Modularity is strongly related to the fa t that the preferential relations repre-

```
sents distan
             e 
               omparisons. This is the key property to enfor
                                                               e the equivalen
                                                                               e
                                                                                

```

with distan e models. Centering states that w is the unique minimal element for its preferential relation ≺w , and 
an be seen as the preferential 
ounterpart of (ID). The Limit Assumption states that ea h non-empty set has at least one minimal element wrt. a preferential relation (i.e it does not ontain an innitely

```
des
   ending 
        hain), and 
                 orresponds to (MIN).
                                

```

Denition 3 (CSL-preferential model). A CSL-preferential model is a triple

## M = (∆,(≺W)W∈∆, .M) Where:

 ∆M is a non-empty set of ob je
ts (or possible worlds).

 (≺w)w∈∆ is a family of preferential relation, ea
h one being modular, 
entered, and satisfying the limit assumption.

 .M is the evaluation fun
tion dened as in denition 1, ex
ept for ⇔:
(A ⇔ B)M =w ∈ ∆∃x ∈ AM su
h that ∀y ∈ BM, x ≺w y Validity is dened as in denition 1.

```
We now show the equivalen
                 e between preferential models and distan
                                           e minspa
                                                 e
                                                 

```

models. We say that a CSL-preferential model I and a CSL-distan
e minspa
e model J are equivalent i they are based on the same set ∆, and for all formulas A ∈ LCSL , AI = AJ .

## Theorem 4 (Equivalen E Between Csl-Preferential Models And Csl- Distan E Models).

1. For ea h CSL-distan
e minspa
e model, there is an equivalent CSL-preferential model.

2. For ea h CSL-preferential model, there is an equivalent CSL-distan
e minspa
e model.

Proof. 1. ([8℄): given I = (∆I, d, .I) a CSL-distan
e minspa
e model, just dene a preferential model J by stipulating x ≺w y i d(w, x) < d(**w, y**),
and for all propositional variable Vi
, V
J
i = V
I
i
. It is to he k that ≺w is modular, entered, and satises the limit assumption, and that I and J are equivalent.

2. Sin e the relation ≺w is modular, we 
an assume that there exists a ranking fun tion rw : ∆ → R
≥0 su h that x ≺w y i rw(x) < rw(y). Therefore, given a CSL-preferential model J = (∆J ,(≺w)w∈∆J , .

J ), we 
an dene a CSL-
distan e minspa e model I = (∆J **, d, .**J ), where the distan
e fun
tion d is dened as follow: if w = x then d(w, x**) = 0**, and d(w, x) = rw(x) otherwise.

We an easily he k that (i) I is a minspa
e be
ause of the limit assumption, and that (ii) I and J are equivalent; this is proved by indu
tion on the omplexity of formulas.