1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

## 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, *N*1, with *O*(*n*) weights, which implements a function from R to {0*,*1}*n *and satisfies the following property. For every x ∈ {0*,*1}*n *, if we feed the network with the real number 0. *x*1*x*2 *. . . **xn*, then the output of the network will be x.

*Hint: *Denote *α *=0. *x*1*x*2 *. . . **xn *and observe that 10*k**α*−0.5 is at least 0.5 if *xk *=1 and is at most −0.3 if *xk *=−1.

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

3. Let *α*1*, . . .,α**n *be *n *real numbers such that every *α**i *is of the form 0.*a*(*i *) 1 *a*(*i *) 2 *. . .**a*(*i *) *n *, with *a*(*i *) *j *∈ {0*,*1}. Construct a network, *N*3, with *O*(*n*) weights, which implements a function from [*n*] to R, and satisfies *N*2(*i *) = *α**i *for every *i *∈ [*n*].