The addition of partial objects has much the same signi cance as in recursion theory, where we shift from total to partial functions: for example, to the integers (represented by singletons) we add the \unde ned" ?.

One should not, however, attach too much importance to this rst intuition. For example, it is misguided to seek to identify the total points of an arbitrary coherence space A. One might perhaps think that the total points of A are the maximal points, i.e. such that:

8 2 jAj (8 0 2 a _^ 0 (mod A)) ) 2 a

which indeed they are | in the simple cases (integers, booleans, etc.). However we would like to de ne totality in coherence spaces which are the interpretations of complex types, using formulae analogous to the ones for reducibility (see 6.1). These are of greater and greater logical complexity4, and altogether unpredictable, whilst the notion of maximality remains desperately 02, so one cannot hope for a coincidence. In fact, for any given coherence space there are many notions of totality, just as there are many reducibility candidates (chapter 14) for the same type. In fact the semantics partialises everything, and the total objects get a bit lost inside it.

The functions from A to B will be seen as functions de ned uniquely by their approximants, and in this way \continuous". Here it is possible to use a topological language where the subsets fa : a ag of A, for a nite, are open. However whereas in Scott-style domain theory the functions between domains are exactly those which are continuous for this topology, this will no longer be so here.

8.3Stable functions

Given two coherence spaces A and B, a function F from A to B is stable if:

i) a0 a 2 A ) F (a0) F (a)

ii) F (S"i2I ai) = S"i2I F (ai) (directed union)

iii) a1 [ a2 2 A ) F (a1 \ a2) = F (a1) \ F (a2) (St)

4The logical complexity of a formula is essentially determined by the number of alternations of quanti ers. In particular, we say that a formula 8x: 9x0: 8x00: : : : P (x; x0; x00; : : :) where P

is a primitive recursive predicate, is of logical complexity 0n, where n is the number of quanti ers. Similarly, 9x: 8x0: 9x00: : : : P (x; x0; x00; : : :) is of logical complexity 0n.

The rst condition says that F preserves approximation: if we provide more information to start o with (a rather than a0) then we get more back at the end. Alternatively, F only uses positive information about its arguments.

The second states continuity:

F (a) = S"fF (a ) : a a; a niteg

This special case of (ii) is in fact equivalent to it.

Considering a coherence space as a category in which the morphisms from a0 to a are inclusions a0 a, the rst condition states that a stable function is a functor and the second that this preserves ltered colimits. These two conditions are entirely familiar from the topological setting; this is no longer true of the last condition | the stability property itself | which has no obvious topological signi cance. It looks a bit peculiar at rst sight, but in terms of categories it just says that the pullback

a1 [ a2

I@

@

@

a1 a2

@I

a1 \ a2

must be preserved. The intention is that this should hold for any set fa1; a2; : : :g which is bounded above, not just nite ones, but in the context of strongly nite approximation (i.e. the fact that the approximating elements have only nitely many elements below them, which is not in general true in Scott's theory) we don't need to say this.

Let us give an example to show that the hypothesis of coherence between a1 and a2 cannot be lifted. We want to be able to represent all functions from N to N as stable functions from Int to Int, in particular f(0) = f(1) = 0, f(n + 2) = 1. This forces F (f0g) = F (f1g) = f0g, F (fn + 2g) = f1g, and by monotonicity, F (?) = ?. Now, F (f0g \ f1g) = F (?) = ? 6= F (f0g) \ F (f1g); we are saved by the incoherence of 0 and 1, which makes f0g [ f1g 2= Int.

We shall see that this property forces the existence of a least approximant in certain cases, simply by taking the intersection of a set which is bounded above.