Integrity constraints are rules that should guarantee the integrity of a database. Provided an adequate mechanism to express them is available, the following question arises: is there any way to populate a database which satisfies the constraints supplied by a database designer? That is, does the database schema, including constraints, admit at least a nonempty model? This work answers the above question in a complex object database environment, providing a theoretical framework, including the following ingredients: (1) two alternative formalisms, able to express a relevant set of state integrity constraints with a declarative style; (2) two specialized reasoners, based on the tableaux calculus, able to check the consistency of complex objects database schemata expressed with the two formalisms. The proposed formalisms share a common kernel, which supports complex objects and object identifiers, and which allow the expression of acyclic descriptions of: classes, nested relations and views, built up by means of the recursive use of record, quantified set, and object type constructors and by the intersection, union, and complement operators. Furthermore, the kernel formalism allows the declarative formulation of typing constraints and integrity rules. In order to improve the expressiveness and maintain the decidability of the reasoning activities, we extend the kernel formalism into two alternative directions. The first formalism, OLCP, introduces the capability of expressing path relations. Because cyclic schemas are extremely useful, we introduce a second formalism, OLCD, with the capability of expressing cyclic descriptions but disallowing the expression of path relations. In fact, we show that the reasoning activity in OLCDP (i.e., OLCP with cycles) is undecidable.
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