An Operational Measure of Similarity for Belief Revision Systems

Many belief revision applications require a measure of similarity. We have described a simple, yet powerful, operational measure of similarity based on change. The advantages of this measure are that it: (i) is nonprobabilistical, (ii) is quantitative in character, (iii) not set theoretical, (iv) takes the internal structure of the information state into consideration, (v) is not necessarily symmetric, (vi) can model accumulated eeects of change, (vii) is operational, and (viii) is implementable. We showed that the graphical representation of information space and the similarity measure based on transmutation schemas provides a useful framework for modeling of goal-directed belief revision, for evaluating the plausibility of belief revision strategies as models for human belief change, for assessing design decisions for belief revision, for comparing possible explanations, and for evaluating surprising information. The proposed framework enhances our understanding of the concept of similarity in the present context, and it provides the necessary practical machinery for computing a natural form of epistemic distance. There are vastly too many possible information states to implement the framework in full, and we outlined two approaches that can be used to address the complexity problem: the use of orthogonal arrays, and the use of partial entrenchment rankings together with heuristic search. Similarity has been of considerable interest in Knowledge Representation over the years, unfortunately it has proved to be a diicult notion to capture formally. For example, Lewis' seminal work 1973] on the subject of counterfactuals relies on a notion of comparative similarity, a measure which according to him is either well-understood and vague, or ad hoc and limited in scope. There are scads of measures of similarity based on, for the want of a better foundation, set theoretical measures. This type of measure is used in various forms of nonmontonic reasoning, and some interpretations of the principle of Minimal Change in the area of Belief Revision. The problem with such measures is that they do not take the internal structure of the information state into consideration, and as a consequence they are superrcial indicators of similarity. The quasi-metric, dist, is able to gauge the diierence between losing one important belief and losing several less important beliefs. Set theoretic measures neglect the property that similarity of system laws have a tendency to outweigh similarities in particular facts Lewis, 1973 p75]. Hansson argues convincingly that minimal change is deenable in terms of similarity. In particular, he says \to …