DOCSLIB.ORG

Constraint Satisfaction, Databases, and Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Constraint Satisfaction, Databases, and Logic Constraint Satisfaction, Databases, and Logic Phokion G. Kolaitis* Computer Science Department University of California, Santa Cruz Santa Cruz, CA 95064, U.S.A [email protected] 1 Introduction is to determine whether there is an assignment of values to Constraint satisfaction problems constitute a broad class of the variables so that the constraints are satisfied. More pre­ algorithmic problems that are ubiquitous in several differ­ cisely, a CSP-instance is a triple (V,, D,C), where V of a ent areas of artificial intelligence and computer science. finite set of variables, D is a finite domain of values for the In their full generality, constraint satisfaction problems are variables, and C is a set constraints {t, R), where t is a tuple NP-complete and, thus, presumed to be algorithmically in­ of variables for some m and R is a rela­ tractable. To cope with the intractability of these prob­ tion on D of arity m. A solution is a mapping h : V -> D lems, researchers have devoted considerable research efforts such that, for every constraint (t,R) C, we have that to both the design of heuristic algorithms for constraint sat­ isfaction and the pursuit of "islands of tractability", that is, Consider the Boolean satisfiability problem 3-SAT: given special cases of constraint satisfaction problems for which a 3CNF-formula with variables and clauses polynomial-time algorithms exist. is satisfiable? Such an instance of 3-SAT can During the past decade, the pursuit of "islands of tractabil- be thought of as the CSP-instance in which the set of vari­ ity" of constraint satisfaction has been intensified and has ables is V = ■ }, the domain is D — {0,1}, and led to a number of discoveries that have also unveiled tight the constraints are determined by the clauses of For ex­ connections between constraint satisfaction, database theory, ample, a clause of the form gives rise to the logic, and universal algebra. Our goal in this paper is to constraint In an analogous present an overview of the current state of affairs in the study manner, 3-COLORABILITY can be modeled as a constraint of the computational complexity of constraint satisfaction satisfaction problem. Indeed, an instance H = {V,E) of with emphasis on the connections of this area of research 3-COLORABILITY can be thought of as the CSP-instance in with database theory and logic. The paper is organized as which the set of variables is the set V of the nodes of the follows: Section 2 contains the precise definition of the CON- graph H, the domain is the set of three colors, STRAINT SATISFACTION PROBLEM and its reformulation as and the constraints are the pairs ((u, u), Q), where E the HOMOMORPHISM PROBLEM; Section 3 contains some of and is the inequal­ the connections between constraint satisfaction problems and ity relation on D. database theory; the remaining Sections 4, 5, and 6 contain A vocabulary is a finite set of relational symbols a high-level account of some of the main results about the RI,... , Rm each of which has a fixed arity. A rela• computational complexity of constraint satisfaction and the tional structure over some vocabulary is a tuple A = pursuit of tractable cases of this problem. such that A is a non-empty set, called the universe of A, and each RA is a relation on A having the 2 The Constraint Satisfaction Problem and same arity as the symbol Rt. Let A and B be two rela­ the Homomorphism Problem tional structures over the same vocabulary. A homomorphism h from A to B is a mapping h : A -> B from the uni­ Constraint satisfaction problems were originally introduced verse A of A to the universe B of B such that, for every by Montanari [Mon74l to model problems in computer vi­ relation RA of A and every tuple we sion. Since that time, however, it has been realized that nu­ have that ,...,Feder and Vardi [FV98] merous important problems in artificial intelligence and com­ were the first to point out that the CONSTRAINT SATISFAC­ puter science can be modeled as constraint satisfaction prob­ TION PROBLEM can be identified with the HOMOMORPHISM lems (see [Dec92a; PJ971). An instance of the CONSTRAINT PROBLEM: given two relational structures A and B, is there SATISFACTION PROBLEM (CSP) consists of a set of vari­ a homomorphism h from A to The intuition behind this ables, a set of possible values for the variables, and a set identification is that the structure A represents the variables of constraints on tuples of variables that restrict the combi­ and the tuples of variables that occur in the constraints, while nations of values that the variables may take; the question the structure B represents the domain of values and the tu­ * Partially supported by NSF Grant No. IIS-9907419 ples of values that these constrained tuples of variables are INVITED SPEAKERS 1587 allowed to take. Moreover, the homomorphisms from A to Every finite relational structure A gives rise to a canonical B are precisely the assignments of values to variables that Boolean conjunctive query the positive existential first- satisfy the constraints. For instance, 3-COLORABILITY is order sentence defining asserts that there exist as many equivalent to the problem of deciding whether there is a ho- elements as the cardinality of the universe of A and states all momorphism h from a given graph H to the complete graph atomic facts satisfied by tuples from the universe of A. For with 3 example, if A = (A, E) is the 3-cycle with A = {1,2,3} and nodes. More generally, k-COLORABILITY, 2, amounts E =(3,1)}, then the canonical conjunctive to the existence of a homomorphism from a given graph query w is definable by the sentence H to the complete graph Kk with A: nodes (also known as the k-clique). Numerous other important NP-complete prob­ lems can be viewed as special cases of the HOMOMORPHISM The following basic result, due to Chandra and Merlin PROBLEM (and, hence, also of the CONSTRAINT SATISFAC­ [CM77], establishes a strong connection between homomor­ TION PROBLEM). For example, consider the CLIQUE prob­ phisms and conjunctive queries. lem: given a graph H and an integer k, does H contain a Theorem 3.1: [CM771 The following are equivalent for finite clique of size A;? A moment's reflection shows that this is relational structures A and B. equivalent to the question: given a complete graph Kk and a graph H, is there a homomorphism from Kk to H? The conceptual insights gained from the identification of the CONSTRAINT SATISFACTION PROBLEM with the HO­ MOMORPHISM PROBLEM have facilitated the use of tech­ niques from universal algebra in the study of constraint As an example, we saw earlier that a graph G = (V, E) is 3- satisfaction [Jea98; FV98]). Moreover, they have clari­ colorable if and only if there is a homomorphism from G to fied the tight connections between constraint satisfaction and K . Consequently, Theorem 3.1 implies that G is 3-colorable database theory. We discuss some of these connections in the 3 if and only if K satisfies the canonical query QG of G. next section. 3 In view of the identification of the CONSTRAINT SATIS­ FACTION PROBLEM with the HOMOMORPHISM PROBLEM, 3 Constraint Satisfaction and Relational the preceding Theorem 3.1 implies that the CONSTRAINT Databases SATISFACTION PROBLEM can also be identified with two fundamental problems in database theory: conjunctive query The most frequently asked queries in relational database sys­ evaluation and conjunctive query implication (or contain• tems involve the computation of the join of two or more re­ ment). This fundamental connection between constraint sat­ lations in a database. Instead of spelling out the formal def­ isfaction and database theory was brought to front stage and inition of the join operation, let us consider for concreteness further investigated in [KVOOal. a database relation R(A, B, C) with A, B, C as attributes and a database relation S{B,C,D,E) with B, C, D, E as 4 Computational Complexity of Constraint attributes. Then the join R S consists of all quintuples such that R and 5. Sev­ Satisfaction eral different researchers, including [Bib88], have pointed out The CONSTRAINT SATISFACTION PROBLEM is NP- that computing the set of all solutions of a CSP-instance can complete, because it is clearly in NP and also contains be viewed as a join evaluation problem. Indeed, if (V, D, C) is NP-hard problems as special cases, including 3-SAT, 3- a CSP-instance, then, for every constraint COLORABILITY, and CLIQUE. As explained in Garey and in C, let be the relation R itself viewed as a Johnson's classic monograph [GJ79], one of the main ways database relation with attributes to cope with NP-completeness is to identify polynomial- is the collection of all database relations obtained this way, time solvable cases of the problem at hand that are ob­ then the join consists of all solutions to the tained by imposing restrictions on the possible inputs. For CSP-instance instance, HORN 3-SAT, the restriction of 3-SAT to Horn Join evaluation is a special case of conjunctive query 3CNF-formulas, is solvable in polynomial-time using a evaluation. Specifically, an n-ary conjunctive query unit-propagation algorithm. Similarly, it is known that 3- Q over a relational vocabulary is a query definable COLOR ABILITY restricted to graphs of bounded treewidth is by a positive existential first-order formula of the solvable in polynomial time (see IDF99]). In the case of con­ formwhere straint satisfaction, the pursuit of tractable cases has evolved is a conjunction of atomic formu­ over the years from the discovery of isolated cases to the dis­ las.
Load more

Recommended publications
  • Optimizing Select-Project-Join Expressions
    Optimizing select-project-join expressions Jan Van den Bussche Stijn Vansummeren 1 Introduction The expression that we obtain by translating a SQL query q into the re- lational algebra is called the logical query plan for q. The ultimate goal is to further translate this logical query plan into a physical query plan that contains all details on how the query is to be executed. First, however, we can apply some important optimizations directly on the logical query plan. These optimizations transform the logical query plan into an equivalent log- ical query plan that is more efficient to execute. In paragraph 16.3.3, the book discusses the simple (but very important) logical optimizations of \pushing" selections and projections; and recogniz- ing joins, based on the algebraic equalities from section 16.2. In addition, in these notes, we describe another important logical opti- mization: the removal of redundant joins. 2 Undecidability of the general problem In general, we could formulate the optimization problem of relational algebra expressions as follows: Input: A relational algebra expression e. Output: A relational algebra expression e0 such that: 1. e0 is equivalent to e (denoted as e0 ≡ e), in the sense that e(D) = e0(D) for every database D. Here, e(D) stands for the relation resulting from evaluating e on D. 2. e0 is optimal, in the sense that there is no expression e0 that is shorter than e0, yet still equivalent to e. Of course, we have to define what the length of a relational algebra expression is. A good way to do this is to count the number of times a relational algebra operator is applied in the expression.
    [Show full text]
  • Conjunctive Queries
    Ontology and Database Systems: Foundations of Database Systems Part 3: Conjunctive Queries Werner Nutt Faculty of Computer Science Master of Science in Computer Science A.Y. 2013/2014 Foundations of Database Systems Part 3: Conjunctive Queries Looking Back . We have reviewed three formalisms for expressing queries Relational Algebra Relational Calculus (with its domain-independent fragment) Nice SQL and seen that they have the same expressivity However, crucial properties ((un)satisfiability, equivalence, containment) are undecidable Hence, automatic analysis of such queries is impossible Can we do some analysis if queries are simpler? unibz.it W. Nutt ODBS-FDBs 2013/2014 (1/43) Foundations of Database Systems Part 3: Conjunctive Queries Many Natural Queries Can Be Expressed . in SQL using only a single SELECT-FROM-WHERE block and conjunctions of atomic conditions in the WHERE clause; we call these the CSQL queries. in Relational Algebra using only the operators selection σC (E), projection πC (E), join E1 1C E2, renaming (ρA B(E)); we call these the SPJR queries (= select-project-join-renaming queries) . in Relational Calculus using only the logical symbols \^" and 9 such that every variable occurs in a relational atom; we call these the conjunctive queries unibz.it W. Nutt ODBS-FDBs 2013/2014 (2/43) Foundations of Database Systems Part 3: Conjunctive Queries Conjunctive Queries Theorem The classes of CSQL queries, SPJR queries, and conjunctive queries have all the same expressivity. Queries can be equivalently translated from one formalism to the other in polynomial time. Proof. By specifying translations. Intuition: By a conjunctive query we define a pattern of what the things we are interested in look like.
    [Show full text]
  • Conjunctive Queries
    DATABASE THEORY Lecture 5: Conjunctive Queries Markus Krotzsch¨ TU Dresden, 28 April 2016 Overview 1. Introduction | Relational data model 2. First-order queries 3. Complexity of query answering 4. Complexity of FO query answering 5. Conjunctive queries 6. Tree-like conjunctive queries 7. Query optimisation 8. Conjunctive Query Optimisation / First-Order Expressiveness 9. First-Order Expressiveness / Introduction to Datalog 10. Expressive Power and Complexity of Datalog 11. Optimisation and Evaluation of Datalog 12. Evaluation of Datalog (2) 13. Graph Databases and Path Queries 14. Outlook: database theory in practice See course homepage [) link] for more information and materials Markus Krötzsch, 28 April 2016 Database Theory slide 2 of 65 Review: FO Query Complexity The evaluation of FO queries is • PSpace-complete for combined complexity • PSpace-complete for query complexity • AC0-complete for data complexity { PSpace is rather high { Are there relevant query languages that are simpler than that? Markus Krötzsch, 28 April 2016 Database Theory slide 3 of 65 Conjunctive Queries Idea: restrict FO queries to conjunctive, positive features Definition A conjunctive query (CQ) is an expression of the form 9y1, ::: , ym.A1 ^ ::: ^ A` where each Ai is an atom of the form R(t1, ::: , tk). In other words, a conjunctive query is an FO query that only uses conjunctions of atoms and (outer) existential quantifiers. Example: “Find all lines that depart from an accessible stop” (as seen in earlier lectures) 9ySID, yStop, yTo.Stops(ySID, yStop,"true") ^ Connect(ySID, yTo, xLine) Markus Krötzsch, 28 April 2016 Database Theory slide 4 of 65 Conjunctive Queries in Relational Calculus The expressive power of CQs can also be captured in the relational calculus Definition A conjunctive query (CQ) is a relational algebra expression that uses only the operations select σn=m, project πa1,:::,an , join ./, and renaming δa1,:::,an!b1,:::,bn .
    [Show full text]
  • Conjunctive Queries Complexity & Decomposition Techniques
    Conjunctive Queries Complexity & Decomposition Techniques G. Gottlob Technical University of Vienna, Austria This talk reports about joint work with I. Adler, M. Grohe, N. Leone and F. Scarcello For papers and further material see: http://ulisse.deis.unical.it/~frank/Hypertrees/ Three Problems: CSP: Constraint satisfaction problem BCQ: Boolean conjunctive query evaluation HOM: The homomorphism problem Important problems in different areas. All these problems are hypergraph based. But actually: CSP = BCQ = HOM CSP Set of variables V={X1,...,Xn}, domain D, Set of constraints {C1,...,Cm} where: Ci= <Si, Ri> scope relation (Xj1,...,Xjr) 1 6 7 3 1 5 3 9 2 4 7 6 3 5 4 7 Solution to this CSP: A substitution h: VÆD such that ∀i: h(Si ∈ Ri) Associated hypergraph: {var(Si) | 1 ≤ i ≤ m } Example of CSP: Crossword Puzzle 1h: P A R I S 1v: L I M B O P A N D A L I N G O and so on L A U R A P E T R A A N I T A P A M P A P E T E R Conjunctive Database Queries are CSPs ! DATABASE: Enrolled Teaches Parent John Algebra 2003 McLane Algebra March McLane Lisa Robert Logic 2003 Kolaitis Logic May Kolaitis Robert Mary DB 2002 Lausen DB June Rahm Mary Lisa DB 2003 Rahm DB May ……… ….. ……. ……… ….. ……. ……… ….. QUERY: Is there any teacher having a child enrolled in her course? ans Å Enrolled(S,C,R) ∧ Teaches(P,C,A) ∧ Parent(P,S) Queries and Hypergraphs ans Å Enrolled(S,C’,R) ∧ Teaches(P,C,A) ∧ Parent(P,S) C’ C R A S P Queries, CSPs, and Hypergraphs Is there a teacher whose child attends some course? Enrolled(S,C,R) ∧ Teaches(P,C,A) ∧ Parent(P,S) C R A S P The Homomorphism Problem Given two relational structures A = (U , R1, R2,..., Rk ) B = (V , S1, S 2 ,..., Sk ) Decide whether there exists a homomorphism h from A to B h : U ⎯⎯→ V such that ∀x, ∀i x ∈ Ri ⇒ h(x) ∈ Si HOM is NP-complete (well-known) Membership: Obvious, guess h.
    [Show full text]
  • Certain Conjunctive Query Answering in SQL
    Certain Conjunctive Query Answering in SQL Alexandre Decan, Fabian Pijcke, and Jef Wijsen Universit´ede Mons, Mons, Belgium, falexandre.decan, [email protected], [email protected] Abstract. An uncertain database db is defined as a database in which distinct tuples of the same relation can agree on their primary key. A repair (or possible world) of db is then obtained by selecting a maxi- mal number of tuples without ever selecting two distinct tuples of the same relation that agree on their primary key. Given a query Q on db, the certain answer is the intersection of the answers to Q on all re- pairs. Recently, a syntactic characterization was obtained of the class of acyclic self-join-free conjunctive queries for which certain answers are definable by a first-order formula, called certain first-order rewriting [15]. In this article, we investigate the nesting and alternation of quantifiers in certain first-order rewritings, and propose two syntactic simplifica- tion techniques. We then experimentally verify whether these syntactic simplifications result in lower execution times on real-life SQL databases. 1 Introduction Uncertainty can be modeled in the relational model by allowing primary key vi- olations. Primary keys are underlined in the conference planning database db0 in Fig. 1. There are still two candidate cities for organizing SUM 2016 and SUM 2017. The table S shows controversy about the attractiveness of Mons, while information about the attractiveness of Gent is missing. A repair (or pos- S Town Attractiveness R Conf Year Town Charleroi C SUM 2012 Marburg Marburg A SUM 2016 Mons Mons A SUM 2016 Gent Mons B SUM 2017 Rome Paris A SUM 2017 Paris Rome A Fig.
    [Show full text]
  • Queries with Guarded Negation
    Queries with Guarded Negation Vince Bar´ any´ Balder ten Cate Martin Otto TU Darmstadt, Germany UC Santa Cruz, CA, USA TU Darmstadt, Germany [email protected] [email protected] [email protected] ABSTRACT a single record in the database. For instance, if a A well-established and fundamental insight in database the- database schema contains relations Author(AuthID,Name) ory is that negation (also known as complementation) tends and Book(AuthID,Title,Year,Publisher), the query that to make queries difficult to process and difficult to reason asks for authors that did not publish any book with Elsevier about. Many basic problems are decidable and admit prac- since “not publishing a book with Elsevier” is a property tical algorithms in the case of unions of conjunctive queries, of an author. The query that asks for pairs of authors and but become difficult or even undecidable when queries are book titles where the author did not publish the book, on the allowed to contain negation. Inspired by recent results in fi- other hand, is not allowed, since it involves a negative con- nite model theory, we consider a restricted form of negation, dition (in this case, an inequality) pertaining to two values guarded negation. We introduce a fragment of SQL, called that do not necessarily co-occur in a record in the database. GN-SQL, as well as a fragment of Datalog with stratified The requirement of guarded negation can be formally stated negation, called GN-Datalog, that allow only guarded nega- most easily in terms of the Relational Algebra: we allow the tion, and we show that these query languages are compu- use of the difference operator E1 − E2 provided that E1 is a tationally well behaved, in terms of testing query contain- projection of a relation from the database.
    [Show full text]
  • 4Conjunctive Queries
    4 Conjunctive Queries Alice: Shall we start asking queries? Sergio: Very simple ones for the time being. Riccardo: But the system will answer them fast. Vittorio: And there is some nice theory. n this chapter we embark on the study of queries for relational databases, a rich topic Ithat spans a good part of this book. This chapter focuses on a limited but extremely natural and commonly arising class of queries called conjunctive queries. Five equivalent versions of this query family are presented here: one from each of the calculus and datalog paradigms, two from the algebra paradigm, and a final one that has a more visual form. In the context of conjunctive queries, the three nonalgebraic versions can be viewed as minor syntactic variants of each other; but these similarities diminish as the languages are generalized to incorporate negation and/or recursion. This chapter also discusses query composition and its interaction with user views, and it extends conjunctive queries in a straightforward manner to incorporate union (or disjunction). The conjunctive queries enjoy several desirable properties, including, for example, decidability of equivalence and containment. These results will be presented in Chapter 6, in which a basic tool, the Homomorphism Theorem, is developed. Most of these results extend to conjunctive queries with union. In the formal framework that we have developed in this book, we distinguish between a query, which is a syntactic object, and a query mapping, which is the function defined by a query interpreted under a specified semantics. However, we often blur these two concepts when the meaning is clear from the context.
    [Show full text]
  • Constraint Satisfaction Complexity and Logic Phokion G. Kolaitis IBM Almaden Research Center & UC Santa Cruz
    Constraint Satisfaction Complexity and Logic Phokion G. Kolaitis IBM Almaden Research Center & UC Santa Cruz 1 Goals Goals: ² Show that Constraint Satisfaction Problems (CSP) form a broad class of algorithmic problems that arise naturally in several di®erent areas. ² Present an overview of CSP with emphasis on the computational complexity aspects of CSP and on the connections with database theory, univeral algebra, and logic 2 Course Outline Constraint Satisfaction Problems (CSP) ² De¯nition & Examples of CSP ² CSP & the Homomorphism Problem ² CSP & Database Theory ² Computational Complexity of CSP { Uniform & Non-Uniform CSP { Schaefer's Dichotomy Theorem for Boolean CSP { The Feder-Vardi Dichotomy Conjecture for general CSP 3 Course Outline ² The Pursuit of Tractable Cases of CSP { Universal Algebra and Closure Properties of Constraints { Datalog, Finite-variable Logics, and Pebble Games { Bounded Treewidth and Finite-variable Log- ics 4 Algorithmic Problems ² k-SAT:(k ¸ 2 is ¯xed) Given a k-CNF formula, is it satis¯able? ² k-Colorability:(k ¸ 2 is ¯xed) Given a graph H = (V; E), is it k-colorable? ² Clique: Given a graph H = (V; E) and a positive inte- ger k, does H contain a k-clique? ² Hamiltonian Cycle: Given a graph H = (V; E), does it contain a Hamiltonian cycle? 5 Algorithmic Problems Latin Squares: L. Euler { 1783 A Latin square of order n is an n £ n matrix of n symbols such that each symbol occurs exactly once in each row and each column of the matrix. A Latin Square of Order 4 2 3 1 4 3 4 2 1 4 1 3 2 1 2 4 3 Latin Square
    [Show full text]
  • Block-Sorted Quantified Conjunctive Queries
    Block-Sorted Quantified Conjunctive Queries Hubie Chen1? and D´anielMarx2?? 1 Universidad del Pa´ısVasco and IKERBASQUE, E-20018 San Sebasti´an,Spain 2 Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary Abstract. We study the complexity of model checking in quantified conjunctive logic, that is, the fragment of first-order logic where both quantifiers may be used, but conjunction is the only permitted connec- tive. In particular, we study block-sorted queries, which we define to be prenex sentences in multi-sorted relational first-order logic where two variables having the same sort must appear in the same quantifier block. We establish a complexity classification theorem that describes precisely the sets of block-sorted queries of bounded arity on which model check- ing is fixed-parameter tractable. This theorem strictly generalizes, for the first time, the corresponding classification for existential conjunc- tive logic (which is known and due to Grohe) to a logic in which both quantifiers are present. 1 Introduction Model checking, the problem of deciding if a logical sentence holds on a struc- ture, is a fundamental computational task that appears in many guises through- out computer science. Witness its appearance in areas such as computational logic, verification, artificial intelligence, constraint satisfaction, and computa- tional complexity. The case where one wishes to evaluate a first-order sentence on a finite structure is a problem of principal interest in database theory and is the topic of this article. This problem is well-known to be quite intractable in general: it is PSPACE-complete.
    [Show full text]
  • Relational Databases, Logic, and Complexity
    Relational Databases, Logic, and Complexity Phokion G. Kolaitis University of California, Santa Cruz & IBM Research-Almaden [email protected] 2009 GII Doctoral School on Advances in Databases 1 What this course is about Goals: Cover a coherent body of basic material in the foundations of relational databases Prepare students for further study and research in relational database systems Overview of Topics: Database query languages: expressive power and complexity Relational Algebra and Relational Calculus Conjunctive queries and homomorphisms Recursive queries and Datalog Selected additional topics: Bag Semantics, Inconsistent Databases Unifying Theme: The interplay between databases, logic, and computational complexity 2 Relational Databases: A Very Brief History The history of relational databases is the history of a scientific and technological revolution. Edgar F. Codd, 1923-2003 The scientific revolution started in 1970 by Edgar (Ted) F. Codd at the IBM San Jose Research Laboratory (now the IBM Almaden Research Center) Codd introduced the relational data model and two database query languages: relational algebra and relational calculus . “A relational model for data for large shared data banks”, CACM, 1970. “Relational completeness of data base sublanguages”, in: Database Systems, ed. by R. Rustin, 1972. 3 Relational Databases: A Very Brief History Researchers at the IBM San Jose Laboratory embark on the System R project, the first implementation of a relational database management system (RDBMS) In 1974-1975, they develop SEQUEL , a query language that eventually became the industry standard SQL . System R evolved to DB2 – released first in 1983. M. Stonebraker and E. Wong embark on the development of the Ingres RDBMS at UC Berkeley in 1973.
    [Show full text]
  • Conjunctive Query Answering Under Existential Rules - Decidability, Complexity, and Algorithms Michaël Thomazo
    Conjunctive Query Answering Under Existential Rules - Decidability, Complexity, and Algorithms Michaël Thomazo To cite this version: Michaël Thomazo. Conjunctive Query Answering Under Existential Rules - Decidability, Complexity, and Algorithms. Artificial Intelligence [cs.AI]. Université Montpellier II - Sciences et Techniques du Languedoc, 2013. English. tel-00925722 HAL Id: tel-00925722 https://tel.archives-ouvertes.fr/tel-00925722 Submitted on 8 Jan 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 Délivré par l’Université Montpellier II Préparée au sein de l’école doctorale I2S Et de l’unité de recherche LIRMM Spécialité: Informatique Présentée par Michaël Thomazo Conjunctive Query Answering Under Existential Rules Decidability, Complexity, and Algorithms Soutenue le 24 octobre 2013 devant le jury composé de : Co-directeurs de thèse M. Jean-François BAGET Chargé de recherche Inria Mme Marie-Laure MUGNIER Professeur Université de Montpellier Rapporteurs M. Georg GOTTLOB Professeur Université d’Oxford M. Carsten LUTZ Professeur Université de Brême Mme Marie-Christine ROUSSET Professeur Université de Grenoble Examinateur M. Christophe PAUL Directeur de recherche CNRS Contents Contents i List of Figures iii List of Tables v Remerciements 1 1 Introduction 3 2 The landscape of OBQA 7 2.1 Representing data and queries .........................
    [Show full text]
  • Answering Conjunctive Queries with Inequalities∗
    Answering Conjunctive Queries with Inequalities∗ Paraschos Koutris1, Tova Milo2, Sudeepa Roy3, and Dan Suciu4 1 University of Washington [email protected] 2 Tel Aviv University [email protected] 3 University of Washington [email protected] 4 University of Washington [email protected] Abstract In this parer, we study the complexity of answering conjunctive queries (CQ) with inequalities (6=). In particular, we compare the complexity of the query with and without inequalities. The main contribution of our work is a novel combinatorial technique that enables the use of any Select-Project-Join query plan for a given CQ without inequalities in answering the CQ with inequalities, with an additional factor in running time that only depends on the query. To achieve this, we define a new projection operator that keeps a small representation (independent of the size of the database) of the set of input tuples that map to each tuple in the output of the projection; this representation is used to evaluate all the inequalities in the query. Second, we generalize a result by Papadimitriou-Yannakakis [18] and give an alternative algorithm based on the color-coding technique [4] to evaluate a CQ with inequalities by using an algorithm for the CQ without inequalities. Third, we investigate the structure of the query graph, inequality graph, and the augmented query graph with inequalities, and show that even if the query and the inequality graphs have bounded treewidth, the augmented graph not only can have an unbounded treewidth but can also be NP-hard to evaluate. Further, we illustrate classes of queries and inequalities where the augmented graphs have unbounded treewidth, but the CQ with inequalities can be evaluated in poly-time.
    [Show full text]