A First Course in FUZZY and NEURAL CONTROL

3.4. SENSITIVITY OF FUNCTIONS 107

Example 3.5Here are some examples for logical connectives on[0,1].

Connective Average sensitivity

∧or∨ S(∧)=S(∨)=1

x 4 y=xy S( 4 )=^23

x 4 y=x+y−xy S( 4 )=^23

x 4 y=(x+y)∧ 1 S( 4 )=1

x 4 y=0∨(x+y−1) S( 4 )=1

α(x)=1−xS(α)=1

A t-norm and its dual t-conorm with respect toα(x)=1−xhave the
same average sensitivity. The functions∧and∨in the examples above are
differentiable at all points in the unit square except for the linex=y,sothere
is no problem calculating the integrals involved. In certain situations, one may
need to use more general notions of derivative.

Theorem 3.5The connectivesx 4 y=xy,x 5 y=x+y−xy,andα(x)
=1−xhave the smallest average sensitivity among t-norms, t-conorms, and
negations, respectively.

Proof.We need to show, for example, thatx 4 y=xyminimizes

Z 1

0

Z 1

0

√μ

∂ 4

∂x

∂ 2

+

μ

∂ 4

∂y

∂ 2!

dxdy

A standard fact from analysis is that 4 minimizes this expression if it satisfies
the Laplace equation

∂^24

∂x^2

+

∂^24

∂y^2

=0

and of course it does. Similar arguments apply in the other two cases.

As in the case of extreme measure of sensitivity, one can use the notion
of average sensitivity as a factor in choosing membership functions for fuzzy
concepts. When facing a fuzzy concept such as a linguistic label, one might have
a class of possible membership functions suitable for modeling the concept. The
membership function within this class that minimizes average sensitivity can be
a good choice.

For a membership functionA:[a,b]→[0,1], the average sensitivity is

S(A)=

1

(b−a)

Zb

a

μ

d

dx

A(x)

∂ 2

dx