Conditional Splittings of Belief Bases and Nonmonotonic Inference with c-Representations

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The concept of conditional syntax splitting for inductive inference from conditional belief bases has been proposed as a generalization of syntax splitting which also covers cases where the conditionals in the subbases share some atoms. p-Entailment and system Z fail to satisfy conditional syntax splitting, and up to now, only two inductive inference operators, lexicographic inference and system W, have been shown to satisfy this property. In this paper, we introduce the concept of conditional semantic splitting. We show that c-representations satisfy a core postulate relating conditional splittings on the syntax and the semantic level. Based on these findings, we investigate conditional syntax splitting for nonmonotonic inference with c-representations. Regarding single c-representations, we utilize the concept of selection strategies, and show that a straightforward property of the selection strategy leads to inference operators satisfying conditional syntax splittings. Furthermore, we show that c-inference taking all c-representations of a belief base into account also fully complies with conditional syntax splitting.