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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Implication graph</span></span>
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<p>In <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a> and <a href="Graph_theory" title="Graph theory">graph theory</a>, an <b>implication graph</b> is a <a href="Skew-symmetric_graph" title="Skew-symmetric graph">skew-symmetric</a>, <a href="Directed_graph" title="Directed graph">directed graph</a> <span class="texhtml"><i>G</i> = (<i>V</i>, <i>E</i>)</span> composed of <a href="Vertex_(graph_theory)" title="Vertex (graph theory)">vertex</a> set <span class="texhtml mvar" style="font-style:italic;">V</span> and directed edge set <span class="texhtml mvar" style="font-style:italic;">E</span>. Each vertex in <span class="texhtml mvar" style="font-style:italic;">V</span> represents the truth status of a <a href="Boolean_literal" class="mw-redirect" title="Boolean literal">Boolean literal</a>, and each directed edge from vertex <span class="texhtml mvar" style="font-style:italic;">u</span> to vertex <span class="texhtml mvar" style="font-style:italic;">v</span> represents the <a href="Material_conditional" title="Material conditional">material implication</a> "If the literal <span class="texhtml mvar" style="font-style:italic;">u</span> is true then the literal <span class="texhtml mvar" style="font-style:italic;">v</span> is also true". Implication graphs were originally used for analyzing complex <a href="Boolean_expression" title="Boolean expression">Boolean expressions</a>.
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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<p>A <a href="2-satisfiability" title="2-satisfiability">2-satisfiability</a> instance in <a href="Conjunctive_normal_form" title="Conjunctive normal form">conjunctive normal form</a> can be transformed into an implication graph by replacing each of its <a href="Disjunction" class="mw-redirect" title="Disjunction">disjunctions</a> by a pair of implications. For example, the statement <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 (x_{0}\lor x_{1})}">
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<annotation encoding="application/x-tex">{\displaystyle (x_{0}\lor x_{1})}</annotation>
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</math></span><img src="./7fd9ba255774770c73022245dc42855f87f1403b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.16ex; height:2.843ex;" alt="{\displaystyle (x_{0}\lor x_{1})}" loading="lazy"></span> can be rewritten as the pair <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 (\neg x_{0}\rightarrow x_{1}),(\neg x_{1}\rightarrow x_{0})}">
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<annotation encoding="application/x-tex">{\displaystyle (\neg x_{0}\rightarrow x_{1}),(\neg x_{1}\rightarrow x_{0})}</annotation>
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</math></span><img src="./73525b090007d4ea7908ec4be1974cb72762198b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.517ex; height:2.843ex;" alt="{\displaystyle (\neg x_{0}\rightarrow x_{1}),(\neg x_{1}\rightarrow x_{0})}" loading="lazy"></span>. An instance is satisfiable <a href="If_and_only_if" title="If and only if">if and only if</a> no literal and its <a href="Negation" title="Negation">negation</a> belong to the same <a href="Strongly_connected_component" title="Strongly connected component">strongly connected component</a> of its implication graph; this characterization can be used to solve <span class="nowrap">2-satisfiability</span> instances in <a href="Linear_time" class="mw-redirect" title="Linear time">linear time</a>.<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>
</p><p>In <a href="CDCL" class="mw-redirect" title="CDCL">CDCL</a> <a href="Boolean_satisfiability_problem" title="Boolean satisfiability problem">SAT</a>-solvers, <a href="Unit_propagation" title="Unit propagation">unit propagation</a> can be naturally associated with an implication graph that captures all possible ways of deriving all implied literals from decision literals,<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> which is then used for clause learning.
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<div class="mw-references-wrap"><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"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFAspvall,_BengtPlass,_Michael_F.Tarjan,_Robert_E.1979" class="citation journal cs1">Aspvall, Bengt; Plass, Michael F.; <a href="Robert_Tarjan" title="Robert Tarjan">Tarjan, Robert E.</a> (1979). "A linear-time algorithm for testing the truth of certain quantified boolean formulas". <i>Information Processing Letters</i>. <b>8</b> (3): <span class="nowrap">121–</span>123. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0020-0190%2879%2990002-4">10.1016/0020-0190(79)90002-4</a>.</cite></span>
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<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="CITEREFPaul_BeameHenry_KautzAshish_Sabharwal2003" class="citation conference cs1">Paul Beame; Henry Kautz; Ashish Sabharwal (2003). <a rel="nofollow" class="external text" href="https://www.cs.cornell.edu/~sabhar/publications/learnIJCAI03.pdf"><i>Understanding the Power of Clause Learning</i></a> <span class="cs1-format">(PDF)</span>. IJCAI. pp. <span class="nowrap">1194–</span>1201.</cite></span>
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