Construct a network, N1, with O(n) weights, which implements a function from R to {0,1}n and satisfies the following property.

1. Construct a network, N1, with O(n) weights, which implements a function from R to {0,1}and satisfies the following property. For every x ∈ {0,1}, if we feed the network with the real number 0. x1x. . . xn, then the output of the network will be x.

Hint: Denote α =0. x1x. . . xn and observe that 10kα−0.5 is at least 0.5 if xk =1 and is at most −0.3 if xk =−1.

2. Construct a network, N2, with O(n) weights, which implements a function from [n] to {0,1}such that N2()=efor all . That is, upon receiving the input i, the network outputs the vector of all zeros except 1 at the ’th neuron.

3. Let α1, . . .,αbe real numbers such that every αis of the form 0.a() 1 a() 2 . . .a(, with a(∈ {0,1}. Construct a network, N3, with O(n) weights, which implements a function from [n] to R, and satisfies N2() = αfor every ∈ [n].

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