#+TITLE: HW 07
#+AUTHOR: Elizabeth Hunt (A02364151)
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* Problem One

\begin{verbatim}
1. [A1] Y  <- Y  - 1
2.      IF Y  != 0 GOTO A
3. [B1] IF X1 != 0 GOTO C
4.      GOTO E
5. [C1] X1 <- X1 - 1
6.      Y  <- Y  + 1
7.      Y  <- Y  + 1
8.      Y  <- Y  + 1
9.      GOTO B1
\end{verbatim}

* Problem Two
1. $(1, \sigma) | \sigma = \{X_1 = 2, Y = 0, Z_1 = 0\}$
2. $(4, \sigma) | \sigma = \{X_1 = 2, Y = 0, Z_1 = 0\}$
3. $(5, \sigma) | \sigma = \{X_1 = 1, Y = 0, Z_1 = 0\}$
4. $(6, \sigma) | \sigma = \{X_1 = 1, Y = 1, Z_1 = 0\}$
5. $(7, \sigma) | \sigma = \{X_1 = 1, Y = 1, Z_1 = 1\}$
6. $(1, \sigma) | \sigma = \{X_1 = 1, Y = 1, Z_1 = 1\}$
7. $(4, \sigma) | \sigma = \{X_1 = 1, Y = 1, Z_1 = 1\}$
8. $(5, \sigma) | \sigma = \{X_1 = 0, Y = 1, Z_1 = 1\}$
9. $(6, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 1\}$
10. $(7, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 2\}$
11. $(1, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 2\}$
12. $(2, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 2\}$
13. $(3, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 3\}$
14. $(8, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 3\}$

* Problem Three
\begin{verbatim}
1. [A1] Y  <- Y
2.      Y  <- Y
3.      Y  <- Y
4.      Y  <- Y
5.      Y  <- Y
6.      GOTO E
\end{verbatim}

* Problem Four
Let $P$ be a program in $L$ that computes $g(x_1, x_2, \cdots, x_n)$; a list of instructions $[I_1, I_2, \cdots, I_k]$,
where $I_1$ is the first instruction and $I_k$ the last.

Then, define $P^i | i \in N$ to be a new program such that each instruction $I_n$ replaces $I_{n+i}$ (when $n=0$
we perform no operation), appending
to the end of the instruction list if necessary. We then replace the sublist $[I_1, \cdots, I_i]$ with
$[Y \leftarrow Y]^i$ in the program $P$. As $Y \leftarrow Y$ produces no side effects then $P^i$ still computes $g$.

Finally, for all $i \in N$ the length of $P^i$ is greater than $k$ and thus there are countably infinitely
many $L$ -programs to compute $g$.