Definition:Unlimited Register Machine/Program/Termination

Definition
A URM program terminates when there are no more instructions to execute.

This can happen in either of two ways:
 * $(1): \quad$ If the program executes the last instruction, and this does not involve a Jump to an earlier instruction, the program will stop.
 * $(2): \quad$ If the program executes a Jump instruction to a non-existent instruction, the program will stop.

Exit Jump
Such a Jump instruction is known as an exit jump.

Exit Line
The line on which a particular run of a URM program stops is called the exit line.

Endless Loop
If a URM program, when running, never reaches a state where it terminates, then it is said to be in an endless loop and will never terminate.

Note that whether a program terminates or not may depend on its input.

It may terminate perfectly well for one input, but go into an endless loop on another.

Also known as
When a URM program terminates, it can also be said that it stops or halts.

Input
The input to a URM program is:
 * either an ordered $k$-tuple $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$
 * or a natural number $n \in \N$.

In the latter case, it is convenient to consider a single natural number as an ordered $1$-tuple $\tuple {n_1} \in \N^1 = \N$.

Hence we can discuss inputs to URM programs solely as instances of tuples, and not be concerned with cumbersome repetition for the cases where $k = 1$ and otherwise.

The convention usually used is for a URM program $P$ to start computation with:
 * the input $\left({n_1, n_2, \ldots, n_k}\right)$ in registers $R_1, R_2, \ldots, R_k$
 * $0$ in all other registers used by $P$.

That is, the initial state of the URM is:
 * $\forall i \in \closedint 1 k: r_i = n_i$
 * $\forall i > k: r_i = 0$.

It is usual for the input (either all or part) to be overwritten during the course of the operation of a program. That is, at the end of a program, $R_1, R_2, \ldots, R_k$ are not guaranteed still to contain $n_1, n_2, \ldots, n_k$ unless the program has been explicitly written so as to ensure that this is the case.

Output
At the end of the running of a URM program, the output will be found in register $R_1$.

Null Program
A null program or empty program is a URM program which contains no instructions.