Representation Invariance for Epistemic Revision Operators

In this article, we investigate the issue of representing total preorders by ranking functions in belief revision scenarios. Both total preorders and Spohn’s ranking functions are most popular semantic structures in nonmonotonic reasoning and belief revision. While each ranking function uniquely induces a total preorder, total preorders can usually be represented by infinitely many ranking functions which only differ in the position of their empty layers. In order to work towards representation invariance of total preorders by ranking functions, we first introduce the notion of revision equivalence which postulates that equivalence is preserved during (most general) revision operations. Moreover, we single out so-called linearly equivalent ranking functions as prototypes of ranking functions with regularly inserted empty layers. We show that, in general, revision equivalence is hard to achieve, which prompts us to take a more operator-focused perspective. We introduce the postulates Preservation of Equivalence (PoE) and Preservation of Scale (PoS) in order to axiomatize operators which can guarantee that the equivalence resp. linear equivalence including the scaling factor between ranking functions is preserved. Afterwards, we evaluate various iterated revision operators from the literature with respect to these postulates. While (PoE) and (PoS) do not generally hold for revision operators in the Darwiche–Pearl framework, we show that strategic c-revisions with suitably chosen strategies are scale-preserving with respect to linearly equivalent ranking functions. Furthermore, we present approaches to defining equivalence- and scale-preserving revision operators for ranking functions from arbitrary existing revision operators.