We study the problem of computing certain answers to a query over a target schema for a source instance under constraints which relate the source and target schemas. Prior work has shown that, for restricted constraints (source-to-target and target-to-target embedded dependencies) and unions of conjunctive queries, there is a special instance, known as a universal solution, such that running the query on it essentially yields the certain answers. Such a universal solution does not always exist for even slight extensions of these classes of constraints and queries. We show that there may be a finite set of instances, which we call a universal set solution, which suffices to compute the certain answers. We also introduce the notion of a k-universal set solution, which is sufficient to compute the certain answers to queries with at most k variables, even when no universal set solution exists. We show how to compute such universal and k-universal set solutions for universal-existential constraints and existential queries. The main algorithm for computing the universal set solution is an extended chase. We provide a completeness result for this chase and sufficient conditions for termination, which strictly extend the best previously known conditions (such as weak acyclicity). We also introduce a new kind of chase to compute k-universal set solutions.
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