formulae of that are in all maximal subsets of that are consistent with ; Williams solved an open problem by developing a new representation for finite bases that allowed for AGM revision and contraction operations to be performed. This representation was translated to a computational model and an anytime algorithm for belief revision was developed. 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 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. Model-based revision and update A number of proposals for revision and update based on the set of models of the involved formulae were developed independently of the AGM framework. The principle behind this approach is that a knowledge base is equivalent to a set of possible worlds, that is, to a set of scenarios that are considered possible according to that knowledge base. Revision can therefore be performed on the sets of possible worlds rather than on the corresponding knowledge bases. The revision and update operators based on models are usually identified by the name of their authors: Winslett, Forbus, Satoh, Dalal, Hegner, and Weber. According to the first four of these proposal, the result of revising/updating a formula by another formula is characterized by the set of models of that are the closest to the models of . Different notions of closeness can be defined, leading to the difference among these proposals. Peppas and Williams provided the formal relationship between revision and update. They introduced the Winslett Identity in Dalal the models of having a minimal Hamming distance to models of are selected to be the models that result from the change; Satoh similar to Dalal, but distance between two models is defined as the set of literals that are given different values by them; similarity between models is defined as set containment of