Objetivos

6.3

MAS theories as combined logics

In Section 6.2.2 we have examined some of the most successful techniques to combine modal logics. In this section we investigate the extent to which the MAS theories we discussed in Section 1.3.2 can be seen as particular examples of logic combination.

6.3.1 Knowledge and other single mental state theories

Let us consider once again the logic S5 that we have been using for most of this thesis as a base for an epistemic theory9_{. Recall from Section 1.3.2 that S5 was motivated as the}

indexed extension of the logic S5 which was proposed long ago to model knowledge. Alternately, let us consider

-copies of the the mono-modal logic S5, one for every a- gent of the group: S5

S

. Having done so, consider now the fusion (as defined

in Section 6.2.2.2) of these. We obtain the logic S5

S5

. Note that the fusion is

well-defined because all the logics involved are normal logics. We can then apply the re- sults of Theorem 6.2.2.2 and Theorem 6.2.2.3. We have that the fusion can be axiomatised by taking the union of the axiomatisations of the

copies of S5, one for every modal box. We now notice that the union of the

axiomatisations gives exactly the logic S5 . In other words S5 can effectively be seen as the fusion (precisely as defined in Section 6.2.2.2) of

S5-components, one for every agent of the model: S5 S5 S5

Indeed, by Theorem 6.13, it follows that the logic S5 is complete and decidable, which is something that had already been proved by more standard means.

Consider now the operators of distributed knowledge and common knowledge of Sec- tion 1.3.2.2. The semantics of these is defined by using the accessibility relations of the pri- vate knowledge operators and so they cannot be seen as independent as in the above case of S5 . Moreover, interaction axioms between common knowledge and private knowledge are also present and so we are unable to recognise the system S5

as either a fusion or as an independent combination of logics. That is certainly not the result of an embedding because of the syntactic limitations of this technique. It might be the result of a fibring or dovetailing but that is not clear either.

So, we conclude that the logic S5 is a fusion or independent combination of

copies of S5, but it is far from being obvious whether any of its extensions devised to deal with group properties can be thought of as the result of a logic combination.

6.3.2 BDI logics

In Section 1.3.2.3.3 we briefly discussed a family of logics commonly known as BDI logics. There we introduced the very basic BDI logic. This was a multi-modal logic for belief, desire and intention skimmed of the temporal dimension (usually CTL ).

In Section 1.3.2.3.3 we discussed that the basic BDI logic is defined from three logics: KD45 for belief, and KD for desires and intentions. In top of this, the two interaction axioms 9

It should be noted that what follows actually applies equally well to any logic modelling a single mental state of a group of agents, e.g. all the logics discussed in Section 1.3.

148 CHAPTER 6. DISCUSSION: COMBINING LOGICS FOR MAS THEORIES that regulate the relation between desires, intentions and beliefs of every agent are imposed. Following the observation regarding S5 of the previous section we can now see the three different component logics as being fusions of

copies for each logic. The basic BDI logic L can then be seen as the fusion of these plus the two interaction axioms. In symbols:

KD45 KD KD

We can now apply the results of Section 6.2.2.2 or Section 6.2.2.3 to prove that the logic

KD45 KD KD

is sound, complete and decidable, but note that these results do not transfer to the logic when we we add the required interaction axioms.

So, combining logics for this case is of limited use. We have the transfer of properties for the three components fused together but we do not have a result that we can apply in the case of interaction axioms being present in the combination.

6.3.3 Knowledge and belief

In Section 1.3.2.3.1 we discussed the system introduced by Kraus and Lehmann to model knowledge and belief in a community of ideal agents. As noted there this system includes operators for common knowledge and common belief. By the same considerations of Sec- tion 6.3.2, we are unable to identify this system as the result of a logic combination.

In this case, differently from the BDI case, it is not even possible to recognise in a straight- forward way evena fragmentof the system, private belief and knowledge for example, as a fusion plus interaction axioms. To see this, note that in the axiomatisation of [KL88] the in- ference rule of necessitation for belief is not present. Still, the rule is actually sound as we can apply necessitation for common knowledge that via Modus Ponens implies necessitation for common belief, from which we can derive “everybody believes” which in turn implies the belief of any agent. Therefore by restricting the axiomatisation to the fragment defined only on the bi-modal family of operators

and

, we would not be getting all the theorems that we have in the combined system, restricted to the language of

and

.

Our conclusion must then be that this system,as it was presentedby Kraus and Lehmann, is so interconnected that it cannot even be easily recognised as a fusion of mono-modal logics plus extra interaction axioms. It may be that the logic defined on the language of private knowledge and belief as

KD45 KD45 S5 S5

is complete with respect to the class of models considered in [KL88] but in order to prove that, as things stand, it looks easier to refer to traditional methods.

6.3.4 Knowledge and time

So far we have had limited success in applying the techniques of Section 6.2.2 to our MAS theories. As a last example, we analyse the system for knowledge and time the we explored in Section 1.3.2.3.2.

Although also in this case the logic is clearly a combination between the epistemic S5 and a basic temporal logic, this time it is even less obvious than before whether the system

6.4. CONCLUSION 149