NÚMERO TOTAL DE INDIVIDUOS POR TIPO DE TRAMPA

Recall from Section 1.2.4, Definition 1.25 of decidability. As we saw in Theorem 1.26 a suffi- cient condition for proving decidability is that the logic has the finite model property (Def- inition 1.27) and is finitely axiomatisable. In our case, the logic S5WD is clearly finitely axiomatisable, so all we need to show in order to prove decidability is that it has the fmp.

In order to prove that S5WD has the fmp, we use filtrations. The idea of filtrations is the following. If a logic is complete, we know that if a formula

is a non-theorem of (i.e. if

is -consistent), then

is invalid on some model for . The modelmight be infinite.

Filtrations enable us to produce a model

from, such that

is finite. If we can further prove that

is also a model for (and therefore does not validate

), then we have proved that the logic has the finite model property.

Formally we proceed as follows. Given a formula

, consider the set , composed by formulae such thatis a well-formed sub-formula of

or the negation of a well-formed sub-formula of

. The set is obviously finite for any formula

3.5. DECIDABILITY 67

Definition 3.21. Consider a model . Two points

are equivalent with respect to

(written as or simply

if it is not ambiguous) if for any we have if and only if .

We can now definefiltrationsas follows.

Definition 3.22. Given a formula

and a model , afiltrationthrough is a model built as follows: , where

is the equivalence relation defined by Definition 3.21. For each the relation

issuitable, i.e. it satisfies the two properties:

1. For all

, if there exists a point such that and , then 6_. 2. For all , if

, then for all formulaesuch that , we have that if , then . For any , . Note that

as defined above is finite because the set is.

It can be proved by induction (see for example [HC84] page 139) that if all the relations

are “suitable” then the following holds.

Theorem 3.23. Given a model, and any formula

, a filtration

ofthrough is such that

for any point

and and for any formula ,

holds if and only if

We now proceed to the case of interest here: the logic S5WD .

Consider the canonical model for S5WD , we know (see Theorem 3.12 and Lemma

3.13) that is an equivalence model and that if we consider the model generated by any

point of it, this is directed. Consider any formula

and the model

defined as follows:

Definition 3.24. Given a model and a formula

define the model

by: , where

is the equivalence relation defined by Definition 3.21.

if for all formulaesuch that , we have if and only if . For any , .

Indeed the model

defined by Definition 3.24 is a filtration as the following shows (stated in [HC84] page 145 for the mono-modal case).

Lemma 3.25. Given an equivalence modeland a formula , the model as described in Defini- tion 3.24 is a filtration. 6

Note that this condition is actually equivalent to implies

. Here we follow the definition given in [HC84].

68 CHAPTER 3. AXIOMATISATION OF HYPERCUBE SYSTEMS

Proof. All we need to prove is that the relations

are suitable.

Property 1. Consider worlds and a world such that and

. We need to prove that

, i.e. that for all formulae such that , we have if and only if

. We prove it from left to right; the other

direction is similar. Note that if and only if because is an

equivalence model; but and so . But and , so ,

which is what we wanted to prove. Property 2. Consider worlds

such that

. This means that for all , we have if and only if . Since is an equivalence

model it follows that

.

We now prove that the filtration defined above produces models for S5WD .

Lemma 3.26. Ifis an equivalence directed model, then for all such that , the model

as defined in Definition 3.24 is such that

.

Proof. We prove that

is a frame for

, i.e. it is an equivalence directed frame. The relations

are clearly equivalence relations. All it remains to show is that

is directed. To do that, consider any

. Since is directed, there exists such that for . But each

is suitable and so, by a consequence

of property 1 of suitability we have that , for

. Therefore the frame

is directed.

We are finally in the position to prove fmp.

Theorem 3.27. The logic

has the finite model property. Proof. Suppose

. Since by the proof of Theorem 3.12 the logic is canonical, the canonical model for

is an equivalence model, it is weakly-directed and there is a point

, such that

. Consider the model

generated (accord-

ing to Definition 1.17) by

from; by Lemma 1.18 we have . The model is

clearly an equivalence model and, since it is connected, by Lemma 3.13, it is also directed. Consider now the filtration

of

through according to Definition 3.24; by Lem-

ma 3.26,

is an equivalence directed model and it is finite by construction because is a finite set. But

is a filtration, and by Theorem 3.23,

, which is what we needed to prove.

Corollary 3.28. The logic

is decidable.

Proof.

has the fmp and it is finitely axiomatisable. The result then follows from Theorem 1.26.