8.5. THE FUNCTION-SPACE

If A and B are two coherence spaces, we de ne A N B by:

jA N Bj = jAj + jBj = f1g jAj [ f2g jBj

In particular, the points of A N B (coherent subsets of jA N Bj) can be written uniquely as f1g a [ f2g b with a 2 A, b 2 B. The reader is invited to show that this is the product in the categorical sense (we shall return to this in the next chapter when we de ne the interpretation).

G(f1g a [ f2g b) = F (a; b)

It is immediate that G is stable; conversely the same formula de nes, from a stable unary function G, a stable binary function F , and the two transformations are inverse.

8.5The Function-Space

We started with the idea that \type = coherence space". The previous section de nes a product of coherence spaces corresponding to the product of types, but what do we do with the arrow? We would like to de ne A ! B as the set of stable functions from A to B, but this is not presented as a coherence space. So we shall give a particular representation of the set of stable functions in such a way as to make it a coherence space.

8.5.1The trace of a stable function

Lemma Let F be a stable function from A to B, and let a 2 A, 2 F (a); then

i)it is possible to nd a a nite such that 2 F (a ).

ii)if a is chosen minimal for the inclusion among the solutions to (i), then a is least, and is in particular unique.

The trace Tr(F ) is the set of pairs (a ; ) such that:

i)a is a nite point of A and 2 jBj

ii)2 F (a )

iii)if a0 a and 2 F (a0) then a0 = a .

Tr(F ) determines F uniquely by the formula

(App) F (a) = f : 9a a (a ; ) 2 Tr(F )g

which results immediately from the lemma. In particular the function F 7!rT(F ) is 1{1.

Consider for example the stable function F1 from Bool NBool to Bool introduced in 8.3.2. The elements of its trace Tr(F1) are:

We can read this as the speci cation:

if the rst argument is true, the result is true;

if the rst argument is false and the second true, the result is true;

if the rst argument is false and the second false, the result is false.

8.5.2Representation of the function space

Proposition As F varies over the stable functions from A to B, their traces give the points of a coherence space, written A ! B.

Proof Let us de ne the coherence space C by jCj = An jBj (An is the set ofnite points of A) where (a1; 1) _^ (a2; 2) (mod C) if

i)a1 [ a2 2 A ) 1 _^ 2 (mod B)

ii)a1 [ a2 2 A ^ a1 6= a2 ) 1 6= 2 (mod B)

In 12.3, we shall see a more symmetrical way of writing this.

If F is stable, then Tr(F ) is a subset of jCj by construction. We verify the coherence modulo C of (a1; 1) and (a2; 2) 2 Tr(F ):

i) If a1 [ a2 2 A then f 1; 2g F (a1 [ a2) so 1 _^ 2 (mod B).

ii) If 1 = 2 and a1 [ a2 2 A, then the lemma applied to 1 2 F (a1 [ a2) gives us a1 = a2.

Conversely, let f be a point of C. We de ne a function from A to B by the

iii) If a1 [ a2 2 A, then F (a1 \ a2) F (a1) \ F (a2) by monotonicity. Conversely, if 2 F (a1) \ F (a2), this means that (a01; ); (a02; ) 2 f for some appropriate a01 a1 and a02 a2. But (a01; ) and (a02; ) are coherent and a01 [ a02 a1 [ a2 2 A, so a01 = a02, a01 a1 \ a2 and 2 F (a1 \ a2).

iv) We nearly forgot to show that F maps A into B: F (a), for a 2 A, is a subset of jBj, of which it is again necessary to show coherence! Now, if0; 00 2 F (a), this means that (a0; 0); (a00; 00) 2 f for appropriate a0; a00 a; but then a0 [ a00 a 2 A, so, as (a0; 0) and (a00; 00) are coherent, 0 _^ 00 (mod B).

Finally, it is easy to check that these constructions are mutually inverse.

In fact, the same application formula occurs in Scott's domain theory [Scott76], but the corresponding notion of \trace" is more complicated.