Definition:URM Computability
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Definition
Let $P$ be a URM program, and let $k$ be any positive integer.
Program
$P$ is said to compute the function $f: \N^k \to \N$ if and only if:
- for all ordered $k$-tuples $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$, the computation of a URM using the program $P$ with input $\tuple {n_1, n_2, \ldots, n_k}$ produces the output $\map f {n_1, n_2, \ldots, n_k}$.
If there are any inputs such that either of the following happens:
- the output fails to equal $\map f {n_1, n_2, \ldots, n_k}$
- the program will never terminate,
then the program does not compute the function $f: \N^k \to \N$.
Function
The function $f: \N^k \to \N$ is said to be URM computable if and only if there exists a URM program which computes it.
Partial Function
$P$ is said to compute the partial function $f: \N^k \to \N$ if and only if:
- For all ordered $k$-tuples $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$:
The partial function $f: \N^k \to \N$ is said to be URM computable if there exists a URM program which computes it.
Note that a URM program can be used with any number of input variables. For any positive integer $k$, the input consists of the state of the registers $R_1, R_2, \ldots, R_k$.
Thus a given URM program $P$ computes a partial function $f: \N^k \to \N$ for each positive integer $k$.
In this context, it is convenient to use the notation $f^k_P$ to denote the partial function of $k$ variables computed by $P$.
Set
Let $A \subseteq \N$.
Then $A$ is a URM computable set if and only if its characteristic function $\chi_A$ is a URM computable function.
Relation
Let $\RR \subseteq \N^k$ be an $n$-ary relation on $\N^k$.
Then $\RR$ is a URM computable relation if and only if its characteristic function $\chi_\RR$ is a URM computable function.