Can a vector space based learning model discover inductive class generalization in a symbolic environment?

Abstract

We outline a general framework for inductive learning based on the recently proposed evolving transformation system model. Mathematical foundations of this framework include two basic components: a set of operations ( on objects) and the corresponding geometry defined by means of these operations. According to the framework, to perform inductive learning in a symbolic environment, the set of operations ( class features) may need to be dynamically updated, and this requires that the geometric component allows for an evolving topology. In symbolic systems, as defined in this paper, the geometric component allows for a dynamic change in topology, whereas finite-dimensional numeric systems (vector spaces) can essentially have only one natural topology. This fact should form the basis of a complete formal proof that, in a symbolic setting, the vector space based models, e. g. artificial neural networks, cannot capture inductive generalization. Since the presented argument indicates that the symbolic learning process is more powerful than the numeric process, it appears that only the former should be properly called an inductive learning process.