ANÁLISIS DE LA DEMANDA

Above we have briefly described the machinery of interpreted environments and observed these generate a Kripke frame in a way which is similar to interpreted systems. But environ- ments offer more expressive potentialities than interpreted systems, because of the expres- siveness they offer.

From the definitions of the previous section it is possible to define a subclass of environ- ments, calledhomogeneous broadcasting environmentin [LMR99] that enjoy special properties. The details are not fully reported here and we refer the reader to that paper.

Briefly, a homogeneous broadcasting environment is an environment with the following additional constraints.

Actions consist of pairs of internal and external actions,

The states of the environment consist of tuples incorporating the external actions being performed on that state together with the private states of the agents;

In initial states no external action is present and agents are ignorant of each other’s state and of the state of the environment;

Agents observe their own private states and the external actions performed by the other agents;

Every agent updates its private state depending on its own private state and on the external actions performed by the agents;

The protocol of the environment depends only on its own internal state and on the last performed external action; the agents run a perfect recall protocol.

Under these assumptions, it was proven in [Mey98] that the class of perfect recall frames generated by homogeneous broadcasting agents has the following interesting property.

82 CHAPTER 3. AXIOMATISATION OF HYPERCUBE SYSTEMS

Theorem 3.53 ([Mey98]). The logic

is sound and complete with respect to the class of perfect recall frames generated by homogeneous broadcasting agents.

Theorem 3.53 provides a characterisation of hypercubes on a low-level semantics like the one of environments. The broadcasting activity that they perform is the act responsible for the sharing of knowledge that they exhibit (Observation 3.47).

We have now completed our analysis of hypercube systems. We defined them as a special class of interpreted systems, we found a semantic correspondence in the class of Kripke frames, we provided some complete and decidable axiomatisations for them and in this section we reported a low-level description of their communication ability.

Chapter 4

A spectrum of degrees of knowledge

sharing

4.1

Introduction

In Chapter 3 we explored the axiomatisation of hypercube systems and proved that the log- ic S5WD is sound and complete with respect to that semantic class. The logic S5WD is axiomatised by extending S5 with the axiomWD.

For the case

the axiomWDbecomes the formula:

which can be read as “If agent 1 considers possible that agent 2 knows

then agent 2 must know that agent 1 considers possible that

is the case”. In Section 3.8 we showed that this logic can be seen as modelling a particular class of MAS that exchange information by broadcasting.

Axiom 2WDis an interaction axiom that models a particular class of agents of knowl- edge; but surely there must be other interesting classes of interactions between agents that can be modelled by extensions of S5 . For example, on page 38 we presented two other examples of agents sharing knowledge.

In the first example we described a collective map making scenario from [dMAE 97] in which an agent is told any knowledge of any other agent

and therefore knows everything that is known by any other agent. Assuming all the agents being ideal, this scenario can be represented by S5 enriched by the interaction axiom:

; for all

The second scenario of page 38 concerns a MAS whose agents have computation capa- bilities that can be ordered. If the agents are executing the same program on the same data, then it is reasonable to model the MAS by enriching the logic S5 by:

; for all The relation

expresses the linear order in the computational power at disposal to the a- gents. In this as in the previous case some information is being shared among the agents of the group.

84 CHAPTER 4. A SPECTRUM OF DEGREES OF KNOWLEDGE SHARING It is easy to imagine other meaningful axioms that expressinteractionsbetween the agents in the system; clearly there is a spectrumof possible degrees of knowledge sharing. At one end of the spectrum is S5 , with no sharing at all. At the other end, there is S5 together with

; for all

saying that the agents have precisely the same knowledge (total sharing). The three exam- ples mentioned above exist somewhere in the (partially ordered) spectrum between these two extremes.

Our aim in this chapter is to explore the spectrum systematically. We restrict our attention to the case of two agents (i.e. to extensions of S5

), and explore axioms of the forms

where each occurrence of is in the set .

Technically we will prove correspondence properties and completeness for extensions of S5

with axioms of these forms. Naturally, this will not give the complete picture: there may be interesting axioms of other forms than those listed above. However, analysis of the literature certainly suggests that most axioms studied for this purpose are of one of these forms. They are sufficient for expressing how knowledge and facts considered possible are related to each other up to a level of nesting of two, which is about the maximum that human intuition can grasp. Note also that the examples above, including the case of bi-dimensional hypercubes, are included in the axiom patterns.

Although our analysis is limited both from considering the case of two agents and from considering only interaction axioms of the shape above, we will see that some non trivial technical problems are present here. Indeed, we will leave two completeness problems as open.

The rest of this chapter is organised as follows. In Section 4.2 we analyse and discuss interaction axioms of the form

. We will then extend these results in Section 4.3 where we discuss the case of the consequent being composed by two modal operators. In Section 4.4 we will analyse the interaction axioms resulting from two nested modalities both in the antecedent and in the consequent. Finally in Section 4.5 we present the lattice gener- ated by the logics and discuss our results.

This chapter is devoted to extensions of S5

and so in the following we will use the prov- ability symbol for . Given this context we will always be working in the classof

equivalence frames

built on two equivalence relations on

. Many of these results can be translated for the system K

; given that the chapter is devoted exclusively to knowledge agents we will not discuss this.