An Inconsistency Tolerant Model for Belief Representation and Belief Revision

This paper is motivated by several concerns in the area of belief representation and belief revision. M i n i m a l Change: When a belief structure is revised in the face of new information, we would like the new belief structure to be as similar to the old one as possible. This requirement goes back to Quine and has been repeatedly emphasized by Gardenfors as the preservation criterion (cf. [Gardenfors, 1988].) C o m p u t a t i o n a l T r a c t a b i l i t y : Since we are seeking to represent the beliefs of real people, and since the satisfiability problem is NP-complete, even at the propositional level we need to be aware of the computational limitations of real people or even real computers and try, if possible, to take them into account in our belief representation models. E x p l i c i t vs I m p l i c i t Beliefs: There are beliefs which are explicit they may be actually asserted or at least agreed to. Other beliefs are implicit in the sense that an agent can be made to agree to them after a discussion. If the set of explicit beliefs is consistent, then we can define the set of implicit beliefs as just their logical consequences. However, this becomes implausible if unknown to the agent the explicit beliefs are inconsistent. Inconsistency Tolerance: The beliefs of real individuals are usually inconsistent from a global point of view. This is also likely to be the case wi th large data bases. So we need to model how an agent deals with possible inconsistencies in its belief base. In this paper we propose a model which comes to grips with all four issues. Our work is related to the splitting languages framework of [Parikh, 1996]. That paper showed how an agent's beliefs can be (uniquely) subdivided into sub-areas and how this division can be used in belief revision. However, it did not address the issue of explicit vs implicit beliefs, not did it tackle the question of inconsistent explicit beliefs. In the current paper we do both. We take the celebrated A G M axioms [Alchourron et. al, 1985] as our starting point, but modify them somewhat and represent an agent's beliefs, not by a theory as is usual, but by what we call a B-structure, a notion which generalizes the notion of a theory. We begin by giving some notation and re-stating the A G M axioms. N o t a t i o n : In the following, L is a finite propositional language. (However, most results continue to hold for a countably infinite first order language without equality.) The constants true, false are in L. The letter L stands both for a set of propositional symbols and for the formulae generated by that set. It wi l l be clear from the context which is meant. A B means that A B is a tautology, i.e. true under all t ru th assignments. Similarly, A B means that A -4 B is a tautology. If X is a set of formulae then is the logical closure of X. Thus X is a theory Letters T,T' denote theories. T A is the revision of T by A, and