Computability considerations regarding neural nets
In mathematics and logic, computable functions = Turing computable functions
Question#1: Neural network computable → Turing computable?
- Presumably Turing machines can simulate any neural network to arbitrary finite precision given enough memory.
Question#2: Turing computable → neural network computable?
- Siegelmann and Sontag (1991). Turing Computability With Neural Nets. Applied Mathematics Letters.
@This paper shows the existence of a finite neural network, made up of sigmoidal neurons, which simulates a universal Turing machine. It is composed of less than 105 synchronously evolving processors, interconnected linearly. High-order connections are not required.@
- J. Pedro Neto, Hava T. Siegelmann, J. Félix Costa, C.P. Suarez Araujo. (1997). Turing Universality of Neural Nets (Revisited). Lecture Notes in Computer Science – 1333, 361-366. Springer-Verlag.
@We show how to use recursive function theory to prove Turing universality of finite analog recurrent neural nets, with a piecewise linear sigmoid function as activation function.@
- Such mathematical results do not show that if X is computational equivalent to Y, then X is just as efficient and practical to implement as Y.
- The systems can differ at the level of algorithm and at the level of hardware implementation.