An Algorithm for Computing the Maximum Entropy Ranking for Variable Strength Defaults

A new algorithm for computing the maximum entropy ranking (me-ranking) over a set of variable strength defaults is given. Although this requires more information than its predecessor in terms of strengths for defaults, it has a much wider applicability, and allows for greater expressiveness in encoding default knowledge. The algorithm is shown to be sound and complete to the extent that it always produces an me-ranking when one exists. The reasons for multiple rankings and for no ranking are explored. A suucient condition for uniqueness of the me-ranking is given. 1 Background Defaults like bird) fly are natural rules which can be used to describe the normal behaviour of some domain. Default knowledge can be used to make inferences about a given situation so that, for example, an arbitrary bird would be assumed to y unless there were information to the contrary. The question arises of what should be inferred from any given set of defaults and research into this has led to the development of many very diierent nonmonotonic reasoning formalisms 8, 12, 13, 14]. A probabilistic model of default reasoning has proven to capture the most basic requirements of nonmonotonic behaviour 1, 7, 11]. A natural extension to this model is to apply the principle of maximum entropy to nd a distinguished consequence relation which captures the default knowledge but makes no additional assumptions, i.e., is the least biased 6, 10]. In this paper, the original work of 5] on applying maximum entropy to default reasoning is extended to cater for arbitrary sets of variable strength defaults. In doing so, it is hoped both to clarify the assumptions required by the me-approach, and to demonstrate that this exible method can help to explain why there has been disagreement among researchers regarding the nature of default inference. 2 Deriving the me-ranking First, it is necessary to look at what a default represents and how defaults are connected to ranking functions and consequence relations. Deenition 2.1 A default is a natural rule of the form a) b such that the antecedent , a, and the consequent, b, are formull of a nite propositional language, L.