzzy information quantitative processing method, finally establishes the corresponding integrated fuzzy models. Generally speaking, the arranged thinking activity process could be divided to the types of judgment, expression, storage and decision according to the thinking target. 2.1. Judgment thinking process The object of judgment thinking process is some objective state of affairs, its terminal is judgment. However, such thinking process also could be divided into some specific types, three thinking links are confirmed. Question Inference Judgment Here, the first question is the suggested question, including initial simple judgment, or by directly proposed, or by collection. Moreover including the related condition and elementary matter, we should arrange these factors.
77 Inference itself is a great deal of front-to-back joined complicated links as well as a single link. We should employ different thinking inference forms according to some specific problem. The conclusion of inference is the judgment, in which the target desires, but this judgment should be integrated into a harmonious result of several thinking forms. 2.2. Decision thinking process Decision thinking process is the thinking process with some actions for object, also could be divided into some specific patterns, for instance: 1) Selecting The main thinking frame is the following links' combination: Collection Comparison Corresponding Decision Examination Such a collection should be various input forms. Input information represents a possible combination of two or more actions. Then we should compare these several possibilities, specially taking the long-range or current interest as the standard, or we could select other standards to compare. Finally we should take steps according to the selective results, and select the feedback information to proof in the practical action. 2) Feedback The main frame form is: Determination Gathering Correcting Proving On some condition one might not think the object factor and measure over before the action. So in the beginning one should make decision in a hurry. Then one starts to gather the information and improve the action in practice. 3. The fuzzy model base of intelligent information disposal Human beings have the perfect thinking organization that could use their own knowledge flexibly to solve the problem with different thinking forms. Commonly human conscious purposeful thinking forms include: (1) abstract thinking; (2) imagination thinking; (3) intuitive thinking. We should use various methods to establish some corresponding fuzzy model in terms of different characteristics of three thinking forms. 3.1. Fuzzy logic inference Model of abstract thinking Abstract thinking is also called logic thinking. It bases sensibility understanding, that reflects nature of a thing correctly or incorrectly, and reveals its internal relation of thinking forms by concept, judgment and inference. In practical
78 abstract thinking people apply the fuzzy concept and fuzzy inference more, so we could employ fuzzy logic method to establish the corresponding fuzzy inference model. Generally speaking, consider the case with two-input, one-output fuzzy system: Input: x equals A' and y equals B' R1: if we impose x equals A1 and y equals B1, then z equals C1; also R2: if we impose x equals A2 and y equals B2, then z equals C2; also Rn: if we impose x equals An and y equals Bn, then z equals Cn. Output: z equals C\ where x, y and z might represent the variables of the state and control quantity of system, Ai, Bi and Ci are the value of x, y and z, respectively. Consider a series of fuzzy control regulation, we should have an overview of fuzzy inclusion relation:
Finally, we get C"=Cc). E x a m p l e 2.1. Considering dom(T)={more true, more false, true, false, less true, less false}, more true strengthen the meaning of true, that is, (more, true)> (I, true) = true; more false strengthen the meaning of false, so, (I, false) = false> (more, false); less false strengthen the negative meaning of false, so, (less, false)> false. L e m m a 2.1. 15 Let C and H be finite linear ordered sets according to natural linguistic positive meaning, respectively. T = 77 x C = {(h, c)\h £ 77,c £ C} such that\/(h\,c\),(h2,C2) £ T, [h\,C\) < (h2,C2) if and only if hi (g(y;h2v),t2)
(21)
^
(p(x; u) -> q(y, v), tx -> t2) P H i :
(P(^. u) ~* g(2/. v ), h), (P(x, u), t2) {q{y,v)M®t2) '
p ( p ( x ' " ) . *i) -* (g(y» •")' f2)» ( p f a u)^s) K5 . —; r—^
.
,
. W>
,0As (24)
Where, operator " " is the lukasiewicz conjunction. Rq Hb
. (p(x;u),t) • (p(a;u),ty
P~Q,(F(P),ht) Hh
-
(F(Q/P),ht)
'
{ 5)
E x a m p l e 5.1. Suppose the following linguistic sentences: 1. If a student works rather hard is very true and his university is very high-ranking is rather true , then he will be a good employee is possibly very true . 2. The university where Robert studies is very high-ranking is possibly true. 3. Robert is studying very hard is rather true. The question: how about is Robert will be a good employee? By formalizing, 1, 2 and 3 can be rewritten as following: 4. (studying(x; rather hard), (very, true)) A (is(Univ(x); very highranking), (rather, true)) —>(employee(x; good), (possibly, (very, true))). 5. (is(Univ(Robert); very high-ranking), (possibly, true)). 6. (studying(Robert; very hard), (rather, true)). Deduction process is as follows:
93 (1) (('studying(x; hard), (r, (v, t))) A (is(Univ(x); high-ranking), (v, (r, t)))) ~+ (employee(x; good), (p, (v, t))), (2) (studying(x; hard) A is(Univ(x); high-ranking), (r, (v, t)) A (v, (r, t))) —> (employee(x; good), (p, (v, t))), (by 1 and Eq.(20)) (3) ((studying(x; hard) A is(Univ(x); high-ranking))—* employ ee(x; good), ((r, (v, t)) A (v, (r, t))) -> (p, (v, t))), (by 2 and Eq.(22)) (4) (is(Univ(Robert); high-ranking), (v, (p, t))), (by v ) and Eq.(17)) (5) (studying(Robert; hard), (v, (r, t))) (by vi) and Eq.(17)) (6) (studying(Robert; hard) A is (Univ (Robert); high-ranking), (v, (r, t)) A (v, (p, t))), (by 4, 5 and Eq.(20)) (7) ((studying(Robert; hard) A is (Univ(Robert); high-ranking))-^ employee (Robert; good), ((r, (v, t)) A (v, (r, t))) -> (p, (v, t))), (8) (employee(Robert; good), (((r, (v, t)) A (v, (r, t))) -> (p, (v, t))) ® ((v, (r, t)) A (v, (p, t)))). (by 6, 7 and Eq.(23)) According to Example 4.1, we obtain (r 6 , (v, t)B) A (v7, (r, t) B ) = (r 6 , (r, t ) 5 ) ,
(26)
((re, (u, t)6) A (v 7 , (r, i) 6 )) -> (p5, (v, t)6) = (re, (r, t) B ) ~> (PB, (V, t) 6 ) = (r fl , («, t)e),
(27)
(W7, (r, t)s) A (v7, (p, t)t) = (v7, (p, t)i),
(28)
(re, (v, t)a) ® (v 7 , (p, t)4) = (r 6 , (p, t) 4 ).
(29)
Hence, the conclusion is (employee(Robert; good), (r, (p, t))), i.e., Robert will be a good employee is rather possibly true. 6. C o n c l u s i o n In this paper, based on H x C, algebra structures of linguistic domain of Truth are discussed. On the other hand, based on f(HxC) and Lukasiewicz product algebra, we obtain Lukasiewicz algebra model of linguistic values of Truth. Hence, reasoning of linguistic values is embedded in logic system, and the deduction of linguistic values has reached high confidence. The conclusions of this paper can be applied to linguistic decision, linguistic information retrieval, and so on. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 60474022). Education Department Foundation of Sichuan Province (Grant No. 2005A121).
94
References 1. Paul P. Wang, Computing with words (John Wiley and Sons, Inc, 2001). 2. Hajek. Petr., Methamathematic of fuzzy logic (Kluwer Academic Publishers, 1998). 3. V. Novak, I. Perfilieva, J. Mockof, Mathematical principles of fuzzy logic (Kluwer Academic Publishers, 1999). 4. L. A. Zadeh, "Fuzzy logic = computing with words", IEEE Trans. Fuzzy Systems. 4 (1996) 103-111. 5. L. A. Zadeh, "Toward a theory of fuzzy information granulation and its centrality in houman reasoning and fuzzy logic", Fuzzy Sets and Systems. 9 0 (1997) 103-111. 6. S. Ben-David, R.Ben-Eliyahu-Zohary, "A modal logic for subjective default reasoning",Artificial Intelligence. 116 (2000) 217-236. 7. P.Cholewinski, V.M.Marek, A.Mikitiuk and M.Truszczynski, "Computing with default logic", Artificial Intelligence. 1 1 2 ( 2 - 3 ) (1999) 105-146. 8. J.L.Castro, "Fuzzy logics as families of bivalued logics", Fuzzy Sets and Systems. 6 4 (1994) 321-332. 9. Yang Xu, Da Ruan, Etienne E. Kerre and J u n Liu, "a—Resolution principle based on first-order lattice-valued logic LF(X)", Information Sciences. 1 3 2 (2001) 221-239. 10. Yang Xu, Da Ruan, Keyun Qin and J u n Liu, Lattice-Valued Logic (SpringerVerlag Berlin Heidelberg, 2003). 11. M.Chechik, S.Easterbrook and V.Petrovykh "Model-checking over multivalued logics", In Proc. of the 2001 FME. LNCS. 2 0 2 1 (2001) 72-98. 12. R.O.D'Aquila, C.Crespo, J.L.Mate and J.Pazos, "An inference engine based on fuzzy logic for uncertain and imprecise expert reasoning", Fuzzy Sets and Systems. 1 2 9 (2002) 187-202. 13. F.Herrera, E.Herrera-Viedma, "Aggregation operators for linguistic weighted information", IEEE Trans.System,Man, Cybernet.-Part A:Systems Humans. 2 7 (1997) 646-656. 14. Nguyen Cat Ho, Tran Dinh Khang, Huynh Van N a m and Nguyen Hai Chau, "Hedge algebras, linguistic valued logic and their application to fuzzy reasoning", International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. 7 (1999) 347-361. 15. Z. Pei and Y. Xu, " Lattice implication algebra model of linguistic variable T r u t h and its inference", in Applied Computational Intelligence, eds Da Ruan, et al (World Scientific, 2004) p p . 93-98. 16. N. C. Ho and W. Walfgang, "Hedge algebras, linguistic valued logic and their application to fuzzy reasoning", Fuzzy Sets and Systems. 3 5 (1990) 281-293. 17. N. C. Ho and W. Walfgang, "Extended hedge algebras and their application to fuzzy logic", Fuzzy Sets and Systems. 52 (1992) 259-281. 18. L. A. Zadeh, " T h e concept of linguistic variable and its application to approximate reasoning (I), (II)", Information Sciences. 8 (1975) 199-149, 3 0 1 357.
P R O P O S I T I O N A L L O G I C L6P(X) BASED ON SIX-LINGUISTIC-TERM LATTICE IMPLICATION ALGEBRA *
WANG WEI Department of Electrical Engineering, Southwest Jiaotong University Chengdu 610031, China E-mail: [email protected] XU YANG AND ZOU LI Department of Mathematics, Southwest Jiaotong University Chengdu 610031, China E-mail: xayangQhome. swjtu. edu. en
In this paper, we present mathematical properties of lattice implication algebra LQ with six linguistic terms: true, false, more true, more false, less true, less false. We also discuss the properties of propositional logic LeP(X) and present the structure of generalized literals and the standardization of formulae.
1. I n t r o d u c t i o n Since 1990, there have been some important conclusions on linguistic truthvalue inference. In 1990, Ho * constructed a distributive lattice-Hedge algebra, which can be used to deal with linguistic truth-value propositions. In 1996, Zadeh 3 discussed the formalization of some words and the proposition of natural languages. He presented the standardization forms of language propositions and production rules, and discussed the fuzzy inference, which has some linguistic truth-values, based on the fuzzy set theory and the fuzzy inference method. Between 1998 and 1999, Turksen 4 , s studied the formalization and the inference about nature languages which have descriptive words and substantive words. It is only a formalized statement that the research has been carried "This work is supported by the Natural Science Foundation of P. R. China (Grant No. 60474022)
95
96 out on inference which has linguistic truth-values, and it isn't on a logical system. Because there are many differences between the world which we described by the two-valued logic and the world which we are living in, it is unreasonable to solve the problem by using the inference which has uncertainty information by the two-valued logic, so we use the lattice-valued logic as a basis of inference which has some linguistic truth-values. The problem about automated theory proof on M-V logic and fuzzy logic has been researched by many people. Many developments and generalizations on classical resolution have been given. But the logical implication used by researchers is Kleene implication (p —> q = p' V q). In syntax, an M-V logical (or fuzzy logical) formula is equivalent to itself in the classical logic. It is important to give an implication connective which is different from Kleene's, and to give a resolute method which is different from the classical resolute method in order to deal with uncertainty information. In 1993, Xu 7 conbined lattice and implication algebra, and found new algebra-lattice implication algebra. Because the implication in lattice implication algebra is a generalized one described by axioms, and it can be used to solve the problem of the inference in the real world, the lattice-valued propositional logical system LP(X) and the lattice-valued first-order logical system LF(X), which are based on the lattice implication algebra, were constructed by Xu 8 ' 9 . Based on the lattice-valued logical algebra, the a—resolution principle on logical system LP{X) and LF(X), which are generalization of the classical resolution principle, were given by Xu 11>12. The a—resolution principle can be used to prove a set of lattice-valued logical formulae being a—false (a £ L). For studying the uncertainty and automated inference with the linguistic truth-value, Pei and Xu 1 3 presented a lattice implication algebra model of a kind of linguistic variable truth, and discussed its inference. In this paper, based on Xu's work of linguistic truth-value implication algebra, the properties of LQ and the properties of LQP(X) were discussed. 2. T h e p r o p e r t i e s of LQ and
LQP(X)
The lattice implication algebra LQ is denned in Fig.l. Clearly, less true, less false and more true are dual numerators of LQ. In LQ, the operation "/" is equivalent to the negative connection of natural language and is defined as follows, (true)' = false, (false)' = true, (more true)' = more false, (more false)' — more true, (less true)' = less false, (less false)' = less true. The implication operation "—>" is equivalent to the implication connection of natural language and is defined in Table 1.
97 a true less false ,)
Va £ Le, a —»false = a', false —> a = true, true —* a = a. Va £ LQ, less false —> a = (less true) V a. Va £ LQ, less true —» a = (less false) V a. Va £ Lg, a —> /ess /rue = a' V /see /rwe. Va £ LQ, a —» /ess /o/se = a' V /ess /a/se. Va £ L$, if a ^ more false, and a ^ more true, Zfoen aV a' = true, a A a' = false.
Since LgP(X) is a lattice implication algebra. For arbitrary logical formulae F,G £ L6P(X), let F V G = ( F - * G) -» G, F A G = ( F ' V G')'. T h e o r e m 2.2. For even/ logical formula F £ LQP(X),
(1) f2j (s) ^J (5j
less true —• F = (/ess /rue)' V F. less false -+ F = (/ess /a/se)' V F. /roe - 4 F = F . F —* less true = (/ess true) V F ' . F -> less false = (less false) V F ' .
then
98 In the lattice implication algebra LQ, the implication operations are Kleene implications except for more true —> more false and more false —» more true. According to Theorem 2.2, if a ^ more false, a ^ more true, then o —» F and F —> a are Kleene implications. Here, F is a logical formula of L§P{X). It is well know that a logical formula, which only contains Kleene implications, can be translated into a conjunctive (disjunctive) normal form. T h e o r e m 2 . 3 . For every variable x more false, x A (x —> x') < more false. 3. Generalized literals of
£ LQP(X),
we have x A x'
x, x —> c. Here, x is a propositional variable. If c ^ {more true, more false, false}, F is equivalent to the formula of following: x, I, c' V x, x' V c. This contradicts that F is a 1—generalized literal. Suppose the conclusion holds for n < k. If n = k + 1, according to the Lemma 11.3.2 6 , there exist k\— generalized Fi and fo—generalized literal
99 F j , such that F = Fi —• F2, fci + &2 + 1 = fc + 1. If there is a constant c occurring in F, then c occurring in F\ or F 2 . If c occurring in F x , and k\ = 0, then F = c —• F2, and c ^ {more true, more false, false}, then F is the one of the following forms: F = F2, F = c' \/ F2. This contradicts that F is an n—generalized literal. If c occurring in F\, and fci > 0, by the proof about n = 1 and the hypothesis of induction, we have c 6 {more true, more false, false). If c occurring in F2, we can prove that the conclusion holds similarly. According to the mathematics induction, the conclusion holds for every number n. For convenience, we let A = {more true, more false, false}, A* — {more true, more false}. T h e o r e m 3.2. For arbitrary a, b g A*, we have (1) b -> (6 -> a) = b -> a. (2) {a->b)-+a = a. T h e o r e m 3 . 3 . For every propositional formula F of LQP(X), b £ A*, we have
and every
(1) {F^b)^F = F. (2) F -» ( F ^ 6) = F -» 6. P r o o f We only give the proof of conclusion (1), the proof of conclusion (2) can be given similarly. For an arbitrary valuation of LQP{X). If v{F) € A*, according to Theorem 3.2, (v{F) -» 6) -> v{F) = v(F). If v(F) v(F) = (v(F)' V b) -
v(F) = {v(F)' -» v{F)) A (6 -» v(F))
= v(F)A(b-*v(F))
= v(F).
Then the conclusion (1) holds by the arbitrarily of valuation v. T h e o r e m 3.4. For arbitrary propositional have
variable x and a,b £ A*, we
(1) (a^F)^(F^b) = F-+b. (2) ( F -+ a) -» ( F -* 6) = F -» (a -» b). (3) ( ( F -» a) -> 6) -» a = ( F -> a) V (6 -> a).
P r o o f We only give the proof of conclusion (1), the proof of conclusions (2) and (3) can be given similarly. If a = 6, we know that the conclusion (1) holds by properties of lattice implication algebra. If a ^ 6, for an arbitrary valuation v of LQP(X), if v(F) € A*, we can prove (o -> v(F)) -* (v{F) -> 6) = w(F) -> b easily. If u(F) ^ A*, according to the definition of implication operation, and a,b € A*, and a 7^ 6, we know o' = 6, o' —> 6 = true, (a -> u(F)) -» (u(F) -» 6) = (u(F) V a') -> ( u ( F ) ' V 6) = ( u ( F ) ' V (t»(F)' V 6)) A ((a' - • u(F)') V (a' -> 6)) ( v ( F ) ' V 6) A true = v(F)' V 6 = v(F) -> 6. Hence, (1) holds by the arbitrarily of valuation v. Definition 3.2. A logical formula F is called an implicative formula, if F only contains a non-Kleene's implicative connection. T h e o r e m 3.5. For every logical formula F of LQP(X), there exist implicative formulae Fjk, j S J, k € K, where J, K are finite index sets, such thatF= V A Fik. P r o o f If F is an implicative formula, it is clear that the conclusion holds. If F has 1 Kleene's implicative connective, then F is one of the following forms: a —• x, x —> a. Here, a j= more true, and a =^ more false. We know that the conclusion holds by Theorem 2.1. Suppose the conclusion holds when F contains k implicative connectives. If F contains fc + 1 implicative connectives, and F = a —> F j , where this implicative connective is Kleene's implicative connective, is a subformula of F , then a —> F\ = a' V Fx by Theorem 2.1, and F can be translated into a conjunction of subformulae Fj* and F£ which have a' and Fi as its subformulae respectively. Since the numbers of implicative connectives occurring in Ff and F2* are all less than k, according to the hypothesis of induction, there exist logical formulae Fjk, j € J i , k £ K\, and F?fc, j € J2, k e K2, where J\,K\,J2, K2 are finite index sets, such that F;=
V
A Fh,Fj=
V
A F?fc.
101 Let J = JlUJ2,K
=
K1UK2 ' Fjk, jGJr,keKi; _ _ I true, j G Juk £ K2; jk | true, j £ J2,k€ Ki; . Ffk, j£j2,keK2.
Clearly F = V
A F,t., and the conclusion holds.
j€Jk£K
3
According to the mathematical induction, for arbitrary logical formula F, the conclusion holds. By Theorem 3.5, we only need to study the implicative formula, and obtain the following theorem. T h e o r e m 3.6. (1) x —> a, (x —* a) —> y, (y —> x) —• a, y —> (x —» a) are generalized literals if and only if a £ A; (2) b —> x, (6 —> x) —> y, (x —> 6) —-> y, b —» (x —> z) ore generalized literals if and only if b 6 ^4*; f3j y - t i , (y -+ x) ->• z, (j/ -+ x) -» y, y -> (a; —> z) and x -+ ( i -> 2) are generalized literals; (4) {x —> b) —» a is a generalized literal if and only if a £ A, b £ A* and a ^ b. (5) (b —• 2) —> a is a generalized literal if and only if a Q A, b £ A*, and a 7^ b'. (6) a —> (b —+ x) is a generalized literal if and only if a = b e A*.
4.
Conclusions
In this paper, the mathematical properties of lattice implication algebra L6 with six linguistic terms were presented. The properties of propositional logic LQP(X), the structure of generalized literals and the standardization of formulae were proposed. These results will be a basis for automated inference dealing with uncertainty information under six linguistic truth-values. Our future work is to study how to select the resolution level a, and how to find the a - resolution field of a generalized literal of LgP(X). Another future work is to prove the soundness theorem and completeness theorem about this a—resolution principle, and present the automated reasoning method based on this a—resolution principle.
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WEIGHTING QUALITATIVE FUZZY FIRST-ORDER LOGIC AND ITS RESOLUTION METHOD* LI ZOU 1 2 , BAIHUA LI 2 , WEI WANG 1 , YANG XU 1 'Center of Intelligent Control and Development, Southwest Jiaotong University, Chengdu, 610031, Sichuan, P.R.China 2 School of Computer and Information Technology, Liaoning Normal University, Dalian, 116029, China
Using Kripke-style semantics, a kind of qualitative fiizzy logic system that can reflect the "elastic" of fuzzy proposition is proposed. The truth value of fuzzy proposition is not a singleton but depends on the context in the real world. Consider a fuzzy proposition one will choose the equivalence relation to get different classes. Based on an equivalence class and its weight, a qualitative fuzzy proposition can hold. Some properties of this system are also discussed. Considering the weight of different class, a method to aggregate the losing information is presented. With the alternation of the possible world and its weight, a dynamic resolution method is introduced.
1. Introduction In classical logic the truth value of a proposition is true or false. Since there are many fuzzy concepts in the real world, the truth value of fuzzy propositions is a real number in the interval [0, 1] [1]. Is it a singleton truth value for a given proposition? As we know, a fuzzy proposition will be given different truth values because of different people or different circumstances. Base on the qualitative fuzzy set [2~7\ the fuzzy characteristic of a fuzzy concept a qualitative fuzzy propositional logic which can reflect the "elastic" of a fuzzy proposition was presented I8]. For example a proposition "5000$ is much money" is it true? Some poor man say: "It is very true!" but a few millionaires say: "It is a little false." If now the poor man is main component of the real world then the truth value should be true. This paper aims to build another fuzzy logic system that can reflect the "elastic" of a fuzzy proposition. The truth value of fuzzy propositions is not a singleton which depends on the context in the real world. Consider a fuzzy proposition one will choose the equivalence relation and get different classes. ' This work is supported by the National Nature Science Foundation of China with granted No. 6047402 and the Innovation Fund of Southwest Jiaotong University for PhD. Candidate.
103
104 Each class will be given a weight. Then based on equivalence classes, qualitative fuzzy propositions can be obtained. The resolution method is discussed. In order to solve the problem of losing information, the weight of possible worlds can be considered so that all the information will be aggregated in the process of resolution.
2. Qualitative fuzzy logic system We consider a fuzzy proposition with different individuals in the real world. For a fuzzy proposition different people will give different truth values. Hence we should consider certain kind of people at first. Then we consider the truth value of the proposition with the given real world. We attempt to represent fuzzy logic using Kripke-style semantic with a little changing. The modal logic interpretation of various uncertainty theories is based on the fundamental semantics of modal logic [9~'l]. That is a model of modal logic M=<W, R, V>, where W, R, V denote a set of possible worlds, a binary relation on W and a value assignment function respectively. The truth (T) or the falsity (F) is assigned to each atomic proposition, le.,V;WxQ->{T,F} where Q is the set of all atoms. Definition 2.1 Let W be the collectivity of possible worlds and 4> be an atom symbol. {w,u) is u in the real world w. Let O be a weighting function, Q : W— [0, 1] such that V £2. =\. The model of qualitative fuzzy logic with the weighting function is a quaternion M=< W, R, I, Q>, where W is a non-empty set of possible worlds; J? is a binary equivalence relation on W and O is the weight of possible world. Let / be a mapping: I:W* { (w\ it1) and V (w", u") be atom formulae in WQFL where w'e [w"] R. The valuation of the logical connectives is defined as follows, for a given w e W: (1) V£~ 4>(w',u'))=l-u' (2) FA ««')V w(w",u"))=max{u',u"} (3) V£
