IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006

709

Error Estimation of Perturbations Under CRI
Guosheng Cheng and Yuxi Fu

Abstract—The analysis of stability and robustness of fuzzy reasoning is an important issue in areas like intelligent systems andfuzzy control. An interesting aspect is to what extent the perturbation of input in a fuzzy reasoning scheme causes the oscillationof the output. In particular, when the error limits (restrictions)of the input values are given, what the error limits of the output
values are. In this correspondence, we estimate the upper and lower
bounds of the output error affected by the perturbation parameters of the input, and obtain the limits of the output values when
the input values range over some interval in many fuzzy reasoning
schemes under compositional rule of fuzzy inference (CRI).
Index Terms—Error estimation, fuzzy reasoning, fuzzy set, interval perturbation, simple perturbation.

I. INTRODUCTION
INCE Zadeh’s compositional rule of fuzzy inference[1] (CRI) was proposed, many other methods of fuzzy
reasoning have been known [2]–[10]. Applications of thesefuzzy reasoning techniques have been successful in various
areas, especially in fuzzy control [11]. When fuzzy reasoning
is applied, the stability and robustness of the fuzzy reasoning
becomes one of the prominent problems. The fuzzy controllers
are available to transform human expertise and subjectivity toquantitative terms, so the deviation of human expertise fromits corresponding quantitative representations gives rise to the
problem of the stability of fuzzy controllers [11]. The corresponding problem in fuzzy reasoning is the variance of output
caused by perturbations of input. Here, the analysis of thestability of a fuzzy reasoning scheme consists of two aspects:
One is how the output values of the scheme are changed by the
perturbation parameters of input values. The other is how toestimate corresponding limits of output values of the scheme
when oscillation limits of input values are given.
For the first question, many researchers have provided theiranswers [12]–[15]. Their approaches to perturbations of input
are based on some proximity of fuzzy sets, using proximity measure in [12], or -similarity measure in [13], or maximum perturbation in [14], or -equality in [15]. In a sense, the problem ofstability of fuzzy reasoning has been well studied. However, theeffects of the perturbation parameters of the input in fuzzy reasoning schemes still call for investigation. For one thing, when asequence of the perturbations of the input for a fuzzy reasoning
scheme has an asymptotic limit, it is obvious that the corresponding sequence of the output cannot be precisely demonstrated in the above approaches.
The correspondence is structured as follows: After introducing the concepts of the simple perturbation and the interval
perturbation of the fuzzy sets, we obtain an estimation of theupper and the lower bounds of the output error affected by thesimple perturbation of the input under CRI. The stability of
fuzzy reasoning schemes is characterized. Also the asymptotic
performance in fuzzy reasoning schemes is exhibited. Next weinvestigate interval perturbation of input in fuzzy reasoning and
get the interval estimation of fuzzy sets inferred from CRI withsome abstract implication operators and conjunction operators.
Some final remarks are made in Section V.

S

II. PRELIMINARIESThe basic form of the CRI methods is as follows:
Antecedent
Fact

If

Conclusion
where , are fuzzy sets on
sets on usually defined as

is

then

is

is
is
and

are fuzzyc
In the previous definition c is some conjunction operator,
is an implication operator. The pair c
is calledandisa scheme of fuzzy reasoning. The ordered triad
an output of fuzzy reasoning.
called an input and, or product, or a Lukasiewicz conjuncUsually c is either,), ortion operator (i.e., cor a -conorm .
a -norm
A. Some Notions

Manuscript receivedAugust 9, 2003; revised April 13, 2005 and April 19,
2006. The work was supported by the National Distinguished Young Scientist
Fund of NNSFC under Grant 60225012, by the National 973 Project under
2003CB317005, by the National Nature Science Foundation of China underGrant 60473006, by the BoShiDian Research Fund under 20010248033, by the
Jiangsu Education Office Research Fund under Grant 05KJD110123, and by
the Nanjing University of Information Science and Technology Research Fundunder Grant QD39.
G. Cheng is with the Department of Mathematics, Nanjing Universityof Information Science and Technology, Nanjing 210044, China (e-mail:gshcheng@sohu.com).
Y. Fu is with the Department of Computer Science, Shanghai Jiaotong University, Shanghai 200030, China (e-mail: fu-yx@cs.sjtu.edu.cn).
Digital Object Identifier 10.1109/TFUZZ.2006.877333

The -norms , -conorms , and the fuzzy complemenwill be fundamental for the present cortary operations in
is called
respondence. A function
t-norm if and only if
is nondecreasing in each argument;1)
is commutative;2)
is associative;3)
has 1 as unit.
4)
is called -conorm if andA function
satisfies 1)–3), as well as 4’) given as follows:
only if
has 0 as unit.
4')1063-6706/$20.00 © IEEE

710

A function
is called fuzzy complement if andonly if,;
1)2) is nonincreasing.
In this correspondence the implication operator will stand:for one of the following, where;• Mamdani:;• Kleene–Dienes:;• Lukasiewicz:;• Reichenbach:;• Zadeh:in [16]:•;, where is a fuzzy• -implications:is a -conorm;
complement,-implications:,•is a t-norm;
where-implications:, where•is required to be a de Morgan triplet;.
• -norm implications:Now, we introduce the concepts of simple perturbation, interval perturbation of a fuzzy set and stability of a fuzzy reasoning scheme.
Definition 1: Let be a universe of discourse, and two
fuzzy sets defined on . If there exists a mapping :, such that for all,, thenis called a simple perturbation of and is a factor of theperturbation of .
The following definition will use the concept of fuzzy inare two fuzzy sets on , and for all,
terval. If ,, thenis called a fuzzy interval on .
be a fuzzy interval on , a fuzzy
Definition 2: Let,, then has
set on . If for all
on , written.
an interval perturbation
Definition 3: Let and be fuzzy sets on , a fuzzy set
andare perturbations of ,andon . Suppose ,with factor ,and , respectively. A fuzzy reasoning
is said to be stable if, given, there exists
scheme csuch thatonwhenever
,
and
.
of an
A perturbation sequencewith a corresponding sequence
input
of factors in some fuzzy reasoning,
scheme is said to be (asymptotic) stable if, given, andsuch that
there exist some natural number
on for all
,
,
, and
.
,
be fuzzy sets on
Definition 4: Let,a fuzzy set on . A fuzzy reasoning schemecis said to be stable if, given, there exist a fuzzy
on and, such that for each,interval, andwhenever,, and.
B. Related Work
1) Pappis’ Work: Pappis introduced the approximately equal
andbe fuzzy setsof two fuzzy sets on in [12], i.e., leton , if

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006

then

and
are said to be approximately equal, denoted by. is said to be a proximity measure ofand.
The Pappis’ result is as follows.
andbe fuzzy sets on , and
and
fuzzy
Let
relations from to . Then
implies;•
implies;•
where is the max–min composition.
We see that this result actually addresses the stability ofsome fuzzy reasoning schemes when , , and of the input
are perturbed, respectively.
2) Hong and Hwang’s Work: Hong and Hwang defined the-similarity of two fuzzy sets on in [13], i.e., letand
be fuzzy sets on , if

then
and
are said to be -similar, denoted by.
Hong and Hwang generalized the Pappis’ result to be thefollowing.
andbe fuzzy sets on , and
and
fuzzy
Let
to . Ifand, thenrelations from.
By means of this result, the stability of some fuzzy reasoning
schemes is obtained when , , and of the input
are all perturbed.
3) Ying’s Work: Ying introduced the concept of maximum
perturbation of fuzzy set in [14] as follows.
andbe fuzzy sets on and. If for eachLet,, thenis called a maximum
perturbation of.
One of the main results in [14] was to obtain the maximumof the output
when
perturbation
andof the input
have the maxall of ,imum perturbations. See [14] for more details.
Evidently, the stability of some fuzzy reasoning schemes may
be precisely characterized by the Ying’s result.
4) Cai’s Work: Cai used the term “ -equality” in [15] asfollows.
andbe fuzzy sets on . Then,andare saidLet,.
to be -equal, if
Cai investigated -equalities for some implication operators,-conorm, fuzzy relations and generalized modus pollens in[15].
The stability and instability of some fuzzy reasoning schemes
are easily addressed by the Cai’s results.
C. Two Lemmas
Lemma 1 can be easily established.
Lemma 1: Let and be bounded, real-valued functions defined on (or ), and fuzzy sets on . Then, the following
properties hold:
;
1)
;
2)
;
3)
;
4)
;
5)
;
6)
,
7)
;

CHENG AND FU: ERROR ESTIMATION OF PERTURBATIONS UNDER CRI

711

8)

,
.
If some implication operators are perturbed, one gets thefollowing.
Lemma 2: Let be a fuzzy set on , a fuzzy set on .
, and are factors of perturbation of and , respectively.
stand for
Let
Then, the following inequalities hold.
is Mamdani implication, then
1) If

2) If

is Kleene–Dienes implication, then

3) If

is Lukasiewicz implication, then

4) If

is Reichenbach implication, then5) If

is Zadeh implication, then

6) If

is

in [16], thenAgain by 3) of Lemma 1, one has thatand that

Therefore by 1)–3) of Lemma 1, one has the equation shownat the bottom of the page, and that

Proof: We only prove 5). The proofs of the other cases are,, one has, by 2) of Lemma 1, thatsimilar. For
It follows that

III. SIMPLE PERTURBATIONIn this section, when the input values are simply perturbed,
we estimate the upper and lower bounds of the output values in712IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006

fuzzy reasoning according to some choice of conjunction andimplication operators.
be, whereLetandare the perturbations of fuzzy sets and on , andis the perturbation of the fuzzy set
on . For simplicity,
,
,
, and
we denote
by
,
,
and
, respectively.
The next three theorems describe the main results.
Theorem 1: Let and be fuzzy sets on , and a fuzzy
and are perturbations of , andset on . Suppose ,with factors , and , respectively. If c isthen thefollowing properties hold.
is Mamdani implication, then1) IfBy 3) of Lemma 2, one has2) If

is Kleene–Dienes implication, then

3) If

is Lukasiewiz implication, then

4) If

is Reichenbach implication, then

5) If

6) If

7) If

Obviously,. We are done by applying (7)and (8) of Lemma.
When c is product, we have the following results.
Theorem 2: Let c be product. The other conditions are thesame as in Theorem 1. Then, the following inequalities hold.
is Mamdani implication, then
1) If

2) If

is Kleene–Dienes implication, then

3) If

is Lukasiewiz implication, then

4) If

is Reichenbach implication, then5) If

is Zadeh implication, then

6) If

is

is Zadeh implication, thenisin [16], thenis Zadeh implication, then

Proof: We only provide the proof of 3). The proofs of theother cases are similar. By 4)–6) of Lemma 1, one obtains that

and

and

as defined in [16], then

Proof: We only give the proof of 2). To start with, one hasby 2) of Lemma 2 that

CHENG AND FU: ERROR ESTIMATION OF PERTURBATIONS UNDER CRI

713

6) If

is

as defined in [16], then

Then
Proof: Now, we only verify that 4) holds. Since the equation shown at the bottom of the page holds, one has
and. Therefore

The rest of the proof of is similar.
If c is the Lukasiewicz conjunction, then we have the following result.
Theorem 3: Let c be Lukasiewicz conjunction. The otherconditions are the same as in Theorem 1. Then, the following
properties hold.
is Mamdani implication, then
1) If

2) If

is Kleene–Dienes implication, thenIt is clear that

. ThusWe omit the details of the rest of the proof of 4) because ofsimilarity.
The proofs of the other assertions are similar.
Let ,be fuzzy sets on ,a fuzzy set on . Suppose that there exists a perturbation sequenceof an input
with a corresponding sequenceof factors in some fuzzy reasoning,,
satisfyscheme. If
where

3) If

or

is Lukasiewiz implication, thenThen, by previous theorems, we have

4) If

5) If

is Reichenbach implication, thenis Zadeh implication, thenccOn the other hand, if there are a small positive real number ,fororand mappings
such that, where, and, then714

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006

Now it is clear that the aformentioned fuzzy reasoning schemes
using Zadeh’s CRI methods are stable. Their asymptotic performance has been demonstrated.

are gradually contracted, i.e., there exists a fuzzy interval perturof the input
bation sequencesuch that

IV. INTERVAL PERTURBATIONc
In this section,, where c is some -norm or -conorm, andis one-implications, or -normof -implications, -implications,
implications. We give the estimations of the limits of the output
with the interval perturbation of the input in CRI methods.
be fuzzy intervals on ,a fuzzyLetandstand for,interval on ,,,, respectively. The
main results are as follows.
,are
Theorem 4: Suppose thatis a fuzzy set on , and c is
fuzzy sets on ,either a -norm or a -conorm.
1) If
is one of -implications, -implications, then
c
c
2) If

is an

-implication, thenccwhere ,
and are, respectively, a -norm, a -conorm
and a fuzzy complement.
is a -norm
implication, then3) Ifchold, then the output values converge stably to some value.
On the other hand, the fuzzy reasoning schemes using
Zadeh’s CRI methods with c ( -norm or -conorm) and some
-implications are not stable.
V. CONCLUSION
We know that a fuzzy reasoning scheme applied to practiceis probably perturbed by “noises” in various ways. In addition
to many proximity measures of fuzzy sets, the simple perturbation and interval perturbation may effectively simulate such“noises” as well. In fact, Ying’s maximum perturbation of a
fuzzy set [14] is a simple perturbation, some error estimationof the output values in fuzzy reasoning is more precise using
the approach of this correspondence. Pappis’ proximity measure of two fuzzy sets [12], Hong and Hwang’s -similarity of
two fuzzy sets [13], Cai’s -equality of two fuzzy sets [15] andYing’s maximum perturbation of a fuzzy set are all formulated by the interval perturbation of the fuzzy sets. Therefore,
in certain sense this correspondence is a further development of
the previous work. On the other hand, we take into account theeffects of realistic noise and accurately evaluate the output errors of fuzzy reasoning. Therefore, we may choose a fuzzy reasoning scheme according to the requirement of the output errors
in applications.

c
Proof:
1) Using the properties of -implications,and -norms, or -conorms, we have

ACKNOWLEDGMENT-implications,

The authors would like to thank the referees for the invaluablecomments and suggestions. The proof of 1) in Theorem 4 is dueto the anonymous referee.

c
c
c

c
c
The inequalities of 2) and 3) are verified directly from themonotonicity of the -norm, -conorm, and the fuzzy complement . We omit the details.
When the -norm, -conorm, and the fuzzy complement arecontinuous in Theorem 4, the fuzzy reasoning scheme using theZadeh’s CRI method with c (some -norm or -conorm) and-implication, or a -norm implicaan -implication or antion is stable. When the oscillation limits of an input
in some fuzzy reasoning scheme using Zadeh’s CRI methods

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Guosheng Cheng received the B.S. degree from
Huaibei Coal Industry Teacher’s College, Huaibei,
China, in 1986, and the M.S. and Ph.D. degrees fromXi’an Jiaotong University, Xi’an, China, in 1996 and2000, respectively.
He is a Professor in the Department of Mathematics, Nanjing University of Information Scienceand Technology, Nanjing, China. His researchinterests include intelligent computation and computability of the measures.

Yuxi Fu received the B.S. degree from Tongji University, Shangai, China, and the Ph.D. degree fromthe University of Manchester, Manchester, U.K., in1986 and 1992, respectively.
He is a Full Professor with the Department ofComputer Science, Shanghai Jiaotong University,
Shangai, China. His research interest lies in theoretical computer science, especially in type theory andconcurrency theory. His recent work has focusedon the studies of process calculi, such as pi-calculus
and chi-calculus.