The constructive 2-variable granular system with universal approximation

An important task of learning is to establish relations among granules such as classes, clusters, sets, groups, etc. The relations can be represented by granular If-Then rules. How to quickly discover the granular If-Then rules becomes a major long-term problem. Conventional training-based approaches such as neural networks and neuro-fuzzy systems have the learning speed bottleneck problem. The new constructive 2-variable granular system was proposed based on soft computing and granular computing to highly speed up granular knowledge discovery. Now the important question is “is the constructive 2-variable granular system a universal approximator?” The constructive 2-variable granular system is proved to be a universal approximator. According to the proof, we can construct a granular constructive a-variable granular system with any required accuracy and a near optimal number of granular rules. In the future, the granular constructive n-variable fuzzy system will be investigated in general