Continuous Symbol Systems: The Logic of Connectionism
It has been long assumed that knowledge and thought are most naturally represented as discrete symbol systems (calculi). Thus a major contribution of connectionism is that it provides an alternative model of knowledge and cognition that avoids many of the limitations of the traditional approach. But what idea serves for connectionism the same unifying role that the idea of a calculus served for the traditional theories? We claim it is the idea of a continuous symbol system. This chapter presents a preliminary formulation of continuous symbol systems and indicates how they may aid the understanding and development of connectionist theories. It begins with a brief phenomenological analysis of the discrete and continuous; the aim of this analysis is to directly contrast the two kinds of symbol systems and identify their distinguishing characteristics. Next, based on the phenomenological analysis and on other observations of existing continuous symbol systems and connectionist models, I sketch a mathematical characterization of these systems. Finally the chapter turns to some applications of the theory and to its implications for knowledge representation and the theory of computation in a connectionist context. Specific problems addressed include decomposition of connectionist spaces, representation of recursive structures, properties of connectionist categories, and decidability in continuous formal systems.