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The set of answers to a query may be very large, potentially overwhelming users when presented with the entire set. In such cases, presenting only a small subset of the answers to the user may be preferable. A natural requirement for this subset is that it should be as diverse as possible to reflect the variety of the entire population. To achieve this, the diversity of a subset is measured using a metric that determines how different two solutions are and a diversity function that extends this metric from pairs to sets. In the past, several studies have shown that finding a diverse subset from an explicitly given set is intractable even for simple metrics (like Hamming distance) and simple diversity functions (like summing all pairwise distances). This complexity barrier becomes even more challenging when trying to output a diverse subset from a set that is only implicitly given such as the query answers of a query and a database. Until now, tractable cases have been found only for restricted problems and particular diversity functions. To overcome these limitations, we focus on the notion of ultrametrics, which have been widely studied and used in many applications. Starting from any ultrametric $d$ and a diversity function $\delta$ extending $d$, we provide sufficient conditions over $\delta$ for having polynomial-time algorithms to construct diverse answers. To the best of our knowledge, these conditions are satisfied by all diversity functions considered in the literature. Moreover, we complement these results with lower bounds that show specific cases when these conditions are not satisfied and finding diverse subsets becomes intractable. We conclude by applying these results to the evaluation of conjunctive queries, demonstrating efficient algorithms for finding a diverse subset of solutions for acyclic conjunctive queries when the attribute order is used to measure diversity.

This document is an introduction to two related formalisms to define Boolean functions: binary decision diagrams, and Boolean circuits. It presents these formalisms and several of their variants studied in the setting of knowledge compilation. Last, it explains how these formalisms can be connected to the notions of automata over words and trees., Comment: 26 pages

The formal XAI community has studied a plethora of interpretability queries aiming to understand the classifications made by decision trees. However, a more uniform understanding of what questions we can hope to answer about these models, traditionally deemed to be easily interpretable, has remained elusive. In an initial attempt to understand uniform languages for interpretability, Arenas et al. (2021) proposed FOIL, a logic for explaining black-box ML models, and showed that it can express a variety of interpretability queries. However, we show that FOIL is limited in two important senses: (i) it is not expressive enough to capture some crucial queries, and (ii) its model agnostic nature results in a high computational complexity for decision trees. In this paper, we carefully craft two fragments of first-order logic that allow for efficiently interpreting decision trees: Q-DT-FOIL and its optimization variant OPT-DT-FOIL. We show that our proposed logics can express not only a variety of interpretability queries considered by previous literature, but also elegantly allows users to specify different objectives the sought explanations should optimize for. Using finite model-theoretic techniques, we show that the different ingredients of Q-DT-FOIL are necessary for its expressiveness, and yet that queries in Q-DT-FOIL can be evaluated with a polynomial number of queries to a SAT solver, as well as their optimization versions in OPT-DT-FOIL. Besides our theoretical results, we provide a SAT-based implementation of the evaluation for OPT-DT-FOIL that is performant on industry-size decision trees.

Formal XAI (explainable AI) is a growing area that focuses on computing explanations with mathematical guarantees for the decisions made by ML models. Inside formal XAI, one of the most studied cases is that of explaining the choices taken by decision trees, as they are traditionally deemed as one of the most interpretable classes of models. Recent work has focused on studying the computation of "sufficient reasons", a kind of explanation in which given a decision tree $T$ and an instance $x$, one explains the decision $T(x)$ by providing a subset $y$ of the features of $x$ such that for any other instance $z$ compatible with $y$, it holds that $T(z) = T(x)$, intuitively meaning that the features in $y$ are already enough to fully justify the classification of $x$ by $T$. It has been argued, however, that sufficient reasons constitute a restrictive notion of explanation, and thus the community has started to study their probabilistic counterpart, in which one requires that the probability of $T(z) = T(x)$ must be at least some value $\delta \in (0, 1]$, where $z$ is a random instance that is compatible with $y$. Our paper settles the computational complexity of $\delta$-sufficient-reasons over decision trees, showing that both (1) finding $\delta$-sufficient-reasons that are minimal in size, and (2) finding $\delta$-sufficient-reasons that are minimal inclusion-wise, do not admit polynomial-time algorithms (unless P=NP). This is in stark contrast with the deterministic case ($\delta = 1$) where inclusion-wise minimal sufficient-reasons are easy to compute. By doing this, we answer two open problems originally raised by Izza et al. On the positive side, we identify structural restrictions of decision trees that make the problem tractable, and show how SAT solvers might be able to tackle these problems in practical settings., Comment: Preliminary version. Abstract slightly trimmed due to arxiv character limit

In this systems paper, we present MillenniumDB: a novel graph database engine that is modular, persistent, and open source. MillenniumDB is based on a graph data model, which we call domain graphs, that provides a simple abstraction upon which a variety of popular graph models can be supported. The engine itself is founded on a combination of tried and tested techniques from relational data management, state-of-the-art algorithms for worst-case-optimal joins, as well as graph-specific algorithms for evaluating path queries. In this paper, we present the main design principles underlying MillenniumDB, describing the abstract graph model and query semantics supported, the concrete data model and query syntax implemented, as well as the storage, indexing, query planning and query evaluation techniques used. We evaluate MillenniumDB over real-world data and queries from the Wikidata knowledge graph, where we find that it outperforms other popular persistent graph database engines (including both enterprise and open source alternatives) that support similar query features.

Several queries and scores have recently been proposed to explain individual predictions over ML models. Given the need for flexible, reliable, and easy-to-apply interpretability methods for ML models, we foresee the need for developing declarative languages to naturally specify different explainability queries. We do this in a principled way by rooting such a language in a logic, called FOIL, that allows for expressing many simple but important explainability queries, and might serve as a core for more expressive interpretability languages. We study the computational complexity of FOIL queries over two classes of ML models often deemed to be easily interpretable: decision trees and OBDDs. Since the number of possible inputs for an ML model is exponential in its dimension, the tractability of the FOIL evaluation problem is delicate but can be achieved by either restricting the structure of the models or the fragment of FOIL being evaluated. We also present a prototype implementation of FOIL wrapped in a high-level declarative language and perform experiments showing that such a language can be used in practice., Comment: Accepted as Spotlight for NeurIPS'2021

In the last decade, substantial progress has been made towards standardizing the syntax of graph query languages, and towards understanding their semantics and complexity of evaluation. In this paper, we consider temporal property graphs (TPGs) and propose temporal regular path queries (TRPQs) that incorporate time into TPG navigation. Starting with design principles, we propose a natural syntactic extension of the MATCH clause of popular graph query languages. We then formally present the semantics of TRPQs, and study the complexity of their evaluation. We show that TRPQs can be evaluated in polynomial time if TPGs are time-stamped with time points, and identify fragments of the TRPQ language that admit efficient evaluation over a more succinct interval-annotated representation. Finally, we implement a fragment of the language in a state-of-the-art dataflow framework, and experimentally demonstrate that TRPQ can be evaluated efficiently.

Graphs have become the best way we know of representing knowledge. The computing community has investigated and developed the support for managing graphs by means of digital technology. Graph databases and knowledge graphs surface as the most successful solutions to this program. The goal of this document is to provide a conceptual map of the data management tasks underlying these developments, paying particular attention to data models and query languages for graphs.

In Machine Learning, the $\mathsf{SHAP}$-score is a version of the Shapley value that is used to explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is an intractable problem, we prove a strong positive result stating that the $\mathsf{SHAP}$-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits are studied in the field of Knowledge Compilation and generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees and Ordered Binary Decision Diagrams (OBDDs). We also establish the computational limits of the SHAP-score by observing that computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider. It also implies that computing $\mathsf{SHAP}$-scores is intractable as well over the class of propositional formulas in DNF. Based on this negative result, we look for the existence of fully-polynomial randomized approximation schemes (FPRAS) for computing $\mathsf{SHAP}$-scores over such class. In contrast to the model counting problem for DNF formulas, which admits an FPRAS, we prove that no such FPRAS exists for the computation of $\mathsf{SHAP}$-scores. Surprisingly, this negative result holds even for the class of monotone formulas in DNF. These techniques can be further extended to prove another strong negative result: Under widely believed complexity assumptions, there is no polynomial-time algorithm that checks, given a monotone DNF formula $\varphi$ and features $x,y$, whether the $\mathsf{SHAP}$-score of $x$ in $\varphi$ is smaller than the $\mathsf{SHAP}$-score of $y$ in $\varphi$., Comment: Up to the formatting, this is the exact content of the paper in Journal of Machine Learning Research (JMLR)

Graphs are by nature unifying abstractions that can leverage interconnectedness to represent, explore, predict, and explain real- and digital-world phenomena. Although real users and consumers of graph instances and graph workloads understand these abstractions, future problems will require new abstractions and systems. What needs to happen in the next decade for big graph processing to continue to succeed?, Comment: 12 pages, 3 figures, collaboration between the large-scale systems and data management communities, work started at the Dagstuhl Seminar 19491 on Big Graph Processing Systems, to be published in the Communications of the ACM

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 and polynomial-time computability for these problems when $q$ is a self-join-free conjunctive query, and study the impact on the complexity of the following two restrictions: (1) every null occurs at most once in $D$ (what is called Codd tables); and (2) the domain of each null is the same. Roughly speaking, we show that counting completions is much harder than counting valuations: for instance, while the latter is always in \#P, we prove that the former is not in \#P under some widely believed theoretical complexity assumption. Moreover, we find that both (1) and (2) can reduce the complexity of our problems. We also study the approximability of these problems and show that, while counting valuations always has a fully polynomial-time randomized approximation scheme (FPRAS), in most cases counting completions does not. Finally, we consider more expressive query languages and situate our problems with respect to known complexity classes., Comment: 51 pages, including 43 pages of main text. Extended version of arXiv:1912.11064. Up to the stylesheet, page/environment numbering, minor formatting, and publisher-induced changes, this is the exact content of the paper in ACM Transactions on Computational Logic

Scores based on Shapley values are widely used for providing explanations to classification results over machine learning models. A prime example of this is the influential SHAP-score, a version of the Shapley value that can help explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is a computationally intractable problem, it has recently been claimed that the SHAP-score can be computed in polynomial time over the class of decision trees. In this paper, we provide a proof of a stronger result over Boolean models: the SHAP-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits, also known as tractable Boolean circuits, generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees, Ordered Binary Decision Diagrams (OBDDs) and Free Binary Decision Diagrams (FBDDs). We also establish the computational limits of the notion of SHAP-score by observing that, under a mild condition, computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider, as removing one or the other renders the problem of computing the SHAP-score intractable (namely, #P-hard)., Comment: 17 pages, including 8 pages of main text. arXiv version of the AAAI'21 conference paper. Except from the addition of the technical appendix, the content is the same as the AAAI one

Conjunctive queries are one of the most common class of queries used in database systems, and the best studied in the literature. A seminal result of Grohe, Schwentick, and Segoufin (STOC 2001) demonstrates that for every class $G$ of graphs, the evaluation of all conjunctive queries whose underlying graph is in $G$ is tractable if, and only if, $G$ has bounded treewidth. In this work, we extend this characterization to the counting problem for conjunctive queries. Specifically, for every class $C$ of conjunctive queries with bounded treewidth, we introduce the first fully polynomial-time randomized approximation scheme (FPRAS) for counting answers to a query in $C$, and the first polynomial-time algorithm for sampling answers uniformly from a query in $C$. As a corollary, it follows that for every class $G$ of graphs, the counting problem for conjunctive queries whose underlying graph is in $G$ admits an FPRAS if, and only if, $G$ has bounded treewidth (unless $\text{BPP} \neq \text{P}$)}. In fact, our FPRAS is more general, and also applies to conjunctive queries with bounded hypertree width, as well as unions of such queries. The key ingredient in our proof is the resolution of a fundamental counting problem from automata theory. Specifically, we demonstrate the first FPRAS and polynomial time sampler for the set of trees of size $n$ accepted by a tree automaton, which improves the prior quasi-polynomial time randomized approximation scheme (QPRAS) and sampling algorithm of Gore, Jerrum, Kannan, Sweedyk, and Mahaney '97. We demonstrate how this algorithm can be used to obtain an FPRAS for many hitherto open problems, such as counting solutions to constraint satisfaction problems (CSP) with bounded hypertree-width, counting the number of error threads in programs with nested call subroutines, and counting valid assignments to structured DNNF circuits.

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-hardness and polynomial-time computability for these problems when $q$ is a self-join--free conjunctive query, and study the impact on the complexity of the following two restrictions: (1) every null occurs at most once in $D$ (what is called Codd tables); and (2) the domain of each null is the same. Roughly speaking, we show that counting completions is much harder than counting valuations (for instance, while the latter is always in #P, we prove that the former is not in #P under some widely believed theoretical complexity assumption). Moreover, we find that both (1) and (2) reduce the complexity of our problems. We also study the approximability of these problems and show that, while counting valuations always has a fully polynomial randomized approximation scheme, in most cases counting completions does not. Finally, we consider more expressive query languages and situate our problems with respect to known complexity classes., Comment: 29 pages, including 12 pages of main text. This is the arXiv version of the PODS'20 paper. Except from minor differences that could be introduced by the publisher, the only difference should be the addition of the appendix, which contains all the proofs that do not appear in the main text

In this work, we study two simple yet general complexity classes, based on logspace Turing machines, which provide a unifying framework for efficient query evaluation in areas like information extraction and graph databases, among others. We investigate the complexity of three fundamental algorithmic problems for these classes: enumeration, counting and uniform generation of solutions, and show that they have several desirable properties in this respect. Both complexity classes are defined in terms of non-deterministic logspace transducers (NL transducers). For the first class, we consider the case of unambiguous NL transducers, and we prove constant delay enumeration, and both counting and uniform generation of solutions in polynomial time. For the second class, we consider unrestricted NL transducers, and we obtain polynomial delay enumeration, approximate counting in polynomial time, and polynomial-time randomized algorithms for uniform generation. More specifically, we show that each problem in this second class admits a fully polynomial-time randomized approximation scheme (FPRAS) and a polynomial-time Las Vegas algorithm for uniform generation. Interestingly, the key idea to prove these results is to show that the fundamental problem $\text{#NFA}$ admits an FPRAS, where $\text{#NFA}$ is the problem of counting the number of strings of length $n$ (given in unary) accepted by a non-deterministic finite automaton (NFA). While this problem is known to be $\text{#P}$-complete and, more precisely, $\text{SpanL}$-complete, it was open whether this problem admits an FPRAS. In this work, we solve this open problem, and obtain as a welcome corollary that every function in $\text{SpanL}$ admits an FPRAS.

Descriptive Complexity has been very successful in characterizing complexity classes of decision problems in terms of the properties definable in some logics. However, descriptive complexity for counting complexity classes, such as FP and #P, has not been systematically studied, and it is not as developed as its decision counterpart. In this paper, we propose a framework based on Weighted Logics to address this issue. Specifically, by focusing on the natural numbers we obtain a logic called Quantitative Second Order Logics (QSO), and show how some of its fragments can be used to capture fundamental counting complexity classes such as FP, #P and FPSPACE, among others. We also use QSO to define a hierarchy inside #P, identifying counting complexity classes with good closure and approximation properties, and which admit natural complete problems. Finally, we add recursion to QSO, and show how this extension naturally captures lower counting complexity classes such as #L.

We report on a community effort between industry and academia to shape the future of graph query languages. We argue that existing graph database management systems should consider supporting a query language with two key characteristics. First, it should be composable, meaning, that graphs are the input and the output of queries. Second, the graph query language should treat paths as first-class citizens. Our result is G-CORE, a powerful graph query language design that fulfills these goals, and strikes a careful balance between path query expressivity and evaluation complexity.

In April 2016, a community of researchers working in the area of Principles of Data Management (PDM) joined in a workshop at the Dagstuhl Castle in Germany. The workshop was organized jointly by the Executive Committee of the ACM Symposium on Principles of Database Systems (PODS) and the Council of the International Conference on Database Theory (ICDT). The mission of this workshop was to identify and explore some of the most important research directions that have high relevance to society and to Computer Science today, and where the PDM community has the potential to make significant contributions. This report describes the family of research directions that the workshop focused on from three perspectives: potential practical relevance, results already obtained, and research questions that appear surmountable in the short and medium term.

We survey foundational features underlying modern graph query languages. We first discuss two popular graph data models: edge-labelled graphs, where nodes are connected by directed, labelled edges; and property graphs, where nodes and edges can further have attributes. Next we discuss the two most fundamental graph querying functionalities: graph patterns and navigational expressions. We start with graph patterns, in which a graph-structured query is matched against the data. Thereafter we discuss navigational expressions, in which patterns can be matched recursively against the graph to navigate paths of arbitrary length; we give an overview of what kinds of expressions have been proposed, and how they can be combined with graph patterns. We also discuss several semantics under which queries using the previous features can be evaluated, what effects the selection of features and semantics has on complexity, and offer examples of such features in three modern languages that are used to query graphs: SPARQL, Cypher and Gremlin. We conclude by discussing the importance of formalisation for graph query languages; a summary of what is known about SPARQL, Cypher and Gremlin in terms of expressivity and complexity; and an outline of possible future directions for the area.

Although RDF graphs have schema information associated with them, in practice it is very common to find cases in which data do not fully conform to their schema. A prominent example of this is DBpedia, which is RDF data extracted from Wikipedia, a publicly editable source of information. In such situations, it becomes interesting to study the structural properties of the actual data, because the schema gives an incomplete description of the organization of a dataset. In this paper we have approached the study of the structuredness of an RDF graph in a principled way: we propose a framework for specifying structuredness functions, which gauge the degree to which an RDF graph conforms to a schema. In particular, we first define a formal language for specifying structuredness functions with expressions we call rules. This language allows a user or a database administrator to state a rule to which an RDF graph may fully or partially conform. Then we consider the issue of discovering a refinement of a sort (type) by partitioning the dataset into subsets whose structuredness is over a specified threshold. In particular, we prove that the natural decision problem associated to this refinement problem is NP-complete, and we provide a natural translation of this problem into Integer Linear Programming (ILP). Finally, we test this ILP solution with two real world datasets, DBpedia Persons and WordNet Nouns, and 4 different and intuitive rules, which gauge the structuredness in different ways. The rules give meaningful refinements of the datasets, showing that our language can be a powerful tool for understanding the structure of RDF data., Comment: 18 pages, 8 figures. To be published in PVLDB Vol. 8, No. 9

Knowledge base exchange is an important problem in the area of data exchange and knowledge representation, where one is interested in exchanging information between a source and a target knowledge base connected through a mapping. In this paper, we study this fundamental problem for knowledge bases and mappings expressed in OWL 2 QL, the profile of OWL 2 based on the description logic DL-Lite_R. More specifically, we consider the problem of computing universal solutions, identified as one of the most desirable translations to be materialized, and the problem of computing UCQ-representations, which optimally capture in a target TBox the information that can be extracted from a source TBox and a mapping by means of unions of conjunctive queries. For the former we provide a novel automata-theoretic technique, and complexity results that range from NP to EXPTIME, while for the latter we show NLOGSPACE-completeness.

Mapping relational databases to RDF is a fundamental problem for the development of the Semantic Web. We present a solution, inspired by draft methods defined by the W3C where relational databases are directly mapped to RDF and OWL. Given a relational database schema and its integrity constraints, this direct mapping produces an OWL ontology, which, provides the basis for generating RDF instances. The semantics of this mapping is defined using Datalog. Two fundamental properties are information preservation and query preservation. We prove that our mapping satisfies both conditions, even for relational databases that contain null values. We also consider two desirable properties: monotonicity and semantics preservation. We prove that our mapping is monotone and also prove that no monotone mapping, including ours, is semantic preserving. We realize that monotonicity is an obstacle for semantic preservation and thus present a non-monotone direct mapping that is semantics preserving.

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. It is easy to show that the composition of an arbitrary number of standard schema mappings is equivalent to the composition of only two standard schema mappings. We show that surprisingly, the analogous result holds also for schema mappings specified by just st-tgds (no target constraints). This is proven by showing that every SO tgd is equivalent to an unnested SO tgd (one where there is no nesting of function symbols). Similarly, we prove unnesting results for st-SO dependencies, with the same types of consequences., Comment: This paper is an extended version of: M. Arenas, R. Fagin, and A. Nash. Composition with Target Constraints. In 13th International Conference on Database Theory (ICDT), pages 129-142, 2010

In the recent years, a lot of attention has been paid to the development of solid foundations for the composition and inversion of schema mappings. In this paper, we review the proposals for the semantics of these crucial operators. For each of these proposals, we concentrate on the three following problems: the definition of the semantics of the operator, the language needed to express the operator, and the algorithmic issues associated to the problem of computing the operator. It should be pointed out that we primarily consider the formalization of schema mappings introduced in the work on data exchange. In particular, when studying the problem of computing the composition and inverse of a schema mapping, we will be mostly interested in computing these operators for mappings specified by source-to-target tuple-generating dependencies.

Nested words are a structured model of execution paths in procedural programs, reflecting their call and return nesting structure. Finite nested words also capture the structure of parse trees and other tree-structured data, such as XML. We provide new temporal logics for finite and infinite nested words, which are natural extensions of LTL, and prove that these logics are first-order expressively-complete. One of them is based on adding a "within" modality, evaluating a formula on a subword, to a logic CaRet previously studied in the context of verifying properties of recursive state machines (RSMs). The other logic, NWTL, is based on the notion of a summary path that uses both the linear and nesting structures. For NWTL we show that satisfiability is EXPTIME-complete, and that model-checking can be done in time polynomial in the size of the RSM model and exponential in the size of the NWTL formula (and is also EXPTIME-complete). Finally, we prove that first-order logic over nested words has the three-variable property, and we present a temporal logic for nested words which is complete for the two-variable fragment of first-order., Comment: revised and corrected version of Mar 03, 2011

SPARQL is the W3C candidate recommendation query language for RDF. In this paper we address systematically the formal study of SPARQL, concentrating in its graph pattern facility. We consider for this study a fragment without literals and a simple version of filters which encompasses all the main issues yet is simple to formalize. We provide a compositional semantics, prove there are normal forms, prove complexity bounds, among others that the evaluation of SPARQL patterns is PSPACE-complete, compare our semantics to an alternative operational semantics, give simple and natural conditions when both semantics coincide and discuss optimizations procedures.

A relational database is inconsistent if it does not satisfy a given set of integrity constraints. Nevertheless, it is likely that most of the data in it is consistent with the constraints. In this paper we apply logic programming based on answer sets to the problem of retrieving consistent information from a possibly inconsistent database. Since consistent information persists from the original database to every of its minimal repairs, the approach is based on a specification of database repairs using disjunctive logic programs with exceptions, whose answer set semantics can be represented and computed by systems that implement stable model semantics. These programs allow us to declare persistence by defaults and repairing changes by exceptions. We concentrate mainly on logic programs for binary integrity constraints, among which we find most of the integrity constraints found in practice., Comment: 34 pages