Abstract: We describe CARIN, a novel family of representation languages, which integrate the expressive power of Horn rules and of description logics. We address the key issue in designing such a language, namely, providing a sound and complete inference procedure. We identify existential entailment as a core problem in reasoning in CARIN, and describe an existential entailment algorithm for CARIN languages whose description logic component is ALCNR. This algorithm entails several important results for reasoning in CARIN, most notably: (1) a sound and complete inference procedure for non recursive CARIN-ALCNR, and (2) an algorithm for determining rule subsumption over ALCNR.