Recall from Section 1.3.2.3.2 the definition of interpreted system. These are defined to give an account of the temporal evolution of a system by defining runs over global states. If we fix the time of an interpreted system to analyse static properties of its global states, we simply obtain a pair of global states with an interpretation, i.e. what we called static interpreted systems (Definition 1.5). In Section 1.3.1.1 we discussed how to represent knowledge in a static interpreted system. This essentially involves generating an equivalence model by using the construction given in Definition 1.6 for the case that the predicate
is the
equality on local states for agent
, as we discussed in more detail on page 24.
In order to introduce hypercube systems we discuss an example of a static interpreted system. Figure 2.2 shows the underlying structures of a static interpreted system and its corresponding generated model2. As expected, we find that two global states are related for agent
if agent
has the same local state in the two global states. In the example we can see that at any state no agent knows the local state of the environment. In fact, for any global state each agent considers the possibility of another global state in which the environment
2
For simplicity the reflexive links are not illustrated. Also in Figure 2.2 the relations are the transitive closure of the ones depicted.
2.2. DEFINITION OF HYPERCUBE SYSTEMS 45 2 1,2 1,2 1,2 1 1
Figure 2.2: An example of the underlying structure of a static interpreted system and its corresponding generated frame. For simplicity the reflexive links are not illustrated and intended to be the transitive closure of the ones depicted.
has a different local state. This is not surprising at all, as we assumed that agents know only about their local states. Local states can encode some information about the environment but this has to be considered not totally observable.
More importantly, in the global state
, the top left state in the figure, agent 1 considers two global states possible:
and
. In both of these global states agent 2’s configuration is the same. So in that global state agent 1 has complete information about agent 2, i.e. when in the system is in that global state agent 1 knowsagent 2’s local state.
The converse of this situation is when an agent has no information at all about another. This happens in the example for agent 2 in the global state
. In this case not only agent 2 considers the possibility of another global state where agent 1 is in the local state
, but also agent 1 has no information about possible dependencies between agent 1’s local state and the environment’s. As it can be verified in the example all the global states of the Cartesian product
, where the local states are defined in the figure, are
effectively present.
Hypercube systems are static interpreted systems in which all the agents at all global states are in the situation of agent 2 at
. Interestingly, although they have no infor- mation about each other’s local states, we will note at the end of Chapter 3 that they still share some knowledge. Formally, hypercube systems result by considering the admissible state space of the MAS to be described by the full Cartesian product of its sets of local s- tates. This means that every global state is in principle possible, i.e. there are no mutually exclusive configurations between local states. With hypercubes systems we are imposing a
46 CHAPTER 2. HYPERCUBE SYSTEMS further simplification on the notion of static interpreted system presented in Definition 1.5: in the tuples representing the configuration of the system we do not consider a slot for the environment. The presence of the environment in the notion of Fagin et al. [FHMV95] is motivated in order to keep track of the changes in the system and in general to represent everything that cannot be captured by the local states of the single agents (most important- ly messages in transit, etc.). This restriction is equivalent to analysing tuples in which the environment is constant. We will reintroduce the environment in Section 3.8 where we will discuss a low-level formal model for hypercube systems, in which we will introduce actions for the agents and the environment.
So, the following are our formal definitions for hypercube system.
Definition 2.1 (Hypercube states). A hypercube state, or hypercube, is a Cartesian product
, where
are non-empty sets of local states. The class of hypercube systems
is denoted by.
Definition 2.2 (Hypercube systems). A hypercube system is a static interpreted system
, where is a hypercube state. The class of hypercube systems is denoted by
.
Static interpreted systems were defined in Definition 1.5.