On the Suitability of Yager’s Implication for Fuzzy Systems

P. Balaji, C.Jagan Mohan Rao, J.Balasubramaniam

Department of Mathematics and Computer Sciences,

Sri Sathya Sai Institute of Higher Learning, 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 these properties.

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 is a set of if-then rules that maps inputs to outputs. 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. 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 these properties.

2. The Desirable Properties of Fuzzy Implication Operators

In this section, we list out some of the desirable properties of Fuzzy Implication Operators employed in Fuzzy Systems.

2.1 Goodness of Inference

A fuzzy if-then rule, which is of the form , is given by the relation

,, (1)

where is a fuzzy implication, expresses the relationship between variables and , involved in the given proposition, and and are fuzzy sets on and respectively. According to fuzzy inference rule, given 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 2.1.

If / Then
/ or
/ or

Table 2.1. Desirable Properties of Generalised Modus Ponens

[Klir and Yuan 2000] and [Fukami et al 1978] have done extensive work in studying various Fuzzy Implication Operators along the above framework.

2.2 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. Hence rule reduction has emerged as one of the most important areas of research in fuzzy control. [JB and CJM 2002] have shown that the following properties

(2)

(3)

(4)

(5)

where ,and are fuzzy implication, t- and s- norms respectively, 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.

2.3 Universal Approximation

A fuzzy system is said to be a universal approximator, if for any continuos function over a compact set and an arbitrary , 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.

2.4 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. According to the Cordon et al., implication operators with the following properties can be considered robust:

a).

b).

c)

3. Suitability of Yager’s Implication

In this work we investigate the suitability of Yager’s Implication operator for fuzzy systems in the light of the criteria mentioned above.

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.

3.2 Goodness of Inference of Yager’s Implication

We study the goodness of inference of 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. From Table 3.1 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.

Control Input / is N or SN / Nature of
Max-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
Table 3.1 Modus Ponens property of Yager’s Implication under various compositions

Note: N-Normal Fuzzy Sets and SN- Sub Normal Fuzzy Sets

3.3 Robustness of Yager’s Implication

Yager’s implication operator satisfies two of the robustness properties as elucidated in Section 2.4:

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

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 (2),(3) and (5), 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 [JB and CJM 2002] in the setting of Fuzzy Systems – both in Fuzzy Logic Controllers and Approximate Reasoning.

4. Conclusions

In this work we have listed a few desirable properties of Fuzzy Implication Operators to be 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 [Turksen et al 1998]. Work is already underway along these lines. All the results in this work pertaining to Yager’s Implication can be exported to A-Implication Operators in a straightforward manner.

Reference:

[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.

[Fukami et al 1978] S. Fukami, M.Mizumoto and K.Tanaka, “On Fuzzy Reasoning”, Systems, Computers, Controls, Vol. 9 (4) (1978) 44 – 53.

[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).

[Nguyen 2000] H.T.Nguyen, E.A.Walker, “ A First Course in Fuzzy Logic”, CRC Press (2000).

[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.