Approximation capabilities of hierarchical neural-fuzzy systems for function approximation on discrete spaces

This paper investigates function approximation on discrete input spaces by both neural networks and neural-fuzzy systems. Rather than use existing neural networks for function approximation on continuous input spaces, this paper proposes, based on a hierarchical systematic perspective, four simplified approximation schemes: simplified neural networks, extended simplified neural networks, simple hierarchical neural-fuzzy systems and hierarchical neural-fuzzy systems. Each scheme is proven to be a universal approximator (i.e., each can approximate any function on discrete input spaces to any degree of accuracy). The results provide both several new and simpler approximation schemes for function approximation on discrete spaces and show that there exist simpler and more effective methods for function approximation on discrete spaces compared with continuous spaces.