# Relational semantics

## Contents |

## Relational semantics

This is the simplest denotational semantics of linear logic. It consists in interpreting a formula *A* as a set *A*^{ * } and a proof π of *A* as a subset π^{ * } of *A*^{ * }.

### The category of sets and relations

It is the category whose objects are sets, and such that . Composition is the ordinary composition of relations: given and , one sets and the identity morphism is the diagonal relation .

An isomorphism in the category is a relation which is a bijection, as easily checked.

#### Monoidal structure

The tensor product is the usual cartesian product of sets (which *is not* a cartesian product in the category in the categorical sense). It is a bifunctor: given (for *i* = 1,2), one sets . The unit of this tensor product is where * is an arbitrary element.

For defining a monoidal category, it is not sufficient to provide the definition of the tensor product functor and its unit , one has also to provide natural isomorphisms , (left and right neutrality of for ) and (associativity of ). All these isomorphisms have to satisfy a number of commutations. In the present case, they are defined in the obvious way.

This monoidal category is symmetric, meaning that it is endowed with an additional natural isomorphism , also subject to some commutations. Here, again, this isomorphism is defined in the obvious way (symmetry of the cartesian product). So, to be precise, the SMCC (symmetric monoidal closed category) is the tuple , but we shall simply denote it as .

The SMCC is closed. This means that, given any object *X* of (a set), the functor (from to ) admits a right adjoint (from to ). In other words, for any objects *X* and *Y*, we are given an object and a morphism with the following universal property: for any morphism , there is a unique morphism such that .

The definition of all these data is quite simple in : , and .

Let . Then we have and hence . It is clear that and hence η is a natural isomorphism: one says that the SMCC is a *-autonomous category, with as dualizing object.

#### Additives

Given a family , let . Let given by . Then is the cartesian product of the *X*_{i}s in the category .

#### Exponentials

One defines as the set of all finite multisets of elements of *X*. Given , one defines where is the multiset containing , taking multiplicities into account. This defines a functor , that we endow with a comonad structure as follows:

- the counit, called dereliction, is the natural transformation , given by
- the comultiplication, called digging, is the natural transformation , given by

### Interpretation of propositional linear logic (*L**L*_{0})

The structure described above gives rise to the following interpretation of formulas and proofs of linear logic.

For all propositional variable *X*, fix a set .
Then with each formula *A*, we associate a set as follows:

- ;
- ;
- ;
- .

We then interpret the proofs of *L**L*_{0} as follows: with each proof
π of sequent , we associate a
subset .

- Identity group:
- Multiplicative group:
- Additive group:
- Exponential group:

**Theorem**

If proof π' is obtained from π by eliminating a cut,
then .