theory of computing

Blum and Rivest showed that any possible neural net learning algorithm based on fixed architectures faces severe computational barriers. This paper extends their NP-completeness result, which applied only to nets based on hard threshold activations, to nets that employ a particular continuous activation. In view of neural network practice, this is a more relevant result to understanding the limitations of backpropagation and related techniques.

This paper deals with finite size networks which consist of interconnections of synchronously evolving processors. Each processor updates its state by applying a "sigmoidal" function to a rational-coefficient linear combination of the previous states of all units. We prove that one may simulate all Turing Machines by such nets. In particular, one can simulate any multi-stack Turing Machine in real time, and there is a net made up of 886 processors which computes a universal partial-recursive function. Products (high order nets) are not required, contrary to what had been stated in the literature. Non-deterministic Turing Machines can be simulated by non-deterministic rational nets, also in real time. The simulation result has many consequences regarding the decidability, or more generally the complexity, of questions about recursive nets.