Equality-generating Dependencies Research Articles

Reasoning on property graphs with graph generating dependencies

Data dependencies are a key concept in data management and have been researched in data integration, data quality and query optimization. With the increasing use of graph-structured data in diverse applications, there is also an increasing interest in the study of graph data dependencies. In this scenario, different classes of graph data dependencies have been proposed in the literature. In this work, we study the class of Graph Generating Dependencies (GGDs). Graph Generating Dependencies (GGDs) informally express constraints between two (possibly different) graph patterns which enforce relationships on both graph's data (via property value constraints) and its structure (via topological constraints). While most of previously proposed classes of graph data dependencies focus on generalizing equality-generating dependencies for graph data, Graph Generating Dependencies (GGDs) can express tuple- and equality-generating dependencies on property graphs, both of which find broad application in graph data management. Given this new class of dependency, in this paper, we discuss the reasoning behind GGDs on Property Graphs. We propose algorithms to solve three main reasoning problems: the satisfiability, implication, and validation problems for GGDs and analyze their complexity. By studying these problems, we can understand the expressiveness and the limitations of GGDs in practical applications. To demonstrate the practical use of GGDs, we propose an algorithm that finds inconsistencies in data through validation of GGDs. Our experiments show that even though the validation of GGDs has high computational complexity, GGDs can be used to find data inconsistencies in a feasible execution time on both synthetic and real-world data.

Query answering over inconsistent knowledge bases: A probabilistic approach

Consistent query answering (CQA) is a widely accepted paradigm for querying inconsistent knowledge bases (KBs). A consistent answer to a query is a tuple that is an answer to the query over every repair of the KB, which is in turn a consistent KB whose extensional knowledge “minimally” differs from the original one's. This coarse-grained classification of answers into consistent and non-consistent ones lacks any information about their degree of consistency, i.e., how likely it is that a tuple is an answer to the query, when considering all the repaired KBs. To overcome this limitation, we consider a fine-grained notion of repair for KBs with equality-generating dependencies (EGDs), based on attribute-level updates, and exploit this notion to propose a probabilistic CQA approach, which associates a confidence to each answer, thereby providing more informative query answers. We first show that computing the query answer confidence is #P-hard. Then, in the light of this intractability result, we study the existence of efficient randomized, approximation schemes. In particular, we show that absolute error approximation schemes always exist in the general case, while more refined relative error approximation schemes, i.e., fully polynomial-time, randomized approximation schemes (FPRAS) exist when assuming that the constraints of the knowledge base are primary keys. Finally, we extend our framework to knowledge bases with tuple-generating dependencies (TGDs) and generalize our approximability results to the new setting, and prove additional inapproximability results.

Exploiting the Power of Equality-Generating Dependencies in Ontological Reasoning

Equality-generating dependencies (EGDs) allow to fully exploit the power of existential quantification in ontological reasoning settings modeled via Tuple-Generating Dependencies (TGDs), by enabling value-assignment or forcing the equivalence of fresh symbols. These capabilities are at the core of many common reasoning tasks, including graph traversals, clustering, data matching and data fusion, and many more related real-world scenarios. However, the interplay of TGDs and EGDs is known to lead to undecidability or intractability of query answering in tractable Datalog+/- fragments, like Warded Datalog+/-, for which, in the sole presence of TGDs, query answering is PTIME in data complexity. Restrictions of equality constraints, like separable EGDs, have been studied, but all achieve decidability at the cost of limited expressive power, which makes them unsuitable for the mentioned tasks. This paper introduces the class of "harmless" EGDs, that subsume separable EGDs and allow to model a very broad class of tasks. We contribute a sufficient syntactic condition for testing harmlessness, an undecidable task in general. We argue that in Warded Datalog+/- with harmless EGDs, ontological reasoning is decidable and PTIME. From such theoretical underpinnings, we develop novel chase-based techniques for reasoning with harmless EGDs and present an implementation within the Vadalog system, a state-of-the-art Datalog-based reasoner. We provide full-scale experimental evaluation and comparative analysis.

Semantic Optimization of Conjunctive Queries

This work deals with the problem of semantic optimization of the central class of conjunctive queries (CQs). Since CQ evaluation is NP-complete, a long line of research has focussed on identifying fragments of CQs that can be efficiently evaluated. One of the most general restrictions corresponds to generalized hypetreewidth bounded by a fixed constant k ≥ 1; the associated fragment is denoted GHW k . A CQ is semantically in GHW k if it is equivalent to a CQ in GHW k . The problem of checking whether a CQ is semantically in GHW k has been studied in the constraint-free case, and it has been shown to be NP-complete. However, in case the database is subject to constraints such as tuple-generating dependencies (TGDs) that can express, e.g., inclusion dependencies, or equality-generating dependencies (EGDs) that capture, e.g., key dependencies, a CQ may turn out to be semantically in GHW k under the constraints, while not being semantically in GHW k without the constraints. This opens avenues to new query optimization techniques. In this article, we initiate and develop the theory of semantic optimization of CQs under constraints. More precisely, we study the following natural problem: Given a CQ and a set of constraints, is the query semantically in GHW k , for a fixed k ≥ 1, under the constraints, or, in other words, is the query equivalent to one that belongs to GHW k over all those databases that satisfy the constraints? We show that, contrary to what one might expect, decidability of CQ containment is a necessary but not a sufficient condition for the decidability of the problem in question. In particular, we show that checking whether a CQ is semantically in GHW 1 is undecidable in the presence of full TGDs (i.e., Datalog rules) or EGDs. In view of the above negative results, we focus on the main classes of TGDs for which CQ containment is decidable and that do not capture the class of full TGDs, i.e., guarded, non-recursive, and sticky sets of TGDs, and show that the problem in question is decidable, while its complexity coincides with the complexity of CQ containment. We also consider key dependencies over unary and binary relations, and we show that the problem in question is decidable in elementary time. Furthermore, we investigate whether being semantically in GHW k alleviates the cost of query evaluation. Finally, in case a CQ is not semantically in GHW k , we discuss how it can be approximated via a CQ that falls in GHW k in an optimal way. Such approximations might help finding “quick” answers to the input query when exact evaluation is intractable.

Semantic Optimization in Tractable Classes of Conjunctive Queries

This paper reports on recent advances in semantic query optimization. We focus on the core class of conjunctive queries (CQs). Since CQ evaluation is NP-complete, a long line of research has concentrated on identifying fragments of CQs that can be efficiently evaluated. One of the most general such restrictions corresponds to bounded generalized hypertreewidth, which extends the notion of acyclicity. Here we discuss the problem of reformulating a CQ into one of bounded generalized hypertreewidth. Furthermore, we study whether knowing that such a reformulation exists alleviates the cost of CQ evaluation. In case a CQ cannot be reformulated as one of bounded generalized hypertreewidth, we discuss how it can be approximated in an optimal way. All the above issues are examined both for the constraint-free case, and the case where constraints, in fact, tuple-generating and equality-generating dependencies, are present

Exchange-Repairs

In a data exchange setting with target constraints, it is often the case that a given source instance has no solutions. In such cases, the semantics of target queries trivialize. The aim of this paper is to introduce and explore a new framework that gives meaningful semantics in such cases by using the notion of exchange-repairs. Informally, an exchange-repair of a source instance is another source instance that differs minimally from the first, but has a solution. Exchange-repairs give rise to a natural notion of exchange-repair certain answers (XR-certain answers) for target queries. We show that for schema mappings specified by source-to-target GAV dependencies and target equality-generating dependencies (egds), the XR-certain answers of a target conjunctive query can be rewritten as the consistent answers (in the sense of standard database repairs) of a union of conjunctive queries over the source schema with respect to a set of egds over the source schema, making it possible to use a consistent query-answering system to compute XR-certain answers in data exchange. We then examine the general case of schema mappings specified by source-to-target GLAV constraints, a weakly acyclic set of target tgds and a set of target egds. The main result asserts that, for such settings, the XR-certain answers of conjunctive queries can be rewritten as the certain answers of a union of conjunctive queries with respect to the stable models of a disjunctive logic program over a suitable expansion of the source schema.

Taming the Infinite Chase: Query Answering under Expressive Relational Constraints

The chase algorithm is a fundamental tool for query evaluation and for testing query containment under tuple-generating dependencies (TGDs) and equality-generating dependencies (EGDs). So far, most of the research on this topic has focused on cases where the chase procedure terminates. This paper introduces expressive classes of TGDs defined via syntactic restrictions: guarded TGDs (GTGDs) and weakly guarded sets of TGDs (WGTGDs). For these classes, the chase procedure is not guaranteed to terminate and thus may have an infinite outcome. Nevertheless, we prove that the problems of conjunctive-query answering and query containment under such TGDs are decidable. We provide decision procedures and tight complexity bounds for these problems. Then we show how EGDs can be incorporated into our results by providing conditions under which EGDs do not harmfully interact with TGDs and do not affect the decidability and complexity of query answering. We show applications of the aforesaid classes of constraints to the problem of answering conjunctive queries in F-Logic Lite, an object-oriented ontology language, and in some tractable Description Logics.

The complexity of embedded axiomatization for a class of closed database views

It is well known that the complexity of testing the correctness of an arbitrary update to a database view can be far greater than the complexity of testing a corresponding update to the main schema. However, views are generally managed according to some protocol which limits the admissible updates to a subset of all possible changes. The question thus arises as to whether there is a more tractable relationship between these two complexities in the presence of such a protocol. In this paper, this question is addressed for closed update strategies, which are based upon the constant-complement approach of Bancilhon and Spyratos. The approach is to address a more general question --- that of characterizing the complexity of axiomatization of views, relative to the complexity of axiomatization of the main schema. For schemata constrained by denial or consistency constraints, that is, statements which rule out certain situations, such as the equality-generating dependencies (EGDs) or, more specifically, the functional dependencies (FDs) of the relational model, a broad and comprehensive result is obtained in a very general framework which is not tied to the relational model in any way. It states that every such schema is governed by an equivalent set of constraints which embed into the component views, and which are no more complex than the original set. For schemata constrained by generating dependencies, of which tuple-generating dependencies (TGDs) in general and, more specifically, both join dependencies (JDs) and inclusion dependencies (INDs) are examples within the relational model, a similar result is obtained, but only within a context known as meet-uniform decompositions, which fails to recapture some important situations. To address the all-important case of relational schemata constrained by both FDs and INDs, a hybrid approach is also developed, in which the general theory regarding denial constraints is blended with a focused analysis of a special but very practical subset of the INDs known as fanout-free unary inclusion dependencies (fanout-free UINDs), to obtain results parallel to the above-mentioned cases: every such schema is governed by an equivalent set of constraints which embed into the component views, and which are no more complex than the original set. In all cases, the question of view update complexity is then answered via a corollary to this main result.

A top-down proof procedure for generalized data dependencies

Data dependencies are well known in the context of relational database. They aim to specify constraints that the data must satisfy to model correctly the part of the world under consideration. The implication problem for dependencies is to decide whether a given dependency is logically implied by a given set of dependencies. A proof procedure for the implication problem, called “chase”, has already been studied in the generalized case of tuple-generating and equality-generating dependencies. The chase is a bottom-up procedure: from hypotheses to conclusion, and thus is not goal-directed. It also entails in the case of TGDs the dynamic creation of new constants, which can turn out to be a costly operation. This paper introduces a new proof procedure which is top-down: from conclusion to hypothesis, that is goal-directed. The originality of this procedure is that it does not act as classical theorem proving procedures, which require a special form of expressions, such as clausal form, obtained after Skolemization. We show, with our procedure, that this step is useless, and that the notion of piece allows infering directly on dependencies, thus saving the cost of Skolemizing the dependencies set and, morever, that the inference can be performed without dynamically creating new constants. Although top-down approaches are known to be less efficient in time than bottom-up ones, the notion of piece cuts down irrelevant goals usually generated, leading to a usable top-down method. With the more recent introduction of constrained and ordered dependencies, some interesting perspectives also arise.

Constraint-Generating Dependencies

Traditionally, dependency theory has been developed for uninterpreted data. Specifically, the only assumption that is made about the data domains is that data values can be compared for equality. However, data is often interpreted and there can be advantages in considering data as such, for instance, obtaining more compact representations as is done in constraint databases. This paper considers dependency theory in the context of interpreted data. Specifically, it studies constraint-generating dependencies. These are a generalization of equality-generating dependencies where equality requirements are replaced by constraints on an interpreted domain. The main technical results in the paper are a general decision procedure for the implication and consistency problems for constraint-generating dependencies and complexity results for specific classes of such dependencies over given domains. The decision procedure proceeds by reducing the dependency problem to a decision problem for the constraint theory of interest and is applicable as soon as the underlying constraint theory is decidable. The complexity results are, in some cases, directly lifted from the constraint theory; in other cases, optimal complexity bounds are obtained by taking into account the specific form of the constraint decision problem obtained by reducing the dependency implication problem.

On the Boundedness of Constant-Time-Maintainable Database Schemes

Constant-time-maintainable database schemes are highly desirable with respect to constraint enforcement, since it is possible to determine whether any of their consistent states plus an inserted tuple is consistent in time independent of the state size. Several proper subclasses of constant-time-maintainable database schemes are known to be bounded with respect to dependencies and hence very desirable with respect to query answering. However, whether the whole class of constant-time-maintainable database schemes is bounded is not known for sure. In this paper, it is proven that the entire class of constant-time-maintainable database schemes is bounded with respect to dependencies and thus very desirable with respect to query answering in the following cases: (1) only cover-embedded functional dependencies appear as constraints; (2) only equality-generating dependencies appear as constraints and the database scheme has a lossless join. In particular, it is shown that total projections of representative instances can be computed via unions of projections of simple chase join expressions. Since it is known how to optimize these expressions, it is possible to compute total projections optimally. These results show that the class of constant-time-maintainable database schemes is the largest class of database schemes, which are highly desirable with respect to both constraint enforcement and query answering. This class of schemes can be effectively recognized by known algorithms. The previously known largest class of database schemes with these desirable properties is the class of independent database schemes, which is a proper subclass of constant-time-maintainable schemes.

Preservation of integrity constraints in definite DATALOG programs

Given a program P and two sets IC and IC′ of integrity constraints, IC uniformly implies IC′ in P if whenever an (input) database I defined on predicates of P satisfies IC then the (output) database computed by P from I satisfies IC′. P preserves IC if IC uniformly implies IC in P. We consider only definite DATALOG programs. We show that testing preservation of downward-closed integrity constraints can be simplified to the case of single-rule programs that are computed in a “nonrecursive manner”. Specially, we present an efficient test algorithm when constraints are equality-generating dependencies. We also show that the uniform implication problem can be similarly simplified if IC is preserved and all given constraints are downward-closed. Our results generalize dependency-preserving database schemes by considering mappings defined by arbitrary definite DATALOG programs and more general integrity constraints.

Constant-time maintainability

The maintenance problem of a database scheme is the following decision problem: Given a consistent database state ρ and a new tuple u over some relation scheme of ρ, is the modified state ρ ∪ { u } still consistent? A database scheme is said to be constant-time-maintainable(ctm) if there exists an algorithm that solves its maintenance problem by making a fixed number of tuple retrievals. We present a practically useful algorithm, called the canonical maintenance algorithm , that solves the maintenance problem of all ctm database schemes within a "not too large" bound. A number of interesting properties are shown for ctm database schemes, among them that non-ctm database schemes are not maintainable in less than a linear time in the state size. A test method is given when only cover embedded functional dependencies (fds) appear. When the given dependencies consist of fds and the join dependency (jd) ⋈ R of the database scheme, testing whether a database scheme is ctm is reduced to the case of cover embedded fds. When dependency-preserving database schemes with only equality-generating dependencies (egds) are considered, it is shown that every ctm database scheme has a set of dependencies that is equivalent to a set of embedded fds, and thus, our test method for the case of embedded fds can be applied. In particular, this includes the important case of lossless database schemes with only egds.

On characterizing boundedness of database schemes with bounded dependencies

We investigate the effect of bounded dependencies on the boundedness of database schemes. The following results are proved. A database scheme with only bounded equality-generating dependencies is always bounded with respect to dependencies; a lossless database scheme with bounded full implicational dependencies is bounded w.r.t. dependencies if and only if the implicational dependencies are equivalent to a single join dependency and some equality-generating dependencies. By a known method, this condition can be tested effectively. These results are relevant in database theory in that they determine in a rather general case whether queries under the representative instance approach can be expressed in relational algebra.

On Bounded Database Schemes and Bounded Horn-Clause Programs

A lossless database scheme with full implicational dependencies is considered. The following condition is necessary in order that restricted projections of the representative instance could be expressed in relational algebra (i.e., in first-order logic): The dependencies of the database scheme are equivalent to a single join dependency and some equality-generating dependencies. An important special case is when the database scheme has only full tuple-generating dependencies. In this case, restricted projections of the representative instance can be expressed in relational algebra if and only if the following condition is true: The dependencies of the database scheme are equivalent to a single join dependency. Testing this condition is decidable. This result also applies to the class of Horn-clause programs consisting only of typed rules with one predicate symbol R (and without function symbols). A program consisting of rules of this form is equivalent to a nonrecursive program if and only if it is equival...

Notions of dependency satisfaction

Two notions of dependency satisfaction, consistency and completeness , are introduced. Consistency is the natural generalization of weak-instance satisfaction and seems appropriate when only equality-generating dependencies are given, but disagrees with the standard notion in the presence of tuple-generating dependencies. Completeness is based on the intuitive semantics of tuple-generating dependencies but differs from the standard notion for equality-generating dependencies. It is argued that neither approach is the correct one, but rather that they correspond to different policies on constraint enforcement, and each one is appropriate in different circumstances. Consistency and completeness of a state are characterized in terms of the tableau associated with the state and in terms of logical properties of a set of first-order sentences associated with the state. A close relation between the problems of testing for consistency and completeness and of testing implication of dependencies is established, leading to lower and upper bounds for the complexity of consistency and completeness. The possibility of formalizing dependency satisfaction without using a universal relation scheme is examined.