
```
The authors have pointed out that in 
                             ase the distan
                                        e spa
                                            es are assumed to
                                                          

```

be minspa es, that is spa es where the inmum of a set of distan es is a tually their minimum, the logi CSL is naturally related to some 
onditional logi
s.

The semanti s of the latter is often expressed in terms of preferential stru tures, that is to say possible-world stru tures equipped by a family of stri t partial

(pre)-orders ≺x indexed on ob je
ts/worlds [6, 11℄. The intended meaning of the relation y ≺x z is namely that x is more similar to y than to z . It is not hard

```
to see that the semanti
                         s over minspa
                                        es is equivalent to the semanti
                                                                        s over preferential stru
                  tures satisfying the well-known prin
                                                         iple of Limit Assumption a
 
 ording to whi
        h the set of minimal elements of a non-empty set always exists.
                                    

```

The minspa e property entails the restri tion to spa es where the distan e fun tion is dis rete. This requirement does not seem in ompatible with the purpose of representing qualitative similarity omparisons, whereas it might not be reasonable for appli ations of CSL to spatial reasoning.

In this paper we ontribute to the study of CSL over minspa
es. We rst show (unsurprisingly) that the semanti s of CSL on minspa
es 
an be equivalently restated in terms of preferential models satisfying some additional onditions, namely modularity, entering, and limit assumption. We then give a dire t axiomatization of this logi
. This problem was not onsidered in detail in [10℄. In that paper an axiomatization of CSL over arbitrary distan e models is proposed, but it makes use of an additional operator. Our axiomatisation is simpler and only employs ⇔. Next, we dene a tableaux 
al
ulus for 
he
king satisability of CSL formulas. Our tableaux pro
edure makes use of labelled formulas and pseudo-modalities indexed on worlds x , similarly to the 
al
uli for onditional logi s dened in [2, 3℄. Termination is assured by suitable blo king onditions. To the best of our knowledge our al ulus provides the rst known pra ti ally-implementable de ision pro edure for CSL logi
.

## 2 The Logi Of Comparative Con Ept Similarity Csl

The language LCSL of CSL is generated from a set of propositional variables Vi by ordinary propositional onne tives plus ⇔: A, B ::= Vi| ¬A | A⊓B | A ⇔ B .

The semanti s of CSL introdu
ed in [8℄ makes use of distan
e spa
es in order to represent the similarity degree between ob je ts. A distan e spa e is a pair

(**∆, d**) where ∆ is a non-empty set, and d : ∆ × ∆ → R
≥0 is a distan e fun tion satisfying the following ondition:

(ID) ∀x, y ∈ ∆, d(x, y**) = 0** i x = y Two further properties are usually onsidered: symmetry and triangle inequality.

We briey dis uss them below.

The distan e between an ob je t w and a non-empty subset X of ∆ is dened by d(w, X) = inf{d(**w, x**) | x ∈ X}. If X = ∅, then d(w, X) = ∞. If for every ob je t w and for every (non-empty) subset X we have the following property
(MIN) inf {d(w, x) | x ∈ X} = min {d(w, x) | x ∈ X} ,