Epistemic Logic for Multiagent Systems
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bounded in the past, and infinite in the future). Note that this is essentially the same notion of runs that was introduced in Chapter 2, but I have formulated it slightly differently. A run r is thus a function
A point is a run together with a time:
Point
=
R u n x N.
A point implicitly identifies a global state. Points will serve as worlds in the logic
of knowledge to be developed. A system is a set of runs. Now, suppose s and s' are two global states.
1
We now define a relation
-i
on states, for each process i,
s - i s f if and onlyif (li = I : ) .
Note that - i will be an equivalence relation. The terminology is that if s -, s', then s and s' are indistinguishable to i, since the local state of i is the same in each global state. Intuitively, the local state of a process represents the information that the process has, and if two global states are indistinguishable, then it has the same information in each. The crucial point here is that since a processes [sic] [choice of] actions.. .are a function of its local state, if two points are indistinguishable to processor i, then processor i will perform the same actions in each state. (Halpern, 1987, pp. 46, 47) (Again, this is the same notion of indistinguishability that I introduced in Chapter 2, except that there it was with respect to the notion of percepts.) The next step is to define a language for reasoning about such systems. The language is that of the multiagent episternic logic defined earlier (i.e. classical propositional logic enriched by the addition of a set of unary modal operators Ki, for i E (1,.. . , n ) ) .The semantics of the language are presented via the satisfaction relation, 'I=', which holds between triples of the form
and formulae of the language. Here, ( r ,u ) is a point, and M is a structure