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 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 natural numbers are taken to begin with zero, the sequence is instead represented as
- ,
which makes it possible to distinguish the codes for the sequences and .
Read more about this topic: Course-of-values Recursion
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