A Topological Semantics for Rule Extraction with Neural Networks

Rule extraction with neural networks is a subject of increasing interest. Research in this area could benefit from the availability of a formal model of the semantics of the rules. A model of this kind would express the relationship between the application data, the neural network learning model and the extracted rules with mathematical rigor, allowing systematic analysis and modification of rule extraction approaches and the neural network architectures used. However, formal models of this kind are not in common use. This paper proposes a formal semantic model and includes an analysis of an example rule extraction architecture and some issues raised by it and other architectures. In the formal model, the semantics of a neural network is expressed through a form of model theory based upon concepts from topology, including limit points and continuous functions. A state of adaptation of the neural network in which it has learned a set of rules from training data corresponds to a continuous function between topological systems. Topological systems, the domains of inputs to the network, are a generalization of the concept of a topological space. The results of an example analysis with this model suggest a direction for improvements to the example architecture and the desirability of applying the model to other rule extraction approaches.