Closure Operation

2.             Recursion.

Consider the simplest case first. A function h(x₁, x₂) is given, and a number k.   The function f defined by

f(0) = k, f(y+1) = h(y, f(y))

is said to be obtained from h and k by recursion.

Assuming h is defined everywhere then f(y) is well defined for all y.

The next case involves a parameter x.

Two functions g(x) and h(x₁, x₂, x₃) are given: the function f(x₁, x₂) where

f(x, 0) = g(x), f(x, y+1) = h(x, y, f(x, y))

        is said to be obtained from g and h by recursion.

        If g and h are defined everywhere then f(x, y) is defined for all y.

Finally, suppose any number of parameters x₁, ..., x_n. Suppose functions

g(x₁, ..., x_n), h(u₁, ..., u_n+2) are given. The function f defined by

f(x₁, ..., x_n, 0) = g(x₁, …, x_n), f(x₁, ..., x_n, y+l) = h(x₁, ..., x_n, y, f(x₁, ..., x_n, y))

        is said to be obtained from g and h by recursion. In this general definition we

        shall allow n = 0 (i.e. no parameters), with the understanding that a function g of

        no variables means some fixed constant k. (This is to cover the first case above.)