CREATING AND DECOMPOSING FUNCTIONS USING FUZZY
FUNCTIONS
J´ozsef D´aniel Dombi and J´ozsef Dombi
Institute of Informatics, University of Szeged,
´
Arp´ad t´er 2., Szeged, Hungary
Keywords:
Fuzzy System, Approximation, Membership function, Sigmoid function, Conjunctive operator, Aggregation
operator, Dombi operator, Pliant concept.
Abstract:
In this paper we will present a new approach for composing and decomposing functions. This technology is
based on the Pliant concept, which is a subclass of the fuzzy concept. We will create an effect by using the
conjunction of the sigmoid function. After, we will make a proper transformation of an effect in order to define
the neutral value. Then, by repeatedly applying this method, we can create effects. Aggregating the effects,
we can compose the desired function. This tool is also capable of function decomposition as well, and can be
used to solve a variety of real-time problems. Two advantages of our solution are that the parameters of our
algorithm have a semantic meaning, and that it is also possible to use the values of the parameter in any kind
of learning procedure.
1 INTRODUCTION
Functions have a very important role in science and
technology and in our everyday lives. They can be
represented in terms of their coordinates or by using
some mathematical expression. Usually if the coor-
dinates are given then it is important to know what
kind of expression approximately describes it, be-
cause sometimes interpolation or extrapolation ques-
tions have to be addressed. In science and technol-
ogy we can usually get samples to determine the rela-
tionship between the input and output values, which
is called curve fitting, and usually we do not require
an exact fit, but only an approximation. One way to
approximate a function with coordinates is via an in-
terpolation process. We can regard interpolation as
a specific kind of curve fitting, where the function
must go through the data points. It is also possible
to use neural networks to find a function approxima-
tion. Curve fitting can be done by minimising the
error function that measures the misfit between the
function for any given value of the parameters and the
data points. One simple and widely used error func-
tion is the sum of the squares of the errors, so in effect
we have to minimise the ’energy function’.
However, every type of curve fitting method has its
drawbacks and this one is no different. The main
problem is how to choose the order n of the polyno-
mial and this will turn out to be a problem of model
comparison or model selection. These methods are
not accurate enough. The parameters that we get after
optimisation give us no direct information about the
behaviour of the function. It would be useful if we
could modify a certain part of the function by vary-
ing the parameter. And it would be good if we could
characterise a function by its behaviour. Using classi-
cal function construction procedures, it is not so easy
to find a parametrical mathematical expression which
corresponds to the natural language description of the
function, but it would be useful in fields like eco-
nomics and marketing.
In this article we will present a solution that solves
some of these problems. We will introduce positive
and negative effects, whose mathematical descrip-
tion can be realised by using continuous-valued logic.
Here we will use a special one called the Pliant con-
cept, which uses the Dombi operator. After an ag-
gregative procedure we get the derived function. In-
stead of the membership function we shall use soft
inequalities and soft intervals which are called dis-
tending functions. All of the parameters introduced
have a definite meaning. It can be proved that certain
function classes may be uniformly approximated.In
the following section we will concentrate on a certain
structure called the Pliant concept for the construction
of the necessary operators.
400
Dániel Dombi J. and Dombi J. (2010).
CREATING AND DECOMPOSING FUNCTIONS USING FUZZY FUNCTIONS.
In Proceedings of the 12th International Conference on Enterprise Information Systems - Artificial Intelligence and Decision Support Systems, pages
400-403
DOI: 10.5220/0002901204000403
Copyright
c
 SciTePress

2 BASIC OPERATOR OF THE
PLIANT CONCEPT
Pliant conjunctive and disjunctive operators belong to
the strict t-norm and t-conorm classes.
2.1 Conjunctive and Disjunctive
Operators
Definition 1. c(x, y) and d(x, y) are strict monotone
logical operators if the following hold true:
1. Continuity
2. Monotonously increasing
3. Compatibility with logic
4. Associativity
5. Archimedian, which means
c(x, x) < x < d(x, x),where xε(0, 1)
Definition 2. (Acz
´
el) The operators with all the prop-
erties except property 3 can be written in the follow-
ing way:
c(x, y) = f
−1
c
( f
c
(x) + f
c
(y))
d(x, y) = f
−1
d
( f
d
(x) + f
d
(y)),
where f
c
(x) and f
d
(x) are the generator functions of
the operator, which is defined up to a multiplicative
constant.
In an article by Dombi (Dombi, 1982b) there are
sets of logical operators that have the above properties
(1).
Definition 3. An operator belongs to the Pliant sys-
tem if
f
c
(x) f
d
(x) = 1 (1)
A special case of the Pliant system is the Dombi
operator.
2.2 Distending Function
In pliant logic we use a soft inequality and we call it
the distending function.
Definition 4. The distending function:
δ
λ
a
(x) = f
−1

e
−λ(x−a)

λεR, aεR (2)
Here f is the generator function of the logical
connectives, λ is responsible for the sharpness and a
is the threshold value.
The semantic meaning of δ
(λ)
a
is
truth(a <
λ
x) = δ
(λ)
a
(x)
The distending function in the Dombi operator case is
the sigmoid (logistic) function.
2.3 Modeling Interval
In fuzzy logic the membership function is not an open
interval, but in most cases it is a soft interval. We have
to give a mathematical description of
truth(a <
λ
1
x <
λ
2
b)
Definition 5. The distending interval falls within in
the Dombi operator case:
σ
λ
1
,λ
2
a,b
(x) =
1
1+
1−ν
0
ν
0
1
A

A
1
e
−λ
1
(x−a)
+ A
2
e
−λ
2
(b−x)

,
where
A = 1− e
−(λ
1
+λ
2
)(b−a)
A
1
= 1− e
−λ
2
(b−a)
A
2
= 1− e
−λ
1
(b−a)
The following properties hold true for the distending
interval.
δ
a,b
(x) = lim
λ
1
→∞,λ
2
→∞
δ
λ
1
,λ
2
a,b
(x) =













0 if x < a
0.5 if x = a
1 if a < x < b
0.5 if x = b
0 if b < x
2.4 Aggregation Operator
The aggregation concept was first introduced in
(Dombi, 1982a), which is also called the uninorm.
Several articles discuss uninorms (J. Fodor and Ry-
balov, 1997), (Li and Shi, 2000), (D. Dubois, 2000),
(Yager., 2001), but here we will just focus on the ag-
gregation concept. Using Acz´el’s theorem (Acz´el,
1966) about an associative equation we get:
a(x, y) = f
−1
a
( f
a
(x) + f
a
(y)) (3)
Applying the Pliant concept and using the Dombi’s
generator function with multiple variables, we have:
a(x
1
, x
2
. . . x
n
) =
n
∏
i=1
x
i
n
∏
i=1
x
i
+
n
∏
i=1
(1− x
i
)
(4)
This operator can be found in (Dombi, 1982a).
Nowadays it is called the 3π operator for obvious rea-
sons. The main properties of the aggregation operator
are:
1. a(x, n(x)) = ν
∗
2. a(x, ν
∗
) = x
CREATING AND DECOMPOSING FUNCTIONS USING FUZZY FUNCTIONS
401

where ν
∗
is the fixpoint of the negation; that is,
n(ν
∗
) = ν
∗
.
The ν
∗
value plays an important role in the defini-
tion of the aggregation operator (Dombi, 2008). If
x, y < ν
∗
then the aggregation can be viewed as a con-
junction and if ν
∗
< x, y then the aggregation can be
viewed as a disjunction. If ν
∗
is close to 0 then the
operation is disjunctive, and if ν
∗
is close to 1 then
the operation is conjunctive.
3 CONSTRUCTION OF
FUNCTION BY POSITIVE
AND NEGATIVE EFFECTS
Because the aggregation operator has a neutral value
we have to transform the interval to [0,ν] or [ν, 1]. We
will define positive and negative effects using the dis-
tending interval. That is,
P
λ
1
,λ
2
a
1
,a
2
(x) =
1
2

1+ γσ
λ
1
,λ
2
a,b
(x)

(5)
N
λ
1
,λ
2
a
1
,a
2
(x) =
1
2

1− γσ
λ
1
,λ
2
a,b
(x)

(6)
where the scaling factor γ ∈ [0, 1] controls the in-
tensity of the effect. Equations (5) and (6) have a
common form if γ ∈ [−1, 1], namely
E
λ
1
,λ
2
a
1
,a
2
(γ, x) =
1
2

1+ γσ
λ
1
,λ
2
a,b
(x)

(7)
Here if γ > 0 then we have a positive effect and if
γ < 0 we have a negative effect.
Using the aggregation operator we get a function that
models certain positive and negative effects.
The goodness of this construction will now be de-
scribed. If the functions are integrable function in the
Riemannian sense, then there exist upper or lower ap-
proximations of rectangles. Because δ
a,b
(x) is a rect-
angle and the aggregation of the rectangles are rectan-
gles, we can define an intervalwhere 0 < a
1
< a
2
. . . <
a
n
< 1. The discretisation of an interval rectangle ap-
proximation will be sufficiently good if the a
i
, a
j
in-
tervals are small enough. So our method can be ap-
plied on any function that is integrable in the Rieman-
nian sense.
4 FUNCTION DECOMPOSITION
In the previous section we saw that we can construct
a desired function using the aggregation operator and
functions that model the effects. When applying this
method, the reverse case may sometimes be helpful
too. We will show that we can do this by using an
optimisation method. We can find a wide variety of
optimisation techniques. If the initial values are prop-
erly chosen it is not hard to get the global minimum
by using a local search algorithm. Here we will apply
the well-known BFGS method (Ruszczynski, 2001).
Because we can define initial points that are not far
away from the optimum, the BFGS method can find
the optimal solution in a few iterations.
In our experiment we will use a function with a
dense sampling procedure. In each example we will
use 100 equidistant coordinates on the given interval.
Let us define a function F : R → [0, 1] that we
would like to approximate. Our task is then to
decompose it into positive and negative effects.
This can be done via our approximation method or
any interpolation procedure. In order to get a good
result, we shall first smooth the function F(x). The
algorithm has the following steps:
1. Find the the local minimum and maximum value
of the function F(x):
F(c
i
) = A
i
, where F(x) < A
i
, if xε(c
i
− ε, c
i
+ ε)
F(c
j
) = A
j
, where F(x) > A
j
, if xε(c
i
− ε, c
i
+ ε)
2. Define the [a
i
, b
i
] intervals
a
n
=
c
n−1
+ c
n
2
, b
n
= c
n
+
c
n−1
+ c
n
2
where
c
1
< c
2
< c
3
< . . . < c
k
3. Define the initial value of λ
i
1
and λ
i
2
λ
i
1
= 2
f(c
i
) − f(a
i
)
c
i
− a
i
λ
i
2
= 2
f(b
i
) − f(c
i
)
b
i
− a
i
Here there is a multiplicative constant of 2, be-
cause we have to transform the distending interval
up and down.
4. Build the initial effects of the function:
E
i
(x) = E
λ
i
1
,λ
i
2
a
i
,b
i
(γ, x)
Using these effects we can create the approxi-
mated function with the help of the aggregation
operator:
G
λ
1
,λ
2
a,b
(γ, x) =
1
1+
n
∏
i=1
1−E
i
(x)
E
i
(x)
ICEIS 2010 - 12th International Conference on Enterprise Information Systems
402

5. Find the optimal solution of the a
i
, b
i
, γ, λ
i
1
, λ
i
2
values with the suggested initial values.
min
a,b,γ,λ
1
,λ
2
∑

G
λ
1
,λ
2
a,b
(γ, x
i
) − F(x
i
)

2
6. If the difference between the original function and
the approximated function is too large then extract
the approximated function from the original func-
tion and repeat the procedure in order to define
additional effects.
It is not easy to minimise this problem, because a
minimum may not be the global minimum. However,
because G
λ
1
,λ
2
a,b
(γ, x) is a continuous function of its
parameters and the initial values are well-chosen, we
can get good results.
The results of this are shown pictorially in figures 1
and 2. Our solution has an error of less than |0.04|.
We performed some additional tests, but the results
turned out to be the same as those presented below.
Figure 1: The function and its approximation.
Figure 2: Plot of the difference between the function and its
approximation.
5 CONCLUSIONS
In this article we developed a new type of non-linear
regression method which is based on positive and neg-
ative effects and provides a natural description of the
function. Our algorithm uses the BFGS method for
getting accurate effects. We showed that this proce-
dure is effective if all the data points are given. We
found that this method is fast (only a few iteration
steps are required for the optimisation procedure) and
it is easy to use. Moreover, it is not necessary to mod-
ify the whole of the function.
REFERENCES
Acz´el, J. (1966). Lectures on Functional Equations and
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D. Dubois, H. P. (2000). Fundamentals of fuzzy sets (the
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Dombi, J. (1982a). Basic concept for a theory of evalua-
tion: the aggregation operator. Europian Journal of
Operations Research, 10.
Dombi, J. (1982b). A general class of fuzzy operators, the
de morgan class of fuzzy operators and fuzziness mea-
sures induced by fuzzy operators. Fuzzy Sets and Sys-
tems, 8.
Dombi, J. (2008). Towards a general class of operators for
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J. Fodor, R. Y. and Rybalov, A. (1997). Structure of uni-
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403