Models of Computation Primitive Recursive Functions

1 The primitive recursive functions 1.1 Intuitive syntax and semantics In informal mathematical notation we often define the addition function in the following way: 0 + n = n m + n = (m + n) We have used the notation m for the successor of the number m. In a functional language we can use a similar definition: add 0 n = n add (s m) n = s (add m n) We know that this is a meaningful definition since the addition function for the argument s n is defined using the value of the function for the argument n. This kind of recursion is well defined since n is smaller than s n. The recursion scheme is called primitive recursion. In the simple case that the function being defined has only one argument the scheme looks like: f(0) = g f(y + 1) = h(y, f(y)) where g is a natural number and h is a given primitive recursive function of two arguments. We notice that in order to define what a primitive recursive function of one argument is, we have to know what a primitive recursive function of two arguments is. We therefore have to generalize and define what a primitive function 1