Novel Neural Algorithms for Solving Fuzzy Relation Equations

Due to the difficulty in a given system of using neural networks to solve fuzzy relation equations, the best learning rate sometimes cannot be decided easily and strict theoretical analyses on convergence of algorithms are not given. To overcome these problems, we present in this chapter some novel neural algorithms based on fuzzy δ rules. We first describe such algorithms for max-min operator networks, then we demonstrate that these algorithms can also be extended to max-times operator network. Important results include some improved fuzzy δ rules, a convergence theorem, and an equivalence theorem which reflects that fuzzy theory and neural networks can reach the same goal by different routes. We also discuss the fuzzy bidirectional associative memory network and its training algorithms, and prove all important theorems with additional simulation and comparison results. Furthermore, we propose a more powerful algorithm for solving many types of fuzzy relation equations. The algorithm is based on a specially designed fuzzy neuron. This fuzzy neuron is found by replacing the operators of the traditional neuron by a pair of abstract fuzzy operators such as $$\left( {\hat{ + },\hat{ \bullet }} \right)$$ , which we call fuzzy neuron operators. Afterwards, we discuss the relationship between the fuzzy neuron operators and the t-norm and t-conorm, and point out that fuzzy neuron operators are based on the t-norm, but are much wider than the t-norm. And finally, we report some simulation results of some pairs of typical compositional operators.