# Coherent semantics

*Coherent semantics* was invented by Girard in the paper *The system F, 15 years later*^{[1]}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.

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## The cartesian closed structure of coherent semantics

There are three equivalent definitions of coherent 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. The last one, *cliqued spaces* is a particular example of a more general scheme that one could call "symmetric reducibility"; this scheme is underlying lots of constructions in linear logic such as phase semantics or the proof of strong normalisation for proof-nets.

### Coherent spaces

A coherent space *X* is a collection of subsets of a set satisfying some conditions that will be detailed shortly. The elements of *X* are called the *cliques* of *X* (for reasons that will be made clear in a few lines). The set is called the *web* of *X* and its elements are called the *points* of *X*; thus a clique is a set of points. Note that the terminology is a bit ambiguous as the points of *X* are the elements of the web of *X*, not the elements of *X*.

The definitions below give three equivalent conditions that have to be satisfied by the cliques of a coherent space.

#### As domains

The cliques of *X* have to satisfy:

- subset closure: if then ,
- singletons: for .
- binary compatibility: if
*A*is a family of pairwise compatible cliques of*X*, that is if for any , then .

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

#### As graphs

There is a reflexive and symetric relation on (the *coherence relation*) such that any subset *x* of is a clique of *X* iff . In other terms *X* is the set of complete subgraphs of the simple unoriented graph of the relation; this is the reason why elements of *X* are called *cliques*.

The *strict coherence relation* on *X* is defined by: iff and .

A coherent space in the domain sense is seen to be a coherent space in the graph sense by setting iff ; conversely one can check that cliques in the graph sense are subset closed and satisfy the binary compatibility condition.

A coherent space is completely determined by its web and its coherence relation, or equivalently by its web and its strict coherence.

#### As cliqued spaces

**Definition** (Duality)

Let be two sets. We will say that they are dual, written if their intersection contains at most one element: . As usual, it defines an orthogonality relation over .

The last way to express the conditions on the cliques of a coherent space *X* is simply to say that we must have .

#### Equivalence of definitions

Let *X* be a cliqued space and define a relation on by setting iff there is such that . This relation is obviously symetric; it is also reflexive because all singletons belong to *X*: if then {*a*} is dual to any element of (actually {*a*} is dual to any subset of ), thus {*a*} is in , thus in *X*.

Let . Then ; indeed there is an such that . This *x* is dual to any , that is meets any in a most one point. Since this is also true of {*a*,*b*}, so that {*a*,*b*} is in thus in *X*.

Now let *x* be a clique for and *y* be an element of . Suppose , then since *a* and *b* are coherent (by hypothesis on *x*) we have and since we must have that {*a*,*b*} and *y* meet in at most one point. Thus *a* = *b* and we have shown that *x* and *y* are dual. Since *y* was arbitrary this means that *x* is in , thus in *X*. Finally we get that any set of pairwise coherent points of *X* is in *X*. Conversely given its points are obviously pairwise coherent so eventually we get that *X* is a coherent space in the graph sense.

Conversely given a coherent space *X* in the graph sense, one can check that it is a cliqued space. Call *anticlique* a set of pairwise incoherent points: for all *a*,*b* in *y*, if then *a* = *b*. Any anticlique intersects any clique in at most one point: let *x* be a clique and *y* be an anticlique, then if , since we have and since *y* is an anticlique we have *a* = *b*. Thus . Conversely given any and , suppose . Then , thus which entails that {*a*,*b*} has at most one point so that *a* = *b*: we have shown that any two elements of *y* are incoherent.

Thus the collection of anticliques of *X* is the dual of *X*. Note that the incoherence relation defined above is reflexive and symetric, so that is a coherent space in the graph sense. Thus we can do for exactly what we've just done for *X* and consider the anti-anticliques, that is the anticliques for the incoherent relation which are the cliques for the in-incoherent relation. It is not difficult to see that this in-incoherence relation is just the coherence relation we started with; we thus obtain that , so that *X* is a cliqued space.

### Stable functions

**Definition** (Stable function)

Let *X* and *Y* be two coherent spaces. A function is *stable* if it satisfies:

- it is non decreasing: for any if then ;
- it is continuous (in the Scott sense): if
*A*is a directed family of cliques of*X*, that is if for any there is a such that , then ; - it satisfies the stability condition: if are compatible, that is if , then .

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

**Theorem**

Let 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 , , if then there is a finite clique such that:

- ,
- for any if then (
*x*_{0}is*the*minimum sub-clique of*x*such that ).

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 space.

**Definition** (The space of stable functions)

Let *X* and *Y* be coherent spaces. We denote by *X*_{fin} the set of *finite* cliques of *X*. The function space is defined by:

- ,
- iff .

One could equivalently define the strict coherence relation on by: iff when then (equivalently or ).

**Definition** (Trace of a stable function)

Let be a function. The *trace* of *F* is the set:

.

**Theorem**

*F* is stable iff Tr(*F*) is a clique of the function space

In particular the continuity of *F* entails that if *x*_{0} is minimal such that , then *x*_{0} is finite.

**Definition** (The evaluation function)

Let *f* be a clique in . We define a function by: .

**Theorem** (Closure)

If *f* is a clique of the function space then we have . Conversely if is a stable function then we have .

### Cartesian product

**Definition** (Cartesian product)

Let *X*_{1} and *X*_{2} be two coherent spaces. We define the coherent space (read *X*_{1} *with* *X*_{2}):

- the web is the disjoint union of the webs: ;
- the coherence relation is the serie composition of the relations on
*X*_{1}and*X*_{2}: iff either or*i*=*j*and .

This definition is just the way to put a coherent space structure on the cartesian product. Indeed one easily shows the

**Theorem**

Given cliques *x*_{1} and *x*_{2} in *X*_{1} and *X*_{2}, we define the subset of by: . Then is a clique in .

Conversely, given a clique , for *i* = 1,2 we define . Then π_{i}(*x*) is a clique in *X*_{i} and the function is stable.

Furthemore these two operations are inverse of each other: and . In particular any clique in is of the form .

Altogether the results above (and a few other more that we shall leave to the reader) allow to get:

**Theorem**

The category of coherent spaces and stable functions is cartesian closed.

In particular this means that if we define by: then Eval is stable.

## The monoidal structure of coherent semantics

### Linear functions

**Definition** (Linear function)

A function is *linear* if it is stable and furthemore satisfies: for any family *A* of pairwise compatible cliques of *X*, that is such that for any , , we have .

In particular if we take *A* to be the empty family, then we have .

The condition for linearity is quite similar to the condition for Scott continuity, except that we dropped the constraint that *A* is *directed*. Linearity is therefore much stronger than stability: most stable functions are not linear.

However most of the functions seen so far are linear. Typically the function is linear from wich one may deduce that the *with* construction is also a cartesian product in the category of coherent spaces and linear functions.

As with stable function we have an equivalent and much more tractable caracterisation of linear function:

**Theorem**

Let be a continuous function. Then *F* is linear iff it satisfies: for any clique and any there is a unique such that .

Just as the caracterisation theorem for stable functions allowed us to build the coherent space of stable functions, this theorem will help us to endow the set of linear maps with a structure of coherent space.

**Definition** (The linear functions space)

Let *X* and *Y* be coherent spaces. The *linear function space* is defined by:

- ,
- iff

Equivalently one could define the strict coherence to be: iff entails .

**Definition** (Linear trace)

Let be a function. The *linear trace* of *F* denoted as LinTr(*F*) is the set:

such that .

**Theorem**

If *F* is linear then LinTr(*F*) is a clique of .

**Definition** (Evaluation of linear function)

Let *f* be a clique of . We define the function by: such that there is an satisfying .

**Theorem** (Linear closure)

Let *f* be a clique in . Then we have . Conversely if is linear then we have .

It remains to define a tensor product and we will get that the category of coherent spaces with linear functions is monoidal symetric (it is actually *-autonomous).

### Tensor product

**Definition** (Tensor product)

Let *X* and *Y* be coherent spaces. Their tensor product is defined by: and iff and .

**Theorem**

The category of coherent spaces with linear maps and tensor product is monoidal symetric closed.

The closedness is a consequence of the existence of the linear isomorphism:

that is defined by its linear trace: .

### Linear negation

**Definition** (Linear negation)

Let *X* be a coherent space. We define the *incoherence relation* on by: iff entails *a* = *b*. The incoherence relation is reflexive and symetric; we call *dual* or *linear negation* of *X* the associated coherent space denoted , thus defined by: and iff .

The cliques of are called the *anticliques* of *X*. As seen in the section on cliqued spaces we have .

**Theorem**

The category of coherent spaces with linear maps, tensor product and linear negation is *-autonomous.

This is in particular consequence of the existence of the isomorphism:

defined by its linear trace: .

## Exponentials

In linear algebra, bilinear maps may be factorized through the tensor product. Similarly there is a coherent space that allows to factorize stable functions through linear functions.

**Definition** (Of course)

Let *X* be a coherent space; recall that *X*_{fin} denotes the set of finite cliques of *X*. We define the space (read *of course X*) by: and iff is a clique of

*X*.

Thus a clique of is a set of finite cliques of *X* the union of wich is a clique of *X*.

**Theorem**

Let *X* be a coherent space. Denote by the stable function whose trace is: . Then for any coherent space *Y* and any stable function there is a unique *linear* function such that .

Furthermore we have .

**Theorem** (The exponential isomorphism)

Let *X* and *Y* be two coherent spaces. Then there is a linear isomorphism:

.

The iso is defined by its trace: .

This isomorphism, that sends an additive structure (the web of a with is obtained by disjoint union) onto a multiplicative one (the web of a tensor is obtained by cartesian product) is the reason why the of course is called an *exponential*.

## Dual connectives and neutrals

By linear negation all the constructions defined so far () have a dual.

### The direct sum

The dual of is defined by: . An equivalent definition is given by: and .

**Theorem**

Let *x*' be a clique of ; then *x*' is of the form where , or .

Denote the function defined by and by the function defined by . Then inl and inr are linear.

If and are *linear* functions then the function defined by *H*(inl(*x*)) = *F*(*x*) and *H*(inr(*y*)) = *G*(*y*) is linear.

In other terms is the direct sum of *X* and *Y*. Note that in the theorem all functions are *linear*. Things doesn't work so smoothly for stable functions. Historically it was after noting this defect of coherent semantics w.r.t. the intuitionnistic implication that Girard was leaded to discover linear functions.

### The par and the why not

We now come to the most mysterious constructions of coherent semantics: the duals of the tensor and the of course.

The *par* is the dual of the tensor, thus defined by: . From this one can deduce the definition in graph terms: and iff or . With this definition one sees that we have:

for any coherent spaces *X* and *Y*. This equation can be seen as an alternative definition of the par: .

Similarly the dual of the of course is called *why not* defined by: . From this we deduce the definition in the graph sense which is a bit tricky: is the set of finite anticliques of *X*, and given two finite anticliques *x* and *y* of *X* we have iff there is and such that .

Note that both for the par and the why not it is much more convenient to define the strict coherence than the coherence.

With these two last constructions, the equation between the stable function space, the of course and the linear function space may be written:

.

### One and bottom

Depending on the context we denote by or the coherent space whose web is a singleton and whose coherence relation is the trivial reflexive relation.

**Theorem**

is neutral for tensor, that is, there is a linear isomorphism .

Similarly is neutral for par.

### Zero and top

Depending on the context we denote by or the coherent space with empty web.

**Theorem**

is neutral for the direct sum , is neutral for the cartesian product .

*Remark:*
It is one of the main defect of coherent semantics w.r.t. linear logic that it identifies the neutrals: in coherent semantics and . However there is no known semantics of LL that solves this problem in a satisfactory way.

## After coherent semantics

Coherent semantics was an important milestone in the modern theory of logic of programs, in particular because it leaded to the invention of Linear Logic, and more generally because it establishes a strong link between logic and linear algebra; this link is nowadays aknowledged by the customary use of monoidal categories in logic. In some sense coherent semantics is a precursor of many forthcoming works that explore the linear nature of logic as for example geometry of interaction which interprets proofs by operators or finiteness semantics which interprets formulas as vector spaces and resulted in differential linear logic...

Lots of this work have been motivated by the fact that coherent semantics is not complete as a semantics of programs (technically one says that it is not *fully abstract*). In order to see this, let us firts come back on the origin of the central concept of *stability* which as pointed above originated in the study of the sequentiality in programs.

### Sequentiality

Sequentiality is a property that we will not define here (it would diserve its own article). We rely on the intuition that a function of *n* arguments is sequential if one can determine which of these argument is examined first during the computation. Obviously any function implemented in a functionnal language is sequential; for example the function *or* defined à la CAML by:

` ``or = fun (x, y) -> if x then true else y`

examines its argument x first. Note that this may be expressed more abstractly by the property: for any boolean *x*: the function *or* needs its first argument in order to compute anything. On the other hand we have : in some case (when the first argument is true), the function doesn't need its second argument at all.

The typical non sequential function is the *parallel or* (that one cannot define in a CAML like language).

For a while one may have believed that the stability condition on which coherent semantics is built was enough to capture the notion of *sequentiality* of programs. A hint was the already mentionned fact that the *parallel or* is not stable. This diserves a bit of explanation.

#### The parallel or is not stable

Let *B* be the coherent space of booleans, also know as the flat domain of booleans: where *t**t* and *f**f* are two arbitrary distinct objects (for example one may take *t**t* = 0 and *f**f* = 1) and for any , define iff *b*_{1} = *b*_{2}. Then *B* has exactly three cliques: the empty clique that we shall denote , the singleton {*t**t*} that we shall denote *T* and the singleton {*f**f*} that we shall denote *F*. These three cliques are ordered by inclusion: (we use for to enforce the idea that coherent spaces are domains).

Recall the definition of the with, and in particular that any clique of has the form where *x* and *y* are cliques of *B*. Thus has 9 cliques: that are ordered by the product order: iff and .

With these notations in mind one may define the parallel or by:

The function is completely determined if we add the assumption that it is non decreasing; for example one must have because the lhs has to be less than both *T* and *F* (because and ).

The function is not stable because , thus whereas .

Another way to see this is: suppose *x* and *y* are two cliques of *B* such that , which means that ; according to the caracterisation theorem of stable functions, if Por were stable then there would be a unique minimum *x*_{0} included in *x*, and a unique minimum *y*_{0} included in *y* such that . This is not the case because both and are minimal such that their value is *T*.

In other terms, knowing that doesn't tell which of *x* of *y* is responsible for that, although we know by the definition of Por that only one of them is. Indeed the Por function is not representable in sequential programming languages such as (typed) lambda-calculus.

So the first genuine idea would be that stability caracterises sequentiality; but...

#### The Gustave function is stable

The Gustave function, so-called after an old joke, was found by Gérard Berry as an example of a function that is stable but non sequential. It is defined by:

The last clause is for all cliques *x*, *y* and *z* such that is incompatible with the three cliques , and , that is such that the union with any of these three cliques is not a clique in . We shall denote *x*_{1}, *x*_{2} and *x*_{3} these three cliques.

We furthemore assume that the Gustave function is non decreasing, so that we get .

We note that *x*_{1}, *x*_{2} and *x*_{3} are pairwise incompatible. From this we can deduce that the Gustave function is stable: typically if then exactly one of the *x*_{i}s is contained in .

However it is not sequential because there is no way to determine which of its three arguments is examined first: it is not the first one otherwise we would have and similarly it is not the second one nor the third one.

In other terms there is no way to implement the Gustave function by a lambda-term (or in any sequential programming language). Thus coherent semantics is not complete w.r.t. lambda-calculus.

The research for a right model for sequentiality was the motivation for lot of
work, *e.g.*, *sequential algorithms* by Gérard Bérry and Pierre-Louis
Currien in the early eighties, that were more recently reformulated as a kind
of game model, and the theory of *hypercoherent spaces* by
Antonio Bucciarelli and Thomas Ehrhard.

### Multiplicative neutrals and the mix rule

Coherent semantics is slightly degenerated w.r.t. linear logic because it identifies multiplicative neutrals (it also identifies additive neutrals but that's yet another problem): the coherent spaces and are equal.

The first consequence of the identity is that the formula becomes provable, and so does the formula . Note that this doesn't entail (as in classical logic or intuitionnistic logic) that linear logic is incoherent because the principle for any formula *A* is still not provable.

The equality has also as consequence the fact that (or equivalently the formula ) is provable. This principle is also known as the mix rule

as it can be used to show that this rule is admissible:

None of the two principles and are valid in linear logic. To correct this one could extend the syntax of linear logic by adding the mix-rule. This is not very satisfactory as the mix rule violates some principles of Polarized linear logic, typically the fact that as sequent of the form where *P*_{1} and *P*_{2} are positive, is never provable.

On the other hand the mix-rule is valid in coherent semantics so one could try to find some other model that invalidates the mix-rule. For example Girard's Coherent Banach spaces were an attempt to address this issue.

## References

- ↑ Girard, Jean-Yves.
*The System F of Variable Types, Fifteen Years Later*. Theoretical Computer Science. Volume 45, Issue 2, pp. 159-192, doi:10.1016/0304-3975(86)90044-7, 1986.