A Unified View of Consequence Relation, Belief Revision and Conditional Logic

The notion of minimality is widely used in three different areas of Artificial Intelligence: non-monotonic reasoning, belief revision, and conditional reasoning. However, it is difficult for the readers of the literature in these areas to perceive the similarities clearly, because each formalization in those areas uses its own language sometimes without referring to other for-malizations. We define ordered structures and families of ordered structures as the common ingredient of the semantics of all the works above. We also define the logics for ordered structures and families. We present a uniform view of how minimality is used in these three areas, and shed light on deep reciprocal relations among different approaches of the areas by using the ordered structures and the families of ordered structures. 1 Introduction The notion of minimality is proving to be a key unifying idea in three different areas of Artificial Intelligence: nonmonotonic reasoning, belief revision, and conditional reasoning. However, it is difficult for the readers of the literature in these areas to perceive the similarities clearly. The models used differ, sometimes superficially, sometimes in depth, and the notation is different, making it hard to apply results on, say, conditional logic to, say, belief revision. Even within the same area there is confusion, as, for example, different authors use different formalisms for conditional logic, sometimes without relating their proposals to the literature. We present a uniform view of how minimality is used in these three areas , shedding light on deep connections among the areas. We clarify differences and similarities between different approaches by classifying them according to the notion of minimality that they are based on. The first field in which minimality plays a crucial role is nonmonotonic reasoning. Shoham [1987] proposes a uniform approach to subsuming various formalisms of nonmonotonic reasoning in terms of preftrenttal relations among interpretations, Kraus, Lehmann and Magidor [1990] propose several consequence relations that capture general patterns of nonmonotonic reasoning. A consequence relation, denoted by means that is a good enough reason to believe or that is a plausible consequence of We can regard their work as an extension of Shoham's work since some consequence relations can be characterized in terms of preferential relations among possible worlds. The second field in which minimality is discussed is knowledge base revision and update. Alchourron, Gardenfors and Makinson [3985] propose, on philosophical grounds, a set of rationality postulates that belief revision operators must satisfy, …