Observational Equivalence of Conditional Belief Bases

Abstract

In nonmonotonic reasoning, a conditional of the form ‘If A then usually B’ is typically accepted if a situation where both A and B hold is deemed to be more plausible, more probable, or less surprising, etc., than a situation where A holds, but B does not hold. In a propositional setting, this leads to a relation on the propositional interpretations, also called worlds, by comparing worlds according to their plausibility. In this paper, we address the question of which kind of relations on the set of worlds can be obtained by completing a conditional belief base via an inductive inference operator. As a key concept for our investigations, we introduce and employ the notion of observational equivalence of belief bases that takes an inductive inference operator and a set of queries into account. This leads to the notion of the observational normal form (ONF) and, by focussing on so-called base conditionals, to the base conditional normal form (BCNF). The acceptance of base conditionals corresponds to the plausibility ordering on possible worlds induced by an inference method. Both normal forms ONF and BCNF are combined with renamings as an additional dimension. We establish the interrelationships among the normal forms and evaluate them empirically with respect to systematically generated belief bases.