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decision processes involving incomplete or uncertain data. The concept in question
will be called fuzzy algorithm because it may be viewed as a generalization, through
the process of fuzzification, of the conventional (nonfuzzy) conception of an algo-
rithm. [106, p. 94] To illustrate, fuzzy algorithms may contain fuzzy instructions
such as: (a) 'Set
y
approximately equal to 10 if
x
is approximately equal to 5,' or
(b) 'If
x
is large, increase
y
by several units,' or (c) 'If
x
is large, increase
y
by sev-
eral units; if
x
is small, decrease
y
by several units; otherwise keep
y
unchanged.'
The sources of fuzziness in these instructions are fuzzy sets which are identified
by their underlined names. [106, p. 94f] All people function according to fuzzy
algorithms in their daily life, Zadeh wrote - they use recipes for cooking, consult
the instruction manual to fix a TV, follow prescriptions to treat illnesses or heed the
appropriate guidance to park a car. Even though activities like this are not normally
called algorithms: “For our point of view, however, they may be regarded as very
crude forms of fuzzy algorithms.” [106, p. 95f]
One year later, in “Toward a Theory of Fuzzy Systems”, written in 1969, Zadeh
clarified “Roughly speaking, a fuzzy algorithm is an algorithm in which some of the
instructions are fuzzy in nature. Examples of such instructions are:
(a) increase
x
slightly if
y
is slightly larger than 10;
(b) decrease
u
until it becomes much smaller than
v
;
(c) reduce speed if the road is slippery.”
“More generally, we may view a fuzzy algorithm as a fuzzy system
A
characterized
by equations of the form:
X
t
+
1
X
t
U
t
U
t
=
(
,
)
,
=
(
)
,
F
H
X
(3.14)
where
X
t
is a fuzzy state of
A
at time
t
,
U
t
is a fuzzy input (representing a fuzzy
instruction) at time
t
,and
X
t
+
1
1 resulting from the
execution of the fuzzy instruction represented by
U
t
. .... The function
F
defines
the dependence of the fuzzy state at time
t
is the fuzzy state at time
t
+
1 on the fuzzy state at time
t
and the
fuzzy input at time
t
, whereas the function
H
describes the dependence of the fuzzy
input at time
t
on the fuzzy state at time
t
.
To illustrate ..., we shall consider a very simple example. Suppose that
X
t
+
is
and
U
t
a fuzzy subset of a finite set
X
=
{
α
1
,
α
2
,
α
3
,
α
4
}
is a fuzzy subset of a
. Since the membership functions of
X
t
and
U
t
are mappings
from, respectively,
X
and
U
to the unit interval, these functions can be represented
as points in unit hypercubes
R
4
finite set
U
=
{
β
1
,
β
2
}
and
R
2
, which we shall denote for convenience
by
C
4
and
C
2
.
Thus,
F
may be defined by a mapping from
C
4
C
2
to
C
4
×
and
H
by a mapping from
C
4
to
C
2
.
For example, if the membership function of
X
t
and that of
U
t
is represented by the vector
(
0
.
5
,
0
.
8
,
1
,
0
.
6
)
by the vector
(
1
,
0
.
2
)
,
then the membership function of
X
t
+
1 would be defined by
F
as a vector say
whereas that of
U
t
(
0
.
2
,
1
,
0
.
8
,
0
.
4
)
would be defined by
H
as a vector
(
0
.
3
,
1
)
,say.”
([109], p. 485f)
In Lotfi Zadeh's opus we don't find another remark on a connection of fuzzy sets
and hypercubes even though he pointed out in 1969 that there exists a representation
