Course-of-values recursion

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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+1) 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.

Definition and examples[edit]

The factorial function n! is recursively defined by the rules

0! = 1,
(n+1)! = (n+1)*(n!).

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

Fib(0) = 0,
Fib(1) = 1,
Fib(n+2) = Fib(n+1) + Fib(n).

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

g(0) = 0,
.

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).

Equivalence to primitive recursion[edit]

In order to convert a definition by course-of-values recursion into a primitive recursion, an auxiliary (helper) function is used. Suppose that one wants to have

.

To define f using primitive recursion, first define the auxiliary course-of-values function that should satisfy

where the right hand side is taken to be a Gödel numbering for sequences.

Thus encodes the first n values of f. The function can be defined by primitive recursion because is obtained by appending to the new element :

,

where append(n,s,x) computes, whenever s encodes a sequence of length n, a new sequence t of length n + 1 such that t[n] = x and t[i] = s[i] for all i < n. This is a primitive recursive function, under the assumption of an appropriate Gödel numbering; h is assumed primitive recursive to begin with. Thus the recursion relation can be written as primitive recursion:

where g is itself primitive recursive, being the composition of two such functions:

Given , the original function f can be defined by , which shows that it is also a primitive recursive function.

Application to primitive recursive functions[edit]

In the context of primitive recursive functions, it is convenient to have a means to represent finite sequences of natural numbers as single natural numbers. One such method, Gödel's encoding, represents a sequence of positive integers as

,

where pi represent the ith prime. It can be shown that, with this representation, the ordinary operations on sequences are all primitive recursive. These operations include

  • Determining the length of a sequence,
  • Extracting an element from a sequence given its index,
  • Concatenating two sequences.

Using this representation of sequences, it can be seen that if h(m) is primitive recursive then the function

.

is also primitive recursive.

When the sequence is allowed to include zeros, it is instead represented as

,

which makes it possible to distinguish the codes for the sequences and .

Limitations[edit]

Not every recursive definition can be transformed into a primitive recursive definition. One known example is Ackermann's function, which is of the form A(m,n) and is provably not primitive recursive.

Indeed, every new value A(m+1, n) depends on the sequence of previously defined values A(i, j), but the i-s and j-s for which values should be included in this sequence depend themselves on previously computed values of the function; namely (i, j) = (m,A(m+1,n)). Thus one cannot encode the previously computed sequence of values in a primitive recursive way in the manner suggested above (or at all, as it turns out this function is not primitive recursive).

References[edit]

  • Hinman, P.G., 2006, Fundamentals of Mathematical Logic, A K Peters.
  • Odifreddi, P.G., 1989, Classical Recursion Theory, North Holland; second edition, 1999.