#+TITLE: HW 08
#+AUTHOR: Elizabeth Hunt (A02364151)
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* Problem One
Let $m$ be the number of statements that don't produce an effect on $Y$ i.e.
$\{ Y \leftarrow Y, X1 \leftarrow X1 + 1, \cdots \}$, $n$ be the number of statements $Y \leftarrow Y - 1$, and
$o$ be the number of statements $Y \leftarrow Y + 1$. A straightline program of length $k$
must have $m + n + o = k$ instructions. Then, the terminal snapshot ends with
$Y = o - n + 0m$. Thus, $Y$ is at a maximum if and only if $o = k$, thus $Y = k$,
and at a minimum if and only if $o = n$, thus $Y = 0$ (we cannot have $n > o$ as
by definition of $L$, for a valid program $P$ $Y$ is a non-negative number).

* Problem Two
Let $P$ be a program in $L$ that computes $f$. Then $f$ is partially computable
by definition, and is partially computable in $L++$ trivially since $L++$ is a
"superset" of $L$. ($L++ \Leftarrow L$).

Then let $Q$ be a program in $L++$ that computes $f$, $f$ is partially computable
if we can convert $Q$ into $L$. For each instruction $i \in Q$, if $i$ is in the form
$V \leftarrow k | k \in N$ then we replace $i$ with the sequence of instructions and unique labels
(with the notation \cdots representing a reapeating instruction until the instruction is the
following index), else write $i$ ($L++ \Rightarrow L$).

\begin{verbatim}
1. [A1] IF V != 0 GOTO B1
2.      GOTO C1
3. [B1] V <- V - 1
4.      GOTO A1
5. [C1] V <- V + 1
...     V <- V + 1
(5 + k - 1). V <- V + 1
\end{verbatim}

And thus any program in $L++$ is also partially computable.

* Problem Three
For $n=0$, $i(n, x) = f^0(x) = x$ is primitive recursive since it can be represented as
a projection function $u_1^1(x) = x$. If $f^n$ is primitive recursive, then so is $f^{n+1}$, as $f^{n+1}$
is a finite composition $f(f^n)$. By induction, for any $n \in N$, $i(n, x)$ is primitive recursive, and thus computable.

* Problem Four
** One
Let $m \in N$ represent the number of compositions in a function such that the set containing all $n$ -ary
functions with "composition number" $m$ are of the form

$f(x_1, \cdots, x_n) = g(f_1(x_1, \cdots, x_n), \cdots, f_j(x_1, \cdots, x_n))$
where each $f_i$ is in the set of functions with a "composition number" $\le m - 1$. Except for the case in which
$m = 0$ - the initial functions; which are all of the form $k$ or $x_i + k$.

Then, let $F^m | m \in N$ represent the set of all functions of composition number $m$ or less,
and assume that $\forall f \in F^m$ are of either form $k$ or $x_i + k$. Then, $g \in F^{m+1} | m \in N$'s elements
are $g(x_1, \cdots, x_n) = f(f_1(x_1, \cdots, x_n), \cdots, f_j(x_1, \cdots, x_n))$ where $f$ is an initial function, by definition, and each
$f_i | 1 \le i \le j \in F^m$.
1. If $f$ is the Zero function $g$ is in the form $k = 0$
2. If $f$ is the Successor function $g$ is some $f_i(\cdots) + 1$ which by assumption is thus $(x_i + k) + 1 \Rightarrow x_i + k$ or $k + 1 \Rightarrow k$.
3. If $f$ is the Projection function $g$ is some $f_i(\cdots)$ which by assumption is $(x_i + k)$ or $k$.

Finally, by induction, $\forall f(x_1, \cdots, x_n) \in$ ~COMP~, $f$ is thus in the form $k$ or $x_i + k | 1 \le i \le n$.

** Two
By what we showed in One, $f(x_1, \cdots, x_n)$ is dependent on a single $x_i \in {x_1, \cdots, x_n}$ and produces $r = (x_i + k)$ or $r = k$. Then $f(y_1, \cdots, y_n) = (y_i + k) | k$
for the same $i$. So if for all $i | 1 \le i \le n$, $x_i \le y_i$ then $(y_i + k) \ge (x_i + k)$ or $k = k$. Thus in both cases $f(y_1, \cdots, y_n) \ge f(x_1, \cdots, x_n)$
and is thus monotonic.

** Three
Yes. Firstly, every $f \in$ ~COMP~ is primitive recursive, just without the recursion, so it is a subset of the set of all primitive recursive
functions.

Consider the (proven in class) primitive recursive function $f(x_1, x_2) = x_1 * x_2$, and the equivalent $g \in$ ~COMP~. $g(x_1, x_2)$ is equivalent to
$x_1 + k$, $x_2 + k$, or $k$. Then there all elements of $\{f(x_1, x_2 + 1), f(x_1 + 1, x_2), f(x_1, x_2)\} | x_1, x_2 > 1$ cannot be present in
$\{g(x_1, x_2 + 1), g(x_1 + 1, x_2), g(x_1, x_2)\}$ for the same $x_1, x_2$ due to the dependence relation, which is a contradiction to $f$ and $g$
being equivalent.

** Four
Yes. By the fact that all primitive recursive functions are computable, ~COMP~ being a set of primitive recursive functions from Three, is a subset of computable functions.
However, we also showed in Three that there exists a computable (primitive recursive) function which is not in ~COMP~, so ~COMP~ is not equivalent to the set
of all computable functions.

* Problem Five
1. Let $a(x_1, x_2) = x_1 + x_2$ and $m(x_1, x_2) = x_1 * x_2$ which are primitive recursive by proofs in class.
2. Let $t(x_1)$ be the composition of the successor function $s$ 10 times on the zero function $z$ function: $t(x_1) = s(s(\cdots(n(x_1))))$, and is primitive recursive.
3. Let $v(x_1)$ be $v(x_1) = a(x_1, a(x_1, a(x_1, a(x_1, u_1^1(x_1)))))$, which is primitive recursive.
4. Let $w(x_1)$ be $w(x_1) = a(t(x_1), v(x_1))$, which is primitive recursive.
5. Let $y(x_1)$ be $y(x_1) = m(m(x_1, x_1), s(s(n(x_1))))$, which is primitive recursive.
6. Then $f(x_1)$ is $a(w(x_1), y(x_1))$, which is primitive recursive.