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Suitability of fuzzy reasoning methods
Fuzzy Sets and Systems 108 (1999) 299–311 www.elsevier.com/locate/fss
Suitability of fuzzy reasoning methods Zhenquan Li Department of Mathematics & Computing, University of Southern Queensland, Toowoomba, Queensland 4350, Australia Received May 1995; received in revised form August 1997
Abstract Similarity, commutativity, continuity and computational times of the currently existing six reasoning methods are discussed. All six have commutativity and continuity. It is found that the reasoning precision of a reasoning method has c 1999 Elsevier relations with similarity degree. The computational complexities of the six reasoning methods are given. Science B.V. All rights reserved. Keywords: Fuzzy reasoning method; Similarity; Commutativity; Continuity
1. Introduction 1.1. Overview The fuzzy reasoning methods play a key role in applications of fuzzy sets. There have been six fuzzy reasoning methods and many successful application examples of such reasoning methods so far. These reasoning methods include: • Zadeh’s CRI-Mamdani version: Compositional Rule of Inference (CRI). • Turksen’s versions: 1. Interval-valued CRI (IVCRI). 2. Point-valued Approximate Analogical Reasoning (PVAAR). 3. Interval-valued Approximate Analogical Reasoning (IVAAR). • Sugeno’s versions: 1. Position type reasoning method (P). 2. Position Gradient type reasoning method (P-G). Informally, fuzzy reasoning method can be viewed as a process by which a possible imprecise conclusion is deduced from a collection of imprecise premises [18]. The imprecise can be described by linguistic variables E-mail address: [email protected] (Z. Li) c 1999 Elsevier Science B.V. All rights reserved. 0165-0114/99/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 2 9 6 - 0
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and the collection is the set of some linguistic rules. There exists a large number of studies concerning the construction of the rules and the design of the linguistic variables. However, a good result not only depends upon the design of linguistic variables and the construction of the rules, but also depends upon the fuzzy reasoning method used. There have been a few studies on fuzzy reasoning methods. This leads to a lack of an ecient explanation on fuzzy reasoning methods and hence, our ability appears to be limited in convincing ‘users’. Usually, the choice of a better fuzzy reasoning method for a physical problem is dicult, as the comparison of currently existing reasoning methods is expensive, especially for a complex problem. Therefore, a theoretical discussion for choice of a reasoning method is needed. In this paper, I discuss theoretically the similarity which is relative to the precision of reasoning methods, computational time and two operation properties that a reasoning method must satisfy from practical consideration for a reasoning method. I hope my discussion will help the ‘users’ to choose a suitable fuzzy reasoning method or develop his= her own, and also serve as a reference for the development of a mathematical axiom system for fuzzy reasoning methods. 1.2. Reasoning methods In general, fuzzy reasoning is based on linguistic rules. The rules are with the following forms: R1 :
If X1 is A11 and · · · and XI is A1I then Y is B1 ;
R2 :
If X1 is A21 and · · · and XI is A2I then Y is B2 ; .. .. . .
RJ :
(1)
If X1 is AJ1 and · · · and XI is AJI then Y is BJ ;
where Rj ( j = 1; 2; : : : ; J ) is implication relation, Xi (i = 1; 2; : : : ; I ) are antecedent linguistic variables. Aji , B j are fuzzy variables, i.e. the linguistic values of corresponding linguistic variables. The fact is with the following form: X1 is A01 · · · XI is A0I and the conclusion is as follows: Y is B0 ; where Y is consequent linguistic variable, A0i (i = 1; 2; : : : ; I ) are observations, and B0 is a resultant linguistic value by an approximate reasoning method. 1.2.1. Zadeh’s method [5, 6, 13] According to Zadeh, Rj means Rj = A j → B j ;
(2)
where A j is de ned by Aj =
I \ i=1
Aji
(3)
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an approximate reasoning is carried out by 0
B j = A0 ◦ Rj
(4)
here ‘◦’ is a compositional operator and A0 is de ned by A0 =
I \
A0i :
(5)
i=1
Then a consequence, B0 is derived by combining the consequence of all the rules: B0 =
J ]
Bj
0
(6)
j=1
where ] ∈ {∨; ∧}, and ] is dependent on the type of implication used. When we adopt point-valued sup–min operations in our model, then the membership value of inferred consequence is calculated from (2) – (6): " ! !# J I I ] ^ ^ sup hA0i (xi ) ∧ hAj (xi ) → hB j (y) ; (7) hB0 (y) = x1 ;:::; xI
j=1
i=1
i=1
i
where hA0i (xi ), hAj (xi ), hB j (y) and hB0 (y) are the corresponding membership functions for the fuzzy sets A0i , i Aji , B j and B0 , respectively. The general form of implication, →, is interpreted in various formulae. When we adopt Mamdani’s implication, which is often used in various applications, we have ( ! " I #) J I _ _ ^ ^ hA0i (xi ) ∧ hAj (xi ) ∧ hB j (y) hB0 (y) = x1 ;:::; xI
j=1
=
J _
(
j=1
I ^ i=1
i=1
"
_
i
i=1
#
)
(hA0i (xi ) ∧ hAj (xi ) ∧ hB j (y) :
x1 ;:::; xI
(8)
i
1.2.2. Turksen’s methods [13–16] 1. Interval-valued CRI (IVCRI) For jth rule ( j = 1; : : : ; J ) in (1), the implication means (Rj ) = (A j (I )) ⇒ B j ;
(9)
where A j (I ) is the overall intersection of the antecedents, i.e., j (I )] (A j (I )) = [LjAND (I ); UAND
(10)
j (I ) de ned using recursive expressions. That is with LjAND (I ), UAND
LjAND (1) = Aj1 ; LjAND (2) = LjAND (1) ∩ Aj2 ; .. . LjAND (I ) = LjAND (I − 1) ∩ AjI ;
(11)
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while j (1) = Aj1 ; UAND [ j j j j (2) = [UAND (1) ∪ Aj2 ∩ UAND (1) ∪ (Aj2 )c ∩ (UAND (1))c ∪ Aj2 ; UAND
LjAND (1) ∪ Aj2 ∩ LjAND (1) ∪ (Aj2 )c ∩ (LjAND (1))c ∪ Aj2 ]; .. . j (I ) = UAND
[
j j j [{UAND (I − 1) ∪ Aj2 } ∩ {UAND (I − 1) ∪ (Aj2 )c } ∩ {(UAND (I − 1))c ∪ Aj2 };
{LjAND (I − 1) ∪ Aj2 } ∩ {LjAND (I − 1) ∪ (Aj2 )c } ∩ {(LjAND (I − 1))c ∪ Aj2 }]: The implication (9) is de ned as an internal valued fuzzy set from (A j (I )) ⇒ B j = [Limp (Rj ); Uimp (Rj )]; where Limp (Rj ) = LjAND (I ) ∩ B j ;
j Uimp (Rj ) = UAND (I ) ∩ B j :
Compositional rule of inference is carried out by 0
(B j ) = (A0 (I )) ◦ (Rj ); where (A0 (I )) is an aggregation of observations. That is, 0 (A0 (I )) = [L0AND (I ); UAND (I )]:
When we adopt interval-valued sup–min operations for the composition, reasoning result B0 is calculated by 0
0
0
(B j ) = [Lcri (B j ); Ucri (B j )]; where 0
Lcri (B j ) = sup min{L0AND (I ); Limp (Rj )}; x1 ;:::; xJ
0
0 Ucri (B j ) = sup min{UAND (I ); Uimp (Rj )}: x1 ;:::; xJ
An estimate of the membership function of the consequence for the jth rule ( j = 1; : : : ; J ) in (1) is determined as follows: hB j0 = 12 (hLcri (B j0 ) + hUcri (B j0 ) ): An estimate of the membership function of the nal consequence is derived as follows: hB0 = 12 (hLOR (J ) + hUOR (J ) ); where UOR (J ), LOR (J ) is de ned using recursive expressions. That is, 0
UOR (1) = B1 ; 0
UOR (2) = UOR (1) ∪ B2 ; .. . 0
UOR (I ) = UOR (J − 1) ∪ B J ;
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0
LOR (1) = B1 ; \ 0 0 0 LOR (2) = [{UOR (1) ∩ B2 } ∪ {UOR (1) ∩ (B2 )c } ∪ {(UOR (1))c ∩ B2 }; 0
0
0
{LOR (1) ∩ B2 } ∪ {LOR (1) ∩ (B2 )c } ∪ {(LOR (1))c ∩ B2 }]; .. . LOR (J ) =
\
0
0
0
[{UOR (J − 1) ∩ B J } ∪ {UOR (J − 1) ∩ (B J )c } ∪ {(UOR (J − 1))c ∩ B J }; 0
0
0
{LOR (J − 1) ∩ B J } ∪ {LOR (J − 1) ∩ (B J )c } ∪ {(LOR (J − 1))c ∩ B J }]: 2. Point-valued approximate analogical reasoning (PVAAR) For PVAAR, pattern matching measures between antecedents and observations is calculated for each rule in (1) as follows: SM j =
I ^
SMij ( j = 1; : : : ; J );
i=1
where SMij is, e.g., a similarity measure of the ith variable de ned with a distance measure DM SMij [Aji ; A0i ] = 1 − DM [Aji ; A0i ]: With the values of SM j , the rules which satisfy the following conditions are red: SM j ¿ ; where is an appropriate threshold. Approximate reasoning is carried out through modi cation of each consequent. Two types of modi cation are proposed, namely: (a) Expansion form: hB j : hBexj = min 1; SM j (b) Reduction form: hBrej = SM j · hB j : The average of the expansion and reduction forms can be adopted as an estimate of the membership function of the consequence from the jth rule ( j = 1; : : : ; J ) in (1) as hB j0 = 12 (hBexj + hBrej ): 3. AAR with interval-valued fuzzy set (IVAAR) In addition to AND; OR, etc., an interval-valued armation can be de ned by DNF and CNF . In general, the form is, FAFF = [DNF (F); CNF (F)] = [(E ∩ F) ∪ (E c ∩ F); (E ∪ F) ∩ (E c ∪ F)];
(12)
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where F is the concept in relation to another concept E. For the jth rule ( j = 1; : : : ; J ) in (1), its consequent is de ned as an interval-valued fuzzy set based on (12) as (B j ) = [LAFF (B j ); UAFF (B j )]; where LAFF (B j ) =
\
j j j j [{LAND (I ) ∪ (LAND (I ))c } ∩ B j ; {UAND (I ) ∪ (UAND (I ))c } ∩ B j ];
and UAFF (B j ) =
[
j j j j [{LAND (I ) ∩ (LAND (I ))c } ∪ B j ; {UAND (I ) ∩ (UAND (I ))c } ∪ B j ]
j j (I ), UAND (I ) are the upper bound and the lower bounds, respectively. Let where LAND j 0 (I ); LAND (I )]; SM jL = SM[LAND
j 0 SM jU = SM[UAND (I ); UAND (I )]:
With the values of SM, the rules which satisfy the following conditions are red: SM jL ¿ 0L
and SM jU ¿ U 0 ;
(13)
here 0L and U 0 are appropriate thresholds. Approximate reasoning is carried out through modi cation of each consequent. There are two types of modi cation as PVAAR, (a) Expansion form: 0
0
0
(B j )ex = [Lex (B j ); Uex (B j )]; where
hLex (B j0) = min
hUAFF (B j ) hLAFF (B j ) ; ; SM jU SM jL
hU (B j ) hUex (B j0) = min 1; AFF : SM jU
(b) Reduction form 0
0
0
(B j )re = [Lre (B j ); Ure (B j )]; where hLre (B j0) = min{hUAFF (B j ) · SM jU ; hLAFF (B j ) · SM jL };
hUre (B j0) = SM jU · hUAFF (B j ) :
The average of upper and lower bounds of the expansion and the reduction forms can be adopted as an estimate of the membership function of the consequence for the ( j = 1; : : : ; J ) in (1) as follows: hB j0 = [ 12 (hUre (B j0) + hLre (B j0) ); 12 (hUex (B j0) + hLex (B j0) )] = [hLIVAAR (B j0) ; hUIVAAR (B j0) ]: An estimate of the membership function of the nal consequence is determined by J J ^ 1_ hLIVAAR (B j0) + hUIVAAR (B j0) : hB 0 = 2 j=1
j=1
(14)
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1.2.3. Sugeno’s methods [8–12] 1. Position type reasoning method. If the fuzzy variables B j ( j = 1; : : : ; J ) in the consequent part of rules in (1) are replaced with a center of gravity vector of corresponding output cluster as follows: bj =
N X
(hjn )w yn
X N
n=1
(hjn )w ;
n=1
(see pp. 262–263 in [7]), then the set of derived rules is said Position type fuzzy model. The nal estimate of the consequence for this model is derived: y=
J X j=1
j j
h b
X J
h j;
j=1
where h j is a membership value of j th rule ( j = 1; : : : ; J ) in (1) derived from the algebraic product of all membership values of input variables in antecedent: h j = hA j (x10 ) · · · hA j (xI0 ); 1
I
where xi0 (16i6I ) is an input. 2. Position-gradient type reasoning method. If the fuzzy variables B j (j = 1; : : : ; J ) in the consequent part of rules in (1) are replaced with ‘b j and @Y=@X1 is c1j and · · · and @Y=@XI is cIj ’ (where @Y =@Xi (16i6I ) is the gradient of the output variable Y for input variables Xi , and cij (i = 1; : : : ; I ) is its singleton linguistic value), then the set of derived rules is said Position-gradient type fuzzy model. The nal estimate of the consequence for this model is derived: " !# J J I X X X j (di (n; j) · ci (n; j); (n; j) · b j + y= j=1
i=1
j=1
0 0 ; : : : ; xIn ) and core of the jth where (n; j) is a weight loaded on the distance, D(n; j), between input data (x1n rule ( j = 1; : : : ; J ) in (1): v u I uX (xi0 − xijc ) 2 ; xijc ∈ {xi |hA j (xi ) = 1; xi ∈ Xi }; (15) D(n; j) = min t i=1
and
i
(n; j) = exp{−D(n; j)}di (n; j) is a projection of D(n; j) on xi .
1.2.4. The comparison results from practical problems The performance characteristics of the above six fuzzy reasoning methods were investigated by [7]. The investigation used three sets of real-life system data which supply a reasonable ground for comparing the reasoning precision, the calculation time and the number of valid input cases. The conclusion is that CRI has appropriate precision, but complex computation; IVCRI takes longer time, the reasoning can be carried out for all case data points and gives a representation of the spread of vagueness that may be associated with linguistic connectives if it is to be taken into account; Sugeno’s methods have an advantage in calculation speed and the P-G method is quite useful in the case where rapid reasoning is necessary and no case data point should be missed in reasoning, but these two methods require absolute scale; the PVAAR and IVAAR methods shows comparably good precision and short time for management decision models, but these methods limit the number of case data points for reasoning when the threshold is increased.
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2. Analysis of reasoning methods 2.1. Similarity The performance of a fuzzy reasoning method RM would depend upon rules it drive the result from. An obvious method to judge which reasoning method is suitable for a given set of rules (1) is that the RM should satisfy RM(A1j ; : : : ; AIj ) ≈ B j ( j = 1; : : : ; J ). I introduce a de nition to describe it as follows. Deÿnition 1. A fuzzy reasoning method RM is said to have similarity if RM(A1j ; : : : ; AIj ) = B j for all 16j6 J . The similarity can also be expressed by a Similarity Measure SM SM[RM(A1j ; : : : ; AIj ); B j ] = 1
(16 j6J ):
2.1.1. Zadeh’s method If the implication adopts Mamdani’s formula and J = 1 in (1), then !" I # I _ ^ ^ hA0i (xi ) hAi (xi ) ∧ hB (y) hB0 (y) = x1 ; ::: ; xI
=
i=1
( I ^ _ i=1
i=1
) [hA0i (xi ) ∧ hAi (xi )]
∧ hB (y):
(16)
xi
When A0i = Ai (i = 1; : : : ; I ), then hB0 (y) = hB (y), and Zadeh’s method has similarity. If linguistic rules have more than one value and the values of fuzzy variables of two linguistic rules on overlap of the two linguistic rules are not zero, for example, the rules of three real-time systems in [7], then Zadeh’s method does not have similarity. 2.1.2. Turksen’s methods 1. IVCRI. For J = 1 in (1), it is easy to prove that IVCRI has similarity. But for J ¿1, it may not have similarity. For example, any two rules in the given rules of the rst real-time system [7], X1 is As1 and X2 is As2 then Y is Bs ; X1 is At1 and X2 is At2 then Y is Bt ; where s 6= t; s; t 6= 1; 6, if let A01 = As1 and A02 = As2 , then hBt0 = 12 (hLCRI (Bt0) + hUCRI (Bt0) ) 6= 0: 2. PVAAR. According to the de nition of similarity measure, SM jI 0 [A0i ; Aij0 ] = 1 if A0i = Aij0 SM j0 =
I ^
for all
16i6 I (16j0 6J );
SM ji 0 = 1:
i=1
If we can choose a similarity measure SM and a threshold 0 such that SM j 6 0 (16j6J; j 6= j0 ), then the only red rule is Rj0 and hB0 = hB j0 , and PVAAR has similarity. In general, the higher threshold value, the more probability PVAAR has similarity.
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3. IVAAR. It is easy to compute j0 0 (I ); LAND (I )] = 1; SM j0 L = SM[LAND
j0 0 SM j0 U = SM[UAND (I ); UAND (I )] = 1;
if Aij0 = A0i for all 16i6I and some j0 (16j0 6J ). If we can choose a similarity measure and thresholds 0L j jL jU L U and U 0 such that SM 6 0 and SM 6 0 (16j6J; j 6= j0 ), then the only rule R0 is red and ( ) hUAFF (B j0 ) hLAFF (B j0 ) j0 U ; h × hUAFF (B j0 ) = hB j0 ; = hB j0 ; h j 0 = min j 0 = SM Lex (B 0 ) Ure (B 0 ) SM j0 U SM j0 L h
= 12 (h
j0
LIVAAR (B 0 )
j0
Ure (B 0 )
+h
j0
Lre (B 0 )
) = hB j0 ;
h
j0
UIVAAR (B 0 )
= 12 (h
j0
Uex (B 0 )
+h
j0
Lex (B 0 )
) = hB j0 ;
therefore
J J ^ 1 _ hLIVAAR (B j ) + hUIVAAR (B j ) = hB j0 ; hB 0 = 2 j=1
j=1
IVAAR has similarity. In general, the higher the threshold, the more probably IVAAR has similarity. 2.1.3. Sugeno’s methods Sugeno’s reasoning methods are the methods used in the clustering algorithm, and is used in the modelling stage for variable selection and rule structuring. The similarity of these reasoning methods depends upon the set of rules (1). If the set of rules (1) is biased toward the Sugeno’s reasoning method, then the results have higher similarity. 2.2. Commutativity The results of a problem are independent of the order of the rules in a set of rules as (1) if the problem is solved by the linguistic rule from physical consideration. If the results of a fuzzy reasoning method are independent of the order of the rules in a set of linguistic rules, the method is said to have commutativity. It is easy to see that the six fuzzy reasoning methods discussed above have commutativity from their de nitions. 2.3. Continuity In this section, we consider the continuity of fuzzy reasoning methods. Suppose DM and DM 0 are two distance measures between fuzzy variables Ai , A0i (16i6I ) and B, B0 . Let R: X1 is A1 and · · · and XI is AI then Y is B be a rule, and X1 is A01 and · · · and XI is A0I be an observation and B0 is the WI corresponding resultant linguistic value by a reasoning method RM . If DM 0 (B; B0 ) tend to zero when i=1 DM(Ai ; A0i ) tends to zero, RM is said to have continuity at rule R. If RM has continuity at every rule of considered system, RM is said to have continuity, we also can say RM is continuous. For example, the two distance measures DM and DM 0 of two variables are de ned as DM(A1 ; A2 ) = supx∈X |hA1 (x) − hA2 (x)|, where X is the universe of discourse. It is easy to prove DM is a distance measure. Lemma 1. The union operator ∨ of fuzzy sets is continuous relative to DM.
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A01 ∨ A02 = A0 . Proof. Let A1 ; A01 and A2 ; A02 are fuzzy sets, and A1 ∨ A2 = A; = sup |hA0 ∨A0 (x) − hA1 ∨A2 (x)| DM (A0 ; A) 1 2 x∈X
= sup max[hA01 (x); hA02 (x)] − max[hA1 (x); hA2 (x)] x∈X x∈X x∈X 6 max sup |hA01 (x) − hA1 (x)|; sup |hA02 (x) − hA2 (x)| : x∈X
(17)
x∈X
tends to zero when DM (A01 ; A1 ) and DM (A02 ; A2 ). Therefore, DM (A0 ; A) The proof for continuity of intersection operator ∧ is the same as Lemma 1. Lemma 2. If fuzzy sets A; B are in two universes of discourse U and V; respectively; then A ∨ B and A ∧ B are continuous relative to DM . The proof is similar to Lemma 1. Lemma 3. If fuzzy set A is in U and fuzzy set B is in V; then operator supu∈U hA (u) ∧ hB (v) is continuous. Proof. Let A0 is a fuzzy set in U , B0 is a fuzzy set in V , then sup sup [hA0 (u) ∧ hB0 (v)] − sup [hA (u) ∧ hB (v)] v∈V u∈U
u∈U
6 sup sup |hA0 (u) ∧ hB0 (v) − hA (u) ∧ hB (v)| v∈V u∈U
6 sup sup [|hA0 (u) − hA (u)| ∨ |hB0 (v) − hB (v)|] v∈V u∈U
6 sup |hA0 (u) − hA (u)| ∨ sup |hB0 (v) − hB (v)|: u∈U
v∈V
Lemma 4. Plus; minus and multiplication of fuzzy variables are continuous relative to DM . The operator of fuzzy variable numerator and real number denominator is also continuous relative to DM . Since the fuzzy variables we discussed are bounded, hence Lemma 4 holds. Lemma 5. The composition of continuous operators relative to DM is continuous relative to DM . It is easy to prove. By Lemmas 1–3, we obtain the following. Theorem 1. If the form of implication adopts Mamdani’s method; then Zadeh’s reasoning method is continuous relative to DM . According to Turksen’s de nition of IVCRI and Lemmas 1– 5, we obtain Theorem 2. IVCRI is continuous relative to DM .
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Now let us consider continuities of Turksen’s AAR and IVAAR. Since similarity measure is de ned with a distance measure DM for AAR, we have SMij [Aij ; A0i ] = 1 − DM [Aij ; A0i ]; SM j =
I ^
SMij
(16 j6 J );
i=1
where Aij , A0i (16i6I; 16 j6 J ) are fuzzy variables in (1). SM j is continuous relative to DM by Lemmas 1– 4. For an appropriate threshold , approximate reasoning is carried out through modi cation of each consequent which satisfy SM j ¿ : We know reduction form and expansion form of consequent are continuous relative to DM , so are the nal consequences for AAR and IVAAR by lemmas. We obtain Theorem 3. AAR and IVAAR are continuous relative to DM . The continuity for Sugeno’s reasoning method is followed by Lemma 4. 2.4. Computational times For comparison of computational complexity of six reasoning methods, we consider an operation between twoVfuzzy numbers as one, i.e. consider a fuzzy number as a real number, e.g., the computational times I of i=1 hA0i (xi ) is I-1. Here we only discuss the computational times of hB0 (y) for di erent reasoning methods, because the defuzzi cations of B0 are same for the methods except Sugeno’s methods. We denote the computational times of hB0 (y) by N . For Zadeh’s reasoning method, if we adopt Mamdani’s form of implication, then N = 2IJ + J − 1: For Turksen’s IVCRI N = 15(I − 1)(J + 1) + 21J + 2I − 17: For Turksen’s AAR (at least one rule is red) 3IJ − J + 56 N 6 3IJ + 4J: For Turksen’s IVAAR (at least one rule is red) (I − 1) (15J + 17) + 336 N 6(I − 1)(15J + 17) + 33J: For Sugeno’s P N = IJ + 2J − 1: For Sugeno’s P − G N = 5IJ + 2I + 4J − 1; where the last two are the computational times of nal consequence.
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The order of the computational times shown above are basically the same with the order of the time that are given in [7] for practical examples. The computational time of a method does not fully depend on computational complexity, but it mainly is determined by the computational complexity. 3. Discussion We discussed four properties which re ect di erent aspects of fuzzy reasoning methods. If similarity degree SD is de ned as SD = 1 −
J _
DM (Bj ; B j );
j=1
where Bj is the consequent linguistic value of the j th rule (j = 1; : : : ; J ) in (1) and B j is the resultant linguistic value of antecedent linguistic values of jth rule ( j = 1; : : : ; J ) in (1) by a fuzzy reasoning method, then SD can be applied to measure the precision of a fuzzy reasoning method. The nearer SD to 1 is, the higher the precision of the reasoning method. This is easy to compute. Commutativity is a property that every reasoning method must have according to practical meaning. If a reasoning method does not satisfy this property, the method could not give good results. When a reasoning method does not satisfy continuity, the di erence between the actual output from the method and resultant linguistic value in (1) may be large even though inputs are much nearer to antecedent linguistic values in (1). The computational time of a reasoning method can be used to estimate the reasoning time of the reasoning method. With the increase of CPU speed, the computational complexity is no longer such an important factor. It is easy to see that the above results can be extended to several consequent variables with no diculty. We give a learning algorithm for fuzzy neural networks based on fuzzy number operations which can learn linguistic rules as (1) with desired similarity degree [4]. As a neural network could be used for reasoning, the fuzzy neural network we designed can be used for reasoning in any physical problem that satis es fuzzy number operations and show good results. The computational times of this method may be bigger than the six reasoning methods discussed above, it depends on how the operations of fuzzy numbers are performed by the computer. References [1] F. Bouslama, A. Ichikawa, Fuzzy control rules and their natural control laws, Fuzzy Sets and Systems 48 (1992) 65 – 86. [2] T.C. Chang, K. Hasegawa, C.W. Ibbs, The e ect of membership function on fuzzy reasoning, Fuzzy Sets and Systems 44 (1991) 169 –186. [3] R. Katayama, Y. Kajitani, K. Matsumoto, M. Watanabe, Y. Nishida, An automatic knowledge acquisition and fast self tuning method for fuzzy controller based on hierarchical fuzzy inference mechanisms, Proc. 4th IFSA Congr., Brussels, 1991, pp. 105 –108. [4] Z. Li, Y. Zhang, M. Nishikawa, A. Ichikawa, A learning algorithm for fuzzy neural networks based on fuzzy number operations, Proc. ICONIP96, Springer, Berlin, pp. 260 – 265. [5] E.M. Mamdani, S. Assilian, Applications of fuzzy algorithms for control of simple dynamic plant, Proc. IEE 121 (1974) 1585 –1588. [6] E.M. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. Man–Mach. Studies 7 (1975) 1–13. [7] H. Nakanishi, I.B. Turksen, M. Sugeno, A review and comparison of six reasoning methods, Fuzzy Sets and Systems 57 (1993) 257 – 294. [8] M. Sugeno, An introductory survey of fuzzy control, Inf. Sci. 36 (1985) 59 – 83. [9] M. Sugeno, Fuzzy control, Nikkan Kogyo Shinbunsya, Tokyo, Japan, 1988. [10] M. Sugeno, G.T. Kang, Structure identi cation of fuzzy model, Fuzzy Sets and Systems 28 (1988) 15 – 33. [11] M. Sugeno, T. Yasukawa, A fuzzy logic based approach to qualitative modeling, IEEE Trans. Fuzzy Systems 1 (1993) 1– 24. [12] T. Takagi, M. Sugeno, Fuzzy identi cation of systems and its applications to modeling and control, IEEE Trans. Systems Man Cybernet. 15 (1985) 116 –132.
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[13] I.B. Turksen, Y. Tian, Interval valued fuzzy set representation on multiple antecedents fuzzy s-implications and reasoning, Fuzzy Sets and Systems 52 (1992) 1– 25. [14] I.B. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20 (1986) 191– 210. [15] I.B. Turksen, Four methods of approximate reasoning with interval-valued fuzzy sets, Internat. J. Approx. Reasoning 3 (1989) 121–142. [16] I.B. Turksen, Z. Zhong, An approximate analogical reasoning scheme based on similarity measures and interval valued sets, Fuzzy Sets and Systems 34 (1990) 323 – 346. [17] R.R. Yager, D.P. Filev, On the issue of defuzzi cation and selection based on a fuzzy set, Fuzzy Sets and Systems 55 (1993) 255 – 271. [18] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems Man Cybernet. 3 (1973) 28 – 44.
Suitability of fuzzy reasoning methods Zhenquan Li Department of Mathematics & Computing, University of Southern Queensland, Toowoomba, Queensland 4350, Australia Received May 1995; received in revised form August 1997
Abstract Similarity, commutativity, continuity and computational times of the currently existing six reasoning methods are discussed. All six have commutativity and continuity. It is found that the reasoning precision of a reasoning method has c 1999 Elsevier relations with similarity degree. The computational complexities of the six reasoning methods are given. Science B.V. All rights reserved. Keywords: Fuzzy reasoning method; Similarity; Commutativity; Continuity
1. Introduction 1.1. Overview The fuzzy reasoning methods play a key role in applications of fuzzy sets. There have been six fuzzy reasoning methods and many successful application examples of such reasoning methods so far. These reasoning methods include: • Zadeh’s CRI-Mamdani version: Compositional Rule of Inference (CRI). • Turksen’s versions: 1. Interval-valued CRI (IVCRI). 2. Point-valued Approximate Analogical Reasoning (PVAAR). 3. Interval-valued Approximate Analogical Reasoning (IVAAR). • Sugeno’s versions: 1. Position type reasoning method (P). 2. Position Gradient type reasoning method (P-G). Informally, fuzzy reasoning method can be viewed as a process by which a possible imprecise conclusion is deduced from a collection of imprecise premises [18]. The imprecise can be described by linguistic variables E-mail address: [email protected] (Z. Li) c 1999 Elsevier Science B.V. All rights reserved. 0165-0114/99/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 7 ) 0 0 2 9 6 - 0
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and the collection is the set of some linguistic rules. There exists a large number of studies concerning the construction of the rules and the design of the linguistic variables. However, a good result not only depends upon the design of linguistic variables and the construction of the rules, but also depends upon the fuzzy reasoning method used. There have been a few studies on fuzzy reasoning methods. This leads to a lack of an ecient explanation on fuzzy reasoning methods and hence, our ability appears to be limited in convincing ‘users’. Usually, the choice of a better fuzzy reasoning method for a physical problem is dicult, as the comparison of currently existing reasoning methods is expensive, especially for a complex problem. Therefore, a theoretical discussion for choice of a reasoning method is needed. In this paper, I discuss theoretically the similarity which is relative to the precision of reasoning methods, computational time and two operation properties that a reasoning method must satisfy from practical consideration for a reasoning method. I hope my discussion will help the ‘users’ to choose a suitable fuzzy reasoning method or develop his= her own, and also serve as a reference for the development of a mathematical axiom system for fuzzy reasoning methods. 1.2. Reasoning methods In general, fuzzy reasoning is based on linguistic rules. The rules are with the following forms: R1 :
If X1 is A11 and · · · and XI is A1I then Y is B1 ;
R2 :
If X1 is A21 and · · · and XI is A2I then Y is B2 ; .. .. . .
RJ :
(1)
If X1 is AJ1 and · · · and XI is AJI then Y is BJ ;
where Rj ( j = 1; 2; : : : ; J ) is implication relation, Xi (i = 1; 2; : : : ; I ) are antecedent linguistic variables. Aji , B j are fuzzy variables, i.e. the linguistic values of corresponding linguistic variables. The fact is with the following form: X1 is A01 · · · XI is A0I and the conclusion is as follows: Y is B0 ; where Y is consequent linguistic variable, A0i (i = 1; 2; : : : ; I ) are observations, and B0 is a resultant linguistic value by an approximate reasoning method. 1.2.1. Zadeh’s method [5, 6, 13] According to Zadeh, Rj means Rj = A j → B j ;
(2)
where A j is de ned by Aj =
I \ i=1
Aji
(3)
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an approximate reasoning is carried out by 0
B j = A0 ◦ Rj
(4)
here ‘◦’ is a compositional operator and A0 is de ned by A0 =
I \
A0i :
(5)
i=1
Then a consequence, B0 is derived by combining the consequence of all the rules: B0 =
J ]
Bj
0
(6)
j=1
where ] ∈ {∨; ∧}, and ] is dependent on the type of implication used. When we adopt point-valued sup–min operations in our model, then the membership value of inferred consequence is calculated from (2) – (6): " ! !# J I I ] ^ ^ sup hA0i (xi ) ∧ hAj (xi ) → hB j (y) ; (7) hB0 (y) = x1 ;:::; xI
j=1
i=1
i=1
i
where hA0i (xi ), hAj (xi ), hB j (y) and hB0 (y) are the corresponding membership functions for the fuzzy sets A0i , i Aji , B j and B0 , respectively. The general form of implication, →, is interpreted in various formulae. When we adopt Mamdani’s implication, which is often used in various applications, we have ( ! " I #) J I _ _ ^ ^ hA0i (xi ) ∧ hAj (xi ) ∧ hB j (y) hB0 (y) = x1 ;:::; xI
j=1
=
J _
(
j=1
I ^ i=1
i=1
"
_
i
i=1
#
)
(hA0i (xi ) ∧ hAj (xi ) ∧ hB j (y) :
x1 ;:::; xI
(8)
i
1.2.2. Turksen’s methods [13–16] 1. Interval-valued CRI (IVCRI) For jth rule ( j = 1; : : : ; J ) in (1), the implication means (Rj ) = (A j (I )) ⇒ B j ;
(9)
where A j (I ) is the overall intersection of the antecedents, i.e., j (I )] (A j (I )) = [LjAND (I ); UAND
(10)
j (I ) de ned using recursive expressions. That is with LjAND (I ), UAND
LjAND (1) = Aj1 ; LjAND (2) = LjAND (1) ∩ Aj2 ; .. . LjAND (I ) = LjAND (I − 1) ∩ AjI ;
(11)
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while j (1) = Aj1 ; UAND [ j j j j (2) = [UAND (1) ∪ Aj2 ∩ UAND (1) ∪ (Aj2 )c ∩ (UAND (1))c ∪ Aj2 ; UAND
LjAND (1) ∪ Aj2 ∩ LjAND (1) ∪ (Aj2 )c ∩ (LjAND (1))c ∪ Aj2 ]; .. . j (I ) = UAND
[
j j j [{UAND (I − 1) ∪ Aj2 } ∩ {UAND (I − 1) ∪ (Aj2 )c } ∩ {(UAND (I − 1))c ∪ Aj2 };
{LjAND (I − 1) ∪ Aj2 } ∩ {LjAND (I − 1) ∪ (Aj2 )c } ∩ {(LjAND (I − 1))c ∪ Aj2 }]: The implication (9) is de ned as an internal valued fuzzy set from (A j (I )) ⇒ B j = [Limp (Rj ); Uimp (Rj )]; where Limp (Rj ) = LjAND (I ) ∩ B j ;
j Uimp (Rj ) = UAND (I ) ∩ B j :
Compositional rule of inference is carried out by 0
(B j ) = (A0 (I )) ◦ (Rj ); where (A0 (I )) is an aggregation of observations. That is, 0 (A0 (I )) = [L0AND (I ); UAND (I )]:
When we adopt interval-valued sup–min operations for the composition, reasoning result B0 is calculated by 0
0
0
(B j ) = [Lcri (B j ); Ucri (B j )]; where 0
Lcri (B j ) = sup min{L0AND (I ); Limp (Rj )}; x1 ;:::; xJ
0
0 Ucri (B j ) = sup min{UAND (I ); Uimp (Rj )}: x1 ;:::; xJ
An estimate of the membership function of the consequence for the jth rule ( j = 1; : : : ; J ) in (1) is determined as follows: hB j0 = 12 (hLcri (B j0 ) + hUcri (B j0 ) ): An estimate of the membership function of the nal consequence is derived as follows: hB0 = 12 (hLOR (J ) + hUOR (J ) ); where UOR (J ), LOR (J ) is de ned using recursive expressions. That is, 0
UOR (1) = B1 ; 0
UOR (2) = UOR (1) ∪ B2 ; .. . 0
UOR (I ) = UOR (J − 1) ∪ B J ;
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0
LOR (1) = B1 ; \ 0 0 0 LOR (2) = [{UOR (1) ∩ B2 } ∪ {UOR (1) ∩ (B2 )c } ∪ {(UOR (1))c ∩ B2 }; 0
0
0
{LOR (1) ∩ B2 } ∪ {LOR (1) ∩ (B2 )c } ∪ {(LOR (1))c ∩ B2 }]; .. . LOR (J ) =
\
0
0
0
[{UOR (J − 1) ∩ B J } ∪ {UOR (J − 1) ∩ (B J )c } ∪ {(UOR (J − 1))c ∩ B J }; 0
0
0
{LOR (J − 1) ∩ B J } ∪ {LOR (J − 1) ∩ (B J )c } ∪ {(LOR (J − 1))c ∩ B J }]: 2. Point-valued approximate analogical reasoning (PVAAR) For PVAAR, pattern matching measures between antecedents and observations is calculated for each rule in (1) as follows: SM j =
I ^
SMij ( j = 1; : : : ; J );
i=1
where SMij is, e.g., a similarity measure of the ith variable de ned with a distance measure DM SMij [Aji ; A0i ] = 1 − DM [Aji ; A0i ]: With the values of SM j , the rules which satisfy the following conditions are red: SM j ¿ ; where is an appropriate threshold. Approximate reasoning is carried out through modi cation of each consequent. Two types of modi cation are proposed, namely: (a) Expansion form: hB j : hBexj = min 1; SM j (b) Reduction form: hBrej = SM j · hB j : The average of the expansion and reduction forms can be adopted as an estimate of the membership function of the consequence from the jth rule ( j = 1; : : : ; J ) in (1) as hB j0 = 12 (hBexj + hBrej ): 3. AAR with interval-valued fuzzy set (IVAAR) In addition to AND; OR, etc., an interval-valued armation can be de ned by DNF and CNF . In general, the form is, FAFF = [DNF (F); CNF (F)] = [(E ∩ F) ∪ (E c ∩ F); (E ∪ F) ∩ (E c ∪ F)];
(12)
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where F is the concept in relation to another concept E. For the jth rule ( j = 1; : : : ; J ) in (1), its consequent is de ned as an interval-valued fuzzy set based on (12) as (B j ) = [LAFF (B j ); UAFF (B j )]; where LAFF (B j ) =
\
j j j j [{LAND (I ) ∪ (LAND (I ))c } ∩ B j ; {UAND (I ) ∪ (UAND (I ))c } ∩ B j ];
and UAFF (B j ) =
[
j j j j [{LAND (I ) ∩ (LAND (I ))c } ∪ B j ; {UAND (I ) ∩ (UAND (I ))c } ∪ B j ]
j j (I ), UAND (I ) are the upper bound and the lower bounds, respectively. Let where LAND j 0 (I ); LAND (I )]; SM jL = SM[LAND
j 0 SM jU = SM[UAND (I ); UAND (I )]:
With the values of SM, the rules which satisfy the following conditions are red: SM jL ¿ 0L
and SM jU ¿ U 0 ;
(13)
here 0L and U 0 are appropriate thresholds. Approximate reasoning is carried out through modi cation of each consequent. There are two types of modi cation as PVAAR, (a) Expansion form: 0
0
0
(B j )ex = [Lex (B j ); Uex (B j )]; where
hLex (B j0) = min
hUAFF (B j ) hLAFF (B j ) ; ; SM jU SM jL
hU (B j ) hUex (B j0) = min 1; AFF : SM jU
(b) Reduction form 0
0
0
(B j )re = [Lre (B j ); Ure (B j )]; where hLre (B j0) = min{hUAFF (B j ) · SM jU ; hLAFF (B j ) · SM jL };
hUre (B j0) = SM jU · hUAFF (B j ) :
The average of upper and lower bounds of the expansion and the reduction forms can be adopted as an estimate of the membership function of the consequence for the ( j = 1; : : : ; J ) in (1) as follows: hB j0 = [ 12 (hUre (B j0) + hLre (B j0) ); 12 (hUex (B j0) + hLex (B j0) )] = [hLIVAAR (B j0) ; hUIVAAR (B j0) ]: An estimate of the membership function of the nal consequence is determined by J J ^ 1_ hLIVAAR (B j0) + hUIVAAR (B j0) : hB 0 = 2 j=1
j=1
(14)
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1.2.3. Sugeno’s methods [8–12] 1. Position type reasoning method. If the fuzzy variables B j ( j = 1; : : : ; J ) in the consequent part of rules in (1) are replaced with a center of gravity vector of corresponding output cluster as follows: bj =
N X
(hjn )w yn
X N
n=1
(hjn )w ;
n=1
(see pp. 262–263 in [7]), then the set of derived rules is said Position type fuzzy model. The nal estimate of the consequence for this model is derived: y=
J X j=1
j j
h b
X J
h j;
j=1
where h j is a membership value of j th rule ( j = 1; : : : ; J ) in (1) derived from the algebraic product of all membership values of input variables in antecedent: h j = hA j (x10 ) · · · hA j (xI0 ); 1
I
where xi0 (16i6I ) is an input. 2. Position-gradient type reasoning method. If the fuzzy variables B j (j = 1; : : : ; J ) in the consequent part of rules in (1) are replaced with ‘b j and @Y=@X1 is c1j and · · · and @Y=@XI is cIj ’ (where @Y =@Xi (16i6I ) is the gradient of the output variable Y for input variables Xi , and cij (i = 1; : : : ; I ) is its singleton linguistic value), then the set of derived rules is said Position-gradient type fuzzy model. The nal estimate of the consequence for this model is derived: " !# J J I X X X j (di (n; j) · ci (n; j); (n; j) · b j + y= j=1
i=1
j=1
0 0 ; : : : ; xIn ) and core of the jth where (n; j) is a weight loaded on the distance, D(n; j), between input data (x1n rule ( j = 1; : : : ; J ) in (1): v u I uX (xi0 − xijc ) 2 ; xijc ∈ {xi |hA j (xi ) = 1; xi ∈ Xi }; (15) D(n; j) = min t i=1
and
i
(n; j) = exp{−D(n; j)}di (n; j) is a projection of D(n; j) on xi .
1.2.4. The comparison results from practical problems The performance characteristics of the above six fuzzy reasoning methods were investigated by [7]. The investigation used three sets of real-life system data which supply a reasonable ground for comparing the reasoning precision, the calculation time and the number of valid input cases. The conclusion is that CRI has appropriate precision, but complex computation; IVCRI takes longer time, the reasoning can be carried out for all case data points and gives a representation of the spread of vagueness that may be associated with linguistic connectives if it is to be taken into account; Sugeno’s methods have an advantage in calculation speed and the P-G method is quite useful in the case where rapid reasoning is necessary and no case data point should be missed in reasoning, but these two methods require absolute scale; the PVAAR and IVAAR methods shows comparably good precision and short time for management decision models, but these methods limit the number of case data points for reasoning when the threshold is increased.
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2. Analysis of reasoning methods 2.1. Similarity The performance of a fuzzy reasoning method RM would depend upon rules it drive the result from. An obvious method to judge which reasoning method is suitable for a given set of rules (1) is that the RM should satisfy RM(A1j ; : : : ; AIj ) ≈ B j ( j = 1; : : : ; J ). I introduce a de nition to describe it as follows. Deÿnition 1. A fuzzy reasoning method RM is said to have similarity if RM(A1j ; : : : ; AIj ) = B j for all 16j6 J . The similarity can also be expressed by a Similarity Measure SM SM[RM(A1j ; : : : ; AIj ); B j ] = 1
(16 j6J ):
2.1.1. Zadeh’s method If the implication adopts Mamdani’s formula and J = 1 in (1), then !" I # I _ ^ ^ hA0i (xi ) hAi (xi ) ∧ hB (y) hB0 (y) = x1 ; ::: ; xI
=
i=1
( I ^ _ i=1
i=1
) [hA0i (xi ) ∧ hAi (xi )]
∧ hB (y):
(16)
xi
When A0i = Ai (i = 1; : : : ; I ), then hB0 (y) = hB (y), and Zadeh’s method has similarity. If linguistic rules have more than one value and the values of fuzzy variables of two linguistic rules on overlap of the two linguistic rules are not zero, for example, the rules of three real-time systems in [7], then Zadeh’s method does not have similarity. 2.1.2. Turksen’s methods 1. IVCRI. For J = 1 in (1), it is easy to prove that IVCRI has similarity. But for J ¿1, it may not have similarity. For example, any two rules in the given rules of the rst real-time system [7], X1 is As1 and X2 is As2 then Y is Bs ; X1 is At1 and X2 is At2 then Y is Bt ; where s 6= t; s; t 6= 1; 6, if let A01 = As1 and A02 = As2 , then hBt0 = 12 (hLCRI (Bt0) + hUCRI (Bt0) ) 6= 0: 2. PVAAR. According to the de nition of similarity measure, SM jI 0 [A0i ; Aij0 ] = 1 if A0i = Aij0 SM j0 =
I ^
for all
16i6 I (16j0 6J );
SM ji 0 = 1:
i=1
If we can choose a similarity measure SM and a threshold 0 such that SM j 6 0 (16j6J; j 6= j0 ), then the only red rule is Rj0 and hB0 = hB j0 , and PVAAR has similarity. In general, the higher threshold value, the more probability PVAAR has similarity.
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3. IVAAR. It is easy to compute j0 0 (I ); LAND (I )] = 1; SM j0 L = SM[LAND
j0 0 SM j0 U = SM[UAND (I ); UAND (I )] = 1;
if Aij0 = A0i for all 16i6I and some j0 (16j0 6J ). If we can choose a similarity measure and thresholds 0L j jL jU L U and U 0 such that SM 6 0 and SM 6 0 (16j6J; j 6= j0 ), then the only rule R0 is red and ( ) hUAFF (B j0 ) hLAFF (B j0 ) j0 U ; h × hUAFF (B j0 ) = hB j0 ; = hB j0 ; h j 0 = min j 0 = SM Lex (B 0 ) Ure (B 0 ) SM j0 U SM j0 L h
= 12 (h
j0
LIVAAR (B 0 )
j0
Ure (B 0 )
+h
j0
Lre (B 0 )
) = hB j0 ;
h
j0
UIVAAR (B 0 )
= 12 (h
j0
Uex (B 0 )
+h
j0
Lex (B 0 )
) = hB j0 ;
therefore
J J ^ 1 _ hLIVAAR (B j ) + hUIVAAR (B j ) = hB j0 ; hB 0 = 2 j=1
j=1
IVAAR has similarity. In general, the higher the threshold, the more probably IVAAR has similarity. 2.1.3. Sugeno’s methods Sugeno’s reasoning methods are the methods used in the clustering algorithm, and is used in the modelling stage for variable selection and rule structuring. The similarity of these reasoning methods depends upon the set of rules (1). If the set of rules (1) is biased toward the Sugeno’s reasoning method, then the results have higher similarity. 2.2. Commutativity The results of a problem are independent of the order of the rules in a set of rules as (1) if the problem is solved by the linguistic rule from physical consideration. If the results of a fuzzy reasoning method are independent of the order of the rules in a set of linguistic rules, the method is said to have commutativity. It is easy to see that the six fuzzy reasoning methods discussed above have commutativity from their de nitions. 2.3. Continuity In this section, we consider the continuity of fuzzy reasoning methods. Suppose DM and DM 0 are two distance measures between fuzzy variables Ai , A0i (16i6I ) and B, B0 . Let R: X1 is A1 and · · · and XI is AI then Y is B be a rule, and X1 is A01 and · · · and XI is A0I be an observation and B0 is the WI corresponding resultant linguistic value by a reasoning method RM . If DM 0 (B; B0 ) tend to zero when i=1 DM(Ai ; A0i ) tends to zero, RM is said to have continuity at rule R. If RM has continuity at every rule of considered system, RM is said to have continuity, we also can say RM is continuous. For example, the two distance measures DM and DM 0 of two variables are de ned as DM(A1 ; A2 ) = supx∈X |hA1 (x) − hA2 (x)|, where X is the universe of discourse. It is easy to prove DM is a distance measure. Lemma 1. The union operator ∨ of fuzzy sets is continuous relative to DM.
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A01 ∨ A02 = A0 . Proof. Let A1 ; A01 and A2 ; A02 are fuzzy sets, and A1 ∨ A2 = A; = sup |hA0 ∨A0 (x) − hA1 ∨A2 (x)| DM (A0 ; A) 1 2 x∈X
= sup max[hA01 (x); hA02 (x)] − max[hA1 (x); hA2 (x)] x∈X x∈X x∈X 6 max sup |hA01 (x) − hA1 (x)|; sup |hA02 (x) − hA2 (x)| : x∈X
(17)
x∈X
tends to zero when DM (A01 ; A1 ) and DM (A02 ; A2 ). Therefore, DM (A0 ; A) The proof for continuity of intersection operator ∧ is the same as Lemma 1. Lemma 2. If fuzzy sets A; B are in two universes of discourse U and V; respectively; then A ∨ B and A ∧ B are continuous relative to DM . The proof is similar to Lemma 1. Lemma 3. If fuzzy set A is in U and fuzzy set B is in V; then operator supu∈U hA (u) ∧ hB (v) is continuous. Proof. Let A0 is a fuzzy set in U , B0 is a fuzzy set in V , then sup sup [hA0 (u) ∧ hB0 (v)] − sup [hA (u) ∧ hB (v)] v∈V u∈U
u∈U
6 sup sup |hA0 (u) ∧ hB0 (v) − hA (u) ∧ hB (v)| v∈V u∈U
6 sup sup [|hA0 (u) − hA (u)| ∨ |hB0 (v) − hB (v)|] v∈V u∈U
6 sup |hA0 (u) − hA (u)| ∨ sup |hB0 (v) − hB (v)|: u∈U
v∈V
Lemma 4. Plus; minus and multiplication of fuzzy variables are continuous relative to DM . The operator of fuzzy variable numerator and real number denominator is also continuous relative to DM . Since the fuzzy variables we discussed are bounded, hence Lemma 4 holds. Lemma 5. The composition of continuous operators relative to DM is continuous relative to DM . It is easy to prove. By Lemmas 1–3, we obtain the following. Theorem 1. If the form of implication adopts Mamdani’s method; then Zadeh’s reasoning method is continuous relative to DM . According to Turksen’s de nition of IVCRI and Lemmas 1– 5, we obtain Theorem 2. IVCRI is continuous relative to DM .
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Now let us consider continuities of Turksen’s AAR and IVAAR. Since similarity measure is de ned with a distance measure DM for AAR, we have SMij [Aij ; A0i ] = 1 − DM [Aij ; A0i ]; SM j =
I ^
SMij
(16 j6 J );
i=1
where Aij , A0i (16i6I; 16 j6 J ) are fuzzy variables in (1). SM j is continuous relative to DM by Lemmas 1– 4. For an appropriate threshold , approximate reasoning is carried out through modi cation of each consequent which satisfy SM j ¿ : We know reduction form and expansion form of consequent are continuous relative to DM , so are the nal consequences for AAR and IVAAR by lemmas. We obtain Theorem 3. AAR and IVAAR are continuous relative to DM . The continuity for Sugeno’s reasoning method is followed by Lemma 4. 2.4. Computational times For comparison of computational complexity of six reasoning methods, we consider an operation between twoVfuzzy numbers as one, i.e. consider a fuzzy number as a real number, e.g., the computational times I of i=1 hA0i (xi ) is I-1. Here we only discuss the computational times of hB0 (y) for di erent reasoning methods, because the defuzzi cations of B0 are same for the methods except Sugeno’s methods. We denote the computational times of hB0 (y) by N . For Zadeh’s reasoning method, if we adopt Mamdani’s form of implication, then N = 2IJ + J − 1: For Turksen’s IVCRI N = 15(I − 1)(J + 1) + 21J + 2I − 17: For Turksen’s AAR (at least one rule is red) 3IJ − J + 56 N 6 3IJ + 4J: For Turksen’s IVAAR (at least one rule is red) (I − 1) (15J + 17) + 336 N 6(I − 1)(15J + 17) + 33J: For Sugeno’s P N = IJ + 2J − 1: For Sugeno’s P − G N = 5IJ + 2I + 4J − 1; where the last two are the computational times of nal consequence.
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The order of the computational times shown above are basically the same with the order of the time that are given in [7] for practical examples. The computational time of a method does not fully depend on computational complexity, but it mainly is determined by the computational complexity. 3. Discussion We discussed four properties which re ect di erent aspects of fuzzy reasoning methods. If similarity degree SD is de ned as SD = 1 −
J _
DM (Bj ; B j );
j=1
where Bj is the consequent linguistic value of the j th rule (j = 1; : : : ; J ) in (1) and B j is the resultant linguistic value of antecedent linguistic values of jth rule ( j = 1; : : : ; J ) in (1) by a fuzzy reasoning method, then SD can be applied to measure the precision of a fuzzy reasoning method. The nearer SD to 1 is, the higher the precision of the reasoning method. This is easy to compute. Commutativity is a property that every reasoning method must have according to practical meaning. If a reasoning method does not satisfy this property, the method could not give good results. When a reasoning method does not satisfy continuity, the di erence between the actual output from the method and resultant linguistic value in (1) may be large even though inputs are much nearer to antecedent linguistic values in (1). The computational time of a reasoning method can be used to estimate the reasoning time of the reasoning method. With the increase of CPU speed, the computational complexity is no longer such an important factor. It is easy to see that the above results can be extended to several consequent variables with no diculty. We give a learning algorithm for fuzzy neural networks based on fuzzy number operations which can learn linguistic rules as (1) with desired similarity degree [4]. As a neural network could be used for reasoning, the fuzzy neural network we designed can be used for reasoning in any physical problem that satis es fuzzy number operations and show good results. The computational times of this method may be bigger than the six reasoning methods discussed above, it depends on how the operations of fuzzy numbers are performed by the computer. References [1] F. Bouslama, A. Ichikawa, Fuzzy control rules and their natural control laws, Fuzzy Sets and Systems 48 (1992) 65 – 86. [2] T.C. Chang, K. Hasegawa, C.W. Ibbs, The e ect of membership function on fuzzy reasoning, Fuzzy Sets and Systems 44 (1991) 169 –186. [3] R. Katayama, Y. Kajitani, K. Matsumoto, M. Watanabe, Y. Nishida, An automatic knowledge acquisition and fast self tuning method for fuzzy controller based on hierarchical fuzzy inference mechanisms, Proc. 4th IFSA Congr., Brussels, 1991, pp. 105 –108. [4] Z. Li, Y. Zhang, M. Nishikawa, A. Ichikawa, A learning algorithm for fuzzy neural networks based on fuzzy number operations, Proc. ICONIP96, Springer, Berlin, pp. 260 – 265. [5] E.M. Mamdani, S. Assilian, Applications of fuzzy algorithms for control of simple dynamic plant, Proc. IEE 121 (1974) 1585 –1588. [6] E.M. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. Man–Mach. Studies 7 (1975) 1–13. [7] H. Nakanishi, I.B. Turksen, M. Sugeno, A review and comparison of six reasoning methods, Fuzzy Sets and Systems 57 (1993) 257 – 294. [8] M. Sugeno, An introductory survey of fuzzy control, Inf. Sci. 36 (1985) 59 – 83. [9] M. Sugeno, Fuzzy control, Nikkan Kogyo Shinbunsya, Tokyo, Japan, 1988. [10] M. Sugeno, G.T. Kang, Structure identi cation of fuzzy model, Fuzzy Sets and Systems 28 (1988) 15 – 33. [11] M. Sugeno, T. Yasukawa, A fuzzy logic based approach to qualitative modeling, IEEE Trans. Fuzzy Systems 1 (1993) 1– 24. [12] T. Takagi, M. Sugeno, Fuzzy identi cation of systems and its applications to modeling and control, IEEE Trans. Systems Man Cybernet. 15 (1985) 116 –132.
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[13] I.B. Turksen, Y. Tian, Interval valued fuzzy set representation on multiple antecedents fuzzy s-implications and reasoning, Fuzzy Sets and Systems 52 (1992) 1– 25. [14] I.B. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20 (1986) 191– 210. [15] I.B. Turksen, Four methods of approximate reasoning with interval-valued fuzzy sets, Internat. J. Approx. Reasoning 3 (1989) 121–142. [16] I.B. Turksen, Z. Zhong, An approximate analogical reasoning scheme based on similarity measures and interval valued sets, Fuzzy Sets and Systems 34 (1990) 323 – 346. [17] R.R. Yager, D.P. Filev, On the issue of defuzzi cation and selection based on a fuzzy set, Fuzzy Sets and Systems 55 (1993) 255 – 271. [18] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems Man Cybernet. 3 (1973) 28 – 44.