The Compositional Rule of Inference:
Introduction, theoretical considerations,
and exact calculation formulas∗
Robert Fullér
rfuller@ra.abo.fi
Brigitte Werners
brigitte.werners@rz.ruhr-uni-bochum.de
Abstract
This paper provides a short introduction to the compositional rule of
inference (Zadeh 1973), which is the mainly used inference rule in approximate reasoning. Stability results are given and exact computational formulas
are provided.
Keywords: Compositional rule of inference, fuzzy relation, fuzzy set, fuzzy interval, triangular norm, extension principle
Contents
1. Introduction
2. Preliminaries
3. Stability and continuity
4. Computational formulas
5. Applications
6. References
∗
Working Paper, RWTH Aachen, institut für Wirtschaftswissenschaften, No.1991/7.
1
1
Introduction
During the past ten years, expert systems have drawn tremendous attention from
researchers and practitioners working in the area of fuzzy information processing. At the same time, approximate reasoning gained importance, especially, after
L.A.Zadeh (1983) published ”The role of fuzzy logic in the management of uncertainty in expert systems”.
Informally, approximate reasoning can be viewed as a process by which a possible
imprecise conclusion is deduced from a collection of imprecise premises. When it
is applied to rule-based expert systems, there are two basis issues to be concerned
with:
(i) The use of linguistic variables (see e.g. Werners 1990) in the representation of
experts’ knowledge or rules.
(ii) The deduction of conclusions from observations and rules in a knowledge
base. There are two alternative ways of doing this: Truth values restriction; compositional rule of inference. The latter has been more widely accepted and applied
in development studies (see. e.g. Driankov 1987, Dubois and Prade 1988, 1991,
Gupta and Qi 1991, Hellendoorn 1991, Margrez and Smets 1989, Di Nola, Sessa
and Pedrycz 1985, Turksen 1989, Yager 1984, Zimmermann 1988)
This paper deals with the compositional rule of inference which has the global
scheme
Observation: X has property P
Relation:
X and Y are in relation W
Conclusion: Y has property Q
where X and Y are variables taking their values from fuzzy sets in classical sets U
and V , respectively, P and Q are unary fuzzy predicates in U and V , respectively,
W is a binary fuzzy relation in U ×V . The membership function of the conclusion
Q, denoted by µQ , is determined via a sup-t-norm composition of P and W as
µQ (y) = sup T (µP (x), µW (x, y)),
x∈U
where T is a triangular norm (see next Section).
An example of this rule is the inference (see Section 5)
Observation:
Relation:
Conclusion:
X is close to 3
X and Y are approximately equal
Y is more or less close to 3
2
A very important special case of the global scheme is the generalized modus ponens, (in which the implication operator is considered as a special fuzzy relation)
which reads
Observation:
X has property P
Relation (Rule): If ’X has property P’ then ’Y has property Q’
Conclusion:
Y has property Q
The relationship between two variables X and Y can rarely be represented through
a single relation W. We need several relations because one rule represents just one
sample of this relationship.
The compositional rule of inference scheme with several relations can be stated
as follows
Observation: X has property P
Relation 1:
X and Y are in relation W1
..
.
Relation m:
Conclusion
X and Y are in relation Wm
Y has property Q
where
Q=
m
P ◦ Wi
i=1
or in detail,
µQ (y) = min sup T (µP (x), µWi (x, y)),
i=1,...,m x∈U
i.e. the conclusion is obtained by triggering the antecedents separately, and combining the partial results in the second step (for other approaches, see Gupta and
Qi 1991, Dubois at al. 1988, Magrez and Smets 1989).
An example of the compositional rule of inference with several relations is the
inference (see Section 5)
Observation: X is close to 3
Relation 1:
X and Y are approximately equal
Relation 2:
Y is essentially smaller than X
Conclusion: Y is not far from 2
The goal of this paper is (i) to provide exact calculation formulas for the compositional rule of inference under Archimedean t-norms, when both the observation
3
and the relation parts are given by Hellendoorn’s (1990) φ-function; (ii) to show
some important properties of the compositional rule of inference under t-norms.
In Section 2 we set up the notations and present some lemmas needed in order to
determine the exact calculation formulas and to prove the stability and continuity
properties of the compositional rule of inference under t-norms.
In Section 3 we show that (i) if the t-norm and the membership function of the
observation P are continuous, then the conclusion Q depends continuously on
the observation; (ii) if the t-norm and the membership function of the relation W
are continuous, then the observation Q has a continuous membership function.
Furthermore, we present a similar result for the discrete case.
An important problem is the (approximate) computation of the membership function of the conclusion in these schemes (Da 1990, Dubois at al. 1988, Hellendoorn
1990, R.Martin-Clouarie 1989, Mizumoto and Zimmermann 1982).
Hellendoorn (1990) showed the closure property of the compositional rule of inference under sup-min composition and presented exact calculation formulas for
the membership function of the conclusion when both the observation and relation
parts are given by S-, π-, (Zadeh 1975) or φ-functions.
Our results in Section 4 are connected with those presented by Hellendoorn (1990)
and we generalize them. We shall determine the exact membership function of the
conclusion, when both the observation and the part of the relation (rule) are given
by concave φ-function; and the t-norm is Archimedean with a strictly convex
additive generator function. We present similar results for the compositional rule
of inference with several relations.
The efficiency of our method stems from the fact that the membership functions,
involved in the relation and observation, are represented by a parametrized φfunction. The deduction process then consists of some simple computations performed on the parameters.
2
Preliminaries
In this Section we set up the notations and present some lemmas needed in order
to calculate the exact calculation formulas and to prove the stability and continuity
properties of the compositional rule of inference under t-norms.
Definition 2.1 Let X be a classical set. A fuzzy set A (Zadeh 1965) in the universe
4
X is defined by its membership function mapping X to the unit interval, i.e.
µA : X → [0, 1].
The concept of a membership function generalizes the notation of the characteristic function of a conventional set. For an element x ∈ X, the value µA (x)
represents the degree of membership of x in the fuzzy set A. Unlike conventional
sets where elements either do belong or do not belong to a set, fuzzy sets admit
partial membership ranging from 0 (non-membership) to 1 (full-membership). We
denote by F(X) the set of fuzzy subsets of X.
A fuzzy set A of the real line IR is called a fuzzy quantity and instead of F(IR) we
simple write F.
The usual set-theoretic operations can be extended in a straight- forward way by
fuzzy logic to fuzzy sets. We obtain
µA∩B (x) = min{µA (x), µB (x)} (intersection)
µA∪B (x) = max{µA (x), µB (x)} (union)
µco(A) (x) = 1 − µA (x) (complement),
for all A, B ∈ F(X) and x ∈ X.
We say that A is a subset of B, written A ⊂ B, if µA (x) ≤ µB (x) holds for all
x ∈ X. A fuzzy set A is normalized if ∃ x ∈ X, such that µA (x) = 1.
Definition 2.2 Let A be a fuzzy set in X. The α-level set of the fuzzy set A is a
classical subset of X, denoted by [A]α , and defined by
[A]α = {x ∈ X|µA (x) ≥ α}
if α > 0 and [A]α := supp(µA ) := {x ∈ X|µA (x) > 0} if α = 0.
Definition 2.3 A fuzzy set A is called convex (or fuzzy-convex) if [A]α is a convex
subset of IR for all α ∈ [0, 1].
Definition 2.4 A fuzzy interval A is a fuzzy quantity with a continuous, finitesupported, fuzzy-convex and normalized membership function µA : IR → [0, 1].
5
It is well-known (Dubois and Prade 1980), that any fuzzy interval A can be described by the following membership function:
 a
−
t


 L
if t ∈ [a − α, a]


α




 1
if t ∈ [a, b]
µA (t) =

t − b)



if t ∈ [b, b + β]
R


β




0
otherwise
where [a, b] is the peak of A; α and β are the lower and upper modal values; L and
R are shape functions: [0, 1] → [0, 1], with L(0) = R(0) = 1 and L(1) = R(1) =
0 which are non-increasing, continuous mappings.
These fuzzy intervals are called of type L-R and denoted by A = (a, b, α, β)LR .
The support of A is (a − α, b + β).
A fuzzy interval A is called fuzzy number if there exists a unique a ∈ IR such
that µA (a) = 1. If A is a fuzzy number, then instead of (a, a, α, β)LR we write
(a, α, β)LR .
It should be noted that some authors do not differ fuzzy number from fuzzy interval.
Fuzzy intervals and numbers are often used to represent linguistic variables.
Following Hellendoorn (1990) we use the φ-function for the representation of
linguistic terms


1
if b ≤ x ≤ c




x−a



if a ≤ x ≤ b, a < b,
 φ1
b−c
φ(x; a, b, c, d) =

x−c


φ2
if c ≤ x ≤ d, c < d,



d−c


 0
otherwise
where φ1 : [0, 1] → [0, 1] is continuous, monoton increasing function and φ1 (0) =
0, φ1 (1) = 1; φ2 : [0, 1] → [0, 1] is continuous, monoton decreasing function
and φ2 (0) = 1, φ2 (1) = 0 So φ is a function which is 0 left of a, increases
to 1 in (a, b), is 1 in [b, c], decreases to 0 in (c, d) and is 0 right of d (for the
6
sake of simplicity, we do not consider the cases a = b or c = d). It should be
noted that φ can be considered as the membership function of the fuzzy interval
ã = (b, c, b − a, d − c)LR , with R(x) = φ2 (x) and L(x) = φ1 (1 − x).
Definition 2.5 Let A be a fuzzy number, then for any θ > 0 we define ωA (θ), the
modulus of continuity of A by
ωA (θ) = max | µA (u) − µA (v) | .
|u−v|≤θ
The following statements hold (Henrici, 1962):
If 0 ≤ θ ≤ θ then ω(ã, θ) ≤ ω(ã, θ )
If α > 0, β > 0, then ω(ã, α + β) ≤ ω(ã, α) + ω(ã, β).
lim ω(ã, θ) = 0
θ→0
Definition 2.6 We metricize the set of fuzzy numbers by the metric (Kaleva, 1987)
D(ã, b̃) = sup d([A]α , [B]α ),
α∈[0,1]
where
d([A]α , [B]α ) = max{| a1 (α) − b1 (α) |, | a2 (α) − b2 (α) |},
and
[A]α = [a1 (α), a2 (α)], [B]α = [b1 (α), b2 (α)], α ∈ [0, 1].
Definition 2.7 Let A and B be fuzzy sets in X. If
supp(µA ) = supp(µB ) = {x1 , · · · , xn },
then their Hamming distance is defined as
H(A, B) =
n
| µA (xi ) − µB (xi ) |
i=1
7
(1)
Definition 2.8 A triangular norm (Schweizer and Sklar, 1963) is a two-place
function from the closed unit square [0, 1] × [0, 1] to the closed unit interval [0, 1]
which satisfies the following conditions
Boundary conditions
T (0, 0) = 0, T (a, 1) = T (1, a) = a.
Monotonicity
T (a, b) ≤ T (c, d) whenever a ≤ c, b ≤ d.
Symmetry
T (a, b) = T (b, a).
Associativity
T (T (a, b), c) = T (a, T (b, c)).
Some examples for t-norms:
(i) Minimum norm
T (u, v) = min{u, v}
(ii) Algebraic product
T (u, v) = uv
(iii) Bounded product
T (u, v) = max{0, u + v − 1}
(iv) Drastic product (or weak t-norm)
T (u, v) =
min{a, b} if max{a, b} = 1
0
otherwise
(v) Einstein product
T (u, v) =
uv
1 + (1 − u)(1 − v)
Recall that a t-norm T is Archimedean iff T is continuous and T (x, x) < x for all
x ∈ (0, 1).
8
Every Archimedean t-norm T is representable by a continuous and decreasing
function f : [0, 1] → [0, ∞] with f (1) = 0 and
T (x, y) = f [−1] (f (x) + f (y))
where f [−1] is the pseudo-inverse of f , defined by
−1
f (y) if y ∈ [0, f (0)]
[−1]
f (y) =
0
otherwise
The function f is the additive generator of T .
Some examples for parametrized Archimedean t-norms (see e.g. Mizumoto 1989):
(i) Yager’s t-norm (p > 0)
T (u, v)1 − min{1, [(1 − a)p + (1 − b)p ]1/p }, with generatorf (t) = (1 − t)p
(ii) Hamacher’s t-norm (p > 0)
T (u, v) =
uv
p + (1 − p)(u + v − uv)
having additive generator
f (t) = ln
p + (1 − p)t
t
(iii) Dombi’s t-norm (p > 0)
T (u, v) =
1+
1
p
(1/u − 1)p + (1/v − 1)p
with additive generator
f (t) =
9
p
1
−1 .
t
One of the basic ideas of fuzzy set theory, which provides a general extension of
nonfuzzy mathematical concepts to fuzzy environments, is the extension principle
(Zadeh 1965).
Suppose that f is a mapping from X to Y and A is a fuzzy set of X. Then the
extension principle allows us to induce from A a fuzzy set f (A) ∈ F(Y ) such
that
supx∈f −1 (y) µA (x) if f −1 (y) = ∅
µf (A)(y) =
0
otherwise
where f −1 (y) = {x ∈ X | f (x) = y}. This is a basic identity that allows the
domain of the definition of a mapping or relation to be extended from points in X
to fuzzy sets of X.
If we have an n-ary function f, which is a mapping from the Cartesian product
X1 × · · · × Xn to a universe Y such that y = f (x1 , . . . , xn ), and A1 , . . . , An ,
which are n fuzzy sets in X1 , . . . , Xn , respectively, then the extension principle
reads
sup{min{µA1 (x1 ), . . . , µAn (xn )} | x ∈ f −1 (y)} if f −1 (y) = ∅
µf (A1 ,...,An ) (y) =
0
otherwise
or more generally,
sup{T (µA1 (x1 ), . . . , µAn (xn )) | x ∈ f −1 (y)} if f −1 (y) = ∅
µf (A1 ,...,An ) (y) =
0
otherwise
where T is a triangular norm.
If T is a t-norm and ã1 , ã2 are fuzzy sets of the real line (i.e. fuzzy quantities) then
their T -sum à := ã1 + ã2 is defined by
Ã(z) =
sup T (ã1 (x1 ), ã2 (x2 )), z ∈ IR
x1 +x2 =z
which can be written in the form
Ã(z) = f [−1] (f (ã1 (x1 )) + f (ã2 (x2 )))
supposing that f is the additive generator of T .
Since f is continuous and decreasing, f [−1] is also continuous and non-increasing,
we have
[−1]
Ã(z) = f
inf (f (ã1 (x1 )) + f (ã2 (x2 )) .
x1 +x2 =z
10
Now, we present two lemmas needed in order to calculate the exact calculation
formulas and to prove the stability and continuity properties of the compositional
rule of inference under t-norms.
Lemma 2.1 (Fullér and Keresztfalvi, 1991)
Let T be an Archimedean t-norm with additive generator f and let ãi = (ai , bi , α, β)LR , i =
1, 2 be fuzzy intervals of type L-R. If L and R are twice differentiable, concave
functions, and f is a twice differentiable, strictly convex function, then the membership function of their T-sum à = ã1 + ã2 is

1
if A ≤ z ≤ B





A−z



f [−1] 2 × f L
if A − 2α ≤ z ≤ A


2α
Ã(z) =

z
−
B



f [−1] 2 × f R
if B ≤ z ≤ B + 2β


2β




0
otherwise
where A = a1 + a2 and B = b1 + b2 .
The proof of this lemma is based on the decomposition principle of fuzzy sets into
two separated parts (Dubois and Prade 1981).
Lemma 2.2 (Fedrizzi and Fullér 1991)
Let δ > 0 be a real number and let A, B be fuzzy numbers. If
D(A, B) ≤ δ
then
sup | µA (t) − µB (t) |≤ max{ωA (δ), ωB (δ)},
t∈IR
where D denotes the generalized Hausdorff metric for fuzzy numbers (2); and ωA ,
ωB denote the modulus of continuity of A, B, respectively.
Lemma 2.2. shows that if all α-level sets of two fuzzy numbers are close to each
other, then there can be only a small deviation between their membership grades.
11
3
The compositional rule of inference: stability and
continuity
In this section we show two very important features of the compositional rule of
inference under triangular norms. Namely, we prove that, (i) if the t-norm defining
the composition and the membership function of the observation are continuous,
then the conclusion depends continuously on the observation; (ii) if the t-norm
and the membership function of the relation are continuous, then the observation
has a continuous membership function.
Furthermore, we present a similar result for the discrete case.
We consider the compositional rule of inference with different observations P and
P :
Observation:
Relation:
Conclusion:
X has property P
X and Y have relation R
Y has property Q
X has property P X and Y have relation R
Y has property Q
According to Zadeh’s compositional rule of inference, Q and Q are computed as
Q = P ◦ R
Q = P ◦ R,
i.e.,
µQ (y) = sup T (µP (x), µR (x, y)),
µQ (y) = sup T (µP (x), µR (x, y)).
x∈IR
x∈IR
The following two theorems (Fullér and Zimmermann 1991a) show that when the
observations are close to each other in the metric D, then there can be only a small
deviation in the membership function of the conclusions.
Theorem 3.1 Let δ ≥ 0 and T be a continuous triangular norm, and let P , P be
fuzzy intervals. If
D(P, P ) ≤ δ
then
sup |µQ (y) − µQ (y)| ≤ ωT (max{ωP (δ), ωP (δ)}).
y∈IR
Proof. Let y ∈ IR be arbitrarily fixed. From Lemma 2.1. it follows that
|µQ (y) − µQ (y)| = | sup T (µP (x), µR (x, y)) − sup T (µP (x), µR (x, y))| ≤
x∈IR
x∈IR
12
sup |T (µP (x), µR (x, y)) − T (µP (x), µR (x, y))| ≤ sup ωT (|µP (x) − µP (x)|) ≤
x∈IR
x∈IR
sup ωT (max{ωP (δ), ωP (δ)}) = ωT (max{ωP (δ), ωP (δ)}).
x∈IR
Which proves the theorem.
It is easy to see that Theorem 3.1. remains valid for all continuous fuzzy sets
representing the observation. Namely, we have the following result.
Theorem 3.2 Let δ > 0 and T be a continuous triangular norm, and let P , P be
continuous fuzzy sets. If
||P − P ||∞ = sup | µP (x) − µP (x) |≤ δ
xinIR
then
sup |µQ (y) − µQ (y)| ≤ ωT (max{ωP (δ), ωP (δ)}).
y∈IR
Theorem 3.2. is weaker than Theorem 3.1., because ||P − P ||∞ ≤ 1, for each
fuzzy sets P and P’, but D(P,P’) can be arbitrary large (when they are fuzzy numbers).
In the following theorem we establish the continuity property of the conclusion
under continuous fuzzy relations and continuous t-norms.
Theorem 3.3 Let R be continuous fuzzy relation, and let T be a continuous tnorm. Then Q is continuous and
ωQ (δ) ≤ ωT (ωR (δ)) for each δ ≥ 0.
Proof. Let δ ≥ 0 be a real number and let u, v ∈ IR such that |u − v| ≤ δ. Then
|µQ (u) − µQ (v)| = | sup T (µP (x), µR (x, u)) − sup T (µP (x), µR (x, v))| ≤
x∈IR
x∈IR
sup |T (µP (x), µR (x, u))−T (µP (x), µR (x, v))| ≤ sup ωT (|µR (x, u)−µR (x, v)|) ≤
x∈IR
x∈IR
sup ωT (ωR (|u − v|)) = ωT (ωR (|u − v|)) ≤ ωT (ωR (δ)).
x∈IR
Which ends the proof.
13
Remark 3.1 From
lim ω(δ) = 0
δ→0
and Theorem 3.1. it follows that
sup | µQ (x) − µQ (x) |→ 0asD(P, P ) → 0
x∈IR
which means the stability of the conclusion under small changes of the observation.
The stability property of the conclusion under small changes of the membership
function of the observation guarantees that small rounding errors of digital computation and small errors of measurement of the input data can cause only a small
deviation in the conclusion, i.e. every successive approximation method can be
applied to the computation of the linguistic approximation of the exact conclusion.
Other results along this line have appeared in [J.B.Kiszka, M.M.Gupta, and P.N.Nikiforuk
1985, P.Z.Wang, H.M.Zhang and W.Xu 1990].
Consider now the case when P and W are finite. The following theorem shows
that the stability property of the conclusion remains valid for the discret case.
Theorem 3.4 Let T be a continuous t-norm. If the observation P and relation
matrix W are finite, then
H(Q, Q ) ≤ ωT (H(P, P )),
where Q and Q are the conclusions, and H denotes the Hamming distance (3).
The proof of this theorem is carried out analogously to the proof of Theorem 3.1.
It should be noted that in the case of T = min we get
H(Q, Q ) ≤ H(P, P ).
4
The compositional rule of inference: Computation formulas
An important problem is the (approximate) computation of the membership function of the conclusion (Da 1990, Dubois at al. 1988, Hellendoorn 1990, R.MartinClouarie 1989, Mizumoto and Zimmermann 1982).
14
Hellendoorn (1990) showed the closure property of the compositional rule of inference under sup-min composition and presented exact calculation formulas for
the membership function of the conclusion when both the observation and relation parts are given by S-, π-, or φ-function. Namely, he proved the following
theorem.
Theorem 4.1 In the compositional rule of inference under minimum norm,
Observation: X has property P
Relation:
X and Y are in relation W
Conclusion: Y has property Q
is true that, when µP (x) = φ(x; a1 , a2 , a3 , a4 ) and µW (x, y) = φ(y−x; b1 , b2 , b3 , b4 )
then
µQ (y) = φ(y; a1 + b1 , a2 + b2 , a3 + b3 , a4 + b4 ),
where the function φ is defined by (1).
Our results in this section are connected with those presented by Hellendoorn
(1990) and we generalize them. We shall determine the exact membership function of the conclusion, when both the observation and the part of the relation (rule)
are given by concave φ-function; and the t-norm is Archimedean with a strictly
convex additive generator function.
The efficiency of our method stems from the fact that the distributions, involved
in the relation and observation, are represented by a parametrized φ-function. The
deduction process then consists of some simple computations performed on the
parameters.
We consider the compositional rule of inference,
Observation: X has property P
Relation:
X and Y are in relation W
Conclusion: Y has property Q
where, the membership functions of P and W are defined by means of a particular
φ-function, and the membership function of the conclusion Q is defined by sup-T
composition of P and W
µQ (y) = (µP ◦ µW )(y) = sup T (µP (x), µW (x, y)).
x∈IR
15
The following theorem (Fullér and Zimmermann 1991) presents an efficient method
for the exact computation of the compositional rule of inference under Archimedean
t-norms.
Theorem 4.2 Let T be an Archimedean t-norm with additive generator f and let
µP (x) = φ(x; a, b, c, d), and µW (x, y) = φ(y − x; a + u, b + u, c + v, d + v).
If φ1 and φ2 are twice differentiable, concave functions, and f is a twice differentiable, strictly convex function, then

if 2b + u ≤ y ≤ 2c + v

 1 

y − 2a − u 


]
if 2a + u ≤ y ≤ 2b + u
 f [−1] 2f φ1 [
2(b − a)
µQ (y) =
y − 2c − v 

[−1]

2f
φ2 [
]
if 2c + v ≤ y ≤ 2d + v
f


2(d − c)



0
otherwise
Proof. Using Lemma 2.1. with
ã1 = (b, c, b − a, d − c)LR and ã2 = (b + u, c + v, b − a, d − c)LR ,
where R(x) = φ2 (x) and L(x) = φ1 (1 − x), we have
µQ (y) = sup T (µP (x), µW (x, y)) =
x∈IR
sup T ((φ(x; a, b, c, d), φ(y − x; a + u, b + u, c + v, d + v) = (ã1 + ã2 )(y).
x∈IR
Which proves the theorem.
Theorem 4.2. can be easily extended to the following inference scheme
Observation: X has property P
Relation 1:
X and Y are in relation W1
···
Relation m: X and Y are in relation Wm
Conclusion: Y has property Q
where
µQ (y) = min sup T (µP (x), µWi (x, y)).
i=1,...,m x∈U
Namely, we have the following result:
16
Theorem 4.3 Let T be an Archimedean t-norm with additive generator f and let
µP (x) = φ(x; a, b, c, d),
µWi (x, y) = φ(y − x; ai + ui , bi + ui , ci + vi , di + vi ),
i = 1, . . . , m. If φ1 and φ2 are twice differentiable, concave functions, and f is a
twice differentiable, strictly convex function, then
µQ (y) = min µQi (y),
i=1,...,m
where

1




y − 2a − u 

[−1]

2f φ1 [
]
 f
2(b − a)
µQ (y) =

y
−
2c
−
v


]
f [−1] 2f φ2 [


2(d − c)



0
if 2b + u ≤ y ≤ 2c + v
if 2a + u ≤ y ≤ 2b + u
if 2c + v ≤ y ≤ 2d + v
otherwise
Proof. The proof of this theorem is carried out analogously to the proof of Theorem 4.2.
Remark 4.1 It should be noted that we have calculated the membership function
of Q under the assumption that the left and right spreads of P do not differ from
the left and right spreads of W (the lengths of their tops can be different). To
determine the exact membership function of Q in the general case:
µP (x) = φ(x; a1 , a2 , a3 , a4 ) and µW (x, y) = φ(y − x; b1 , b2 , b3 , b4 ),
seems to be very difficult and it is possible, that there does not exist an explicit
formula for it.
5
Applications
In this Section, using Theorem 4.1., we compute the exact membership function
of the conclusion Q in the compositional rule of inference under Yager’s, Dombi’s
and Hamacher’s parametrized t-norms when both the observation and the part of
the relation (rule) are given by concave φ-functions.
Let us consider the following scheme,
17
µP (x)
= φ(x; a, b, c, d)
µW (y, x) = φ(y − x; a + u, b + u, c + v, d + v)
µQ (y)
= (P ◦T W )(y)
where
µQ (y) = (P ◦ W )(y) = sup T (µP (x), µW (x, y)).
x∈IR
Using the notations
σ :=
(y − 2a − u)
,
2(b − a)
θ :=
y − 2c − v
,
2(d − c)
we get the following formulas for the membership function of the conclusion Q.
(i) Yager’s t-norm with p > 1. Here
p
T (x, y) = 1 − min 1, (1 − x)p + (1 − y)p .
with generator f (t) = (1 − t)p , and

−1 −1/p
1/p
),

 1 − 2 (1 − φ1 (σ)) if 0 < σ < φ1 (2
1
if 2b + u ≤ y ≤ 2c + v,
Q(y) =


1/p
1 − 21/p (1 − φ2 (θ)) if 0 < θ < φ−1
),
2 (1 − 2
(ii) Hamacher’s t-norm with p ≤ 2. Here
T (x, y) =
xy
p + (1 − p)(x + y − xy)
with generator
f (t) = ln
and
p + (1 − p)t
,
t

2

 p/(τ1 − 1 + p) if 0 < σ < 1,
1
if 2b + u ≤ y ≤ 2c + v,
Q(y) =

 p/(τ 2
2 − 1 + p) if 0 < θ < 1,
where
τ1 =
p + (1 − p)φ1 (σ)
,
φ1 (σ)
18
τ2 =
p + (1 − p)φ2 (σ)
φ2 (σ)
1
3
5
x
Figure 1: ”x is close to [3, 4]”
W
-2
2
0
x-y
Figure 2: ”x and y are approximately equal”
(iii) Dombi’s t-norm with p > 1. Here
T (x, y) =
1+
p
1
(1/x − 1)p + (1/y − 1)p
with additive generator
f (t) =
and
p
1
−1 ,
t

1/p

 1/(1 + 2 τ3 ) if 0 < σ < 1,
1
if b + u ≤ y ≤ 2c + v,
Q(y) =


1/p
1/(1 + 2 τ4 ) if 0 < θ < 1,
where
τ3 =
1
− 1,
φ1 (σ)
τ4 =
1
−1
φ2 (σ)
We illustrate Theorem 4.1. by the following example:
Example 5.1 Consider the following compositional rule of inference scheme with
one relation
19
Q
6
3
y
Figure 3: ”y is more or less close to [3, 4]”, Yager’s t-norm.
x is close to 3
x and y are approximately equal
y is more or less close to [3, 4]
where

1



 (x − a)/(b − c)
φ(x; a, b, c, d) =

1 − (x − c)/(d − c)



0
φ(x; 1, 3, 4, 7)
φ(y − x; −2, 0, 0, 3)
Q(y)
if b ≤ x ≤ c
if a ≤ x ≤ b, a < b,
if c ≤ x ≤ d, c < d,
otherwise
We have used the membership function φ(y − x; −2, 0, 0, 3) to describe ”x and
y are approximately equal”. This means that the membership degree is one, iff x
and y are equal in the classical sense. If y − x > 2 or x − y > 3, then the degree
of membership is 0. The conclusion Q has been called ”y is more or less close to
[3, 4]”, because P (t) = Q(T ) = 1, when t ∈ [3, 4] and P (t) < Q(t) otherwise.
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