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<title>Query (complexity)</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Query (complexity)</span></span>
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<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"><p>In <a href="Descriptive_complexity" class="mw-redirect" title="Descriptive complexity">descriptive complexity</a>, a <b>query</b> is a mapping from structures of one <a href="Signature_(logic)" title="Signature (logic)">signature</a> to structures of another vocabulary. <a href="Neil_Immerman" title="Neil Immerman">Neil Immerman</a>, in his book Descriptive Complexity,<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> "use[s] the concept of query as the fundamental paradigm of computation" (p. 17).
</p><p>Given signatures <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
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<mi>σ<!-- σ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">
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<mi>τ<!-- τ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>
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</math></span><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"></span>, we define the set of <a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">structures</a> on each language, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{STRUC}}[\sigma ]}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mtext>STRUC</mtext>
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<mo stretchy="false">[</mo>
<mi>σ<!-- σ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle {\mbox{STRUC}}[\sigma ]}</annotation>
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</math></span><img src="./bbb1b6cb871373070d84dfce92ff90adbb971515.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.726ex; height:2.843ex;" alt="{\displaystyle {\mbox{STRUC}}[\sigma ]}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{STRUC}}[\tau ]}">
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<mrow class="MJX-TeXAtom-ORD">
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<mtext>STRUC</mtext>
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<mo stretchy="false">[</mo>
<mi>τ<!-- τ --></mi>
<mo stretchy="false">]</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\mbox{STRUC}}[\tau ]}</annotation>
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</math></span><img src="./b84dfe41dfb6a45d27366be7dce3a2ee81fdc42f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.598ex; height:2.843ex;" alt="{\displaystyle {\mbox{STRUC}}[\tau ]}" loading="lazy"></span>. A query is then any mapping
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<blockquote>
<p><span class="mwe-math-element mwe-math-element-inline"><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:{\mbox{STRUC}}[\sigma ]\to {\mbox{STRUC}}[\tau ]}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>I</mi>
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<mtext>STRUC</mtext>
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<mo stretchy="false">[</mo>
<mi>σ<!-- σ --></mi>
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<mo stretchy="false">→<!-- → --></mo>
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<mstyle displaystyle="false" scriptlevel="0">
<mtext>STRUC</mtext>
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<mo stretchy="false">[</mo>
<mi>τ<!-- τ --></mi>
<mo stretchy="false">]</mo>
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<annotation encoding="application/x-tex">{\displaystyle I:{\mbox{STRUC}}[\sigma ]\to {\mbox{STRUC}}[\tau ]}</annotation>
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</math></span><img src="./816a8508c68f407dacb4842779a3c229b3a3fd69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.047ex; height:2.843ex;" alt="{\displaystyle I:{\mbox{STRUC}}[\sigma ]\to {\mbox{STRUC}}[\tau ]}" loading="lazy"></span>
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<p><a href="Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a> can then be phrased in terms of the power of the mathematical logic necessary to express a given query.
</p>
<div class="mw-heading mw-heading2"><h2 id="Order-independent_queries">Order-independent queries</h2></div>
<p>A query is <b>order-independent</b> if the ordering of objects in the structure does not affect the results of the query. In databases, these queries correspond to generic queries (Immerman 1999, p. 18). A query is order-independent iff <span class="mwe-math-element mwe-math-element-inline"><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({\mathfrak {A}})\equiv I({\mathfrak {B}})}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>I</mi>
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<annotation encoding="application/x-tex">{\displaystyle I({\mathfrak {A}})\equiv I({\mathfrak {B}})}</annotation>
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</math></span><img src="./5afd7ccd6212d7d5f9624a502d7cc5784aa289be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.784ex; height:2.843ex;" alt="{\displaystyle I({\mathfrak {A}})\equiv I({\mathfrak {B}})}" loading="lazy"></span> for any isomorphic structures <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}}</annotation>
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</math></span><img src="./34aa92fbdb716183c034a2cfc30dafbaa51cfcd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {A}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}}">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}}</annotation>
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</math></span><img src="./f939c87a07b7af23e09792e9edb2c7caebb18864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.054ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {B}}}" loading="lazy"></span>.
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFNeil1999" class="citation book cs1">Neil, Immerman (1999). <a href="Descriptive_Complexity" title="Descriptive Complexity"><i>Descriptive Complexity</i></a>. New York, NY: Springer New York. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781461205395</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/853271745">853271745</a>.</cite></span>
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