Primitive recursive function

Primitive recursive functions are a class of functions which form an important building block on the way to a full formalization of computability. They are defined using recursion and composition as central operations. The primitive recursive functions are a strict subset of the recursive functions (which are exactly those functions which we call "computable"; see Church-Turing thesis).

Definition

Primitive recursive functions take natural numbers or tuples of natural numbers as arguments and produce a natural number. A function which takes n arguments is called n-ary. The basic primitive recursive functions are given by these axioms:

1. The constant function 0 is primitive recursive.
2. The successor function S, which takes one argument and returns the succeeding number as given by the Peano postulates, is primitive recursive.
3. The projection functions P_iⁿ, which take n arguments and return their ith argument, are primitive recursive.

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

1. Composition: Given f, a k-ary primitive recursive function, and k l-ary primitive recursive functions g₀,...,g_k-1, the composition of f with g₀,...,g_k-1, i.e. the function h(x₀,...,x_l-1) = f(g₀(x₀,...,x_l-1),...,g_k-1( x₀,...,x_l-1)), 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 defined as the primitive recursion of f and g, i.e. the function h where h(0,x₀,...,x_k-1) = f(x₀,...,x_k-1) and h(S(n),x₀,...,x_k-1) = g(h(n,x₀,...,x_k-1),n,x₀,..., x_k-1), is primitive recursive.

(Note that the projection functions allow us to get around the apparent rigidity in terms of the arity of the functions above, as via composition we can pass any subset of the arguments.)

A function is primitive recursive if it is one of the basic functions above, or can be obtained from one of the basic functions by applying the operations a finite number of times.

Example primitive recursive function definitions

Addition: Intuitively we would like to define addition recursively as:

add(0,x)=x

add(n+1,x)=add(n,x)+1

In order to fit this into a strict primitive recursive definition, we define:

add(0,x)=P₁¹(x)

add(S(n),x)=S(P₁³(add(n,x),n,x))

(Note: here P₁³ is a function, which takes 3 arguments and returns the first one.)

Note that P₁¹ is simply the identity function; its inclusion is required by the definition of the primitive recursion operator above; it plays the role of h. The composition of S and P₁³, which is primitive recursive, plays the role of g.

Subtraction: We can define limited subtraction, i.e. subtraction that bottoms out at 0 (since we have no concept of negative numbers yet). First we must define the "predecessor" function, which acts as the opposite of the successor function.

Intuitively we would like to define predecessor as:

pred(0)=0

pred(n+1)=n

To fit this in to a formal primitive recursive definition, we write:

pred(0)=0

pred(S(n))=P₂²(pred(n),n)

Now we can define subtraction in a very similar way to how we defined addition.

sub(0,x)=P₁¹(x)

sub(S(n),x)=pred(P₁³(sub(n,x),n,x))

(Note that for the sake of simplicity, the order of the arguments has been switched from the "standard" definition to fit the requirements of primitive recursion, i.e. sub(a,b) corresponds to b-a. This could easily be rectified using composition with suitable projections.)

Many other familiar functions can be shown to be primitive recursive; some examples include conditionals, exponentiation, primality testing, and course-of-values induction, and the primitive recursive functions can be extended to operate on other objects such as integers and rational numbers.

Limitations of the primitive recursive functions

Primitive recursive functions tend to correspond very closely with our intuition of what a computable function must be. Certainly the initial set of functions are intuitively computable (in their very simplicity), and the two operations by which one can create new primitive recursive functions are also very straightforward. However the set of primitive recursive functions does not include every possible computable function --- this can be seen with a variant of Cantor's diagonalization argument. This argument provides a computable function which is not primitive recursive. A sketch of the proof is as follows:

The primitive recursive functions can be computably numbered. This numbering is unique on the definitions of functions, though not unique on the actual functions themselves (as every function can have an infinite number of definitions --- consider simply composing by the identity operator). The numbering is computable in the sense that it can be defined under format models of computability such as recursive functions or Turing machines; but an appeal to the Church-Turing thesis is likely sufficient.

Now consider a matrix where the rows are the primitive recursive functions of one argument under this numbering, and the columns are the natural numbers. Then each element (i, j) correponds to the ith unary primitive recursive function being calculated on the number j. We can write this as f_i(j).

Now consider the function g(x)=S(f_x(x)). g lies on the diagonal of this matrix and simply adds one to the value it finds. This function is computable (by the above), but clearly no primitive recursive function exists which computes it as it differs from each possible primitive recursive function by at least one value. Thus there must be computable functions which are not primitive recursive.

Note that this argument can be applied to any class of computable (total) functions that can be enumerated in this way. Therefore, any such explicit list of computable (total) functions can never be complete. Note however that the partial computable functions (those which need not be defined for all arguments) can be explicitly enumerated, for instance by enumerating Turing machine encodings.

One can also explicitly exhibit a simple 1-ary computable function which is recursively defined for any natural number, but which is not primitive recursive, see Ackermann function.

Relation to the recursive functions

The set of primitive recursive functions does not encompass everything that we think of as computable. Nevertheless they form an important class and many of the functions normally studied in number theory, and approximations to real-valued functions, are primitive recursive.

In order to formalize the full class of computable functions, we must allow for partial functions and introduce an additional operator to the above: the unbounded search operator (see Recursive function).