Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

3 Computation by Discrete Neural Nets

Stan Franklin Max Garzon Institute for Intelligent Systems The University of Memphis


1. INTRODUCTION

Classical computability theory is a branch of theoretical computer science. It concerns itself with determining which algorithmic problems can be solved by computers (or models thereof, e.g., Turing machines) given even the most unrealistically ideal availability of time and memory. The theory probes the outer limits of what can ever be computed with man-made devices. It is a mathematical theory in that formal definitions are given and theorems are then proven concerning computational power.

In an analogous way, this chapter is concerned with what problems we can ever hope to solve using discrete-time neural nets, again under the most ideal conditions of available time and space (nodes and connections). It is the outer reaches of computation using neural nets that are explored. Formal definitions are given, and theorems concerning the computational power of neural nets are stated with pointers to their proofs. The beginnings of a mathematical theory of neural computability emerges. It arises from the mathematical perspective on neural nets offered by classical computability theory.

Yet another mathematical perspective on parallel computation is offered by cellular automata and, more generally, automata nets. These mathematical objects are examples of computation on networks (graphs) rather than on linear tapes. We see that they are intimately related to neural nets and thereby offer insights into neural computability.

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