Mathematical Perspectives on Neural Networks

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

It is not clear that, even if such an extension is possible, it will be useful in the theory of neural networks. Of course, if such as extension turns out to be possible and useful, then problems 1-4 should be restated to accommodate functions of more than one argument.

The digital computer is ubiquitous; the analog computer is barely known. This is unfortunate. There are many tasks where a suitable analog device, if it existed, would be more natural. Suppose, for example, one wishes to transmit a photograph. The image must first be discretized before transmission. The resulting picture, after transmission, is also discrete. It would be more natural to have an analog device perform the task. A suitable definition of computable function of a real variable in terms of basic components abstracted from hardware considerations might be helpful.


APPENDIX

Here we give two definitions: computable real and computable continuous function of a real variable.

To obtain the definition of computable real, one effectivizes the classical Cauchy sequence definition of real number. Classically, a real number is the limit of a sequence of rationals. The effective version goes as follows.

The real number r is computable if there exist four Turing-computable functions b, c, s and h, mapping the nonnegative integers into themselves such that

(Here, of course, c (n) ≠ 0 for all n.)

The definition of computable continuous function on a closed bounded interval I can be given via an effective version of the Weierstrass approximation theorem. Classically, this theorem states that every continuous function on I is the limit of a sequence of polynomials with rational coefficients. The effective version goes as follows.

The continuous function φ (x) is computable on I if there exist five Turing computable functions b, c, s, d, h such that on I

[Of course, c (n, j) ≠ 0 for all nonnegative integers n, j.]

The definition of computable continuous function can be extended to functions of more than one variable in an obvious way. In particular, if the function φ is a function of the n variables x1, . . . , xn, then the associated polynomials are polynomials in n variables x1, . . . , xn. A function of a complex variable is computable

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