• Note: choice of activation functions.

Question#1: Neural network computable → Turing computable?

Question#2: Turing computable → neural network computable?

Turing computable → neural network computable?

Comments

Computability considerations regarding neural nets

In mathematics and logic, computable functions = Turing computable functions

• Turing machines

Neural network computable → Turing computable?

• Presumably Turing machines can simulate any neural network to arbitrary finite precision given enough memory.

Neural networks can compute Turing computable functions

• 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.@

Comment

• 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.

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