Definition:Unlimited Register Machine/Register

Definition

A URM has a sequence of registers which can store natural numbers: $\set {0, 1, 2, \ldots}$.

Any given URM program may make use of only a finite number of these registers.

Registers are usually referred to by the subscripted uppercase letters $R_1, R_2, R_3, \ldots$.

The number held at any one time by a register is usually referred to by the corresponding lowercase letter $r_1, r_2, r_3, \ldots$.

The registers are unlimited in the following two senses:

$(1): \quad$ Although a URM program may make use of only a finite number of registers, there is no actual upper bound on how many a particular URM program can actually use.

$(2): \quad$ There is no upper bound on the size of the natural numbers that may be stored in any register.

The subscript (which is a natural number) appended to a URM register is called the index of that register.

Hence, for example, the index of register $R_5$ is $5$.

1963: John C. Shepherdson and H.E. Sturgis: Computability of Recursive Functions (J. ACM Vol. 10, no. 2: pp. 217 – 255)