A Combined System for Update Logic and Belief Revision
Guillaume Aucher
Abstract:
Roughly speaking, in this thesis we will propose a logical system
merging update logic as conceived by A.Baltag, L.Moss, S.Solecki on
the one hand; and belief revision theory as conceived by C.Alchourron,
P.Gardenfors and D.Mackinson (viewed from the point of view of
W.Spohn) on the other hand. Before tackling the topic, we need to set
out some general assumptions about the type of phenomenon that we
intend to study thanks to these theories. It will also indirectly
provide us a framework for our future work, and give an idea of the
topic of this thesis (and these theories).
We assume that any situation s involving several agents can be
rendered from the point of view of the agents' knowledge and beliefs
of the situation by a mathematical model M. We assume that this
association is correct, in the sense that every intuitive judgement
concerning s corresponds to a formal assertion concerning M. Now in
the situation s, an action a may occur. We also assume that this
action a can be correctly (see above) rendered from the point of view
of the agents' knowledge and beliefs by a mathematical model
\Sigma. Now in reality the agents update their knowledge and beliefs
according to these two pieces of information: action a and situation
s, giving rise to a new actual situation s x a. We assume again that
we can render this update mechanism by a mathematical update (X) such
that, as above, M (X) \Sigma corresponds correctly (see above) to s x
a. Moreover we assume that the update mechanism concerning the agents'
beliefs be the closest possible to a belief revision (conceived by
AGM). Note that in reality, once the agents receive the new
information carried by the action a, update their knowledge and
beliefs. This double process may be done simultaneously in reality by
the agents. Yet we carefully separate it in our approach by
introducing \Sigma because these are two conceptually different
things: apprehension of the new information (corresponding to \Sigma
and update (corresponding to the update (X), on the basis of this
apprehension.
By the very nature of the BMS and AGM theories, merging them seems
one of the best ways to give concrete form to these assumptions. Yet
the resulting system should be a genuine logical system and we must
keep that in mind. So, first we will set out these theories. Second,
we will propose a merge system of these theories. Third we will propose
a proof system for this system (with the introduction of a special
canonical model in the completeness proof). Finally, we will compare
our system with other similar systems, and also the AGM theory.