Course-of-values recursion

In computability theory, course-of-values recursion is a technique for defining number-theoretic functions by recursion. In a definition of a function f by course-of-values recursion, the value of f(n) is computed from the sequence . The fact that such definitions can be converted into definitions using a simpler form of recursion is often used to prove that functions defined by course-of-values recursion are primitive recursive. Contrary to course-of-values recursion, in primitive recursion the computation of a value of a function requires only the previous value; for example, for a 1-ary primitive recursive function g the value of g(n+1) is computed only from g(n) and n. The factorial function n! is recursively defined by the rules This recursion is a primitive recursion because it computes the next value (n+1)! of the function based on the value of n and the previous value n! of the function. On the other hand, the function Fib(n), which returns the nth Fibonacci number, is defined with the recursion equations In order to compute Fib(n+2), the last two values of the Fib function are required. Finally, consider the function g defined with the recursion equations To compute g(n+1) using these equations, all the previous values of g must be computed; no fixed finite number of previous values is sufficient in general for the computation of g. The functions Fib and g are examples of functions defined by course-of-values recursion. In general, a function f is defined by course-of-values recursion if there is a fixed primitive recursive function h such that for all n, where is a Gödel number encoding the indicated sequence. In particular provides the initial value of the recursion. The function h might test its first argument to provide explicit initial values, for instance for Fib one could use the function defined by where s[i] denotes extraction of the element i from an encoded sequence s; this is easily seen to be a primitive recursive function (assuming an appropriate Gödel numbering is used).