Coherent semantics

Coherent semantics was invented by Girard in the paper The system F, 15 years later with the objective of building a denotationnal interpretation of second order intuitionnistic logic (aka polymorphic lambda-calculus).

Coherent semantics is based on the notion of stable functions that was initially proposed by Gérard Berry. Stability is a condition on Scott continuous functions that expresses the determinism of the relation between the output and the input: the typical Scott continuous but non stable function is the parallel or because when the two inputs are both set to true, only one of them is the reason why the result is true but there is no way to determine which one.

A further achievement of coherent semantics was that it allowed to endow the set of stable functions from $X$ to $Y$ with a structure of domain, thus closing the category of coherent spaces and stable functions. However the most interesting point was the discovery of a special class of stable functions, linear functions, which was the first step leading to Linear Logic.

Preliminary definitions and notations

There are two equivalent definitions of coherence spaces: the first one, coherent spaces as domains, is interesting from a historical point of view as it emphazises the fact that coherent spaces are particular cases of Scott domains. The second one, coherent spaces as graphs, is the most commonly used and will be our "official" definition in the sequel.

Coherent spaces

As domains

Definition (Coherent space)

A coherent space $X$ is a family of subsets of a given set $\web X$ (the web of $X$ ), which satisfies:

subset closure: if $x\subset y\in X$ then $x\in X$ ,
singletons: $\{a\}\in X$ for
binary compatibility: if $A$ is a family of pairwise compatible elements of $X$ , that is if $x\cup y\in X$ for any $x,y\in A$ , then $\bigcup A\in X$ .

A coherent space is thus ordered by inclusion; one easily checks that it is a domain. In particular finite elements of $X$ correspond to compact elements.

As graphs

Definition (Coherent space)

A coherent space $X$ is a family of subsets of a given set $\web X$ (the web of $X$ ) which satisfies that there is a reflexive and symetric relation $\coh_X$ on $\web X$ (the coherence relation) such that $x\in X$ iff $\forall a,b\in x,\, a\coh_X b$ , for any $x\subset\web X$ . In other terms $X$ is the set of complete subgraphs of the simple unoriented graph of the $\coh_X$ relation. For this reason, the elements of $X$ are called cliques.

The strict coherence relation $\scoh_X$ on $X$ is defined by: $a\scoh_X b$ iff $a\neq b$ and $a\coh_X b$ .

A coherent space in the domain sense is seen to be a coherent space in the graph sense by setting $a\coh_X b$ iff $\{a,b\}\in X$ ; conversely one easily checks that cliques in the graph sense are subset closed and satisfy the binary compatibility condition.

Stable functions

Definition (Stable function)

Let $X$ and $Y$ be two coherent spaces. A function $F:X\mapsto Y$ is stable if it satisfies:

it is non decreasing: for any $x,y\in X$ if $x\subset y$ then $F(x)\subset F(y)$ ;
it is continuous (in the Scott sense): if $A$ is a directed family of cliques of $X$ , that is if for any $x,y\in A$ there is a $z\in A$ such that $x\cup y\subset z$ , then $\bigcup_{x\in A}F(x) = F(\bigcup A)$ ;
it satisfies the stability condition: if $x,y\in X$ are compatible, that is if $x\cup y\in X$ , then $F(x\cap y) = F(x)\cap F(y)$ .

This definition is admitedly not very tractable. An equivalent and most useful caracterisation of stable functions is given by the following theorem.

Theorem

Let $F:X\mapsto Y$ be a non-decreasing function from the coherent space $X$ to the coherent space $Y$ . The function $F$ is stable iff it satisfies: for any $x\in X$ , $b\in\web Y$ , if $b\in F(x)$ then there is a finite clique $x_0\subset x$ such that:

$b\in F(x_0)$ ,
for any $y\subset x$ if $b\in F(y)$ then $x_0\subset y$ ( $x 0$ is the minimum sub-clique of $x$ such that $b\in F(x_0)$ ).

Note that the stability condition doesn't depend on the coherent space structure and can be expressed more generally for continuous functions on domains. However, as mentionned in the introduction, the restriction to coherent spaces allows to endow the set of stable functions from $X$ to $Y$ with a structure of coherent spaces.

Definition (The space of stable functions)

Let $X$ and $Y$ be coherent spaces. The function space $X\rightarrow Y$ is defined by:

$\web{X\rightarrow Y} = \mathcal{C}_{\mathrm{fin}}(X)\times \web Y$ ,
$(x_0, a)\coh_{X\rightarrow Y}(y_0, b)$ iff $\begin{cases}\text{if } x_0\cup y_0\in X\text{ then } a\coh_Y y,\\ \text{if } x_0\cup y_0\in X\text{ and } x_0\neq y_0 \text{ then } a\scoh_Y b\end{cases}$ .