Databases in real life are often neither entirely closed-world nor entirely open-world. Indeed, databases in an enterprise are typically partially closed, in which a part of the data is constrained by master data that contains complete information about the enterprise in certain aspects. It has been shown that despite missing tuples, such a database may turn out to have complete information for answering a query. This paper studies partially closed databases from which both tuples and attribute values may be missing. We specify such a database in terms of conditional tables constrained by master data, referred to as c-instances. We first propose three models to characterize whether a c-instance T is complete for a query Q relative to master data. That is, depending on how missing values in T are instantiated, the answer to Q in T remains unchanged when new tuples are added. We then investigate three problems, to determine (a) whether a given c-instance is complete for a query Q, (b) whether there exists a c-instance that is complete for Q relative to master data available, and (c) whether a c-instance is a minimal-size database that is complete for Q. We establish matching lower and upper bounds on these problems for queries expressed in a variety of languages, in each of the three models for specifying relative completeness.
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