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Inductive inference operators capture the process of completing a conditional belief base to an inference relation. One such operator is c-inference which is based on the c-representations of a belief base, c-representations being a special kind of ranking functions. c-Inference exhibits many desirable properties put forward for nonmonotonic reasoning; for instance, it fully complies with syntax splitting. A characterization of c-inference as a constraint satisfaction problem (CSP) yields a basis for implementing c-inference. However, the definitions of c-representations and of c-inference only take belief bases into account that satisfy a rather strong notion of consistency requiring every possible world to be at least somewhat plausible. In this paper, we extend the definition of c-representations to belief bases that need to satisfy only a weaker notion of consistency where some worlds may be completely infeasible. Based on these extended c-representations, we also extend the definition of c-inference correspondingly, thus covering all weakly consistent belief bases. Furthermore, we develop an adapted CSP characterizing the such extended c-inference that can be used as a basis for an implementation.

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