217
CORRESPONDENCE
C5[C12]
C7[C11]
=
=
CBC12[C6 + C2 + C3 + C1][C9 + Cul + C5 + C1] order > 2
C7C11.
Prime closed set {C7, Cil} does not cover the state set.
We would continue by examining prime classes with two closure
conditions, but since k =2, Rule 3 indicates that such examination
is futile. All prime classes have been examined, and hence {C3,
Clo, Cui} is the minimum cover found.
REFERENCES
[1] A. Grasselli and F. Luccio, "A method for minimizing the number of internal
states in incompletely specified sequential networks,' IEEE Trans. Electron.
Comput., vol. EC-14, pp. 350-359, June 1965.
[2] S. C. DeSarkar, A. K. Basu, and A. K. Choudhury, 'Simplification of incompletely specified flow tables with the help of prime closed sets," IEEE Trans.
Comput. (Short Notes), vol. C-18, pp. 953-956, Oct. 1969.
[3] -, 'On the determination of irredundant prime closed sets," IEEE Trans.
Comput. (Short Notes), vol. C-20, pp. 933-937, Aug. 1971.
Comments on "Minimization of Fuzzy Functions"
ABRAHAM KANDEL
Abstract-This correspondence points out the insufficiency of the
method for minimization of fuzzy functions as presented in a recent
short note.1 Two lemmas, which are the basis for a new minimization
technique [2 ], are represented.
Index Terms-Fuzzy function, fuzzy logic, minimization, prime
implicant of fuzzy function.
In the above short note,' Siy and Chen proposed an algorithm to
reduce fuzzy functions to their minimal canonical sum-of-products
form. In their note, they presented three theorems (Theorems 2, 3,
and 4) upon which the minimization algorithm is based. We represent
these theorems as presented therein.
Theorem 2: Every prime implicantfk of a fuzzy function
F= Efj
jES
is a term of F, i.e., every k ES.
Theorem 3: The reduction rule generates all the prime implicants
of F.
Theorem 4: The minimum canonical sum-of-products form of
fuzzy function F is the union of all its prime implicants.
These theorems are incorrect, since they are based upon the claim
that any pair of fuzzy variables x and x are incomparable, and therefore, the complement operation, in fuzzy logic, has the affect of considering x and * as two unrelated variables when the relation < is
used. This is wrong since x=1 -x. Due to this relation between x and
x, the system is not a "free" lattice FL(n) as stated by Siy and Chen,
and the results of Whitman [1] cannot be utilized to deduce Theorems 2, 3, and 4.
Hence, these theorems do not present the general criterion for the
minimum expression of fuzzy functions. The above can be illustrated
by the following examples.
Example 1: Let xi and x2 be fuzzy variables, and let' F(xi, x2)
=X1fCX2+xiii12 be a fuzzy function of xi and x2. Using the algorithm,'
the set of fuzzy prime implicants consists of the terms xlisX2 and
xli,12, and the minimized function is given by
F(xi, X2) = X121X2 + XI1XI2.
However, it is clear that xlix <0.5 and x2 +x2 > 0.5. Therefore,
F(x1, X2) = XlXl(X2 + X2) = Xl2s,
and hence, xiii is a fuzzy prime implicant of F that does not appear in
the original function. Moreover, xl,,x2 +xll2 is not the minimized
expression for F.
Example 2: Let a and , be fuzzy variables and let
Manuscript received May 16, 1972.
The author is with the Department of Computer Science, New Mexico Institute
of Mining and Technology, Socorro, N. Mex. 87801.
1 P. Siy and C. S. Chen, IEEE Trans. Comput. (Short Notes), voL C-21, pp.
100-102, Jan. 1972.
F(a,13) =a+a+13B
be a fuzzy function in a and 13. Using the algorithm,1 this is the minimized fuinction. However, ,13i<0.5 and +a>+O.5, and thus, F(a, 13)
=oea+&+131"ae+&.
A complete algorithm for the minimization of fuzzy functions is
presented in [2]. In this correspondence, we will only outline the
principles of the technique, by means of Lemmas 1 and 2. In Boolean
algebra we apply the following identities in order to minimize a
Boolean function:
xx=x
1) x+x=x
x-0=0 2')x+0=x x 1=x
2) x+1=1
xx =0.
3) x+x=1
These identities can be applied to Boolean variables or Boolean functions, and they are the only primitive set of rules by which Boolean
minimization is performed. It is clear that rules 1 and 2 are also true
in fuzzy algebra, and practically, these are the identities that the
algorithms' are based on. In general, one cannot apply the identities
in (3) to fuzzy expressions, as can be easily seen from the following
example.
Example 3: Let F=xl +glx2. Clearly, in Boolean algebra F
= (xI +1) (x +x2) =xI +x2 by applying identity (3). However, this
is not so in fuzzy algebra (e.g., let xi = 0.4, x2= 0.7. This implies that
x1 +x2=0.7 but xl +xIx2 = max [0.4, min (0.6, 0.7) ] = 0.6).
In certain cases, the fuzzy version of identity (3) can be applied.
Before these cases are discussed, we need some preliminary definitions.
Definitions:
1) A clause is a disjunction of one or more literals.
2) A phrase is a conjunction of one or more literals.
3) A form S is said to be in disjunctive normal form if S =Pl +P2
+ * * +Pm, m> 1 and every Pi, 1 < i < m is a phrase.
4) A form S is said to be in conjunctive normal form if S= C1C2
Ck, k > 1 and every Ci, 1 <j<k is a clause.
These special cases are discussed in the following lemmas.
Lemma 1: Let the set {Fs}Fl= be a set of fuzzy functions over
xI, x2, * , xn, and let F be a conjunction of functions from this set.
Iff there exist functions Fi and Pi in F, then a disjunction Fd, of any
function Fk and its complement Fk, can be appended to or deleted
from the conjunction representing F without affecting the value of F.
Proof: 1) Assume Fi and Pi in F. Obviously, FiFJ<0.5, and thus,
F<0.5. However, Fd= Fk+Fk>0.5, and therefore,
FiFiFd = FiFY
for all Fd's of the above form. 2) Assume that no conjunction of a
function Fi and its complement Pi appears in F. Thus, 0< F< 1 and
Fd=Fk+Fk>0.5 can fuzzily imply F. Two cases must be checked.
Assume that we append Fd to the conjunction F. In general, we might
have F> Fd, and then Fd fuzzily implies F, which is incorrect. The
second case is deleting Fd from F. Let F= aFd where a is a conjunction
of functions where 0 <a < 1. F can have the same value as Fd in case
that a> Fd, and thus, F, cannot be deleted from F.
Hence, any appending or deleting of Fd will unequivocally change
Q.E.D.
the valte of F.
Lemma 2: Let the set {Fj} t.=l be a set of fuzzy functions over xI, x2,
x", and let F be a disjunction of functions from this set. Iff there
exist functions F, and Fj in F, then a conjunction F, of any function
F, and its complement F,, can be appended to or deleted from the
disjunction representing F without affecting the value of F,,. This is a
dual lemma to Lemma 1 and a dual proof can be used.
Corollary 1: Let F(xX,2-*- , xn) be expressed in conjunctive
normal form. Iff there exists a variable xi and its complement fi in F,
then a disjunction of any variable xk and its complement ik, 1 <k <n
can be appended to or deleted from F without affecting the value of F.
Corollary 2: Let F(x,, x2, * *, xn) be expressed in disjunctive normal form. Iff there exists a variable xj and its complement xi in F,
then a conjunction of any variable xl and its complement xi, 1 <1 <n
can be appended to or deleted from F without affecting the value of F.
The complete minimization technique and relationships between
fuzzy functions and binary logic are discussed in [2].
-
REFERENCES
11 P. M. Whitman, 'Free lattices," Ann. Math., vol. 42, pp. 325-329, 1941.
121 A. Kandel, "On minimization of fuzzy functions," submitted for publication,
1972.