Course-of-values recursion

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 $\langle f(1),f(2),\ldots ,f(n)\rangle$ .

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

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,

g(n+1)=\sum _{{i=0}}^{{n}}g(i)^{{n-i}}

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,

f(n)=h(n,\langle f(0),f(1),\ldots ,f(n-1)\rangle )

where $\langle f(0),f(1),\ldots ,f(n-1)\rangle$ is a Gödel number encoding the indicated sequence. In particular

f(0)=h(0,\langle \rangle ),

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

h(n,s)={\begin{cases}n&{\text{if }}n<2\\s[n-2]+s[n-1]&{\text{if }}n\geq 2\end{cases}}

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

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

f(n)=h(n,\langle f(0),f(1),\ldots ,f(n-1)\rangle )

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

{\bar {f}}(n)=\langle f(0),f(1),\ldots ,f(n-1)\rangle

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

Thus ${\bar {f}}(n)$ encodes the first $n$ values of $f$ . The function ${\bar {f}}$ can be defined by primitive recursion because ${\bar {f}}(n+1)$ is obtained by appending to ${\bar {f}}(n)$ the new element $h(n,{\bar {f}}(n))$ :

{\bar {f}}(0)=\langle \rangle

{\bar {f}}(n+1)={\mathit {append}}(n,{\bar {f}}(n),h(n,{\bar {f}}(n))),

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:

{\bar {f}}(n+1)=g(n,{\bar {f}}(n))

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

g(i,j)={\mathit {append}}(i,j,h(i,j)),

Given ${\bar {f}}$ , the original function $f$ can be defined by $f(n)={\bar {f}}(n+1)[n]$ , which shows that it is also a primitive recursive function.

Application to primitive recursive functions

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 $\langle n_{0},n_{1},n_{2},\ldots ,n_{k}\rangle$ as

\prod _{i=0}^{k}p_{i}^{n_{i}}

where p_i 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

f(n)=h(\langle f(0),f(1),f(2),\ldots ,f(n-1)\rangle )

is also primitive recursive.

When the sequence $\langle n_{0},n_{1},n_{2},\ldots ,n_{k}\rangle$ is allowed to include zeros, it is instead represented as

\prod _{i=0}^{k}p_{i}^{(n_{i}+1)}

which makes it possible to distinguish the codes for the sequences $\langle 0\rangle$ and $\langle 0,0\rangle$ .

Limitations

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 resursive.

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

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

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