Data exchange heavily relies on the notion of incomplete database instances. Several semantics for such instances have been proposed and include open (OWA), closed (CWA), and open-closed (OCWA) world. For all these semantics important questions are: whether one incomplete instance semantically implies another; when two are semantically equivalent; and whether a smaller or smallest semantically equivalent instance exists. For OWA and CWA these questions are fully answered. For several variants of OCWA, however, they remain open. In this work we adress these questions for Closed Powerset semantics and the OCWA semantics of Libkin and Sirangelo, 2011. We define a new OCWA semantics, called OCWA*, in terms of homomorphic covers that subsumes both semantics, and characterize semantic implication and equivalence in terms of such covers. This characterization yields a guess-and-check algorithm to decide equivalence, and shows that the problem is NP-complete. For the minimization problem we show that for several common notions of minimality there is in general no unique minimal equivalent instance for Closed Powerset semantics, and consequently not for the more expressive OCWA* either. However, for Closed Powerset semantics we show that one can find, for any incomplete database, a unique finite set of its subinstances which are subinstances (up to renaming of nulls) of all instances semantically equivalent to the original incomplete one. We study properties of this set, and extend the analysis to OCWA*.
As the information available to lay users through autonomous data sources continues to increase, mediators become important to ensure that the wealth of information available is tapped effectively. A key challenge that these information mediators need to handle is the varying levels of incompleteness in the underlying databases in terms of missing attribute values. Existing approaches such as QPIAD aim to mine and use Approximate Functional Dependencies (AFDs) to predict and retrieve relevant incomplete tuples. These approaches make independence assumptions about missing values---which critica…
Codd's relational model describes just one possible world. To better cope with incomplete information, extended database models allow several possible worlds. Vague tables are one such convenient extended model where attributes accept sets of possible values (e.g., the manager is either Jill or Bob). However, conceptual database design in such cases remains an open problem. In particular, there is no canonical definition of functional dependencies (FDs) over possible worlds (e.g., each employee has just one manager). We identify several desirable properties that the semantics of such FDs shoul…
This paper addresses the problem of verifying equivalence between a pair of programs that operate over databases with different schemas. This problem is particularly important in the context of web applications, which typically undergo database refactoring either for performance or maintainability reasons. While web applications should have the same externally observable behavior before and after schema migration, there are no existing tools for proving equivalence of such programs. This paper takes a first step towards solving this problem by formalizing the equivalence and refinement checkin…
We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the domains of these unknown values are finite and, for a Boolean query $q$, we consider the following two problems: given as input an incomplete database $D$, (a) return the number of completions of $D$ that satisfy $q$; or (b) return the number of valuations of the nulls of $D$ yielding a completion that satisfies $q$. We obtain dichotomies between \#P-hardness …
We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the domains of these unknown values are finite and, for a Boolean query $q$, we consider the following two problems: given as input an incomplete database $D$, (a) return the number of completions of $D$ that satisfy $q$; or (b) return or the number of valuations of the nulls of $D$ yielding a completion that satisfies $q$. We obtain dichotomies between #P-hardnes…
One of the first attempts to set a solid theoretical foundation for extending the content of relational databases with incomplete information was the fuzzy relational model by Buckles and Petry. This structure was based on two generalizations of the traditional relational model: (1) A tuple component can be any subset of the corresponding domain, rather than a single element and (2) A similarity relation is defined on each domain. This relation satisfies the properties of reflexivity, symmetry and max-min transitivity, thus having the equality relation as a special case. This generalization ke…
Queries with aggregation and arithmetic operations, as well as incomplete data, are common in real-world database, but we lack a good understanding of how they should interact. On the one hand, systems based on SQL provide ad-hoc rules for numerical nulls, on the other, theoretical research largely concentrates on the standard notions of certain and possible answers. In the presence of numerical attributes and aggregates, however, these answers are often meaningless, returning either too little or too much. Our goal is to define a principled framework for databases with numerical nulls and ans…
In this paper we study the problem of reducing the evaluation costs of queries on finite databases in presence of integrity constraints, by designing and materializing views. Given a database schema, a set of queries defined on the schema, a set of integrity constraints, and a storage limit, to find a solution to this problem means to find a set of views that satisfies the storage limit, provides equivalent rewritings of the queries under the constraints (this requirement is weaker than equivalence in the absence of constraints), and reduces the total costs of evaluating the queries. This prob…
Classical approaches for OLAP assume that the data of all tables is complete. However, in case of incomplete tables with missing tuples, classical approaches fail since the result of a SQL aggregate query might significantly differ from the results computed on the full dataset. Today, the only way to deal with missing data is to manually complete the dataset which causes not only high efforts but also requires good statistical skills to determine when a dataset is actually complete. In this paper, we propose an automated approach for relational data completion called ReStore using a new class …
Probabilistic databases play a crucial role in the management and understanding of uncertain data. However, incorporating probabilities into the semantics of incomplete databases has posed many challenges, forcing systems to sacrifice modeling power, scalability, or restrict the class of relational algebra formula under which they are closed. We propose an alternative approach where the underlying relational database always represents a single world, and an external factor graph encodes a distribution over possible worlds; Markov chain Monte Carlo (MCMC) inference is then used to recover this …
Current systems and formalisms for representing incomplete information generally suffer from at least one of two weaknesses. Either they are not strong enough for representing results of simple queries, or the handling and processing of the data, e.g. for query evaluation, is intractable. In this paper, we present a decomposition-based approach to addressing this problem. We introduce world-set decompositions (WSDs), a space-efficient formalism for representing any finite set of possible worlds over relational databases. WSDs are therefore a strong representation system for any relational quer…
In database-as-a-service platforms, automated verification of query equivalence helps eliminate redundant computation in the form of overlapping sub-queries. Researchers have proposed two pragmatic techniques to tackle this problem. The first approach consists of reducing the queries to algebraic expressions and proving their equivalence using an algebraic theory. The limitations of this technique are threefold. It cannot prove the equivalence of queries with significant differences in the attributes of their relational operators. It does not support certain widely-used SQL features. Its verif…
This paper studies the complexity of query evaluation for databases whose relations are partially ordered; the problem commonly arises when combining or transforming ordered data from multiple sources. We focus on queries in a useful fragment of SQL, namely positive relational algebra with aggregates, whose bag semantics we extend to the partially ordered setting. Our semantics leads to the study of two main computational problems: the possibility and certainty of query answers. We show that these problems are respectively NP-complete and coNP-complete, but identify tractable cases depending o…
We present a categorical denotational semantics for a database mapping, based on views, in the most general framework of a database integration/exchange. Developed database category DB, for databases (objects) and view-based mappings (morphisms) between them, is different from Set category: the morphisms (based on a set of complex query computations) are not functions, while the objects are database instances (sets of relations). The logic based schema mappings between databases, usually written in a highly expressive logical language (ex. LAV, GAV, GLAV mappings, or tuple generating dependenc…
Classical algorithms for query optimization presuppose the absence of inconsistencies or uncertainties in the database and exploit only valid semantic knowledge provided, e.g., by integrity constraints. Data inconsistency or uncertainty, however, is a widespread critical issue in ordinary databases: total integrity is often, in fact, an unrealistic assumption and violations to integrity constraints may be introduced in several ways. In this report we present an approach for semantic query optimization that, differently from the traditional ones, relies on not necessarily valid semantic knowled…
CobbleDB: Modelling Levelled Storage by Composition
University of British Columbia（不列颠哥伦比亚大学）；Télécom SudParis（巴黎电信学院）；RPTU University of Kaiserslautern-Landau（凯泽斯劳滕-兰道理工大学）；Sorbonne-Université—LIP6 & Inria（索邦大学—LIP6与法国国家信息与自动化研究所）
ETH Zurich（苏黎世联邦理工学院）；Georgia Institute of Technology（佐治亚理工学院）；Microsoft Research（微软研究院）
Machine learning (ML) applications have been thriving recently, largely attributed to the increasing availability of data. However, inconsistency and incomplete information are ubiquitous in real-world datasets, and their impact on ML applications remains elusive. In this paper, we present a formal study of this impact by extending the notion of Certain Answers for Codd tables, which has been explored by the database research community for decades, into the field of machine learning. Specifically, we focus on classification problems and propose the notion of "Certain Predictions" (CP) -- a tes…
