On Zadeh’s Compositional Rule of
Inference ∗
Robert Fullér
rfuller@abo.fi
Hans-Jürgen Zimmermann
zi@buggi.or.rwth-aachen.de
Abstract
This paper deals with Zadeh’s compositional rule of inference [7] under
triangular norms: IF X is P AND X and Y have relation R, THEN Y is Q,
where P and Q are fuzzy sets of the real line IR, R is a fuzzy relation on IR
and the conclusion Q is defined via sup-T composition of P and R:
πQ (y) = sup T (πP (x), πR (x, y)), y ∈ IR.
x∈IR
It is shown that (i) when the triangular norm T and the membership function
of the observation P are continuous, then the conclusion Q depends continuously on the observation; (ii) when T and the membership function of the
relation R are continuous, then the observation Q has continuous membership
function. Furthermore, we present a similar result for the discrete case
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Preliminaries
A fuzzy set A with membership function πA : IR → [0, 1] = I is called fuzzy number if
A is normal, continuous,fuzzy convex and compactly supported. The fuzzy numbers
will represent the continuous possibility distributions for fuzzy terms.
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 |πA (u) − πA (v)|.
|u−v|≤θ
An α-level set of a fuzzy interval A is a non-fuzzy set denoted by [A]α and is defined
by
[A]α = {t ∈ IR | πA (t) ≥ α}
for α ∈ (0, 1] and [A]α = cl(suppπA ) for α = 0.
We metricize F by the metric [4],
D(A, B) = sup d([A]α , [B]α ),
α∈[0,1]
∗
The final version of this paper appeared in: R.Lowen and M.Roubens eds., Proceedings of the
Fourth IFSA Congress, Vol. Artifical intelligence, Brussels, 1991 41-44.
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where d denotes the classical Hausdorff metric in the family of compact subsets of
IR2 , i.e.
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)].
When the fuzzy sets A and B have finite support {x1 , . . . , xn }, then their Hamming
distance is defined as
H(A, B) =
n
X
|πA (xi ) − πB (xi )|.
i=1
In the sequel we need the following lemma.
Lemma 1 [5] Let δ ≥ 0 be a real number and let A, B be fuzzy intervals. If
D(A, B) ≤ δ,
then
sup |πA (t) − πB (t)| ≤ max{ωA (δ), ωB (δ)}.
t∈IR
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Stability and continuity properties of the compositional rule of inference
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 0
X and Y have relation R
Y has property Q0
According to the compositional rule of inference the membership functions of the
conclusions are computed as
πQ (y) = sup T (πP (x), πR (x, y)),
πQ0 (y) = sup T (πP 0 (x), πR (x, y)).
x∈IR
(1)
x∈IR
The following theorem shows that when the observations are closed to each other in
the metric D, then there can be only a small deviation in the membership functions
of the conclusions.
Theorem 1 (Stability theorem) Let δ ≥ 0 and T be a continuous triangular norm,
and let P , P 0 be fuzzy intervals. If
D(P, P 0 ) ≤ δ
then
sup |πQ (y) − πQ0 (y)| ≤ ωT (max{ωP (δ), ωP 0 (δ)}).
y∈IR
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Proof. Let y ∈ IR be arbitrarily fixed. From Lemma 2.1. it follows that
|πQ (y) − πQ0 (y)| = | sup T (πP (x), πR (x, y)) − sup T (πP (x), πR (x, y))| ≤
x∈IR
x∈IR
sup |T (πP (x), πR (x, y)) − T (πP 0 (x), πR (x, y))| ≤ sup ωT (|πP (x) − πP 0 (x)|) ≤
x∈IR
x∈IR
sup ωT (max{ωP (δ), ωP 0 (δ)}) = ωT (max{ωP (δ), ωP 0 (δ)}).
x∈IR
Which proves the theorem.
Theorem 2 (Continuity theorem) Let R be continuous fuzzy relation, and let T be
a continuous t-norm. 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
|πQ (u) − πQ (v)| = | sup T (πP (x), πR (x, u)) − sup T (πP (x), πR (x, v))| ≤
x∈IR
x∈IR
sup |T (πP (x), πR (x, u)) − T (πP (x), πR (x, v))| ≤ sup ωT (|πR (x, u) − πR (x, v)|) ≤
x∈IR
x∈IR
sup ωT (ωR (|u − v|)) = ωT (ωR (|u − v|)) ≤ ωT (ωR (δ)).
x∈IR
Which ends the proof.
Remark 1 From
lim ω(δ) = 0
δ→0
and Theorem 1 it follows that
sup |πQ (x) − πQ0 (x)| → 0 as D(Q, Q0 ) → 0
x∈IR
which means the stability of the conclusion under small changes of the observation.
Other results along this line have appeared in [1, 2, 3, 7, 8].
Consider now the case when P and R are finite.
The following theorem shows that the stability property of the conclusion remains
valid in the discret case.
Theorem 3 Let T be a continuous t-norm. If the observation P and the relation
matrix R are finite, then
H(Q, Q0 ) ≤ ωT (H(P, P 0 ))
(2)
where H denotes the Hamming distance and the conclusions Q and Q0 are computed
by (1).
The proof of this theorem is carried out analogously to the proof of Theorem 1.
It should be noted that in the case of T (u, v) = min{u, v} (2) yields
H(Q, Q0 ) ≤ H(P, P 0 ).
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References
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