Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview
with finitely many cells are less likely to have the shadowing property. More importantly, the property of observability by shadowing has been generalized by Botelho and Garzon ( 1994b) to a concept of relative shadowing that yields a natural definition of approximate observability (or computability) via neural nets. The reader is referred to Garzon and Botelho ( 1993a) and Garzon ( 1995) for the precise definitions and results.
5. OPEN QUESTIONS
This brief tour apparently covers most of what is known of neural computability. Many interesting questions have been left open. In this last section we will scratch the surface with just a few of them.
1. Is it also true that neural nets, cellular automata, and automata networks have equivalent computational power without the bandwidth restriction? One would still want to insist on finite fan-in and finite fan-out at each node. There are surely some interesting neural nets without finite bandwidth, AMNIAC for one. This leads us to the next question.
2. Is there a universal neural net for some class of neural nets more inclusive than those of finite bandwidth? For all neural nets with finite fan-in and fan-out?
3. Is there a universal neural net with finite bandwidth for the class of all neural nets of finite bandwidth? (The design of AMNIAC is basically two dimensional although it seems three, dimensional at first glance). Can an observable version be produced?
4. What is a good characterization of the real-valued functions on the unit interval that are π computable? This problem is both of interest in itself and as a step toward the solution of the next one.
5. The defining problem of neural computability is, of course, to answer in some useful way the question: what dynetic problems are neurally solvable? This is, perhaps, a formidable undertaking, not to mention the question: what interesting subfamilies are learnable?

These problems appear at least as difficult as the corresponding problems in the classical theory of computability by Turing machines. On the other hand, there are enough hints that characterizing computable self-maps of configuration space (i.e., solvable dynetic problems) is a question of considerable importance for neural computing.


ACKNOWLEDGMENT

The work of S. Franklin was partially supported by the Office of Naval Research Grant N0014-88-K-0110 and by the National Science Foundation Grant CCR-

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