 Neural networks more powerful than TMs may be defined, however, by exploiting once again the expressive power of real numbers. The best known networks of this kind are Analog Recurrent Neural Networks (ARNNs) (Siegelmann 1999). ARNNs should not be confused with analog computers in the traditional sense (Pour-El 1974, Rubel 1993, Mills 2008). Whereas analog computers manipulate real variables without relying on the exact value of arbitrary real-valued quantities, ARNNs manipulate strings of digits by (possibly) relying on the exact value of arbitrary real-valued quantities. Specifically, the weights connecting individual processing units within ARNNs can take exact values of arbitrary real numbers, including values that are Turing-uncomputable, and such exact values are exploited in the computation. When their weights are Turing-uncomputable, ARNNs can go beyond the power of Turing-machines: they compute any function over binary strings. ARNNs more powerful than TMs cannot operated reliably, because they require unboundedly precise weights. In addition, the required weights are Turing-uncomputable, so constructing ARRNs more powerful than TMs requires already possessing the ability to perform hypercomputations (Davis 2004, Schonbein 2005, Siegelmann 1999, 148).