Belief Revision - Foundational Revision

Foundational Revision

In the AGM framework, a belief set is represented by a deductively closed set of propositional formulae. While such sets are infinite, they can always be finitely representable. However, working with deductively closed sets of formulae leads to the implicit assumption that equivalent belief bases should be considered equal when revising. This is called the principle of irrelevance of syntax.

This principle has been and is currently debated: while and are two equivalent sets, revising by should produce different results. In the first case, and are two separate beliefs; therefore, revising by should not produce any effect on, and the result of revision is . In the second case, is taken a single belief. The fact that is false contradicts this belief, which should therefore be removed from the belief base. The result of revision is therefore in this case.

The problem of using deductively closed knowledge bases is that no distinction is made between pieces of knowledge that are known by themselves and pieces of knowledge that are merely consequences of them. This distinction is instead done by the foundational approach to belief revision, which is related to foundationalism in philosophy. According to this approach, retracting a non-derived piece of knowledge should lead to retracting all its consequences that are not otherwise supported (by other non-derived pieces of knowledge). This approach can be realized by using knowledge bases that are not deductively closed and assuming that all formulae in the knowledge base represent self-standing beliefs, that is, they are not derived beliefs. In order to distinguish the foundational approach to belief revision to that based on deductively closed knowledge bases, the latter is called the coherentist approach. This name has been chosen because the coherentist approach aims at restoring the coherence (consistency) among all beliefs, both self-standing and derived ones. This approach is related to coherentism in philosophy.

Foundationalist revision operators working on non-deductively closed belief bases typically select some subsets of that are consistent with, combined them in some way, and then conjoined them with . The following are two non-deductively closed base revision operators.

WIDTIO

(When in Doubt, Throw it Out) the maximal subsets of that are consistent with are intersected, and is added to the resulting set; in other words, the result of revision is composed by and of all formulae of that are in all maximal subsets of that are consistent with ;

Ginsberg-Fagin-Ullman-Vardi

the maximal subsets of that are consistent and contain are combined by disjunction;

Nebel

similar to the above, but a priority among formulae can be given, so that formulae with higher priority are less likely to being retracted than formulae with lower priority.

A different realization of the foundational approach to belief revision is based on explicitly declaring the dependences among beliefs. In the truth maintenance systems, dependence links among beliefs can be specified. In other worlds, one can explicitly declare that a given fact is believed because of one or more other facts; such a dependency is called a justification. Beliefs not having any justifications play the role of non-derived beliefs in the non-deductively closed knowledge base approach.