=Paper=
{{Paper
|id=Vol-1492/Paper_43
|storemode=property
|title=Inverted Fuzzy Implications in Backward Reasoning Without Yager Implication
|pdfUrl=https://ceur-ws.org/Vol-1492/Paper_43.pdf
|volume=Vol-1492
|dblpUrl=https://dblp.org/rec/conf/csp/SurajL15
}}
==Inverted Fuzzy Implications in Backward Reasoning Without Yager Implication==
https://ceur-ws.org/Vol-1492/Paper_43.pdf
Inverted Fuzzy Implications in Backward Reasoning
Without Yager Implication
Zbigniew Suraj1 and Agnieszka Lasek2
1
Chair of Computer Science,
Faculty of Mathematics and Natural Sciences, University of Rzeszow,
Prof. S. Pigonia Str. 1, 35-310 Rzeszow, Poland
zbigniew.suraj@ur.edu.pl
2
Department of Electrical Engineering and Computer Science
Lassonde School of Engineering, York University,
4700 Keele Street, Toronto, Canada
alasek@cse.yorku.ca
Abstract. One of the most popular methods of knowledge representation are the
fuzzy rules. One of the ways of representation of fuzzy rules is the functional
representation. From over eight decades a number of different fuzzy implica-
tions have been described, e.g. [5]-[9]. This leads to the following question: how
to choose the proper function among basic fuzzy implications. This paper is a
continuation of study [15], where we proposed a new method for choosing impli-
cations in backward reasoning. Here we presented a way of simplify the analysis
by skipping Yager fuzzy implication.
Key words: fuzzy logic, fuzzy implications, inverted fuzzy implications, back-
ward reasoning
1 Introduction
One of the most popular methods of knowledge representation are the fuzzy rules.
From imprecise inputs and fuzzy rules imprecise conclusions are obtained. Reason-
ing is mainly classified into two types: forward reasoning and backward reasoning. The
inference mechanism of forward reasoning has a strong forecasting capability, whereas
the aim of backward reasoning generally is to find the most possible causes associated
with the existing reality. Backward reasoning plays an essential role in fault diagnosis,
accident analysis, and defect detection. This kind of reasoning uses fuzzy logic [3] to
reason about data in the inference mechanism instead of many other logics, including
Boolean logic, (non-fuzzy) many-valued logics, non-monotonic logics, etc.
Paper [4] discusses different representations of rules in a non-fuzzy setting and ex-
tends these representations to rules with a fuzzy conclusion part. It introduces the dif-
ferent types of fuzzy rules and put them in the framework of fuzzy sets and possibility
theory.
Fuzzy rules are often presented in the form of implications. In [3] a typology of
fuzzy rules and the problem of multiple-valued implications are discussed. The paper
188
reviews the problem of representing fuzzy knowledge, and ranges from linguistic vari-
ables to conditional if-then rules and qualified statements.
One of the ways of representation of fuzzy rules is the functional representation
(e.g.[11],[12],[17]). The definition of fuzzy implications and their mathematical prop-
erties can be found e.g. in [1] and [16]. One of basic problems in building an inference
system is choosing the relevant fuzzy implication. In [10] authors proposed a method
allowing to choose the most suitable fuzzy implication in an inference system appli-
cation. They introduced an algorithm that calculates the distance between two fuzzy
implications and which is based on generalized modus ponens.
In paper [13] we have presented a fuzzy forward reasoning methodology for rule-
based systems using the functional representation of fuzzy rules. In [15] we extended
this methodology for selecting relevant fuzzy implications for backward reasoning. The
proposed methodology takes full advantage of the functional representation of fuzzy
implications and the algebraic properties of the family of all fuzzy implications. It al-
lows to compare two fuzzy implications. If the truth value of the conclusion and the
truth value of the implication are given, we can easily optimize the truth value of the
implication premise. In particular, in [15] we introduced an algorithm of finding the
fuzzy implication which has the highest truth value of the antecedent when the truth
value of the consequent and the truth value of the implication are given. This method-
ology can be useful for the design of inference engine based on the rule knowledge for
a given rule-based system.
In the solution in [15] we divided the domain of fuzzy implications into areas, in
which it was possible to select appropriate fuzzy implication, and to do that we had
to use the Lambert W function. Lambert W function is a special function used when
solving equations containing unknown to both the base and the exponent power. It is
marked W (z) and defined as the inverse of f (z) = zez , where z belongs to the set
of complex numbers. Thus, for each complex number z holds: z = W (z)eW (z) . The
Lambert W function cannot be expressed in terms of elementary functions.
In this paper we present the way of avoiding this complexity of solution presented
in [15].
The rest of this paper is organized as follows. Sect. 2 contains basic information
on fuzzy implications. In Sect. 3 the research problem is formulated. Sect. 4 presents
the solution of the given research problem. Sect. 5 is devoted to the pseudo-code of an
algorithm for determining a basic fuzzy implication which has the highest truth value
of the antecedent when the truth value of the consequent and the truth value of the
implication are given. Sect. 6 includes summarizing of our research and some remarks.
2 Preliminaries
In this section we recall a definition of a fuzzy implication and we list a few of basic
fuzzy implications known from the subject literature [1].
A function I : [0, 1]2 → [0, 1] is called a fuzzy implication if it satisfies, for all
x, x1 , x2 , y, y1 , y2 ∈ [0, 1], the following conditions:
– if x1 ≤ x2 , then I(x1 , y) ≥ I(x2 , y), i.e., I(., y) is decreasing;
189
Name Year Formula of basic fuzzy implication
Łukasiewicz 1923, [9] ILK (x, y) = min(1,
1 − x + y)
1 if x ≤ y
Gödel 1932, [4] IGD (x, y) =
y if x > y
Reichenbach 1935, [11] IRC (x, y) = 1 − x + xy
Kleene-Dienes 1938, [8]; 1949, [2] IKD (x, y) = max(1 − x, y)
1 if x ≤ y
Goguen 1969, [7] IGG (x, y) = y
x if x > y
1 if x ≤ y
Rescher 1969, [12] IRS (x, y) =
0
if x > y
1 if x = 0 and y = 0
Yager 1980, [18] IY G (x, y) = x
if x > 0 or y > 0
y
1 if x < 1
Weber 1983, [17] IW B (x, y) =
y if x = 1
1 if x ≤ y
Fodor 1993, [3] IF D (x, y) =
max(1 − x, y) if x > y
Table 1. Examples of basic fuzzy implications
– if y1 ≤ y2 , then I(x, y1 ) ≤ I(x, y2 ), i.e., I(x, .) is increasing;
– I(0, 0) = 1; I(1, 1) = 1; I(1, 0) = 0.
There exist uncountably many fuzzy implications. The following Table 1 contains
a few examples of basic fuzzy implications. One of the fuzzy implication in the table
is Yager implication. As we noted in Sect. 1 we skip this implication in our analysis in
this paper to avoid complexity of solution presented in [15].
Figure 1 gives us some plots of these functions.
3 Problem Statement
Our goal is to design an algorithm to find a method of selecting fuzzy implication in
view of the value of the implication antecedent.
Assume that there is given a basic fuzzy implication z = I(x, y), where x, y belong
to [0,1]. y is the truth value of the consequent and is known. z is the truth value of
the implication and is also known. In order to determine the value of the truth of the
implication antecedent x it is needed to compute the inverse function InvI(y, z). In
other words, the inverse function InvI(y, z) has to be determined. Not every of basic
implications can be inverted. The function can be inverted only when it is injective.
4 Results
Table 2 lists inverse fuzzy implications and their domains and in Figure 2 there are some
plots of them.
190
£ ukasiewicz implication Kleene-Dienes implication
1.0 1.0
1.0 1.0
z z
0.5 0.5
0.0 0.0
0.0 0.5 y 0.0 0.5 y
0.5 0.5
x x
0.0 0.0
1.0 1.0
(a) (b)
Fig. 1. Plots of ILK and IKD fuzzy implications
Formula of inverted fuzzy implication Domain of inverted fuzzy implication
InvILK (y, z) = 1 − z + y y ≤ z < 1, y ∈ [0, 1)
1−z
InvIRC (y, z) = 1−y y ≤ z ≤ 1, y ∈ [0, 1)
InvIKD (y, z) = 1 − z y < z ≤ 1, y ∈ [0, 1)
InvIGG (y, z) = yz y ≤ z < 1, y ∈ (0, 1)
InvIF D (y, z) = 1 − z y < z < 1 − y, y ∈ [0, 1)
Table 2. Inverted fuzzy implications
Inverted Kleene-Dienes implication
Inverted £ ukasiewicz implication 1.0
1.0
z
z 0.5
0.5
0.0
1.0
0.0
1.0
0.5
x
0.5 x
0.0 0.0
1.0
1.0
0.5
0.5
y
y
0.0
0.0
(a) (b)
Fig. 2. Plots of InvILK and InvIKD fuzzy implications
191
The domains of every considered inverted fuzzy implications are included in a half
of the unit square, where y ≤ z < 1 and y ∈ (0, 1). Only one inverted fuzzy implication
has a domain which is smaller than this area. This is inverted Fodor implication and in
the whole its domain ( y ≤ z < 1 − y, y ∈ [0, 1] ) this function is equal to inverted
Kleene-Dienes implication.
For y ≤ z < 1 − y there are the following inequalities: InvIF D = InvIKD <
InvIRC < InvILK , InvIRC < InvILK , InvIGG < InvILK . A graphical represen-
tation of the ordering of inverted basic fuzzy implications is given in Figure 3.
InvILK
InvIRC InvIGG
InvIKD = InvIF D
Fig. 3. A graphical representation of the ordering of inverted basic fuzzy implications
for y ≤ z < 1 − y
For 1 − y ≤ z < 1 and y ≤ z there are the same inequalities, but without inverted
Fodor implication, because this function does not exist in this area.
The resulting inverse functions can be compared with each other so that it is possible
to order them. However, some of those functions are incomparable in the whole domain.
By taking into account six inverted fuzzy implications (including inverted Yager impli-
cation) and by dividing their domain into separable areas, we obtained 19 inequalities
between inverted fuzzy implications for any y ≤ z < 1 and y ∈ (0, 1) described in
[15].
To simplify that solution and avoid Lambert W function in this paper we skip Yager
fuzzy implication in our analysis. With this assumption there is only five different area
and inequalities instead of nineteen. The areas are shown in the Figure 4 and the in-
equalities are given in Table 3.
192
Fig. 4. The unit square [0, 1]2 divided into five separable areas
Table 3: Table of inequalities
No Area and inequality Chart of area Graph of inequalities
z InvILK
1.0
0.8
A InvIGG
For z > 1 − y 0.6
A. InvIKD < InvIRC <
InvIGG < InvILK 0.4
InvIRC
0.2
y InvIKD
0.2 0.4 0.6 0.8 1.0
z
1.0
InvILK
0.8
B
For z = 1 − y 0.6
InvIRC = InvIGG
B. InvIKD < InvIRC =
0.4 InvIKD
InvIGG < InvILK
0.2
y
0.2 0.4 0.6 0.8 1.0
Continued on next page
193
Table 3 – Continued from previous page
No Area and inequality Chart of area Graph of inequalities
z InvILK
1.0
√
For (z > 1+ 21−4y or 0.8
InvIRC
√
z < 1− 21−4y or 0.6
C. z ∈ (0.25, 0.5)) and z < 1 − y C
0.4
InvIKD < InvIGG < InvIGG
InvIRC < InvILK 0.2
y InvIKD
0.2 0.4 0.6 0.8 1.0
z
1.0
InvILK
√ 0.8
For z = 1+ 21−4y or
√
D. z = 1− 21−4y 0.6
D InvIRC
InvIGG = InvIKD < 0.4
InvIRC < InvILK InvIGG = InvIKD
0.2
y
0.2 0.4 0.6 0.8 1.0
z InvILK
1.0
√ 0.8
For z > 1− 21−4y and InvIRC
√
E. z < 1+ 21−4y 0.6
E
InvIGG < InvIKD < 0.4
InvIKD
InvIRC < InvILK
0.2
y InvIGG
0.2 0.4 0.6 0.8 1.0
All inequalities given in Table 3 can be proven in a similar way. As examples, we
will consider one of inequalities. Let y ∈ (0, 1) and z ∈ (y, 1). y < z, so obviously
y 2 < yz. By adding and subtracting 1 − z + y to the equation we obtained 1 − z <
1−z
1 − z + y − y + yz − y 2 . And therefore, 1−y < 1 − z + y. This completes the proof of
the inequality: InvIRC < InvILK in domains of these functions.
194
5 Algorithm
Below we present the pseudo-code of the algorithm (DetermineImplicationGTVA) for
determining a basic fuzzy implication which has the highest truth value of the an-
tecedent whereas the truth value of the consequent and the truth value of the implication
are given.
The algorithm uses the results of our research presented in Table 3. The first step in
the algorithm determines to which area (A) − (E) from Table 3 point (y, z) belongs to.
Algorithm DetermineImplicationGTVA
Input: W - a given subset of the basic fuzzy implications;
y - the truth value of the consequent;
z - the truth value of the implication
Output: I ∈ W - fuzzy implication(s) which has (have) the highest truth value of the
antecedent
1. a ← area(y, z) //determines the area from (A) − (E) to which a point (y, z)
belongs to;
2. order the set W with respect to the graph Ga of inequalities from the area a;
3. I ← the maximal element(s) from the ordered set W ;
4. return I;
6 Concluding Remarks
In the paper, we introduced an algorithm for finding the fuzzy implication which has the
highest truth value of the antecedent from a given subset of the basic fuzzy implications,
when the truth value of the consequent and the truth value of the implication are given.
In order to simplify the solution we skipped Yager fuzzy implication in the presented
analysis.
We considered a set of basic implications mentioned in Table 1, because they are
well known and widely used. But considering only these basic implications implied the
solution which does not cover the whole unit square as in the case with the forward
reasoning [13], only one of its halves. It raises the question how to find such a set of im-
plications that could give a solution for a backward reasoning in the whole unit square.
Our future works will focus on answering the question whether such implications could
exist and how they could be defined.
Acknowledgments. This work was partially supported by the Center for Innovation and
Transfer of Natural Sciences and Engineering Knowledge at the University of Rzeszów.
The authors are also very grateful to the anonymous reviewer for giving him precious
and helpful comments.
195
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