Environments

Although interpreted systems (Section 1.3.1.1) are certainly a very expressive formalism for expressing MAS systems, one could argue that they are perhaps a bit too general to represent certain classes of distributed processes. One example for which they seem to be inadequate is the case of protocols. Intuitively, in practical applications we are not interested in the set ofall possible runs of a system, but only in the ones that follow some constraints. Indeed in certain cases we would like to be able to expressexplicitlythese constraints regulating the transitions from a state to another.

This is the reason why Fagin and colleagues introduced the formalism of contexts(see [FHMV97, FHMV95]), in which the notion of protocol is explicitly present. In the following we use a variant of contexts, called environments, presented in [Mey96]9. The interested reader can find more details and motivations in [Mey96]. The aim of the following is to show that executing a protocol in an environment determines a Kripke frame that describes states of knowledge of the agents. In Section 3.8.3 we will discuss under what assumptions these evolutions generate exactly a class of agents which is modelled by the same logic as hypercubes.

Differently from the rest of this thesis, we work with a set of agents. Agent 0 is intuitively the environment of interpreted systems and it will play the special role of modelling the architecture of the systems.

Definition 3.48 (Actions). For any

the non-empty set

is the set of actionsfor agent

. Ajoint actionis a tuple , where

is an action for agent

. The set

is the set of joint actions.

Actions of agent 0 correspond to nondeterministic behaviour of the context in which the agents are situated. The fundamental notion of this model is the definition ofenvironment.

Definition 3.49 (Environment). Aninterpreted environmentis a tuple of the form

where the components are as follows:

is a set ofstates of the environment. Intuitively, states of the environment may encode such information as messages in transit, failure of components, etc.

is a subset of , representing the possibleinitial statesof the environment.

80 CHAPTER 3. AXIOMATISATION OF HYPERCUBE SYSTEMS

is a function, called theprotocol of the environment, mapping states to subsets of the set

of actions that can be performed by the environment. Intuitively,

represents the set of actions that may be performed by the environment when the system is in the state

.

is a function mapping joint actions

to state transition functions

. Intuitively, when the joint action

is performed in the state

, the resulting state of the environment is . is a function from to

for some set

of observations. For each , the function mapping to the th component of

, is called theobservation function of agent .Intuitively,

represents theobservationperformed by agent

in the state .

is a valuation for the atoms.

The definition above defines transitions over states. Given an environment, sequences of states related by transition functions define atrace.

Definition 3.50 (Traces). Atraceof an environmentis afinitesequence of states such that

and for allsuch that

there exists a joint action such that and . Given a trace ,

is the final state of

the trace.

The intuition is that a trace

represents a finite history of the system. Note that in the transitions above, agent 0 follows its own protocol

. This is not the case for the other agents that in principle can perform any possible action. In practice we would like to specify what protocol these agents follow. In the context of this work we will assume the agents follow aperfect recall protocolwhich is defined as follows:

Definition 3.51 (Perfect Recall). Given an environment and a trace

on it, the

perfect recall local stateof an agent in a trace

is defined as the sequence

of observations made by agent

in the trace

. Aperfect recall protocol for agent

is a function

mapping each sequence of observations in

to a non-empty subset of

. A joint perfect recall protocol is a tuple

, where for every agent

is a perfect recall protocol for agent

.

Protocols specify the actions that are allowed for the agents. More precisely, we say that given a trace

on an environmentand an agent , an action is

enabled with respect to a joint protocol if

. We also say that an action

of the environment is enabled at

if . A joint action is enabled at

with respect to a protocolif each of its components

is enabled at

. If all the agents follow their protocol by executing enabled actions we obtain aconsistent

trace. More precisely, given an environment and a joint protocol, a trace

on isenabledif for each

, there exists a joint action

enabled at with respect to, such that .

All the enabled traces define the intended evolutions of the environment according to the joint protocol. Similarly to what we saw in the case of interpreted systems it is possible to ascribe knowledge to agents following a perfect recall protocol. Since agents perform observations it is meaningful to assume two states to be indistinguishable for an agent if the series of observation she has performed in the two states are the same. So, once again we can define a Kripke model from a low-level description.

3.8. HOMOGENEOUS BROADCASTING SYSTEMS 81

Definition 3.52 (Perfect recall frame derived from a protocol and environment). Let be

an environment and let be a joint protocol. The perfect recall frame derived from andis

the structure , where

is the set of all traces of the environmentconsistent with the protocol, For every , the relation is defined by if .

It is also possible to derive a Kripke model by considering the same valuation

of the environmentin question.

From the way the relations

are defined, it is clear that every agent has perfect recall and it is common knowledge that the environment they are operating in is and that the

joint protocol is.

By taking other accessibility relations one can encode different phenomena (for exam- ple one could define two states to be indistinguishable if their latest observation is equal). The assumption of perfect recall is widely used in computer science because it amounts to assuming that the agents use all the information they acquired in an ideal way.