Function Obtained by Minimization from URM Computable Functions
Theorem
Let the function $f: \N^{k+1} \to \N$ be a URM computable function.
Let $g: \N^k \to \N$ be the function obtained by minimization from $f$ thus:
- $g \left({n_1, n_2, \ldots, n_k}\right) \approx \mu y \left({f \left({n_1, n_2, \ldots, n_k}\right) = 0}\right)$.
Then $g$ is also URM computable.
Proof
Let $f: \N^{k+1} \to \N$ be a URM computable function.
Let $P$ be a URM program which computes $f$.
Let $u = \rho \left({P}\right)$ be the number of registers used by $P$.
We can use:
- the registers $R_{u+1}, R_{u+2}, \ldots, R_{u+k}$ to store the input $\left({n_1, n_2, \ldots, n_k}\right)$;
- the register $R_{u+k+1}$ to store the current value of the recursion variable $y$.
We check whether or not $f \left({n_1, n_2, \ldots, n_k}\right) = 0$ by comparing the output from running $P$ (appearing in register $1$ at the end of the running of $P$) with the number in register $R_{u+k+2}$, which remains at $0$ throughout.
The following algorithm can be followed to create a URM program $H$ to compute $h$.
We assume that the input is $\left({n_1, n_2, \ldots, n_k}\right)$, which is held in $R_1, R_2, \ldots, R_k$.
We are to use the following registers:
- $R_{u+1}, R_{u+2}, \ldots, R_{u+k}$ will be used to store the input $\left({n_1, n_2, \ldots, n_k}\right)$ so it does not get overwritten.
- $R_{u+k+1}$ will hold the value of $y$.
- $R_{u+k+2}$ will hold the $0$ throughout.
When $r_{u+k+2} = r_{u+k+1}$, the computation will have ended.
We also define $v = \lambda \left({H}\right)$ to be the number of basic instructions in $H$.
Step | Process | Notes | $\lambda \left({H}\right)$ | |
---|---|---|---|---|
$1$ | Append a Block Copy Program $C \left({1, u, k}\right)$ to $H$^{[1]}. | This stores the input somewhere safe so it can be accessed again later. | $k$ | |
$2$ | Increment the Jumps in $P$ by $k$ lines^{[2]}. Call this amended version $P'$. | As $P$ was written so as to start from line 1, we need to move all the Jumps so as to point to the same lines relative to the start of $P'$. | ||
$3$ | Append $P'$ to $H$. | This will compute $f \left({n_1, n_2, \ldots, n_k, y}\right)$. | $k + s$ | |
$4$ | Append the command $J \left({1, u+k+2, v}\right)$ to $H$. | This Jumps $H$ to the end of the program if the output of $P$ is zero. | $k + s + 1$ | |
$5$ | Append the command $S \left({u + k + 1}\right)$ to $H$. | Increment $y$. | $k + s + 2$ | |
$6$ | Append a Block Copy Program $C \left({u+k+1, 1, k+1}\right)$ to $H$. | Recall the input values, and put $y$ in $R_{k+1}$. | $2 k + s + 3$ | |
$8$ | Append the command $J \left({1, 1, k + 1}\right)$ to $H$. | This makes the program jump back to the start of $P'$. | $2 k + s + 4$ | |
$7$ | Append the command $C \left({1, k+2}\right)$ to $H$. | This copies the output of $P'$ into the last input location for function $g$, that is, the output from $h \left({n_1, n_2, \ldots, n_k, i-1}\right)$. | $2 k + s + 5$ | |
$8$ | Append the command $C \left({u+k+1, 1}\right)$ to $H$. | Put $y$ into the output register. | $v = 2 k + s + 6$ |
It can easily be determined that $H$ computes $g$.
Hence $g$ is URM computable.
$\blacksquare$
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Footnotes
- ↑ $H$, at this point, is a null URM program.
- ↑ To increment the Jumps by $r$ for any normalized URM program is done by changing all Jumps of the form $J \left({m, n, q}\right)$ to $J \left({m, n, q+r}\right)$.