Primitive recursive function
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In computability theory, 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 and are a strict subset of the recursive functions, which are exactly the computable functions. The broader class of recursive functions are defined by additionally allowing partial functions and introducing an unbounded search operator.
Many of the functions normally studied in number theory, and approximations to real-valued functions, are primitive recursive, such as addition, division, factorial, exponential, finding the nth prime, and so on (Brainerd and Landweber, 1974). In fact, it is difficult to devise a function that is not primitive recursive, although some are known (see e.g. Ackermann function). Thus, by studying them, we can discover properties that have wide-reaching consequences.
Primitive recursive functions can be computed by machines that always halt, while recursive functions require Turing-complete systems.
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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:
- The constant function 0 is primitive recursive.
- The successor function S, which takes one argument and returns the succeeding number as given by the Peano postulates, is primitive recursive.
- The projection functions Pin, 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:
- Composition: Given f, a k-ary primitive recursive function, and k l-ary primitive recursive functions g0,...,gk-1, the composition of f with g0,...,gk-1, i.e. the function h(x0,...,xl-1) = f(g0(x0,...,xl-1),...,gk-1(x0,...,xl-1)), is primitive recursive.
- 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,...,xk-1) = f(x0,...,xk-1) and h(S(n),x0,...,xk-1) = g(h(n,x0,...,xk-1),n,x0,...,xk-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.
Examples
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 f. 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 mathematical induction, and the primitive recursive functions can be extended to operate on other objects such as integers and rational numbers.
Limitations
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 diagonal 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 numerated. 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 function). The numbering is computable in the sense that it can be defined under formal 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) corresponds to the ith unary primitive recursive function being calculated on the number j. We can write this as fi(j).
Now consider the function g(x) = S(fx(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, such as those functions computed by a machine that always halts. 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.
Bibliography
- Brainerd, W.S., Landweber, L.H. (1974), Theory of Computation, Wiley, ISBN 0471095850