1852:
1009:-complete with respect to combined complexity and is therefore even harder under widely held complexity-theoretic assumptions). However, in the usual application scenario, databases are large, while queries are very small, and the data complexity model may be appropriate for studying and describing their difficulty.
445:. To give an example, imagine a relational database for storing information about students, their address, the courses they take and their gender. Finding all male students and their addresses who attend a course that is also attended by a female student is expressed by the following conjunctive query:
853:
While any conjunctive query can be written as a
Datalog rule, not every Datalog program can be written as a conjunctive query. In fact, only single rules over extensional predicate symbols can be easily rewritten as an equivalent conjunctive query. The problem of deciding whether for a given Datalog
507:
in which the where-condition uses exclusively conjunctions of atomic equality conditions, i.e. conditions constructed from column names and constants using no comparison operators other than "=", combined using "and". Notably, this excludes the use of aggregation and subqueries. For example, the
206:
1534:
Unrestricted conjunctive queries over tree data (i.e., a relational database consisting of a binary child relation of a tree as well as unary relations for labeling the tree nodes) have polynomial time combined complexity.
448:(student, address) . ∃ (student2, course) . attends(student, course) ∧ gender(student, 'male') ∧ attends(student2, course) ∧ gender(student2, 'female') ∧ lives(student, address)
1402:: a conjunctive query is acyclic if and only if it has hypertree-width 1. For the special case of conjunctive queries in which all relations used are binary, this notion corresponds to the treewidth of the
1375:
for conjunctive queries. In fact, it turns out that the query containment problem for conjunctive queries is exactly the same problem as the query evaluation problem. Since queries tend to be small,
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1358:
1865:
314:
360:
259:
1679:
Bagan, Guillaume; Durand, Arnaud; Grandjean, Etienne (2007). Duparc, Jacques; Henzinger, Thomas A. (eds.). "On
Acyclic Conjunctive Queries and Constant Delay Enumeration".
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1049:
89:
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1471:
1360:. The main application of query containment is in query optimization: Deciding whether two queries are equivalent is possible by simply checking mutual containment.
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438:. This formula cannot be implemented in the select-project-join fragment of relational algebra, and hence should not be considered a conjunctive query.
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can be expressed in this way. Conjunctive queries also have a number of desirable theoretical properties that larger classes of queries (e.g., the
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451:
Note that since the only entity of interest is the male student and his address, these are the only distinguished variables, while the variables
958:
of evaluating conjunctive queries, two problems have to be distinguished. The first is the problem of evaluating a conjunctive query on a
966:, while the complexity of the problem of evaluating a query on a relational database, where the query is assumed fixed, is called
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where both the query and the database are considered part of the input. The complexity of this problem is usually referred to as
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32:
operator. Many first-order queries can be written as conjunctive queries. In particular, a large part of queries issued on
503:(i.e., relational algebra queries that do not use the operations union or difference) and to select-from-where queries in
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here is usually considered acceptable. The query containment problem for conjunctive queries is also equivalent to the
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213:
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Although there are no quantifiers in this notation, variables appearing in the head of the rule are still implicitly
1398:. Acyclicity of conjunctive queries is a structural property of queries that is defined with respect to the query's
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1406:
of the variables in the query (i.e., the graph having the variables of the query as nodes and an undirected edge
1380:
1305:
1741:
Kolaitis, Phokion G.; Vardi, Moshe Y. (2000), "Conjunctive-Query
Containment and Constraint Satisfaction",
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858:(corresponding to a positive relational algebra query, or, equivalently, a formula of positive existential
267:
1866:
Presentation on structural decomposition methods for the efficient evaluation of conjunctive queries (PDF)
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1028:
delay between each solution. Specifically, these are the acyclic conjunctive queries which also satisfy a
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The problem of listing all answers to a non-Boolean conjunctive query has been studied in the context of
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218:
850:, while variables only appearing in the body of the rule are still implicitly existentially quantified.
1282:, the query containment problem is the problem of deciding whether for all possible database instances
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strictly subsume the conjunctive queries and are thus at least as hard (in fact, relational algebra is
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61:
472:
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475:. Conjunctive queries where all variables are distinguished (and no variables are bound) are called
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As an example of why the restriction to domain independent first-order logic is important, consider
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201:{\displaystyle (x_{1},\ldots ,x_{k}).\exists x_{k+1},\ldots x_{m}.A_{1}\wedge \ldots \wedge A_{r}}
1516:
981:, while the data complexity of conjunctive queries is very low, in the parallel complexity class
1386:
An important class of conjunctive queries that have polynomial-time combined complexity are the
1601:
1581:: Undecidable Boundedness Problems for Datalog Programs. J. Log. Program. 25(2): 163-190 (1995)
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1048:
for larger classes of queries are feasible for conjunctive queries. For example, consider the
1819:: Hypertree Decompositions and Tractable Queries. J. Comput. Syst. Sci. 64(3): 579-627 (2002)
1791:(2001). "The complexity of acyclic conjunctive queries". Journal of the ACM 48 (3): 431–498.
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Conjunctive queries can express a large proportion of queries that are frequently issued on
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in the query) and the conjunctive query is acyclic if and only if its dependency graph is
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rules. Many authors in fact prefer the following
Datalog notation for the query above:
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Proceedings of the fourteenth annual ACM symposium on Theory of computing - STOC '82
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1519:, which is a measure of how close to acyclic a hypergraph is, analogous to bounded
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76:∀. Each such formula can be rewritten (efficiently) into an equivalent formula in
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1667:. STOC '77: Proceedings of the ninth annual ACM symposium on Theory of computing
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1195:, we write the result relation of evaluating the query on the instance simply as
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The formal study of all of these extensions is justified by their application in
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above query can be written as an SQL query of the conjunctive query fragment as
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1594:(1982), "The complexity of relational query languages (Extended Abstract)",
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Besides their logical notation, conjunctive queries can also be written as
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conjunctive queries. The query evaluation, and thus query containment, is
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Information integration using logical views
Theoretical Computer Science
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Optimal
Implementation of Conjunctive Queries in Relational Data Bases
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Conjunctive queries also correspond to select-project-join queries in
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1771:: Algorithms for Acyclic Database Schemes . Proc. VLDB 1981: 82-94.
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in that many interesting problems that are computationally hard or
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The conjunctive queries are the fragment of (domain independent)
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between two variables if and only if there is an atomic formula
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Conjunctive queries without distinguished variables are called
1839:: Conjunctive queries over trees. J. ACM 53(2): 238-272 (2006)
862:, or, as a special case, a conjunctive query) is known as the
1020:) of the queries for which enumeration can be performed with
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Conjunctive queries are one of the great success stories of
83:
Thus conjunctive queries are of the following general form:
1553:
Size Bounds for
Factorised Representations of Query Results
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An important generalization of acyclicity is the notion of
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given by the set of formulae that can be constructed from
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of conjunctive queries may appear surprising, since
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1731:: Foundations of Databases. Addison-Wesley, 1995.
1527:. Conjunctive queries of bounded tree-width have
1363:The query containment problem is undecidable for
874:Extensions of conjunctive queries capturing more
1872:
261:being called distinguished variables, and the
1740:
1425:
1413:
491:(when selecting all columns of the result).
427:{\displaystyle x_{1}.\exists x_{2}.R(x_{2})}
80:, thus this form is usually simply assumed.
1353:{\displaystyle Q_{1}(I)\subseteq Q_{2}(I)}
896:conjunctive queries extended by union and
885:, which are equivalent to positive (i.e.,
479:, because they are the equivalent, in the
1754:
1605:
316:being called undistinguished variables.
1743:Journal of Computer and System Sciences
1687:. Springer Berlin Heidelberg: 208–222.
1111:if and only if each tuple occurring in
434:, which is not domain independent; see
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1016:, with a characterization (under some
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1714:
1712:
1683:. Lecture Notes in Computer Science.
1590:
495:Relationship to other query languages
309:{\displaystyle x_{k+1},\ldots ,x_{m}}
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355:{\displaystyle A_{1},\ldots ,A_{r}}
254:{\displaystyle x_{1},\ldots ,x_{k}}
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1018:computational hardness assumptions
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919:, e.g., arithmetic predicates
883:unions of conjunctive queries
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1693:10.1007/978-3-540-74915-8_18
1551:Dan Olteanu, Jakub Závodný,
1071:{\displaystyle R\subseteq S}
866:problem and is undecidable.
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473:boolean conjunctive queries
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62:existential quantification
1050:query containment problem
922:conjunctive queries with
915:conjunctive queries with
985:, which is contained in
973:Conjunctive queries are
956:computational complexity
730:
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463:, i.e. undistinguished.
461:existentially quantified
74:universal quantification
24:is a restricted form of
1517:bounded hypertree-width
1431:{\displaystyle \{x,y\}}
936:and is in the realm of
40:queries) do not share.
1756:10.1006/jcss.2000.1713
1681:Computer Science Logic
1502:
1501:{\displaystyle R(y,x)}
1467:
1466:{\displaystyle R(x,y)}
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1394:-complete and thus in
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1014:enumeration algorithms
848:universally quantified
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1797:10.1145/382780.382783
1616:10.1145/800070.802186
1531:combined complexity.
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1371:but is decidable and
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1271:{\displaystyle Q_{2}}
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1244:{\displaystyle Q_{1}}
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954:For the study of the
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1857:, 2000, 239, 189-210
1600:, pp. 137–146,
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1224:. Given two queries
1217:{\displaystyle Q(I)}
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934:relational databases
856:nonrecursive program
443:relational databases
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34:relational databases
1817:Francesco Scarcello
1789:Francesco Scarcello
1571:Paris C. Kanellakis
1173:relational database
1100:{\displaystyle R,S}
979:combined complexity
964:combined complexity
960:relational database
925:aggregate functions
917:built-in predicates
864:Datalog boundedness
481:relational calculus
30:logical conjunction
1769:Mihalis Yannakakis
1567:Gerd G. Hillebrand
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489:relational algebra
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78:prenex normal form
38:relational algebra
28:queries using the
1295:{\displaystyle I}
1188:{\displaystyle I}
1164:{\displaystyle Q}
1144:{\displaystyle S}
1124:{\displaystyle R}
1036:Formal properties
911:first-order logic
860:first-order logic
477:equi-join queries
64:∃, but not using
50:first-order logic
22:conjunctive query
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1151:. Given a query
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1847:External links
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1781:Georg Gottlob
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1640:on 2011-08-23
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1813:Nicola Leone
1804:
1785:Nicola Leone
1776:
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1746:
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1736:
1729:Victor Vianu
1684:
1680:
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1652:
1642:, retrieved
1638:the original
1596:
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1107:of the same
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989:and thus in
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60:∧ and
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1373:NP-complete
1052:. We write
1046:undecidable
1032:condition.
1030:free-connex
1022:linear time
995:NP-hardness
975:NP-complete
901:, which by
66:disjunction
58:conjunction
26:first-order
1644:2011-05-16
1539:References
1400:hypergraph
944:Complexity
870:Extensions
44:Definition
1602:CiteSeerX
1521:treewidth
1329:⊆
1175:instance
1063:⊆
878:include:
485:equi-join
483:, of the
467:Fragments
459:are only
390:∃
337:…
291:…
236:…
212:with the
186:∧
183:…
180:∧
154:…
132:∃
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68:∨,
1875:Category
1663:, 1977.
1078:for two
1026:constant
987:LOGSPACE
898:negation
887:negation
814:student2
796:student2
457:student2
70:negation
1634:7869248
1510:acyclic
1388:acyclic
889:-free)
838:address
832:student
790:attends
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726:Datalog
720:Datalog
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1007:PSPACE
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576:lives
1697:ISBN
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