It is known that the composition of schema mappings, each specified by source-to-target tgds (st-tgds), can be specified by a second-order tgd (SO tgd). We consider the question of what happens when target constraints are allowed. Specifically, we consider the question of specifying the composition of standard schema mappings (those specified by st-tgds, target egds, and a weakly-acyclic set of target tgds). We show that SO tgds, even with the assistance of arbitrary source constraints and target constraints, cannot specify in general the composition of two standard schema mappings. Therefore, we introduce source-to-target second-order dependencies (st-SO dependencies), which are similar to SO tgds, but allow equations in the conclusion. We show that st-SO dependencies (along with target egds and target tgds) are sufficient to express the composition of every finite sequence of standard schema mappings, and further, every st-SO dependency specifies such a composition. In addition to this expressive power, we show that st-SO dependencies enjoy other desirable properties. In particular, they have a polynomial-time chase that generates a universal solution. This universal solution can be used to find the certain answers to unions of conjunctive queries in polynomial time.

A schema mapping is a specification that describes how data structured under one schema (the source schema) is to be transformed into data structured under a different schema (the target schema). The notion of an inverse of a schema mapping is subtle, because a schema mapping may associate many target instances with each source instance, and many source instances with each target instance. In PODS 2006, Fagin defined a notion of the inverse of a schema mapping. This notion is tailored to the types of schema mappings that commonly arise in practice (those specified by “source-to-target tuple-generating dependencies”, or s-t tgds ). We resolve the key open problem of the complexity of deciding whether there is an inverse. We also explore a number of interesting questions, including: What is the structure of an inverse? When is the inverse unique? How many nonequivalent inverses can there be? When does an inverse have an inverse? How big must an inverse be? Surprisingly, these questions are all interrelated. We show that for schema mappings M specified by full s-t tgds (those with no existential quantifiers), if M has an inverse, then it has a polynomial-size inverse of a particularly nice form, and there is a polynomial-time algorithm for generating it. We introduce the notion of “essential conjunctions” (or “essential atoms” in the full case), and show that they play a crucial role in the study of inverses. We use them to give greatly simplified proofs of some known results about inverses. What emerges is a much deeper understanding about this fundamental and complex operator.

We investigate the question of whether a query Q can be answered using a set V of views. We first define the problem in information-theoretic terms: we say that V determines Q if V provides enough information to uniquely determine the answer to Q . Next, we look at the problem of rewriting Q in terms of V using a specific language. Given a view language V and query language Q , we say that a rewriting language R is complete for V -to- Q rewritings if every Q ∈ Q can be rewritten in terms of V ∈ V using a query in R , whenever V determines Q . While query rewriting using views has been extensively investigated for some specific languages, the connection to the information-theoretic notion of determinacy, and the question of completeness of a rewriting language have received little attention. In this article we investigate systematically the notion of determinacy and its connection to rewriting. The results concern decidability of determinacy for various view and query languages, as well as the power required of complete rewriting languages. We consider languages ranging from first-order to conjunctive queries.

We revisit the standard chase procedure, studying its properties and applicability to classical database problems. We settle (in the negative) the open problem of decidability of termination of the standard chase, and we provide sufficient termination conditions which are strictly less over-conservative than the best previously known. We investigate the adequacy of the standard chase for checking query containment under constraints, constraint implication and computing certain answers in data exchange, gaining a deeper understanding by separating the algorithm from its result. We identify the properties of the chase result that are essential to the above applications, and we introduce the more general notion of F-universal model set, which supports query and constraint languages that are closed under a class F of mappings. By choosing F appropriately, we extend prior results to existential first-order queries and ∀∃-firstorder constraints. We show that the standard chase is incomplete for finding universal model sets, and we introduce the extended core chase which is complete, i.e. finds an F-universal model set when it exists. A key advantage of the new chase is that the same algorithm can be applied for all mapping classes F of interest, simply by modifying the set of constraints given as input. Even when restricted to the typical input in prior work, the new chase supports certain answer computation and containment/implication tests in strictly more cases than the incomplete standard chase.

A schema mapping is a high-level specification that describes the relationship between two database schemas. As schema mappings constitute the essential building blocks of data exchange and data integration, an extensive investigation of the foundations of schema mappings has been carried out in recent years. Even though several different aspects of schema mappings have been explored in considerable depth, the study of schema-mapping optimization remains largely uncharted territory to date.

Data exchange deals with inserting data from one database into another database having a different schema. Fagin et al. [2005] have shown that among the universal solutions of a solvable data exchange problem, there exists—up to isomorphism—a unique most compact one, “the core”, and have convincingly argued that this core should be the database to be materialized. They stated as an important open problem whether the core can be computed in polynomial time in the general setting where the mapping between the source and target schemas is given by source-to-target constraints that are arbitrary tuple generating dependencies (tgds) and target constraints consisting of equality generating dependencies (egds) and a weakly acyclic set of tgds. In this article, we solve this problem by developing new methods for efficiently computing the core of a universal solution. This positive result shows that data exchange based on cores is feasible and applicable in a very general setting. In addition to our main result, we use the method of hypertree decompositions to derive new algorithms and upper bounds for query containment checking and computing cores of arbitrary database instances. We also show that computing the core of a data exchange problem is fixed-parameter intractable with respect to a number of relevant parameters, and that computing cores is NP-complete if the rule bodies of target tgds are augmented by a special predicate that distinguishes a null value from a constant data value.

Composition of mappings between schemas is essential to support schema evolution, data exchange, data integration, and other data management tasks. In many applications, mappings are given by embedded dependencies. In this article, we study the issues involved in composing such mappings. Our algorithms and results extend those of Fagin et al. [2004], who studied the composition of mappings given by several kinds of constraints. In particular, they proved that full source-to-target tuple-generating dependencies (tgds) are closed under composition, but embedded source-to-target tgds are not. They introduced a class of second-order constraints, SO tgds , that is closed under composition and has desirable properties for data exchange. We study constraints that need not be source-to-target and we concentrate on obtaining (first-order) embedded dependencies. As part of this study, we also consider full dependencies and second-order constraints that arise from Skolemizing embedded dependencies. For each of the three classes of mappings that we study, we provide: (a) an algorithm that attempts to compute the composition; and (b) sufficient conditions on the input mappings which guarantee that the algorithm will succeed. In addition, we give several negative results. In particular, we show that full and second-order dependencies that are not limited to be source-to-target are not closed under composition (for the latter, under the additional restriction that no new function symbols are introduced). Furthermore, we show that determining whether the composition can be given by these kinds of dependencies is undecidable.

Data exchange deals with inserting data from one database into another database having a different schema. We study and solve a central computational problem of data exchange, namely, computing the core of a universal solution to a data exchange problem. Fagin, Kolaitis, and Popa [9], have shown that among the universal solutions of a solvable data exchange problem, there exists a most compact one (up to isomorphism), "the core" (of any universal solution), and have convincingly argued that this core should be the solution of choice. They stated as an important open problem whether the core of a universal solution can be computed in polynomial time in the general setting where the source-to-target constraints are arbitrary tuple generating dependencies (TGDs) and the target constraints consist of equation generating dependencies (EGDs) and weakly-acyclic TGDs. In this paper we solve this problem by developing new efficient methods for computing the core of a universal solution. This positive result shows that the core approach of Fagin, Kolaitis, and Popa is feasible and applicable in a very general setting and thus provides a further momentum to the use of cores in data exchange.

Composition of mappings between schemas is essential to support schema evolution, data exchange, data integration, and other data management tasks. In many applications, mappings are given by embedded dependencies. In this paper, we study the issues involved in composing such mappings.

A Web service operation can be seen as a function op: X/sub 1/,..., X/sub n/ /spl rarr/ Y/sub 1/,..., Y/sub m/ having an input message (request) with n arguments (parts), and an output message (response) with m parts. We study the problem of deciding whether a query Q is feasible, i.e., whether there exists a logically equivalent query Q' that can be executed observing the limited access patterns given by the Web service (source) relations. Executability depends on the specific syntactic form of a query, while feasibility is a more "robust" semantic notion, involving all equivalent queries (i.e., reorderings, minimized queries, etc). We show that deciding query feasibility (called "stability") is NP-complete for conjunctive queries (CQ) and for conjunctive queries with union (UCQ).

We study the problem of answering queries over sources with limited access patterns. Given a first-order query Q, the problem is to decide whether there is an equivalent query which can be executed observing the access patterns restrictions. If so, we say that Q is feasible. We define feasible for first-order queries---previous definitions handled only some existential cases---and characterize the complexity of many first-order query classes. For each of them, we show that deciding feasibility is as hard as deciding containment. Since feasibility is undecidable in many cases and hard to decide in some others, we also define an approximation to it which can be computed in NP for any first-order query and in P for unions of conjunctive queries with negation. Finally, we outline a practical overall strategy for processing first-order queries under limited access patterns.

We define and study strong diagonalization and compare it to weak diagonalization, implicit in the work of D. Kozen (1980). Kozen's result shows that virtually every separation can be recast as weak diagonalization. We show that there are classes of languages, which cannot be separated by strong diagonalization and provide evidence that strong diagonalization does not relativize. We also define two kinds of indirect diagonalization and study their power: Since we define strong diagonalization in terms of universal languages, we study their complexity. We distinguish and compare weak and strict universal languages. Finally we analyze some apparently weaker variants of universal languages, which we call pseudouniversal languages, and show that under weak closure conditions they easily yield universal languages.