Although neural networks often achieve impressive learning and generalization performance, their internal workings are typically all but impossible to decipher. This characteristic of the networks, their opacity, is one of the disadvantages of connectionism compared to more traditional, rule-oriented approaches to artificial intelligence. Without a thorough understanding of the network behavior, confidence in a system's results is lowered, and the transfer of learned knowledge to other processing systems - including humans - is precluded. Methods that address the opacity problem by casting network weights in symbolic terms are commonly referred to as rule extraction techniques. This work describes a principled approach to symbolic rule extraction from standard multilayer feedforward networks based on the notion of weight templates, parameterized regions of weight space corresponding to specific symbolic expressions. With an appropriate choice of representation, we show how template parameters may be efficiently identified and instantiated to yield the optimal match to the actual weights of a unit. Depending on the requirements of the application domain. the approach can accommodate n-ary disjunctions and conjunctions with O(k) complexity, simple n-of-m expressions with O(k(2)) complexity, or more general classes of recursive n-of-m expressions with O(k(L+2)) complexity, where ii is the number of inputs to an unit and L the recursion level of the expression class. Compared to other approaches in the literature, our method of rule extraction offers benefits in simplicity, computational performance, and overall flexibility. Simulation results on a variety of problems demonstrate the application of our procedures as well as the strengths and the weaknesses of our general approach. (C) 1999 Published by Elsevier Science Ltd. All rights reserved.
