On The Suitability of Yager’s Implication for Fuzzy Systems
P. Balaji, C.Jagan Mohan Rao, J.Balasubramaniam
Sri Sathya Sai Institute of Higher Learning,
Prasanthi Nilayam, India - 515134.
Abstract
Fuzzy Systems are one of the most important applications of Fuzzy Set Theory. Fuzzy Implication Operators play an important role in both Fuzzy Logic Control Systems and Approximate Reasoning. A lot of work has been done on studying these Implication Operators. In this work we propose a few desirable properties of Implication Operators with respect to their suitability in Fuzzy Systems. We also discuss the suitability of Yager’s implication operator for fuzzy systems based on the properties mentioned above.
Key words: Fuzzy Logic, Universal Approximation, Rule Reduction, Robustness, Goodness of Inference, Yager’s Implication and Fuzzy Logic Control Systems
1. Introduction
Fuzzy Systems are Model-free estimators of input-output relations. A fuzzy system [Kosko 1997] is a set of if-then rules that maps inputs to outputs. The rules define fuzzy patches in the input-output state space . The fuzzy system ,approximates a function by covering its graph with rule patches and averaging patches that overlap.
It is characterised by a set of Fuzzy If-Then rules, as follows:
,
where are the input domains and is the output domain, with as the input fuzzy sets and as the output fuzzy sets, in the rules ,
Fuzzy Implication Operators play an important role in both Fuzzy Systems and Approximate Reasoning, two of the most established applications of Fuzzy Logic. The nature of inference both in Approximate Reasoning and Fuzzy Systems depend to a very large extent on these implication operators, which relate the antecedents and consequents in a fuzzy if-then rule. To select an appropriate fuzzy implication operator for approximate reasoning under each particular situation is difficult. Although some theoretical guidelines are available for some situations, a general solution to this problem is yet to be established. We investigate the different desirable properties of Fuzzy Implication Operators with respect to Fuzzy Systems and Approximate Reasoning.
2. The Desirable Properties of Fuzzy Implication Operators
In this section, we give the definition of a Fuzzy Implication Operator and a few classes of Fuzzy Implication Operators. Then we list out some of the desirable properties of Fuzzy Implication Operators employed in Fuzzy Systems.
2.1 Basic Definitions
Definition 2.1 A Fuzzy Implication Operator is a binary operation from such that the following properties hold:
J1) - Monotonic non-increasing in the first variable
J2) - Monotonic non-decreasing in the second variable
J3) - Neutrality Principle
J4) - Falsity Principle
J5) - Exchange Principle.
Following are two of the well-established classes of Fuzzy Implication Operators.
Definition 2.2 An S-implication is obtained from an s-norm and a strong negation as follows:
Table 2.1 Some of the well known S-implications with their corresponding s-norms
Name / / /Zadeh / / /
Reichenbach / / /
Lukasiawicz / / /
Definition 2.3 An R-implication is obtained from a t-norm as its residuation as follows:
Definition 2.4 A QL-implication (Quantum Logic Implication) isobtained from an s-norm , a strong Negation and a t-norm as follows:
Table 2.2 Some of the well-known R-implications and their corresponding t-norms
Name / t(a,b) /Lukasiawicz / /
Mamdani / /
Larsen / /
Table 2.3. Some well-known QL-implications
Name / / / /Early Zadeh / / / /
Kleene-Dienes / / / /
Klir and Yuan / / / /
2.2 Goodness of Inference
Consider the fuzzy if-then rule given by
(1)
This rule is given by the relation
, (2)
where is a fuzzy implication, expresses the relationship between variables and , involved in the given proposition, and and are fuzzy sets on and respectively.
Then for every and , the membership function represents the truth-value of the proposition
.
Now, the truth values of propositions “” and “” are expressed by the membership grades and respectively. Since the meaning of fuzzy implication is not unique, contrary to the classical logic, we have to select the fuzzy implication, which makes the formula (2) operational, so that the results can be evaluated and compared. It is obvious that the criteria must emerge from the Generalized Modus Ponens rule given by
Rule:
Fact:
Conclusion:
Since in Fuzzy Control we are interested only in forward reasoning, we restrict this study only to the case of Generalized Modus Ponens, the extension of Classical Modus Ponens.
According to fuzzy inference rule, given fuzzy if-then rule of the form
and the fact “”, we conclude that “” by the compositional rule of inference , where is the composition for a t-norm . The generalized modus ponens should coincide with classical modus ponens in special case when , i.e.. The various properties proposed and studied by researchers are given in Table 2.4.
Table 2.4. Desirable Properties of Generalised Modus Ponens
If / Then/ or
/ or
[Klir and Yuan 2000] and [Fukami et al 1978] have done extensive work in studying the Fuzzy Implication Operators along the above framework. Form these studies it can be seen that S- and R-implications have most of the above properties when used with an appropriate composition operator.
2.3 Rule Reduction
For an -input Multi-Input Single-Output (MISO) fuzzy system, with membership functions defined on each of the input domains , there are rules. Thus we observe that an increase in the number of input variables and /or the number of membership functions in the input domain quickly lead to a combinatorial explosion in the number of rules. Hence rule reduction has emerged as one of the most important areas of research in fuzzy control.
Combs and Andrews [Combs 1993] have proposed a rule reduction by employing the well-known classical logic equality
(3)
The ensuing debate on which fuzzy implications can boast of this property led Trillas [Trillas 2002] to explore the following general equation
(4)
and in [JB and CJM 2002] the authors have explored the other general equations (5)
(6)
(7)
where ,and are fuzzy implication, t- and s- norms respectively.
Again R- and S-implications where the only ones that have all the above properties (4) – (7). Also [JB and CJM 2001] have shown that the above properties lead to Lossless Rule Reduction, i.e., the inference obtained from the original rule base and that obtained from the reduced rule base is identical.
2.4 Universal Approximation
A fuzzy system is said to be a universal approximator, if for any continuos function over a compact set and an arbitrarily positive number , there is a fuzzy system, its corresponding system function based on a given implication with an appropriate defuzzifier satisfies the inequality relation .
A fuzzy implication is said to enable universal approximation, if the fuzzy system employing this implication is a universal approximator for suitable choices of membership functions and defuzzifier.
Recently [Y M Li 2002] has proven that both Single-Input Single-Output (SISO) and Multi-Input Single-Output (MISO) fuzzy systems with pseudo-trapezoidal membership functions, employing R-, S- and QL-implications and MOA defuzzifier are Universal Approximators.
2.5 Robustness
Cordon et al. [Cordon et al 1997] have defineda fuzzy implication to be robust, if the fuzzy system employing this implication gives good average behaviour with different applications and when different defuzzification methods are used. They have measured the good average behaviour of the fuzzy system in terms of Medium Square Error, Adaptation Degree associated to the Median Square Error, Mean Adaptation Degree for a fuzzy implication operator, Average Mean Adaptation Degree, etc.
According to the Cordon et al., implication operators with the following properties can be considered robust:
a).
b).
c)
In this work we investigate the suitability of Yager’s Implication operator for fuzzy systems in the light of the criteria mentioned above.
3. Suitability of Yager’s Implication
3.1 Yager’s Implication
Ronald R. Yager in [Yager 1980] introduced a new implication operator defined as
This implication is unique due to its exponential like form, which is, absent in the other implication operators. This implication operator satisfies many of the properties prescribed for implication operator. They are
- Anti-monotonicity in first argument
i.e., implies since.
- Monotonicity in second argument
i.e., implies since .
- Dominance of falsity
i.e., , since .
- Neutrality of truth
i.e., , since
Turksen et al. [Turksen et al 1998] formulated a new class of implication operators called A-Implication operators inspired by Yager’s implication operator. We hope this study on Yager’s implication operator will help in evaluating the suitability of A-Implication operators.
3.2 Goodness of Inference of Yager’s Implication
In this section we study the goodness of inference of the Yager’s implication operator on the basis of Zadeh’s Compositional Rule of Inference, by far the most established method in Approximate Reasoning.
We investigate Yager’s Implication with respect to the criteria listed in Table 2.4, the results of which are presented in the following Table 3.1.
Table 3.1 Modus Ponens property of Yager’s Implication under various compositions
Control Input / is N or SN / Nature ofMax-Min / Max-Product / Max-Lukasiewicz / Max-Drastic
/ N / / / /
SN / / / /
More or Less
i.e. / N / / / /
SN / / / /
Very
i.e. / N / / / /
SN / / / /
Not
i.e. / N / Unknown / Unknown / Unknown / Unknown
SN / Unknown / Unknown / Unknown / Unknown
Note: N-Normal Fuzzy Sets and SN- Sub Normal Fuzzy Sets
From the above table we note that Yager’s Implication can be employed along with different types of compositions to obtain the required output for a given input. Thus Yager’s implication has good inference properties.
3.3 Robustness of Yager’s Implication
We recall that the criterion for robust implication operator is given by
a),
b),
c)
Yager’s implication operator satisfies two of the above properties as shown below:
a).
b) and therefore, in particular, .
Though Yager’s implication doesn’t satisfy the criterion c), this is not a serious draw back, since this criterion contradicts the falsity property of any genuine implication operator .
Thus Yager’s implication operator can be considered robust, i.e., it gives good inference in different applications and with various defuzzification methods.
3.4 Universal Approximation with Yager’s Implication
Being robust alone is not enough to adjudge an implication operator to be good. The implication operator should also Universal Approximation, i.e., fuzzy systems employing this implication operator should be universal approximators.
Along the lines of the proof of the Theorem 8.4.2 from [Nguyen 2000] it can be shown that Mamdani Fuzzy Systems with Yager’s Implication are Universal Approximators.
3.5 Rule Reduction with Yager’s Implication
Though Yager’s implication does not satisfy 3 of the 4 properties required for Lossless Rule Reduction, viz., equations (5) – (7), it does satisfy equation (4) with , viz.,
when the t-norm used is the Algebraic Product, i.e., .
Thus we find that Yager’s Implication
has most of the desirable properties, next only to R- and S-implications in the setting of Fuzzy Systems – both in Fuzzy Logic Controllers and Approximate Reasoning.
4. Conclusions
In this work we have listed the desirable properties of Fuzzy Implication Operators employed in Fuzzy Systems. We have shown that Yager’s Implication has most of these desirable properties and thus can be a very suitable choice to employ both in Fuzzy Logic Controller and Approximate Reasoning Systems. We hope that this work will better enable the study and characterisation of the recently proposed A-implications. Work is already underway along these lines. Recently Baczynski [Baczynski 2001] has dealt with Implication operators that are almost conjugate with Yager’s Implication. All the results pertaining to Yager’s Implication can be exported to these Implication Operators in a straightforward manner.
Reference:
[Baczynski 2001] Michal Baczynski, “ On a class of Distributive Fuzzy Implications”, Intl. Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 9 No. 2, pp. 229-238, 2001.
[Cordon et al 1997] O. Cordón, F. Herrera and A. Peregrín, “Applicability of the Fuzzy Operators in the Design of Fuzzy Logic Controllers”, Fuzzy Sets and Systems 86 (1997) 15-41.
[Driankov et al 1993] D. Driankov, H. Hellendoorn and M. Reinfrank, An Introduction to Fuzzy control, Narosa Publishing, New Delhi (1993).
[JB and CJM 2001] J.Balasubramaniam, C.Jagan Mohan Rao, “R-implication operators and rule reduction in Mamdani-type fuzzy systems”, Proceedings of the 6th Joint Conference on Information Sciences, Fuzzy Theory &Technology, Durham, USA, March 8-12, 2002.
[JB and CJM 2002] J.Balasubramaniam, C.Jagan Mohan Rao, “On the Distributivity of Implication Operators over T and S norms”, IEEE Trans. Fuzzy Systems, Accepted for publication.
[Klir and Yuan 2000] G.J.Klir, B.Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall of India (2000).
[Kosko 1997] Bart Kosko, " Fuzzy Engineering ", Prentice Hall, 1997.
[Nguyen 2000] H.T.Nguyen, E.A.Walker, “ A First Course in Fuzzy Logic”, CRC Press (2000).
[Trillas 2002] E.Trillas, C.Alsina, “ On the Law [(p q) r ] = [(p r) v (q r)]in Fuzzy Logic”, IEEE Trans. Fuzzy Systems, Vol. 10, pp. 84 – 88, 2002.
[Y M Li 2002] Y.M.Li, Z.K.Shi, Z.H.Li, “Approximation Theory of Fuzzy Systems based upon genuine many-valued implication – SISO Case”, Fuzzy Sets and Systems, Vol. 130 No. 2, 147 – 158, 2002.
[Turksen et al 1998] I.B.Turksen, V.Kreinovich and R.Yager, “A new class of Fuzzy Implications: Axioms of Fuzzy Implication revisited”, Fuzzy Sets and Systems, Vol. 100, 267 – 272,1998.
[Yager 1980] R.Yager, “An Approach to Inference in Approximate Reasoning”, Intl. Journal of Man – Machine Studies, Vol. 12, 323 – 338, 1980.
1