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.

Read more about this topic:  Primitive Recursive Function

Famous quotes containing the word definition:

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)

    “Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
    —Nadine Gordimer (b. 1923)