Primitive Recursive Function - Definition

Definition

The primitive recursive functions are among the number-theoretic functions, which are functions from the natural numbers (nonnegative integers) {0, 1, 2, ...} to the natural numbers. These functions take n arguments for some natural number n and are called n-ary.

The basic primitive recursive functions are given by these axioms:

  1. Constant function: The 0-ary constant function 0 is primitive recursive.
  2. Successor function: The 1-ary successor function S, which returns the successor of its argument (see Peano postulates), is primitive recursive. That is, S(k) = k + 1.
  3. Projection function: For every n≥1 and each i with 1≤i≤n, the n-ary projection function Pin, which returns its i-th argument, is primitive recursive.

More complex primitive recursive functions can be obtained by applying the operations given by these axioms:

  1. Composition: Given f, a k-ary primitive recursive function, and k m-ary primitive recursive functions g1,...,gk, the composition of f with g1,...,gk, i.e. the m-ary function is primitive recursive.
  2. Primitive recursion: Given f, a k-ary primitive recursive function, and g, a (k+2)-ary primitive recursive function, the (k+1)-ary function h is defined as the primitive recursion of f and g, i.e. the function h is primitive recursive when
    and

The primitive recursive functions are the basic functions and those obtained from the basic functions by applying these operations a finite number of times.

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