A neuro-fuzzy framework for inferencing

Abstract

Earlier we proposed a connectionist implementation of compositional rule of inference (COI) for rules with antecedents having a single clause. We first review this net, then generalize it so that it can deal with rules with antecedent having multiple clauses. We call it COIN, the compositional rule of inferencing network. Given a relational representation of a set of rules, the proposed architecture can realize the COI. The outcome of COI depends on the choice of both the implication function and the inferencing scheme. The problem of choosing an appropriate implication function is avoided through neural learning. COIN can automatically find a ‘good’ relation to represent a set of fuzzy rules. We model the connection weights so as to ensure learned weights lie in [0,1]. We demonstrate through extensive numerical examples that the proposed neural realization can find a much better representation of the rules than that by usual implication and hence results in much better conclusions than the usual COI.

Introduction

Fuzzy sets (Zadeh, 1965) are generalization of crisp sets and have greater flexibility to capture faithfully various aspects of incompleteness or imperfection in information, and can be used to model human reasoning/thinking process. Let A and B be two fuzzy sets defined on the universes X and Y, respectively. Consider a simple rule: If x is A then y is B. Now given the fact, If x is A′, we like to infer y is B′, such that the closer the A′ to A, the closer would be B′ to B, where A′ and B′ are fuzzy sets on X and Y, respectively. Thus, the problem of fuzzy inferencing is as follows:

• Premise 1: If x is A Then y is B

• Premise 2: x is A′

• Conclusion: y is B′

Neural networks (NNs) (Haykin, 1994), like fuzzy logic systems, are excellent at developing systems that can perform information processing similar to what our brain does. The concept of artificial NNs has been inspired by biological NNs, which enjoy the following characteristics:

• They are non-linear devices, highly parallel, robust and fault tolerant.

• They have a built-in capability to adapt its synaptic weights to changes in the surrounding environment.

• They can easily handle imprecise, fuzzy, noisy and probabilistic information.

• They can generalize from known tasks or examples to unknown ones.

Artificial NN is an attempt to mimic some or all of these characteristics. Although the development of NN is inspired by the model of brains, its purpose is not just to mimic a biological neural net, but to use principles from nervous systems to solve complex problems in an efficient manner.

Both fuzzy systems and NN have been successfully used in many applications (Haykin, 1994, Lee, 1990, Scharf and Mandic, 1985, Self, 1990, Sugeno, 1985, Suh and Kim, 1994, Zadeh, 1988). Apart from the learning ability of NN, it has inherent robustness and parallelism. Fuzzy logic, on the other hand, has the capability of modeling vagueness, handling uncertainty, and can support human type reasoning. Integration of these two soft computing paradigms (often known as neuro-fuzzy computing) is, therefore, expected to result in more intelligent systems (Gupta and Rao, 1994, Pal and Pal, 1996).

In the recent past extensive research work is going on for integration of fuzzy systems with NNs (Hayashi et al., 1992, Ishibuchi et al., 1993, Keller, 1990, Keller, 1993, Keller and Tahani, 1992a, Keller and Tahani, 1992b, Keller and Yager, 1989, Keller et al., 1992a, Keller et al., 1993, Nie and Linkens, 1992, Nie and Linkens, 1995, Shann and Fu, 1995, Takagi and Hayashi, 1991). The objective here is to combine the expert knowledge or operators' experience and reasoning ability of fuzzy systems with the computational capabilities of NNs in an efficient manner to solve complex problems (Hayashi et al., 1992, Ishibuchi et al., 1993, Keller, 1990, Keller, 1993, Keller and Tahani, 1992a, Keller and Tahani, 1992b, Keller and Yager, 1989, Keller et al., 1992b, Keller et al., 1992a, Keller et al., 1993, Nie and Linkens, 1992, Nie and Linkens, 1995, Pal et al., 1998, Shann and Fu, 1995, Takagi and Hayashi, 1991). The integration of fuzzy logic and NN often is done in two ways—a fuzzy system implemented in a neural architecture (neural fuzzy system) and a NN equipped with the capability of handling fuzzy information (fuzzy NN). Several attempts have been made in both of these directions. Of course, there are several hybrid systems which may not be categorized strictly to either of these two classes.

Keller et al. (1992a) proposed a neural implementation of fuzzy reasoning. Pal et al. (1998) analyzed the system by Keller et al. (1992a) and derived learning rules for finding good parameters for this network. For a special case, Pal et al. showed how the optimal parameters can be computed, and demonstrated the method with some examples. A new architecture is also proposed by Pal et al. (1998) which exhibits better characteristics than the network by Keller et al. (1992a).

The philosophy behind the models proposed by Keller et al., 1992a, Pal et al., 1998 is a kind of similarity-based reasoning. The more the similarity between the antecedent of a rule and the given fact, the more close would be the inferred conclusion or the consequent of the rule. Pal and Pal (1999) proposed a scheme for realization of compositional rule of inference (COI) in a neural framework. But the method by Pal and Pal (1999) can deal with only simple rules with one antecedent clause. Here, we generalize the system to implement COI with several clauses in the antecedent, in a connectionist framework. We explain how the connection weights should be modeled so that a learning rule can be designed to ensure the weights to lie in [0,1]. Note that, this is not a multi-layer perceptron which loses the fuzzy reasoning structure and acts as a black-box type function approximator.