SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Embedded dependency</title>

</head>

Embedded dependency

</h1>

</header>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In relational <a href="Database_theory" title="Database theory">database theory</a>, an embedded dependency (ED) is a certain kind of constraint on a <a href="Relational_database" title="Relational database">relational database</a>. It is the most general type of constraint used in practice, including both <a href="Tuple-generating_dependency" title="Tuple-generating dependency">tuple-generating dependencies</a> and <a href="Equality-generating_dependency" title="Equality-generating dependency">equality-generating dependencies</a>. Embedded dependencies can express functional dependencies, join dependencies, multivalued dependencies, inclusion dependencies, foreign key dependencies, and many more besides.

An algorithm known as <a href="Chase_(algorithm)" title="Chase (algorithm)">the chase</a> takes as input an instance that may or may not satisfy a set of EDs, and, if it terminates (which is a priori undecidable), output an instance that does satisfy the EDs.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

An embedded dependency (ED) is a <a href="Sentence_(logic)" class="mw-redirect" title="Sentence (logic)">sentence</a> in <a href="First-order_logic" title="First-order logic">first-order logic</a> of the form:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall x_{1},\ldots ,x_{n}.\phi (x_{1},\ldots ,x_{n})\rightarrow \exists z_{1},\ldots ,z_{k}.\psi (y_{1},\ldots ,y_{m})}</annotation>

</semantics>

</math><img src="./31c2b7111499002a0543ebf096686d3b97a467a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.674ex; height:2.843ex;" alt="{\displaystyle \forall x_{1},\ldots ,x_{n}.\phi (x_{1},\ldots ,x_{n})\rightarrow \exists z_{1},\ldots ,z_{k}.\psi (y_{1},\ldots ,y_{m})}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{z_{1},\ldots ,z_{k}\}=\{y_{1},\ldots ,y_{m}\}\setminus \{x_{1},\ldots ,x_{n}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{z_{1},\ldots ,z_{k}\}=\{y_{1},\ldots ,y_{m}\}\setminus \{x_{1},\ldots ,x_{n}\}}</annotation>

</semantics>

</math><img src="./c7b0abf7c9a4370c19a02f241258faab4dc0ddde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.048ex; height:2.843ex;" alt="{\displaystyle \{z_{1},\ldots ,z_{k}\}=\{y_{1},\ldots ,y_{m}\}\setminus \{x_{1},\ldots ,x_{n}\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> are <a href="Logical_conjunction" title="Logical conjunction">conjunctions</a> of relational and equality atoms.<a href="#cite_note-Kanellakis1990-1">[1]</a> A relational atom has the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(w_{1},\ldots ,w_{h})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(w_{1},\ldots ,w_{h})}</annotation>

</semantics>

</math><img src="./814b22a81a0b8d6023a7a7039c9d86e32607915f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.313ex; height:2.843ex;" alt="{\displaystyle R(w_{1},\ldots ,w_{h})}" loading="lazy"> and an equality atom has the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}=w_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{i}=w_{j}}</annotation>

</semantics>

</math><img src="./0607083be6c6e917f256f04571369cfa5e98c299.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.136ex; height:2.343ex;" alt="{\displaystyle w_{i}=w_{j}}" loading="lazy">, where each of the <a href="Term_(logic)" title="Term (logic)">terms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w,...,w_{h},w_{i},w_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w,...,w_{h},w_{i},w_{j}}</annotation>

</semantics>

</math><img src="./ba28fa8741443890cb8a0096f87bf2bf17696c60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.783ex; height:2.343ex;" alt="{\displaystyle w,...,w_{h},w_{i},w_{j}}" loading="lazy"> are <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> or constants.

Actually, one can remove all equality atoms from the body of the dependency without loss of generality.<a href="#cite_note-Abiteboul95-p217-2">[2]</a> For instance, if the body consists in the conjunction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x,y)\land B(y,z,w)\land y=3\land z=w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(x,y)\land B(y,z,w)\land y=3\land z=w}</annotation>

</semantics>

</math><img src="./edf57c465b0c9b0b84be683303197cd5ef874608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.635ex; height:2.843ex;" alt="{\displaystyle A(x,y)\land B(y,z,w)\land y=3\land z=w}" loading="lazy">, then it can be replaced with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x,3)\land B(3,z,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(x,3)\land B(3,z,z)}</annotation>

</semantics>

</math><img src="./9b0b594855d44cf428c52ed416cf766ebd85d65f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.641ex; height:2.843ex;" alt="{\displaystyle A(x,3)\land B(3,z,z)}" loading="lazy"> (analogously replacing possible occurrences of the variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> in the head). Analogously, one can replace existential variables occurring in the head if they appear in some equality atom.<a href="#cite_note-Abiteboul95-p217-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Restrictions">Restrictions</h2></div>

In literature there are many common restrictions on embedded dependencies, among with:<a href="#cite_note-Kanellakis1990-1">[1]</a><a href="#cite_note-3">[3]</a>

<ul><li>full (or universal) dependencies, which are the ones without <a href="Existential_quantification" title="Existential quantification">existentially-quantified</a> variables (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists z_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exists z_{i}}</annotation>

</semantics>

</math><img src="./554efe8efc47c0072e5a531b7123c1c69cecea45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.173ex; height:2.509ex;" alt="{\displaystyle \exists z_{i}}" loading="lazy">)</li>

<li><a href="Tuple-generating_dependency" title="Tuple-generating dependency">tuple-generating dependencies</a> (TGDs)</li>

<li><a href="Equality-generating_dependency" title="Equality-generating dependency">equality-generating dependencies</a> (EGDs)</li>

<li>single-head (or 1-head) dependencies, which have only one atom in the head</li>

<li>unirelational dependencies, in which only one relation symbol occurs</li></ul>

When all atoms in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> are equalities, the ED is an EGD and, when all atoms in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> are relational, the ED is a TGD. Every ED is equivalent to an EGD and a TGD.

<div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2></div>

A common extension of embedded dependencies are disjunctive embedded dependencies (DEDs),<a href="#cite_note-deutsch-4">[4]</a> which can be defined as follows:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall x_{1},\ldots ,x_{n}.\phi (x_{1},\ldots ,x_{n})\rightarrow \bigvee _{i=1}^{\ell }\exists z_{1}^{i},\ldots ,z_{k}^{i}.\psi (y_{1}^{i},\ldots ,y_{m}^{i})}</annotation>

</semantics>

</math><img src="./cebc7d700eb8c8e1e47216839fdf2de7da982641.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.729ex; height:7.343ex;" alt="{\displaystyle \forall x_{1},\ldots ,x_{n}.\phi (x_{1},\ldots ,x_{n})\rightarrow \bigvee _{i=1}^{\ell }\exists z_{1}^{i},\ldots ,z_{k}^{i}.\psi (y_{1}^{i},\ldots ,y_{m}^{i})}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{z_{1}^{i},\ldots ,z_{k}^{i}\}=\{y_{1}^{i},\ldots ,y_{m}^{i}\}\setminus \{x_{1},\ldots ,x_{n}\}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{z_{1}^{i},\ldots ,z_{k}^{i}\}=\{y_{1}^{i},\ldots ,y_{m}^{i}\}\setminus \{x_{1},\ldots ,x_{n}\}}</annotation>

</semantics>

</math><img src="./06ddf95cb0b8cd4c6ab0aac10d8e55c8c7219948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.048ex; height:3.176ex;" alt="{\displaystyle \{z_{1}^{i},\ldots ,z_{k}^{i}\}=\{y_{1}^{i},\ldots ,y_{m}^{i}\}\setminus \{x_{1},\ldots ,x_{n}\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> are <a href="Logical_conjunction" title="Logical conjunction">conjunctions</a> of relational and equality atoms.

Disjunctive embedded dependencies are more expressive than simple embedded dependencies, because DEDs in general can not be simulated using one or more EDs. A further extension is the disjunctive embedded dependency with inequalities (indicated with DED<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{\neq }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{\neq }}</annotation>

</semantics>

</math><img src="./c282f67f2a384b9135f8c559f1e67c57556bbb13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.511ex; height:2.509ex;" alt="{\displaystyle ^{\neq }}" loading="lazy">), in which every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> may contain also inequality atoms.<a href="#cite_note-deutsch-4">[4]</a> However, it is important to note that this extension does not enhance expressive power, as DEDs are already expressively complete for recursively enumerable Boolean query answering.<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a><a href="#cite_note-:0-7">[7]</a>

All the restriction above can be applied also to disjunctive embedded dependencies. Beside them, DEDs can also be seen as a generalization of disjunctive tuple-generating dependencies (DTGDs).<a href="#cite_note-8">[8]</a>

Unlike the relationship between DEDs and EDs, when considering query answering with conjunctive queries (CQs), DTGDs can always be equivalently rewritten as TGDs.<a href="#cite_note-:0-7">[7]</a> However, if unions of conjunctive queries (UCQs) are allowed in query answering, the expressive power of DTGDs still strictly exceeds that of TGDs.<a href="#cite_note-:0-7">[7]</a> In addition, it is also noteworthy that DEDs are strictly more expressive than DTGDs.<a href="#cite_note-:0-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-Kanellakis1990-1">^ <a href="#cite_ref-Kanellakis1990_1-0">a</a> <a href="#cite_ref-Kanellakis1990_1-1">b</a> (<a href="#CITEREFKanellakis1990">Kanellakis 1990</a>)

</li>

<li id="cite_note-Abiteboul95-p217-2">^ <a href="#cite_ref-Abiteboul95-p217_2-0">a</a> <a href="#cite_ref-Abiteboul95-p217_2-1">b</a> (<a href="#CITEREFAbiteboulHullVianu1995">Abiteboul, Hull & Vianu 1995</a>, p. 217)

</li>

/* start https://en.wikipedia.org/ */

.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFGrecoZumpano2000" class="citation conference cs1">Greco, Sergio; Zumpano, Ester (Nov 2000). Michel Parigot, Andrei Voronkov (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x9yIe1NiWHkC&pg=PA308">Querying Inconsistent Databases</a>. 7th International Conference on Logic for Programming Artificial Intelligence and Reasoning. Reunion Island, France: Springer. pp. 308–325. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-44404-1_20">10.1007/3-540-44404-1_20</a>.</cite>

</li>

<li id="cite_note-deutsch-4">^ <a href="#cite_ref-deutsch_4-0">a</a> <a href="#cite_ref-deutsch_4-1">b</a> (<a href="#CITEREFDeutsch2009">Deutsch 2009</a>)

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFZhangZhangYou2016" class="citation journal cs1">Zhang, Heng; Zhang, Yan; You, Jia-Huai (2016-07-09). <a rel="nofollow" class="external text" href="https://dl.acm.org/doi/10.5555/3060621.3060806">"Expressive completeness of existential rule languages for ontology-based query answering"</a>. Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence. IJCAI'16. New York, New York, USA: AAAI Press: 1330–1337. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-57735-770-4</bdi>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFZhangZhangYouFeng2020" class="citation journal cs1">Zhang, Heng; Zhang, Yan; You, Jia-Huai; Feng, Zhiyong; Jiang, Guifei (2020-04-03). <a rel="nofollow" class="external text" href="https://ojs.aaai.org/index.php/AAAI/article/view/5699">"Towards Universal Languages for Tractable Ontology Mediated Query Answering"</a>. Proceedings of the AAAI Conference on Artificial Intelligence. 34 (3): 3049–3056. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1911.11359">1911.11359</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1609%2Faaai.v34i03.5699">10.1609/aaai.v34i03.5699</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2374-3468">2374-3468</a>.</cite>

</li>

<li id="cite_note-:0-7">^ <a href="#cite_ref-:0_7-0">a</a> <a href="#cite_ref-:0_7-1">b</a> <a href="#cite_ref-:0_7-2">c</a> <a href="#cite_ref-:0_7-3">d</a> <cite id="CITEREFZhangJiang2022" class="citation conference cs1">Zhang, Heng; Jiang, Guifei (Jun 2022). <a rel="nofollow" class="external text" href="https://ojs.aaai.org/index.php/AAAI/article/view/20540">Characterizing the Program Expressive Power of Existential Rule Languages</a>. AAAI Conference on Artificial Intelligence. Vol. 36. pp. 5950–5957. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2112.08136">2112.08136</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1609%2Faaai.v36i5.20540">10.1609/aaai.v36i5.20540</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFDeutschTannen2003" class="citation journal cs1">Deutsch, Alin; Tannen, Val (2003). Calvanese, Diego; Lenzerini, Maurizio; Motwani, Rajeev (eds.). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/3-540-36285-1_15">"Reformulation of XML Queries and Constraints"</a>. Database Theory — ICDT 2003. Berlin, Heidelberg: Springer: 225–241. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-36285-1_15">10.1007/3-540-36285-1_15</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-36285-2</bdi>.</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFKanellakis1990" class="citation conference cs1">Kanellakis, Paris C. (1990). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/B9780444880741500226">"Elements of Relational Database Theory"</a>. Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics. Amsterdam: Elsevier. pp. 1073–1156. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fb978-0-444-88074-1.50022-6">10.1016/b978-0-444-88074-1.50022-6</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-444-88074-1</bdi>.</cite></li>

<li><cite id="CITEREFAbiteboulHullVianu1995" class="citation book cs1"><a href="Serge_Abiteboul" title="Serge Abiteboul">Abiteboul, Serge</a>; Hull, Richard B.; <a href="Victor_Vianu" title="Victor Vianu">Vianu, Victor</a> (1995). Foundations of Databases. Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-53771-0</bdi>.</cite></li>

<li><cite id="CITEREFDeutsch2009" class="citation book cs1">Deutsch, Alin (2009). "FOL Modeling of Integrity Constraints (Dependencies)". Encyclopedia of Database Systems. Boston, MA: Springer US. pp. 1155–1161. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-39940-9_980">10.1007/978-0-387-39940-9_980</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-39940-9</bdi>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-23" href="https://en.wikipedia.org/wiki/?title=Embedded_dependency&oldid=1291863981">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>