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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Database_theory" title="Database theory">database theory</a>, a <b>conjunctive query</b> is a restricted form of <a href="/wiki/First-order_logic" title="First-order logic">first-order</a> queries using the <a href="/wiki/Logical_conjunction" title="Logical conjunction">logical conjunction</a> operator. Many first-order queries can be written as conjunctive queries. In particular, a large part of queries issued on <a href="/wiki/Relational_database" title="Relational database">relational databases</a> can be expressed in this way. Conjunctive queries also have a number of desirable theoretical properties that larger classes of queries (e.g., the <a href="/wiki/Relational_algebra" title="Relational algebra">relational algebra</a> queries) do not share. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunctive_query&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The conjunctive queries are the fragment of (domain independent) <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> given by the set of formulae that can be constructed from <a href="/wiki/Atomic_formula" title="Atomic formula">atomic formulae</a> using <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a> ∧ and <a href="/wiki/Existential_quantification" title="Existential quantification">existential quantification</a> ∃, but not using <a href="/wiki/Disjunction" class="mw-redirect" title="Disjunction">disjunction</a> ∨, <a href="/wiki/Negation" title="Negation">negation</a> ¬, or <a href="/wiki/Universal_quantification" title="Universal quantification">universal quantification</a> ∀. Each such formula can be rewritten (efficiently) into an equivalent formula in <a href="/wiki/Prenex_normal_form" title="Prenex normal form">prenex normal form</a>, thus this form is usually simply assumed. </p><p>Thus conjunctive queries are of the following general form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{k}).\exists x_{k+1},\ldots x_{m}.A_{1}\wedge \ldots \wedge A_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> <mi mathvariant="normal">∃</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>.</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>…</mo> <mo>∧</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{k}).\exists x_{k+1},\ldots x_{m}.A_{1}\wedge \ldots \wedge A_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5265288436eafcb343916497972bb45524c6e192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.221ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{k}).\exists x_{k+1},\ldots x_{m}.A_{1}\wedge \ldots \wedge A_{r}}"></span>,</dd></dl> <p>with the <a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">free variables</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00a6511886c686290cc57b23ab22f45fda9e1c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.981ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{k}}"></span> being called distinguished variables, and the <a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">bound variables</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k+1},\ldots ,x_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k+1},\ldots ,x_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e275f769fb73a02e027aaff8b3823171675039d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.702ex; height:2.009ex;" alt="{\displaystyle x_{k+1},\ldots ,x_{m}}"></span> being called undistinguished variables. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},\ldots ,A_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/362fce4b76afd41de73795da1cb405db9343c818" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.692ex; height:2.509ex;" alt="{\displaystyle A_{1},\ldots ,A_{r}}"></span> are <a href="/wiki/Atomic_formula" title="Atomic formula">atomic formulae</a>. </p><p>As an example of why the restriction to domain independent first-order logic is important, consider <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}.\exists x_{2}.R(x_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> <mi mathvariant="normal">∃</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}.\exists x_{2}.R(x_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/923bc759aabcfe0a9f892e90a0d12db13dd9236a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.085ex; height:2.843ex;" alt="{\displaystyle x_{1}.\exists x_{2}.R(x_{2})}"></span>, which is not domain independent; see <a href="/wiki/Codd%27s_theorem" title="Codd's theorem">Codd's theorem</a>. This formula cannot be implemented in the select-project-join fragment of relational algebra, and hence should not be considered a conjunctive query. </p><p>Conjunctive queries can express a large proportion of queries that are frequently issued on <a href="/wiki/Relational_database" title="Relational database">relational databases</a>. 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: </p> <pre>(student, address) . ∃ (student2, course) . attends(student, course) ∧ gender(student, 'male') ∧ attends(student2, course) ∧ gender(student2, 'female') ∧ lives(student, address) </pre> <p>Note that since the only entity of interest is the male student and his address, these are the only distinguished variables, while the variables <code>course</code>, <code>student2</code> are only <a href="/wiki/Existential_quantification" title="Existential quantification">existentially quantified</a>, i.e. undistinguished. </p> <div class="mw-heading mw-heading2"><h2 id="Fragments">Fragments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunctive_query&action=edit&section=2" title="Edit section: Fragments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Conjunctive queries without distinguished variables are called <a href="/wiki/Boolean_conjunctive_query" title="Boolean conjunctive query">boolean conjunctive queries</a>. Conjunctive queries where all variables are distinguished (and no variables are bound) are called <b>equi-join queries</b>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> because they are the equivalent, in the <a href="/wiki/Relational_calculus" title="Relational calculus">relational calculus</a>, of the <a href="/wiki/Equi-join" class="mw-redirect" title="Equi-join">equi-join</a> queries in the <a href="/wiki/Relational_algebra" title="Relational algebra">relational algebra</a> (when selecting all columns of the result). </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_other_query_languages">Relationship to other query languages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunctive_query&action=edit&section=3" title="Edit section: Relationship to other query languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Conjunctive queries also correspond to select-project-join queries in <a href="/wiki/Relational_algebra" title="Relational algebra">relational algebra</a> (i.e., relational algebra queries that do not use the operations union or difference) and to select-from-where queries in <a href="/wiki/SQL" title="SQL">SQL</a> 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 above query can be written as an SQL query of the conjunctive query fragment as </p> <div class="mw-highlight mw-highlight-lang-sql mw-content-ltr" dir="ltr"><pre><span></span><span class="k">select</span><span class="w"> </span><span class="n">l</span><span class="p">.</span><span class="n">student</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">.</span><span class="n">address</span> <span class="k">from</span><span class="w"> </span><span class="n">attends</span><span class="w"> </span><span class="n">a1</span><span class="p">,</span><span class="w"> </span><span class="n">gender</span><span class="w"> </span><span class="n">g1</span><span class="p">,</span> <span class="w"> </span><span class="n">attends</span><span class="w"> </span><span class="n">a2</span><span class="p">,</span><span class="w"> </span><span class="n">gender</span><span class="w"> </span><span class="n">g2</span><span class="p">,</span> <span class="w"> </span><span class="n">lives</span><span class="w"> </span><span class="n">l</span> <span class="k">where</span><span class="w"> </span><span class="n">a1</span><span class="p">.</span><span class="n">student</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">g1</span><span class="p">.</span><span class="n">student</span><span class="w"> </span><span class="k">and</span> <span class="w"> </span><span class="n">a2</span><span class="p">.</span><span class="n">student</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">g2</span><span class="p">.</span><span class="n">student</span><span class="w"> </span><span class="k">and</span> <span class="w"> </span><span class="n">l</span><span class="p">.</span><span class="n">student</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">g1</span><span class="p">.</span><span class="n">student</span><span class="w"> </span><span class="k">and</span> <span class="w"> </span><span class="n">a1</span><span class="p">.</span><span class="n">course</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a2</span><span class="p">.</span><span class="n">course</span><span class="w"> </span><span class="k">and</span> <span class="w"> </span><span class="n">g1</span><span class="p">.</span><span class="n">gender</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'male'</span><span class="w"> </span><span class="k">and</span> <span class="w"> </span><span class="n">g2</span><span class="p">.</span><span class="n">gender</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'female'</span><span class="p">;</span> </pre></div> <div class="mw-heading mw-heading2"><h2 id="Datalog">Datalog</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunctive_query&action=edit&section=4" title="Edit section: Datalog"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Besides their logical notation, conjunctive queries can also be written as <a href="/wiki/Datalog" title="Datalog">Datalog</a> rules. Many authors in fact prefer the following Datalog notation for the query above: </p> <div class="mw-highlight mw-highlight-lang-prolog mw-content-ltr" dir="ltr"><pre><span></span> <span class="nf">result</span><span class="p">(</span><span class="s s-Atom">student</span><span class="p">,</span> <span class="s s-Atom">address</span><span class="p">)</span> <span class="p">:-</span> <span class="nf">attends</span><span class="p">(</span><span class="s s-Atom">student</span><span class="p">,</span> <span class="s s-Atom">course</span><span class="p">),</span> <span class="nf">gender</span><span class="p">(</span><span class="s s-Atom">student</span><span class="p">,</span> <span class="s s-Atom">male</span><span class="p">),</span> <span class="nf">attends</span><span class="p">(</span><span class="s s-Atom">student2</span><span class="p">,</span> <span class="s s-Atom">course</span><span class="p">),</span> <span class="nf">gender</span><span class="p">(</span><span class="s s-Atom">student2</span><span class="p">,</span> <span class="s s-Atom">female</span><span class="p">),</span> <span class="nf">lives</span><span class="p">(</span><span class="s s-Atom">student</span><span class="p">,</span> <span class="s s-Atom">address</span><span class="p">).</span> </pre></div> <p>Although there are no quantifiers in this notation, variables appearing in the head of the rule are still implicitly <a href="/wiki/Universal_quantification" title="Universal quantification">universally quantified</a>, while variables only appearing in the body of the rule are still implicitly existentially quantified. </p><p>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 program there is an equivalent <a href="/wiki/Recursion" title="Recursion">nonrecursive program</a> (corresponding to a positive relational algebra query, or, equivalently, a formula of positive existential <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>, or, as a special case, a conjunctive query) is known as the <a href="/w/index.php?title=Datalog_boundedness&action=edit&redlink=1" class="new" title="Datalog boundedness (page does not exist)">Datalog boundedness</a> problem and is undecidable.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunctive_query&action=edit&section=5" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Extensions of conjunctive queries capturing more <a href="/wiki/Expressive_power_(computer_science)" title="Expressive power (computer science)">expressive power</a> include: </p> <ul><li><b>unions of conjunctive queries</b>, which are equivalent to positive (i.e., <a href="/wiki/Negation" title="Negation">negation</a>-free) <a href="/wiki/Relational_algebra" title="Relational algebra">relational algebra</a></li> <li><b>conjunctive queries extended by union and <a href="/wiki/Negation" title="Negation">negation</a></b>, which by <a href="/wiki/Codd%27s_theorem" title="Codd's theorem">Codd's theorem</a> correspond to <a href="/wiki/Relational_algebra" title="Relational algebra">relational algebra</a> and <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a></li> <li>conjunctive queries with <b>built-in predicates</b>, e.g., arithmetic predicates</li> <li>conjunctive queries with <b><a href="/wiki/Aggregate_function" title="Aggregate function">aggregate functions</a></b>.</li></ul> <p>The formal study of all of these extensions is justified by their application in <a href="/wiki/Relational_databases" class="mw-redirect" title="Relational databases">relational databases</a> and is in the realm of <a href="/wiki/Database_theory" title="Database theory">database theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Complexity">Complexity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunctive_query&action=edit&section=6" title="Edit section: Complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Query_evaluation" title="Query evaluation">Query evaluation</a></div> <p>For the study of the <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">computational complexity</a> of evaluating conjunctive queries, two problems have to be distinguished. The first is the problem of evaluating a conjunctive query on a <a href="/wiki/Relational_database" title="Relational database">relational database</a> where both the query and the database are considered part of the input. The complexity of this problem is usually referred to as <b>combined complexity</b>, while the complexity of the problem of evaluating a query on a relational database, where the query is assumed fixed, is called <b>data complexity</b>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Conjunctive queries are <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> with respect to <a href="/wiki/Combined_complexity" class="mw-redirect" title="Combined complexity">combined complexity</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> while the data complexity of conjunctive queries is very low, in the parallel complexity class <a href="/wiki/AC0" title="AC0">AC0</a>, which is contained in <a href="/wiki/LOGSPACE" class="mw-redirect" title="LOGSPACE">LOGSPACE</a> and thus in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>. The <a href="/wiki/NP-hard" class="mw-redirect" title="NP-hard">NP-hardness</a> of conjunctive queries may appear surprising, since <a href="/wiki/Relational_algebra" title="Relational algebra">relational algebra</a> and <a href="/wiki/SQL" title="SQL">SQL</a> strictly subsume the conjunctive queries and are thus at least as hard (in fact, relational algebra is <a href="/wiki/PSPACE" title="PSPACE">PSPACE</a>-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. </p><p>The problem of listing all answers to a non-Boolean conjunctive query has been studied in the context of <a href="/wiki/Enumeration_algorithm" title="Enumeration algorithm">enumeration algorithms</a>, with a characterization (under some <a href="/wiki/Computational_hardness_assumption" title="Computational hardness assumption">computational hardness assumptions</a>) of the queries for which enumeration can be performed with <a href="/wiki/Linear_time" class="mw-redirect" title="Linear time">linear time</a> preprocessing and <a href="/wiki/Constant_time" class="mw-redirect" title="Constant time">constant</a> delay between each solution. Specifically, these are the acyclic conjunctive queries which also satisfy a <i>free-connex</i> condition.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formal_properties">Formal properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunctive_query&action=edit&section=7" title="Edit section: Formal properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Conjunctive queries are one of the great success stories of <a href="/wiki/Database_theory" title="Database theory">database theory</a> in that many interesting problems that are computationally hard or <a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a> for larger classes of queries are feasible for conjunctive queries.<sup id="cite_ref-AHV_6-0" class="reference"><a href="#cite_note-AHV-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> For example, consider the <a href="/w/index.php?title=Query_containment_problem&action=edit&redlink=1" class="new" title="Query containment problem (page does not exist)">query containment problem</a>. We write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subseteq S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>⊆</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\subseteq S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ab53336ab789637a720d5ea2ed48097cbf9878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.362ex; height:2.343ex;" alt="{\displaystyle R\subseteq S}"></span> for two <a href="/wiki/Database_relation" class="mw-redirect" title="Database relation">database relations</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369b318eb1a5dcd7aceef09dd85d8641b3c1ac98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.297ex; height:2.509ex;" alt="{\displaystyle R,S}"></span> of the same <a href="/wiki/Database_schema" title="Database schema">schema</a> if and only if each tuple occurring in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> also occurs in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. Given a query <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> and a <a href="/wiki/Relational_database" title="Relational database">relational database</a> instance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>, we write the result relation of evaluating the query on the instance simply as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c49770d6390697d52ea98bd37df1e49fd9bee8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.819ex; height:2.843ex;" alt="{\displaystyle Q(I)}"></span>. Given two queries <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ea6463cb36d8278ff71214fb4d13127039ae53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86e8bff64d5e62fc8f45a35875e78bc9bef74a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{2}}"></span> and a <a href="/wiki/Database_schema" title="Database schema">database schema</a>, the query containment problem is the problem of deciding whether for all possible database instances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> over the input database schema, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}(I)\subseteq Q_{2}(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{1}(I)\subseteq Q_{2}(I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/855b9321d00afbf5729095bc88615501a5403b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.846ex; height:2.843ex;" alt="{\displaystyle Q_{1}(I)\subseteq Q_{2}(I)}"></span>. The main application of query containment is in query optimization: Deciding whether two queries are equivalent is possible by simply checking mutual containment. </p><p>The query containment problem is undecidable for <a href="/wiki/Relational_algebra" title="Relational algebra">relational algebra</a> and <a href="/wiki/SQL" title="SQL">SQL</a> but is decidable and <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> 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.<sup id="cite_ref-AHV_6-1" class="reference"><a href="#cite_note-AHV-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Since queries tend to be small, <a href="/wiki/NP-completeness" title="NP-completeness">NP-completeness</a> here is usually considered acceptable. The query containment problem for conjunctive queries is also equivalent to the <a href="/wiki/Constraint_satisfaction_problem" title="Constraint satisfaction problem">constraint satisfaction problem</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>An important class of conjunctive queries that have polynomial-time combined complexity are the <b>acyclic</b> conjunctive queries.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The query evaluation, and thus query containment, is <a href="/wiki/LOGCFL" title="LOGCFL">LOGCFL</a>-complete and thus in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Acyclicity of conjunctive queries is a structural property of queries that is defined with respect to the query's <a href="/wiki/Hypergraph" title="Hypergraph">hypergraph</a>:<sup id="cite_ref-AHV_6-2" class="reference"><a href="#cite_note-AHV-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> 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 <a href="/wiki/Dependency_graph" title="Dependency graph">dependency graph</a> of the variables in the query (i.e., the graph having the variables of the query as nodes and an undirected edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x,y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x,y\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2611cdc8fecaffa28cb0ea888dbba55f3a31077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.844ex; height:2.843ex;" alt="{\displaystyle \{x,y\}}"></span> between two variables if and only if there is an atomic formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45f0bb548f5fe8470b7257809f044c967628eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.092ex; height:2.843ex;" alt="{\displaystyle R(x,y)}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43098c72ba8e09f98eb72bcbd766f0d9ee3737e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.092ex; height:2.843ex;" alt="{\displaystyle R(y,x)}"></span> in the query) and the conjunctive query is acyclic if and only if its dependency graph is <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">acyclic</a>. </p><p>An important generalization of acyclicity is the notion of <a href="/wiki/Hypertree_decomposition" class="mw-redirect" title="Hypertree decomposition">bounded hypertree-width</a>, which is a measure of how close to acyclic a hypergraph is, analogous to bounded <a href="/wiki/Treewidth" title="Treewidth">treewidth</a> in <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>. Conjunctive queries of bounded tree-width have <a href="/wiki/LOGCFL" title="LOGCFL">LOGCFL</a> combined complexity.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>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.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunctive_query&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Dan Olteanu, Jakub Závodný, <i>Size Bounds for Factorised Representations of Query Results</i>, 2015, DOI 10.1145/2656335, <a rel="nofollow" class="external autonumber" href="http://www.cs.ox.ac.uk/dan.olteanu/papers/oz-tods15.pdf">[1]</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="/w/index.php?title=Gerd_G._Hillebrand&action=edit&redlink=1" class="new" title="Gerd G. 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Program. 25(2): 163-190 (1995)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFVardi1982" class="citation cs2"><a href="/wiki/Moshe_Y._Vardi" class="mw-redirect" title="Moshe Y. 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Proc. VLDB 1981: 82-94.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="/wiki/Georg_Gottlob" title="Georg Gottlob">Georg Gottlob</a>, <a href="/wiki/Nicola_Leone" title="Nicola Leone">Nicola Leone</a>, and <a href="/w/index.php?title=Francesco_Scarcello&action=edit&redlink=1" class="new" title="Francesco Scarcello (page does not exist)">Francesco Scarcello</a> (2001). "The complexity of acyclic conjunctive queries". Journal of the ACM 48 (3): 431–498. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F382780.382783">10.1145/382780.382783</a>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="/wiki/Georg_Gottlob" title="Georg Gottlob">Georg Gottlob</a>, <a href="/wiki/Nicola_Leone" title="Nicola Leone">Nicola Leone</a>, and <a href="/w/index.php?title=Francesco_Scarcello&action=edit&redlink=1" class="new" title="Francesco Scarcello (page does not exist)">Francesco Scarcello</a>: Hypertree Decompositions and Tractable Queries. J. Comput. Syst. Sci. 64(3): 579-627 (2002)</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="/wiki/Georg_Gottlob" title="Georg Gottlob">Georg Gottlob</a>, <a href="/w/index.php?title=Christoph_Koch&action=edit&redlink=1" class="new" title="Christoph Koch (page does not exist)">Christoph Koch</a>, <a href="/w/index.php?title=Klaus_U._Schulz&action=edit&redlink=1" class="new" title="Klaus U. Schulz (page does not exist)">Klaus U. Schulz</a>: Conjunctive queries over trees. J. ACM 53(2): 238-272 (2006)</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunctive_query&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1G-43G4RRN-2&_user=5423704&_coverDate=05/28/2000&_rdoc=2&_fmt=summary&_orig=browse&_srch=doc-info(%23toc%235674%232000%23997609997%23255297%23FLP%23display%23Volume)&_cdi=5674&_sort=d&_docanchor=&_ct=8&_acct=C000007438&_version=1&_urlVersion=0&_userid=5423704&md5=adc35b17aaf28963cc6925a017813785">Ullman, J. D. <i>Information integration using logical views Theoretical Computer Science</i>, 2000, 239, 189-210</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged June 2020">dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></li> <li><a href="/wiki/Georg_Gottlob" title="Georg Gottlob">Georg Gottlob</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070205175730/http://www.dis.uniroma1.it/~lenzerin/homepagine/didattica/viewbasedqueryprocessing/ConjunctiveQueries.pdf">Presentation on structural decomposition methods for the efficient evaluation of conjunctive queries (PDF)</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Conjunctive_query&oldid=1189634016">https://en.wikipedia.org/w/index.php?title=Conjunctive_query&oldid=1189634016</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Database_theory" title="Category:Database theory">Database theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:All_articles_with_dead_external_links" title="Category:All articles with dead external links">All articles with dead external links</a></li><li><a href="/wiki/Category:Articles_with_dead_external_links_from_June_2020" title="Category:Articles with dead external links from June 2020">Articles with dead external links from June 2020</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 December 2023, at 01:57<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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