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We describe the Consistency Extractor System (Cons Ex) that computes consistent answers to Datalog queries with negation posed to relational databases that may be inconsistent with respect to certain integrity constraints. In order to solve this task, Cons Ex uses answers set programming. More precisely, Cons Ex uses disjunctive logic programs with stable models semantics to specify and reason with the repairs, i.e. with the consistent virtual instances that minimally depart from the original database. The consistent information is invariant under all repairs. Cons Ex achieves efficient query evaluation by implementing magic sets techniques. We describe the general methodology, its optimizations for query answering, and the architecture of the system. We also present encouraging experimental results.