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<title>Fresh variable</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Fresh variable</span></span>

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<p>In formal reasoning, in particular in <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a>, <a href="Computer_algebra" title="Computer algebra">computer algebra</a>, and <a href="Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a>, a <b>fresh variable</b> is a variable that did not occur in the context considered so far.<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>

The concept is often used without explanation.<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><p>Fresh variables may be used to replace other variables, to eliminate <a href="Variable_shadowing" title="Variable shadowing">variable shadowing</a> or capture. For instance, in <a href="Alpha-conversion" class="mw-redirect" title="Alpha-conversion">alpha-conversion</a>, the processing of terms in the <a href="Lambda_calculus" title="Lambda calculus">lambda calculus</a> into equivalent terms with renamed variables, replacing variables with fresh variables can be helpful as a way to avoid accidentally capturing variables that should be <a href="Free_variable" class="mw-redirect" title="Free variable">free</a>.<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> Another use for fresh variables involves the development of <a href="Loop_invariant" title="Loop invariant">loop invariants</a> in <a href="Formal_verification" title="Formal verification">formal program verification</a>, where it is sometimes useful to replace constants by newly introduced fresh variables.<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>

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

<p>For example, in <a href="Term_rewriting" class="mw-redirect" title="Term rewriting">term rewriting</a>, before applying a rule <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 l\to r}">

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<annotation encoding="application/x-tex">{\displaystyle l\to r}</annotation>

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</math></span><img src="./e48a11156729bc205e675856d91ecaaf303f513b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.356ex; height:2.176ex;" alt="{\displaystyle l\to r}" loading="lazy"></span> to a given term <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 t}">

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

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</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>, each variable in <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 l\to r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l\to r}</annotation>

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</math></span><img src="./e48a11156729bc205e675856d91ecaaf303f513b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.356ex; height:2.176ex;" alt="{\displaystyle l\to r}" loading="lazy"></span> should be replaced by a fresh one to avoid clashes with variables occurring in <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 t}">

</mstyle>

</mrow>

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

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</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>.

Given the rule

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<annotation encoding="application/x-tex">{\displaystyle append(cons(x,y),z)\to cons(x,append(y,z))}</annotation>

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</math></span><img src="./2893f3c81b2ab920b3f12d4aa5a57c922034aced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.898ex; height:2.843ex;" alt="{\displaystyle append(cons(x,y),z)\to cons(x,append(y,z))}" loading="lazy"></span></dd></dl>

<p>and the term

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<annotation encoding="application/x-tex">{\displaystyle append(cons(x,cons(y,nil)),cons(3,nil))}</annotation>

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</math></span><img src="./1330dbfabba91d361f51535b74b9720a576101d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.923ex; height:2.843ex;" alt="{\displaystyle append(cons(x,cons(y,nil)),cons(3,nil))}" loading="lazy"></span>,</dd></dl>

<p>attempting to find a matching substitution of the rule's left-hand side, <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 append(cons(x,y),z)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle append(cons(x,y),z)}</annotation>

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</math></span><img src="./34e326f0c3c1bfcd2e06f26c90aaa9fb122998ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.142ex; height:2.843ex;" alt="{\displaystyle append(cons(x,y),z)}" loading="lazy"></span>, within <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 append(cons(x,cons(y,nil)),cons(3,nil))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle append(cons(x,cons(y,nil)),cons(3,nil))}</annotation>

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</math></span><img src="./1330dbfabba91d361f51535b74b9720a576101d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.923ex; height:2.843ex;" alt="{\displaystyle append(cons(x,cons(y,nil)),cons(3,nil))}" loading="lazy"></span> will fail, since <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 y}">

</mstyle>

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

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</math></span><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"></span> cannot match <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 cons(y,nil)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle cons(y,nil)}</annotation>

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</math></span><img src="./37d2cdbfa559bd157c96f2698a1c7f839cc2989f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.509ex; height:2.843ex;" alt="{\displaystyle cons(y,nil)}" loading="lazy"></span>.

However, if the rule is replaced by a <i>fresh copy</i><sup id="cite_ref-copy_5-0" class="reference"><a href="#cite_note-copy-5"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle append(cons(v_{1},v_{2}),v_{3})\to cons(v_{1},append(v_{2},v_{3}))}</annotation>

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</math></span><img src="./af16cca0d7afdcc1159ba8c332fad8392b1c5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.843ex; height:2.843ex;" alt="{\displaystyle append(cons(v_{1},v_{2}),v_{3})\to cons(v_{1},append(v_{2},v_{3}))}" loading="lazy"></span></dd></dl>

<p>before, matching will succeed with the answer substitution <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 \{v_{1}\mapsto x,\;v_{2}\mapsto cons(y,nil),\;v_{3}\mapsto cons(3,nil)\}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \{v_{1}\mapsto x,\;v_{2}\mapsto cons(y,nil),\;v_{3}\mapsto cons(3,nil)\}}</annotation>

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</math></span><img src="./b00a89d418f87028cf51afad4801ce6ba6d46701.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.425ex; height:2.843ex;" alt="{\displaystyle \{v_{1}\mapsto x,\;v_{2}\mapsto cons(y,nil),\;v_{3}\mapsto cons(3,nil)\}}" loading="lazy"></span>.

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

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<li id="cite_note-copy-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-copy_5-0">^</a></b></span> <span class="reference-text">that is, a copy with each variable consistently replaced by a fresh variable</span>

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

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</style><cite id="CITEREFCarmen_Bruni2018" class="citation report cs1">Carmen Bruni (2018). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/~cbruni/CS245Resources/lectures/2018_Fall/13_Predicate_Logic_Natural_Deduction_post.pdf">Predicate Logic: Natural Deduction</a> <span class="cs1-format">(PDF)</span> (Lecture Slides). Univ. of Waterloo.</cite> Here: slide 13/26.</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"><cite id="CITEREFMichael_Färber2023" class="citation report cs1">Michael Färber (Feb 2023). Denotational Semantics and a Fast Interpreter for jq (Technical Report). Univ. of Innsbruck. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2302.10576">2302.10576</a></span>.</cite> Here: p.4.</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"><cite id="CITEREFGordonMelham1996" class="citation book cs1">Gordon, Andrew D.; Melham, Thomas F. (1996). "Five axioms of alpha-conversion". In von Wright, Joakim; Grundy, Jim; Harrison, John (eds.). <i>Theorem Proving in Higher Order Logics, 9th International Conference, TPHOLs'96, Turku, Finland, August 26-30, 1996, Proceedings</i>. Lecture Notes in Computer Science. Vol. 1125. Springer. pp. <span class="nowrap">173–</span>190. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFB0105404">10.1007/BFB0105404</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-61587-3</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFCohen1990" class="citation book cs1">Cohen, Edward (1990). "Loops B — On replacing constants by fresh variables". <i>Programming in the 1990s</i>. Monographs in Computer Science. New York: Springer. pp. <span class="nowrap">149–</span>194. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4613-9706-9">10.1007/978-1-4613-9706-9</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781461397069</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1509875">1509875</a>.</cite></span>

</li>

</ol></div></div>

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This article is issued from <a class="external text" title="Last edited on 2025-01-04" href="https://en.wikipedia.org/wiki/?title=Fresh_variable&oldid=1267290149">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.

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