Chapter 15

Representation Theorem

In this chapter we aim to study the \strength" of system F with a view to identifying the class of algorithms which are representable. For example, if f is a closed term of type Int!Int, it gives rise to a function (in the set-theoretic sense) jfj from N to N by

f(n) jfj(n)

The function jfj is recursive, indeed we have a procedure for calculating it, namely:

write the term f(n);

normalise it: any normalisation strategy will do this, since the strong normalisation theorem says that all reduction paths lead to the (same) normal form;

observe that the normal form is a numeral m: we have seen that this is true for system T, and this is also valid for system F, as we shall show next;

put jfj(n) = m.

In the rst part, we shall show that jfj is provably total in second order Peano arithmetic, by close examination of the proof of strong normalisation in the previous chapter.

In the second part, we shall use Heyting's ideas once again, essentially in the form of the realisability method due to Martin-L•of, to show the converse of this, that if a function is provably total then it is representable.

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