Editing Belief revision (section)

==Foundational revision==
In the AGM framework, a belief set is represented by a deductively closed set of [[propositional formula]]e. 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 sets should be considered equal when revising. This is called the ''principle of irrelevance of syntax''.

This principle has been and is currently debated: while <math>\{a, b\}</math> and <math>\{a \wedge b\}</math> are two equivalent sets, revising by <math>\neg a</math> should produce different results. In the first case, <math>a</math> and <math>b</math> are two separate beliefs; therefore, revising by <math>\neg a</math> should not produce any effect on <math>b</math>, and the result of revision is <math>\{\neg a, b\}</math>. In the second case, <math>a \wedge b</math> is taken a single belief. The fact that <math>a</math> is false contradicts this belief, which should therefore be removed from the belief set. The result of revision is therefore <math>\{\neg a\}</math> 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 sets typically select some subsets of <math>K</math> that are consistent with <math>P</math>, combined them in some way, and then conjoined them with <math>P</math>. The following are two non-deductively closed base revision operators.

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

;[[Mary-Anne Williams|Williams]] : solved an open problem by developing a new representation for finite bases that allowed for AGM revision and contraction operations to be performed.<ref>{{Cite book|url=https://dl.acm.org/citation.cfm?id=758987&CFID=831191299&CFTOKEN=25549413|title=On the Logic of Theory Base Change Proceeding JELIA '94 Proceedings of the European Conference on Logics in Artificial Intelligence Pages 86-105|date=5 September 1994|pages=86–105|publisher=ACM Digital Library|isbn=9783540583325|accessdate=November 18, 2017}}</ref> This representation was translated to a computational model and an anytime algorithm for belief revision was developed.<ref>{{Cite web|url=https://www.ijcai.org/Proceedings/97-1/Papers/013.pdf|title=Anytime Belief Revision IJCAI'97 Proceedings of the 15th international joint conference on Artificial intelligence - Volume 1 Pages 74-79|publisher=ijcai.org|accessdate=November 18, 2017}}</ref>

; Ginsberg–Fagin–Ullman–Vardi : the maximal subsets of <math>K \cup \{P\}</math> that are consistent and contain <math>P</math> 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 system]]s, dependence links among beliefs can be specified. In other words, 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.