Total Preorders vs Ranking Functions under Belief Revision – the Dynamics of Empty Layers

Total preorders and Spohn’s ranking functions are most popular semantic structures in nonmonotonic reasoning and belief revision. Each ranking function uniquely induces a total preorder, while each total preorder corresponds to infinitely many ranking functions because of the empty layers that ranking functions may have. In this paper, we adopt a dynamic perspective and investigate the role of empty layers in belief revision scenarios. We strengthen the notion of (inferential) equivalence of ranking functions by introducing revision equivalence which postulates the equivalence of ranking functions after (most general) revision operations. Moreover, we single out so-called linearly equivalent ranking functions as prototypes of ranking functions with regularly inserted empty layers. Such ranking functions are most suitable to provide an invariance property for revision equivalence which claims that linear equivalence should be preserved. We show that strategic c-revisions ensure (conditional) revision equivalence of linearly equivalent ranking functions if the strategies are adequately chosen, whereas the Darwiche-Pearl postulates for iterated revision alone are not enough to guarantee revision equivalence of ranking functions. We evaluate various other iterated revision approaches from the literature with respect to revision equivalence and preserving linear equivalence under revision. Furthermore, we present an approach to defining equivalence preserving revision operators for ranking functions from revision operators for total preorders.