https://en.formulasearchengine.com/w/api.php?action=feedcontributions&feedformat=atom&user=108.68.109.77 2026-08-25T05:00:30Z User contributions MediaWiki 1.47.0-wmf.7 https://en.formulasearchengine.com/w/index.php?title=Continuous_quantum_computation&diff=16290 2011-02-25T04:03:22Z

<p>108.68.109.77: </p> <hr /> <div>[[Image:Implication graph.svg|thumb|360px|An implication graph representing the [[2-satisfiability]] instance &lt;math&gt;\scriptscriptstyle(x_0\lor x_2)\land(x_0\lor\lnot x_3)\land(x_1\lor\lnot x_3)\land(x_1\lor\lnot x_4)\land(x_2\lor\lnot x_4)\land{}\atop\quad\scriptscriptstyle(x_0\lor\lnot x_5)\land (x_1\lor\lnot x_5)\land (x_2\lor\lnot x_5)\land (x_3\lor x_6)\land (x_4\lor x_6)\land (x_5\lor x_6).&lt;/math&gt;]]<br /> In [[mathematical logic]], an '''implication graph''' is a [[skew-symmetric graph|skew-symmetric]] [[directed graph]] ''G''(''V'', ''E'') composed of vertex set ''V'' and directed edge set ''E''. Each vertex in ''V'' represents the truth status of a [[Boolean literal]], and each directed edge from vertex ''u'' to vertex ''v'' represents the [[material implication]] &quot;If the literal ''u'' is true then the literal ''v'' is also true&quot;. Implication graphs were originally used for analyzing complex [[Boolean expression]]s.<br /> <br /> ==Applications==<br /> A [[2-satisfiability]] instance in [[conjunctive normal form]] can be transformed into an implication graph by replacing each of its [[disjunction]]s by a pair of implications. An instance is satisfiable if and only if no literal and its negation belong to the same [[strongly connected component]] of its implication graph; this characterization can be used to solve 2-satisfiability instances in linear time.&lt;ref&gt;{{cite journal|author = Aspvall, Bengt; [[Michael Plass|Plass, Michael F.]]; [[Robert Tarjan|Tarjan, Robert E.]]|title = A linear-time algorithm for testing the truth of certain quantified boolean formulas|journal = Information Processing Letters | volume = 8 | issue = 3 | pages = 121–123|year = 1979|doi = 10.1016/0020-0190(79)90002-4}}&lt;/ref&gt;<br /> <br /> ==References==<br /> &lt;references/&gt;<br /> <br /> [[Category:Boolean algebra]]<br /> [[Category:Application-specific graphs]]<br /> [[Category:Directed graphs]]<br /> [[Category:Graph families]]</div>

108.68.109.77