8.2Coherence Spaces

A coherence space3 is a set (of sets) A which satis es:

i) Down-closure: if a 2 A and a0 a, then a0 2 A.

ii) Binary completeness: if M A and if 8a1; a2 2 M (a1 [ a2 2 A), then

SM 2 A.

In particular, we have the unde ned object, ? 2 A.

The reader may consider a coherence space as a \domain" (partially ordered by inclusion); as such it is algebraic (any set is the directed union of its nite subsets) and satis es the binary condition (ii), so that

are (very basic) coherence spaces, respectively called Bool and Int, but

?

is not. However we shall see that coherence spaces are better regarded as undirected graphs.

8.2.1The web of a coherence space

def S

Consider jAj = A = f : f g 2 Ag. The elements of jAj are called tokens, and the coherence relation modulo A is de ned between tokens by

which is a re exive symmetric relation, so jAj equipped with _^ is a graph, called the web of A.

3The term espace coherent is used in the French text, and indeed Plotkin has also used the word coherent to refer to this binary condition. However coherent space is established, albeit peculiar, usage for a space with a basis of compact open sets, also called a spectral space. Consequently, the term was modi ed in translation.

For example, the web of Bool consists of the tokens t and f, which are incoherent; similarly the web of Int is a discrete graph whose points are the integers. Such domains we call at.

The construction of the web of a coherence space is a bijection between coherence spaces and (re exive-symmetric) graphs. From the web we recover the coherence space by:

a 2 A , a jAj ^ 8 1; 2 2 a ( 1 _^ 2 (mod A))

So in the terminology of Graph Theory, a point is exactly a clique, i.e. a complete subgraph.

8.2.2Interpretation

The aim is to interpret a type by a coherence space A, and a term of this type by a point of A (coherent subset of jAj, in nite in general: we write An for the set of nite points).

To work in an e ective manner with points of A, it is necessary to introduce a notion of nite approximation. An approximant of a 2 A is any subset a0 of a. Condition (i) for coherence spaces ensures that approximants are still in A. Above all, there are enough nite approximants to a:

a is the union of its set of nite approximants.

The set I of nite approximants is directed. In other words,

i)I is nonempty (? 2 I).

ii)If a0; a00 2 I, one can nd a 2 I such that a0; a00 a (take a = a0 [ a00).

This comes from the following idea:

On the one hand we have the true (or total) objects of A. For example, in Bool, the singletons ftg and ffg, in Int, f0g, f1g, f2g, etc.

On the other hand, the approximants, of which, in the two simplistic cases considered, ? is the only example. They are partial objects.