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{\it The Final Exam (Neural Networks) ... 18 Jan 2001}\\
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Name [ \hspace{2in} ] \hspace{.1in} Student-ID [ \hspace{1in} ]\\

\begin{itemize}
\item[1.] We have a feed-forward neural network 
with two neurons in input layer, 
one neuron in hidden layer, and one neuron in output layer.
The task is to realize XOR function.
Assume that all the weights are initialized to 0.2 and
$\epsilon=0.05$. Also assume that transfer function $f(x) = 0.1x^3$
and threshold of all the neurons are fixed to zero,
which are just for the sake of simplicity. 
Of course neurons here are all analog ones, not binary ones.
We now start to train the neural network by the Back-propagation algorithm.
Calculate the following quantities to modify the weights 
when $x_1 = -1$, $x_2 = -1$ are given as a first training sample.
The notations are those we used in the course.
\vspace{7in}
\begin{displaymath}
\begin{array}{lllll}
h^2_1=\hspace{.6in}& y^2_1=\hspace{.6in}& h^3_1=\hspace{.6in}& 
z_1=\hspace{.6in}& e_1= \\
e^2_1=\hspace{.6in}&{\delta}^2_1=\hspace{.6in}& 
\triangle w^1_{12}=\hspace{.6in}& w^1_{12}=
\hspace{.6in}& w^2_{11}=
\end{array}
\end{displaymath}
\clearpage
\item[2.]When we are to store two patterns $\{+1 -1 -1 -1\}$ and
$\{-1 +1 -1 +1\}$ to the fully connected neural network of 
the Hopfield model. Calculate the appropriate value of weight $w_{23}$,
following the notations used in the course.
\vspace{3.5in}
\item[3.]Calculate energy of all the possible states of network with
fully connected three neurons with their weights being
$w_{12} = w_{21} = w_{23} = w_{32} = -1$,
$w_{13} = w_{31} = 1$ and with all threshold being $-0.5$.
Then pick up the state(s) of the Garden-of-Eden and 
the stable state(s).

\end{itemize}
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