Here, x j X j, y Y, A ij F (X j ), B i F (Y ), W i [; ]. In particular, when Y is a nite set fy ; : : : ; y L g, we consider the following types of th

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1 A Proposal of Interpretations on Numerical Degrees of Condence for Fuzzy If{Then Rules and a Mathematical Verication of Properties under Various Reasoning Methods Tatsuya Nomura ATR Human Information Processing Research Laboratories -, Hikaridai, Seika-cho, Soraku-gun, Kyoto 69-, JAPAN Phone: , FAX: nomura@hip.atr.co.jp Abstract Some fuzzy expert systems have used fuzzy rules with numerical values which represent degrees of condence for rules. We discuss two kinds of interpretations for these numerical degrees of condence for rules, called "direct degrees " and "indirect degrees". Then, we apply Zadeh's, Baldwin's, and Tsukamoto's reasoning method to the rules under the two interpretations using general T-norms, and verify their properties. Moreover, in cases where fuzzy sets in descendant parts of rules are dened on a nite set, we present conditions for equivalence between rules with numerical degrees of condence where descendant parts are singleton form and conventional rules, under usage of {max or {sum composition for conclusions of reasoning. Introduction Some of fuzzy expert systems have recently used fuzzy rules with numerical or fuzzy values which represent degrees of condence for rules. In particular, rules with singleton types of the Then{parts and numerical degrees of condence have been used in some cases of automatic fuzzy If{Then rule extraction using neural networks or Genetic Algorithms [][][3][4]. In reasoning and composition of these rules with numerical degrees, max{min reasoning or max{product reasoning by Zadeh's direct method is basically used. However, there has been no discussions on properties in interpretations of numerical degrees of condence in reasoning by general T {norms and other methods, in particular, the relations between the conventional rules and the rules with singleton types of the Then{parts and numerical degrees of condence. In this paper, we verify the conditions for T {norms and implication functions where the conventional fuzzy If{Then rules are equivalent to the rules with singleton types of the Then{parts and numerical degrees of condence in cases where the fuzzy sets of the Then{parts are dened on a nite set, in usage of several interpretations of degrees and Zadeh's, Baldwin's, and Tsukamoto's reasoning methods. For the above purpose, we give two interpretations for numerical degrees of condence for rules (Direct and Indirect degrees) in general cases in Section. Next, we apply Zadeh's, Baldwin's, and Tsukamoto's reasoning methods to the above two interpretations and verify the relations between the conclusions of reasoning by all the methods in Section 3. Then, we derive the conditions under which the conventional rules are equivalent to the rules with singleton types of the Then{parts and numerical degrees of condence in Section 4. Direct and Indirect Degrees of Condence for Rules. Basic Descriptions In this paper, we use the following basic descriptions: F (S): The set which consists of all fuzzy sets on a set S F S : S! [; ]: The membership function of a fuzzy set F S F (S) Let X i be the i{th input set (i = ; : : : ; n), X = X X n, Y be the output set. We consider the following two types of fuzzy If{Then rules. One is the conventional type: H i : If x is A i and : : : and x n is A in ; then y is B i (i = ; : : : ; m) and another is the type made by adding numerical degrees of condence to the above H i : R i : If x is A i and : : : and x n is A in ; then y is B i with W i (i = ; : : : ; m)

2 Here, x j X j, y Y, A ij F (X j ), B i F (Y ), W i [; ]. In particular, when Y is a nite set fy ; : : : ; y L g, we consider the following types of the rules with singleton type of the Then{parts: H ki : If x is A ki and : : : and x n is A kin ; then y is y k R ki : If x is A ki and : : : and x n is A kin ; then y is y k with W ki k = ; : : : ; L; i = ; : : : ; mk ; A kij F (X j ); W ki [; ] In reasoning processes, an implication function I and a T {norm satisfying the following conditions are given : I : [; ] [; ]! [; ] T : [; ] [; ]! [; ] We describe as T (a; b) = a b. In particular, T satises the following (T){(T4): (T) a = a (8a [; ]) (T) a c; b d ) a b c d (T3) a b = b a (8a; b [; ]) (T4) a (b c) = (a b) c (8a; b; c [; ]) Here, b = (8b [; ]) is derived from (T) and (T). We do not assume any special conditions for I unless we note. Moreover, we call a fuzzy set on [; ] Fuzzy Truth Value. For F V F ([; ]), we describe F V : [; ]! [; ]. As examples for Fuzzy Truth Values, we can give the following ones: T rue (r) = r (8r [; ]) (r = ) AbsolutelyT rue (r) = otherwise Now, assume that premise fuzzy sets A i F (X i)(i = ; : : : ; n) are given. Let A = A A n F (X) be the direct product of A ; : : : ; A n by T {norm : A`(x) = A (x ) A (x ) A n (x n ) (8x = (x ; : : : ; x n ) X) In the rules H i ; R i ; H ki ; R ki, we describe as A i = A i A in and A ki = A ki A kin. We consider the following Modus Ponense: Premise : x is A Rule : H i Conclusion : y is Bi In order to derive the conclusion Bi in the above Modus Ponense, Zadeh's, Baldwin's, and Tsukamoto's reasoning methods have been proposed. In Zadeh's method (called Direct Method), the rule H i is assumed to represent a fuzzy relation A i ) B i on XY. The conclusion is given in the following way: B i (y) = sup A (x) I( Ai (x); Bi (y)) () xx In Baldwin's and Tsukamoto's methods (called Indirect Methods, or Truth Value Space Methods), rst, the fuzzy truth value for the If{part of H i "x is A i ", which is described as Ai ( F ([; ])), is determined from the premise "x is A " in the following way (called Converse Truth Qualication): Ai (a) = sup xx A i (x)=a A(x) (8a [; ]) () Next, the fuzzy truth value for the Then{part of H i "y is B i ", which is described as Bi ( F ([; ])), is determined from Ai and the implication function I. Finally, the conclusion is given in the following way (called Truth Quantication): B i (y) = Bi ( Bi (y)) (8y Y ) (3) In Baldwin's method, Bi is given in the following way: Bi (b) = sup Ai (a) I(a; b) (4) In Tsukamoto's method, the fuzzy truth value for a proposition "If x is P; then y is Q" (P F (X); Q F (Y )), which is described as P )Q F ([; ]), is dened as the following form: (r) = P )Q (r) = and Bi is given in the following way: sup (a;b)[;] I(a;b)=r (a) (b) (5) ( ; F ([; ])) Bi (b) = sup Ai (a) Ai)Bi (c) I(a;b)=c = sup Ai (a) (c) (6) I(a;b)=c In composition of a total of m conclusions B, B, : : :, B m, we consider the following two methods, the method by max B max and the method by sum B sum: B (y) = max max B i=;:::;m i (y) (y Y ) (7) B sum (y) = mx i= B i (y) (y Y ) (8). Interpretations for Numerical Degrees of Condence In this paper, we consider the following two interpretations for the degree of condence W i for the rule R i :. Direct Degrees of Condence In this interpretation, the numerical degrees are embedded in the conventional rules; that is, W i directly aects the Then{part of R i in the following way: R i : If x is A i and : : : and x n is A in ; then y is C i = W i B i

3 Here, the fuzzy set W i B i is dened as the following form: WiB i (y) = W i Bi (y) (8y Y ) (9) In this interpretation, R i is regarded as the same type as the conventional type. Thus, we can directly apply Zadeh's, Baldwin's, and Tsukamoto's methods to R i in the reasoning processes.. Indirect Degrees of Condence In this interpretation, the numerical degrees aect the implication function and the fuzzy truth values in the reasoning processes. In Zadeh's method, the conclusion of R i is given by applying the following implication function J Wi instead of I in (): J Wi (a; b) = W i I(a; b) () In Baldwin's and Tsukamoto's methods, the degrees aect the fuzzy truth value Bi in (3). In Baldwin's method, the conclusion is given by applying the above J Wi instead of I in (4). In Tsukamoto's method, the conclusion is given by applying (Ai)Bi)=Wi, dened in the following way, instead of Ai)Bi in (6). Here, on the denition of the fuzzy truth value for a proposition with a numerical degree "If x is P; then y is Q with W " (P F (X); Q F (Y ); W [; ]), (P )Q)=W F ([; ]), we propose the following three forms: W (r) = sup (a;b)[;] J W (a;b)=r (W (a) (b)) () W (r) = P )Q (r) = (r) () W 3 (r) = sup W (c) (3) c[;] W c=r 3 Properties of Several Reasoning Methods by Direct and Indirect Degrees For the premise A, the fuzzy If{Then rule H i, and the rule with the degree of condence R i in Section., we dene in the following way: ZB i : the conclusion by H i and Zadeh's method BB i : the conclusion by H i and Baldwin's method T B i : the conclusion by H i and Tsukamoto's method ZC i : the conclusion by R i and Zadeh's method in direct degrees BC i : the conclusion by R i and Baldwin's method in direct degrees T C i : the conclusion by R i and Tsukamoto's method in direct degrees IZC i : the conclusion by R i and Zadeh's method in indirect degrees IBC i : the conclusion by R i and Baldwin's method in indirect degrees I d T C i : the conclusion by R i and Tsukamoto's method in indirect degrees using the above W d (r) (d = ; ; 3) In this section, we verify the relations between the above conclusions. 3. Properties of Each Reasoning Method by Direct Degrees From the denitions of ZB i, BB i, T B i, ZC i, BC i, and T C i, their membership functions are represented in the following way: ZBi (y) = sup A (x) I( Ai (x); Bi (y)) (4) xx BBi (y) = sup Ai (a) I(a; Bi (y)) (5) T Bi (y) = sup I(a; B i (y))=c Ai (a) (c) (6) ZCi (y) = sup A (x) I( Ai (x); Ci (y)) (7) xx = sup A (x) I( Ai (x); W i Bi (y)) xx BCi (y) = sup Ai (a) I(a; Ci (y)) (8) = sup Ai (a) I(a; W i Bi (y)) T Ci (y) = sup I(a; C i (y))=c Ai (a) (c) (9) = sup I(a;W i B i (y))=c Ai (a) (c) Before the verication of the relations between the above fuzzy sets, we cite the following important lemma and theorem from [5], which show the inclusion relation between the conclusion by Zadeh's method and that by Baldwin's method: Lemma (from [5]) For any set of index M and fa z : z Mg [; ], sup (a z b) ( sup a z ) b (8b [; ]) zm zm In addition, if is left{continuous (i:e:; f b : [; ]! [; ] (f b (a) = a b) is continuous for any b [; ]), sup (a z b) = ( sup a z ) b (8b [; ]) zm zm Theorem (from [5]) ZB i BB i (i:e:; ZBi (y) BBi (y) (8r Y )). In addition, if is left{continuous, ZB i = BB i. As concrete examples of left{continuous T {norms in Lemma and Theorem, we can give "min" and the product "". In other words, the conclusion by Zadeh's method coincides with that by Baldwin's method in the conventional rules when T = min or is used.

4 Note In the above theorem, B i in the subject H i is any fuzzy set on Y. Thus, we also obtain the same results for ZC i and BC i. Moreover, we cite some lemma and theorem from [6] in the following extended form, which show the sucient conditions where the conclusion by Baldwin's method coincides with that by Tsukamoto's method in the conventional rules: Lemma (from [6]) In (5), if the following conditions are satised, then = T rue :. I(; b) I(a; b) (8a; b). I(; b) = b (8b [; ]) 3. = = T rue (Proof) From the condition, I(; r) = r. Thus, we obtain the following equation from the condition 3 and (T)(T3): (r) = sup (a b) r = r I(a;b)=r Moreover, from the conditions, and (T), I(a; b) = r ) b = I(; b) I(a; b) = r ) a b r = r Thus, (r) = r. Theorem (from [6]) If = T rue, then BB i = T B i. (Proof) T Bi (y) = sup a;c[;] I(a; B i (y))=c Ai (a) (c) = sup a;c[;] I(a; B i (y))=c Ai (a) c = sup Ai (a) I(a; Bi (y)) = BBi (y) Furthermore, we give the following lemma. Lemma 3 If the following conditions are satised, then = T rue :. = = T rue. I(a; b) = a b (8a; b [; ]) (Proof) From (T), f(a; b) [; ] : a b = rg 6= /o for 8r [; ]. Thus, we obtain the following equation from the conditions and : (r) = sup a b = r (a;b)[;] ab=r Using Lemma and 3, we obtain the following corollaries of Theorem : Corollary If the following conditions are satised, then BB i = T B i :. I(; b) I(a; b) (8a; b). I(; b) = b (8b [; ]) 3. = = T rue Corollary If the following conditions are satised, then BB i = T B i :. = = T rue. I(a; b) = a b (8a; b [; ]) As a result, the conclusion by Baldwin's method coincides with that by Tsukamoto's method in the conventional rules when = = T rue is used in (5) and, for example, I(a; b) = ab is used as the implication function. Note In the above theorem and corollaries, B i in the subject H i is any fuzzy set on Y. Thus, we also obtain the same results for ZC i and BC i. Next, we give the following theorem, which shows the relation between the conclusions from the conventional rule and the rule with the direct degree of condence by Zadeh's and Baldwin's methods: Theorem 3 If the following conditions are satised, then ZC i = W i ZB i = W i BB i = BC i :. I(a; w b) = w I(a; b) (8a; b; w [; ]) (Proof) From (T3)(T4), Lemma, the conditions,, and the equations (7)(8), we can obtain the following equations: ZCi (y) = W i sup xx n A (x) I( Ai (x); Bi (y)) = W i ZBi (y) BCi (y) = W i sup Ai (a) I(a; Bi (y)) = W i BBi (y) Furthermore, from Theorem and Note, we can obtain ZB i = BB i and ZC i = BC i. We can derive the following corollaries from Theorem, its corollaries, and Note : Corollary If the following conditions are satised, then ZC i = BC i = T C i = W i ZB i = W i BB i = W i T B i :. = T rue 3. I(a; w b) = w I(a; b) (8a; b; w [; ]) Corollary If the following conditions are satised, then ZC i = BC i = T C i = W i ZB i = W i BB i = W i T B i :. = = T rue 3. I(a; b) = a b (8a; b [; ])

5 As a result, the conclusion from the rule with the direct numerical degree is calculated from the T {norm and the conclusion from the conventional rule without the degree, under the usage of any of Zadeh's, Baldwin's, and Tsukamoto's methods, in particular, when T = min or and I(a; b) = a b. 3. Properties of Each Reasoning Method by Indirect Degrees From the denitions of IZC i, IBC i, and T C d i (d = ; ; 3), their membership functions are represented in the following way: IZCi (y) = sup A (x) W i I( Ai (x); Bi (y))() xx n IBCi (y) = sup Ai (a) W i I(a; Bi (y)) () d Id T C i (y) = sup Ai (a) W i (c) W ii(a; B i (y))=c (d = ; ; 3) () From equations ()() and Lemma, we obtain the following theorem, which shows the relation between the conclusions from the conventional rule and the rule with the indirect numerical degree by Zadeh's and Baldwin's methods: Theorem 4 If is left{continuous, then IZC i = W i ZB i = W i BB i = IBC i. Moreover, we obtain the following theorem from Theorem 3 and 4, which shows the relation between the conclusions from the conventional rule and the rules with the direct and indirect numerical degrees by Zadeh's and Baldwin's methods: Theorem 5 If the following conditions are satised, then IZC i = W i ZB i = W i BB i = IBC i = ZC i = BC i :. I(a; w b) = w I(a; b) (8a; b; w [; ]) Furthermore, we give the following theorem, which shows which shows the relation between the conclusions from the conventional rule and the rule with the indirect numerical degree by Tsukamoto's method: Theorem 6 If d W i (r) = T rue (r) = r for r s:t: fc [; ] : W i c = rg 6= /o, then I d T C i = IBC i. In addition, if is left{continuous, then I d T C i = IBC i = W i BB i = W i ZB i = IZC i. (Proof) Id T C i (y) = sup Ai (a) W d i (W i I(a; Bi (y))) = sup Ai (a) W i I(a; Bi (y)) = IBCi The latter half of the statement is trivial from Theorem 4. Here, we prepare the following lemmas in order to derive the corollaries of the above theorem. Lemma 4 If = T rue, then W 3 i (r) = T rue (r) for r s:t: fc [; ] : W i c = rg 6= /o. (Proof) W 3 i (r) = sup W i c = r c[;] W ic=r Lemma 5 If the following conditions are satised, then W i (r) = T rue (r) for r s:t: fc [; ] : W i c = rg 6= /o:. I(; b) I(a; b) (8a; b). I(; b) = b (8b [; ]) 3. = = T rue (Proof) From the condition, if W i c = r, then W i I(; c) = r. Thus, from the condition 3 and (T)(T), W i (r) = sup W i a b W i c = r W ii(a;b)=r From the conditions, and (T){(T4), W i I(a; b) = r ) W i b = W i I(; b) W i I(a; b) = r ) W i a b r = r Thus, (r) = r. Lemma 6 If the following conditions are satised, then W i (r) = T rue (r) for r s.t. fc [; ] : W i c = rg 6= /o:. = = T rue. I(a; b) = a b (8a; b [; ]) (proof) From (T), f(a; b) [; ] : W i a b = rg 6= /o (8r [; ]). Thus, from the conditions and, (r) = sup W i a b = r (a;b)[;] W iab=r From Lemma, 3, 4, 5, 6, and Theorem, 3, 6, we obtain the following corollaries: Corollary If = T rue, then I T C i = I 3 T C i = IBC i. In addition, if is left{continuous, then I T C i = I 3 T C i = IZC i = IBC i = W i ZB i = W i BB i = W i T B i. Corollary If the following conditions are satised, then I T C i = I T C i = I 3 T C i = IBC i :. I(; b) I(a; b) (8a; b). I(; b) = b (8b [; ]) 3. = = T rue In addition, if is left{continuous, then I T C i = I T C i = I 3 T C i = IBC i = IZC i = W i BB i = W i ZB i = W i T B i. Corollary 3 If the following conditions are satised, then I T C i = I T C i = I 3 T C i = IBC i :. = = T rue

6 . I(a; b) = a b (8a; b [; ]) In addition, if is left{continuous, then I T C i = I T C i = I 3 T C i = IBC i = IZC i = W i BB i = W i ZB i = W i T B i = ZC i = BC i = T C i. As a result, the conclusions from the rule with the direct and indirect numerical degree are equal to each other and are calculated from the T {norm and the conclusion from the conventional rule without the degree, under the usage of any of Zadeh's, Baldwin's, and Tsukamoto's methods, when T = min or, and I(a; b) = a b. 4 Comparison between Singleton Rules and Conventional Rules In this section, we assume that Y is a nite set fy ; y ; : : : ; y L g. For the premise A, the conventional fuzzy If{Then rules fh i g m i=, and the rules with the degree of condence fr ki g k=;:::;l in Section., we dene i=;:::;m k in the following way: ZB i : the conclusion by H i and Zadeh's method BB i : the conclusion by H i and Baldwin's method T B i : the conclusion by H i and Tsukamoto's method ZC ki : the conclusion by R ki and Zadeh's method in direct degrees BC ki : the conclusion by R ki and Baldwin's method in direct degrees T C ki : the conclusion by R ki and Tsukamoto's method in direct degrees IZC ki : the conclusion by R ki and Zadeh's method in indirect degrees IBC ki : the conclusion by R ki and Baldwin's method in indirect degrees I d T C ki : the conclusion by R ki and Tsukamoto's method in indirect degrees using the above W d (r) (d = ; ; 3) The membership functions of the above fuzzy sets are given in the following forms: ZBi (y l ) = sup A (x) I( Ai (x); Bi (y l )) (3) xx BBi (y l ) = sup Ai (a) I(a; Bi (y l )) (4) T Bi (y l ) = sup Ai (a) (c) (5) I(a; B i (y l))=c (i = ; : : : ; m; l = ; : : : ; L) ZCki (y l ) = sup A(x) I( Ai (x); W ki kl ) xx n (6) BCki (y l ) = sup Ai (a) I(a; W ki kl ) (7) T Cki (y l ) = sup Ai (a) (c) (8) I(a;W ki kl )=c IZCki (y l ) = sup xx n A(x) W ki I( Aki (x); kl ) (9) IBCki (y l ) = sup Aki (a) W ki I(a; kl ) (3) Id T C ki (y l ) = sup Aki (a) W d ki (c) (3) W ki I(a; kl )=c k = ; : : : ; L; i = ; : : : ; mk ; l = ; : : : ; L Here, kl is Kronecker's Delta. Moreover, Aki is the fuzzy truth value of the proposition "x is A ki " for the premise A, dened in (). P L Since a total of D = k= m k rules are given in the cases of both the conventional rules and the rules with degrees in Section., a total of D conclusions are obtained from their Modus Ponense. When C ki is the conclusion by H ki or R ki (k = ; : : : ; L; i = ; : : : ; m k ), these conclusions are integrated into one conclusion Cmax, or Csum ( F (Y )) based on (7) or (8). Cmax (y) = max Cki (y) (3) k=;:::;l i=;:::;m k Csum (y) = LX k= Xm k i= (8y Y ) Cki (y)! (33) Now, we consider the following special rules with singleton types of the Then{part and degrees of condence, which are derived from the conventional rules fh i g m i=: R ki : If x is A i and : : : and x n is A in ; then y is y k with W ki = Bi (y k ) (k = ; : : : ; L; i = ; : : : ; m) Figure shows the relation between the conventional rule H i and the derived rules fr ki g L. k= Moreover, we dene C max (i) and C sum (i) F (Y ) as the following forms: C (i) (y) = max max C ki (y) (34) k=;:::;l LX C (i) sum (y) = Cki (y) (35) k= (8y Y ) From (3)(33), we obtain the following equations: Cmax (y) = max (y) (36) i=;:::;m C max (i) Csum (y) = mx i= C (i) (y) (37) sum (8y Y ) We give the following theorem which represents the relations between the conventional rules and the rules with degrees derived from them: Theorem 7. If I(a; ) I(a; b) (8a; b [; ]), then C (i) max = ZBi if C pq = ZC pq (8p; q) BB i if C pq = BC pq (8p; q) (i = ; : : : ; m)

7 Ri: If x is Ai, then y is y with Wi= µ Bi( y) Therefore, max ZC ki (y l ) k=;:::;l = ZBi (y l ) max BC ki (y l ) k=;:::;l = BBi (y l ) Hi: If x is Ai, then y is Bi Bi y y y3 y4 y y y y3 y4 y Ri: If x is Ai, then y is y y y y3 y4 y R3i: If x is Ai, then y is y3 with W3i= µ Bi( y3) R4i: y y y3 y4 y If x is Ai, then y is y4 with W4i= µ Bi( y4) with Wi= µ Bi( y) y y y3 y4 y Figure : The Relation between H i and fr ki g L k= Derived from H i (case of L = 4). If at least one of the following conditions (i) and (ii) is satised, then C (i) sum = ZBi if C pq = ZC pq (8p; q) BB i if C pq = BC pq (8p; q) (i = ; : : : ; m) (i) I(a; ) = (8a > ) and Ai (x) > (8x X n ) (ii) I(a; ) = (8a) (Proof) From (6)(7) and W ki = Bi (y k ) (8k; i), ZCli (y l ) = sup xx n A (x) I( Ai (x); Bi (y l )) = ZBi (y l ) BCli (y l ) = sup Ai (a) I(a; Bi (y l )) = BBi (y l ) (Proof of ) I(r; ) I(r; Bi (y)) (8r; y). Thus, for k 6= l, ZBi (y l ) = sup xx n A(x) I( Ai (x); Bi (y l )) sup A (x) I( Ai (x); ) xx n = ZCki (y l ) BBi (y l ) = sup Ai (r) I(r; Bi (y l )) r[;] sup Ai (r) I(r; ) r[;] = BCki (y l ) (Proof of ) If (i) or (ii) is satised, the following equation is obtained for k 6= l: ZCki (y l ) = sup xx n A (x) I( Ai (x); ) = Moreover, if (i) is satised, Ai () = is derived from the denition of Ai. Thus, if (i) or (ii) is satised, the following equation is obtained for k 6= l: Therefore, BCki (y l ) = sup Ai (r) I(r; ) = r[;] LX LX k= k= ZCki (y l ) = ZBi (y l ) BCki (y l ) = BBi (y l ) We can derive the following corollaries from Theorem and Theorem Corollary : Corollary If the following conditions are satis- ed, then the conclusion from the conventional rules fh i g m i= coincides with that from the rules with degrees fr ki g k=;:::;l derived from fh i g, under the usage of the i=;:::;m reasoning by Zadeh's, Baldwin's, and Tsukamoto's methods, the {max composition, and direct degrees of condence:. I(a; ) I(a; b) (8a; b [; ]) 3. I(; b) I(a; b) (8a; b) 4. I(; b) = b (8b [; ]) 5. = = T rue Corollary If the following conditions are satis- ed, then the conclusion from the conventional rules fh i g m i= coincides with that from the rules with degrees fr ki g k=;:::;l derived from fh i g, under the usage of the i=;:::;m reasoning by Zadeh's, Baldwin's, and Tsukamoto's methods, the {sum composition, and direct degrees of condence:. (I(a; ) = (8a > ) and Ai (x) > (8x X)) or, (I(a; ) = (8a)) 3. I(; b) I(a; b) (8a; b) 4. I(; b) = b (8b [; ]) 5. = = T rue

8 Furthermore, we can derive the following corollary from Theorem 6 Corollary 3: Corollary 3 If the following conditions. = = T rue 3. I(a; b) = a b (8a; b [; ]) are satised, then (I): the conclusion from the conventional rules fh i g m i= coincides with that from the rules with degrees fr ki g k=;:::;l derived from fh i g, under the usage of i=;:::;m the reasoning by Zadeh's, Baldwin's, and Tsukamoto's methods, the {max composition, and direct and indirect degrees of condence. (II): the conclusion from the conventional rules fh i g m i= coincides with that from the rules with degrees fr ki g k=;:::;l derived from fh i g, under the usage of i=;:::;m the reasoning by Zadeh's, Baldwin's, and Tsukamoto's methods, the {sum composition, and direct and indirect degrees of condence. As a result, the capacity of the representation in the rules with the numerical degrees of condence is equal to that in the conventional rule under the usage of any of Zadeh's, Baldwin's, and Tsukamoto's methods, when T = min or, I(a; b) = a b, and the {max or {sum composition is used. Note 3 Even if the conditions in Corollary are satis- ed, the conclusion by direct degrees generally diers form that by indirect degrees, if the conditions in Corollary 3 are not satised. For example, let us consider a case of L =,n =, X = [; ], and m =. Now, we assume the following conditions for H : (8x [; )) A (x) = ; (x = ) B (y ) = p; B (y ) = q Moreover, we set = min and I(a; b) = min((?a+b); ). If the premise A ( A (x) = v 8x and v > max(p; q)) is given, then the membership functions are given in the following way: ZB (y ) = ZB (y ) = v ZC (y ) = ZC (y ) = v ZC (y ) = ZC (y ) = v Thus, ZB coincides with the conclusion of ZC and ZC by max composition. However, IZC (y ) = IZC (y ) = p IZC (y ) = IZC (y ) = q are obtained for IZC and IZC. Thus, ZB diers from the conclusion of ZC and ZC by max composition. 5 Conclusion We proposed the two interpretations on numerical degrees of condence for the fuzzy If{Then rules, direct and indirect degrees, and applied Zadeh's, Baldwin's, and Tsukamoto's methods to the reasoning process under the two interpretations using a general T {norm. Finally, we proved that the conventional rules are equivalent to the rules with singleton types of the Then{parts and numerical degrees of condence in cases where the fuzzy sets of the Then{parts are dened on a nite set, the T {norm is left{continuous (ex. min, ), the implication function is given as I(a; b) = a b, the max or sum composition is used in the integration of the conclusions, and Zadeh's, Baldwin's, or Tsukamoto's reasoning method is used, regardless of the interpretations of the degrees of condence. As future works, we consider an extension of the above results to degrees of condence represented by fuzzy truth values. Acknowledgment The author would like to thank Dr. Katsunori Shimohara at ATR Human Information Processing Research Laboratories for support, and Dr. Toru Yamaguchi at Utsunomiya University for discussions. References [] T. Tsuchiya, Y. Matsubara, and M. Nagamachi. A Learning Fuzzy Rule Parameters Using Genetic Algorithm. In Proc. 8th Fuzzy System Symposium, pages 45{48, Hiroshima, Japan, May 99. (Japanese). [] H. Ishibuchi, T. Nakashima, and T. Murata. A Fuzzy Classier System for Generating Linguistic Classi- cation Rules. In Proc. IEEE/Nagoya University WWW'95, pages {7, Nagoya, Japan, Nov 995. [3] T. Nomura and T. Miyoshi. An Adaptive Rule Extraction with the Fuzzy Self-Organizing Map and a Comparison with Other Methods. In Proc. ISUMA{ NAFIPS'95, pages 3{36, Maryland, USA, Sept 995. [4] T. Nomura and M. Miyoshi. Numerical Coding and Unfair Average Crossover in GA for Fuzzy Clustering and Their Applications for Automatic Fuzzy Rule Extraction. In Proc. IEEE/Nagoya University WWW'95, pages 3{, Nagoya, Japan, Nov 995. [5] S. Kawase and N. Yanagihara. On the Truth Space Approach and the Direct Approach in Fuzzy Reasoning. The Transaction of The Institute of Electronics, Information and Communication Engineers, J77{ A(3):53{537, Mar 994. (Japanese). [6] C. Qihao, N. Yanagihara, and S. Kawase. On Relations between Several Methods in Fuzzy Reasoning. Journal of Japan Society for Fuzzy Theory and Systems, 7(6):{8, 995. (Japanese).