Letters

Although they then cite a "slow, rigorous proof" by K. Hornik et al. that implies the superiority of neural systems to digital ones, the contrast needs to be reemphasized. In practice, since neural nets and Turing-equivalent systems are simulated on systems constructed from digital integrated circuits, they are of course equivalent. However, in theory, the fundamental computations of neural networks depend on the arithmetic of real numbers rather than integers. The ideal neural unit computes in a noisefree, infinite precision fashion. These computations can be simulated arbitrarily closely by a Turing machine, yet as the Greek philosopher Zeno observed 2200 years ago, the continuous computation can attain values in a fixed time that the digital approximation with uniform timestep will take infinite time to reach.