The definitions of elementary recursive functions are the same as for primitive recursive functions, except that primitive recursion is replaced by bounded summation and bounded product. All functions work over the natural numbers. The basic functions, all of them elementary recursive, are:
- Zero function. Returns zero: f(x) = 0.
- Successor function: f(x) = x + 1. Often this is denoted by S, as in S(x). Via repeated application of a successor function, one can achieve addition.
- Projection functions: these are used for ignoring arguments. For example, f(a, b) = a is a projection function.
- Subtraction function: f(x, y) = x - y if y < x, or 0 if y ≥ x. This function is used to define conditionals and iteration.
From these basic functions, we can build other elementary recursive functions.
- Composition: applying values from some elementary recursive function as an argument to another elementary recursive function. In f(x1, ..., xn) = h(g1(x1, ..., xn), ..., gm(x1, ..., xn)) is elementary recursive if h is elementary recursive and each gi is elementary recursive.
- Bounded summation: is elementary recursive if g is elementary recursive.
- Bounded product: is elementary recursive if g is elementary recursive.
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