Like Pearl’s System Z, c-representations provide a constructive approach to compute a ranking function from a conditional belief base from which further (conditional) beliefs can be derived, meeting major quality standards of nonmonotonic reasoning. This paper proposes a network-based structure for c-representations that allows for cutting down the complexity of reasoning significantly by decomposing the conditional belief base over a hypertree. We introduce c-LEG networks capturing the interactions among conditionals on a syntactical basis in full compatibility with the semantics of c-representations. This allows for reasoning in much smaller local contexts while still complying with the global information provided by the full conditional belief base. Moreover, we generalize the so-called safety property, which was recently presented in the context of conditional syntax splitting, to ensure that local c-representations of subbases over the hyperedges can be merged to yield global c-representations of the full conditional belief base. This allows for computing global c-representations step by step in local contexts, following the structure of the hypertree.

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.