{"id":28402,"date":"2024-09-15T00:44:51","date_gmt":"2024-09-15T00:44:51","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28402"},"modified":"2024-09-15T00:44:51","modified_gmt":"2024-09-15T00:44:51","slug":"normal-form-algorithm-and-applications","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/normal-form-algorithm-and-applications\/","title":{"rendered":"Normal Form Algorithm and Applications"},"content":{"rendered":"<p><center><\/p>\n<h1>Normal Form Algorithm and Applications<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>SUMMARY<\/strong><br \/><em>In this class, we will review the DNF\/CNF algorithm, which will allow us to find, from any expression in propositional logic, its equivalent expression in conjunctive or disjunctive normal form. We will begin by explaining the three steps that make up this algorithm, which consist of eliminating implications and double implications, eliminating double negations, and applying distribution, depending on whether we want to obtain a CNF or a DNF. Additionally, we will present examples of how to apply this algorithm to concrete expressions. Subsequently, we will address how to obtain the normal form from truth tables using simple and compound switches, and black boxes. For this, concepts such as wires, nodes, simple switches, compound switches, and black boxes will be used. Finally, we will present example exercises in which information must be summarized in a truth table and the DNF and CNF that reproduce the operation of a device must be extracted, as well as designing a compound switch that has the same operation as the device.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nAt the end of this class, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Apply<\/strong> the DNF\/CNF algorithm to concrete expressions to find their conjunctive and disjunctive normal forms.<\/li>\n<li><strong>Understand<\/strong> the use of simple and compound switches in propositional logic.<\/li>\n<li><strong>Identify<\/strong> the structure of compound switches and black boxes.<\/li>\n<li><strong>Use<\/strong> the truth table to summarize information about a device.<\/li>\n<li><strong>Extract<\/strong> the DNF and CNF of a device from its truth table.<\/li>\n<li><strong>Design<\/strong> a compound switch that has the same operation as a given device.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">THE DNF\/CNF ALGORITHM<\/a><br \/>\n<a href=\"#2\">ALGORITHM FOR OBTAINING THE NORMAL FORM FROM TRUTH TABLES: BLACK BOXES AND COMPOUND SWITCHES<\/a><br \/>\n<a href=\"#3\">EXAMPLE EXERCISES<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/aEQ2WFB4dls\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p style=\"text-align: justify;\">Although we have proven that all propositional logic expressions are equivalent to a <strong>normal form,<\/strong> we have not said anything about how to find such normal forms. To achieve this, we will review an algorithm whose goal is to generate expressions in normal form and, finally, we will review the applications that emerge from these topics.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>The DNF\/CNF Algorithm<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=107s\" target=\"_blank\" rel=\"noopener\"><strong>The DNF\/CNF algorithm is a series of steps <\/strong><\/a>that will allow you to find, from any expression in propositional logic, its equivalent expression in DNF or CNF (as appropriate). It is performed as follows:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>STEP 1:<\/strong> Replace all sub-expressions of the form <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\rightarrow G)<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\neg F\\vee G),<\/span><\/span> and similarly with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\leftrightarrow G).<\/span><\/span> Repeat this step until all implications and double implications in the expression are eliminated.<\/li>\n<li><strong>STEP 2:<\/strong> Eliminate double negations and apply De Morgan&#8217;s laws where possible. The following replacements must be applied\n<ul style=\"text-align: justify;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg G \\longmapsto G<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(G\\wedge H) \\longmapsto (\\neg G \\vee \\neg H)<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(G\\vee H) \\longmapsto (\\neg G \\wedge \\neg H)<\/span><\/span>\n<p style=\"text-align: justify;\">When there are no longer sub-expressions of the form <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg\\neg G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(G \\wedge H)<\/span><\/span> or <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(G \\vee H),<\/span><\/span> continue with step 3.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong>Step 3:<\/strong> This step is divided into two parts depending on whether you want to reach a DNF or a CNF\n<ul>\n<li><strong>If you want to reach a CNF:<\/strong>\n<p style=\"text-align: justify;\">Use <span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span>-distribution where possible. That is, the following replacements must be applied:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\begin{matrix}G\\vee(H\\wedge K) \\\\ \\\\ (H\\wedge K)\\vee G \\end{matrix} \\right\\} \\longmapsto (G\\vee H)\\wedge (G\\vee K)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">When there are no longer expressions of the form <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G\\vee(H\\wedge K)<\/span><\/span> or <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(H\\wedge K)\\vee G<\/span><\/span>, a CNF will have been reached.<\/p>\n<\/li>\n<li><strong>If you want to reach a DNF:<\/strong>\n<p style=\"text-align: justify;\">Use <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-distribution where possible. That is, the following replacements must be applied:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\begin{matrix}G\\wedge(H\\vee K) \\\\ \\\\ (H\\vee K)\\wedge G \\end{matrix} \\right\\} \\longmapsto (G\\wedge H)\\vee (G\\vee K)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">When there are no longer expressions of the form <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G\\wedge(H\\vee K)<\/span><\/span> or <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(H\\vee K)\\wedge G<\/span><\/span>, a DNF will have been reached.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3>Examples<\/h3>\n<p style=\"text-align: justify;\">Use the DNF\/CNF Algorithm for the following expressions in their conjunctive and disjunctive normal forms.<\/p>\n<ol style=\"text-align: justify;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(A\\rightarrow (B\\rightarrow A))<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=436s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((A\\vee B)\\rightarrow(\\neg B \\wedge A))<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=665s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/li>\n<\/ol>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Algorithm for Obtaining the Normal Form from Truth Tables: Black Boxes and Compound Switches<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=915s\" target=\"_blank\" rel=\"noopener\"><strong>The DNF\/CNF algorithm allows us to find,<\/strong><\/a> for any expression in propositional logic, its equivalent expression in normal form. But there are situations where we do not have an initial expression to work with in the first place. This is the case when we have the result of a truth table of some expression <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> whose propositional structure we do not know. Explaining this in words is a long process; the technique, however, is much better understood by showing examples of how it is developed, so I will leave some examples that I will develop in a video, but first you must review the following concepts:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Wire:<\/strong> Medium through which a signal circulates<\/li>\n<li><strong>Node:<\/strong> Point where 3 or more wires meet.<\/li>\n<li><strong>Simple switch:<\/strong> A device that admits the states of on (1) and off (0), always being in one, and only one, of those states. The on state allows the passage of a signal and the off state prevents it.<\/li>\n<li><strong>Compound switch:<\/strong> It is a device composed of simple switches and wires.<\/li>\n<li><strong>Black Box:<\/strong> It is any device whose internal structure cannot be observed.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=1361s\" target=\"_blank\" rel=\"noopener\"><strong>Simple switches are modeled<\/strong><\/a> through propositional variables, and compound ones through expressions in propositional logic. The simplest cases are those obtained from the disjunction and conjunction connectors shown below<\/p>\n<h3>Conjunction Scheme<\/h3>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-xK_trhHRmzw\/YHeQhyDxUBI\/AAAAAAAAE48\/VUFuzNDMtBslDTcM7X2UlrmSzJ_evD4-ACLcBGAsYHQ\/s0\/conectorY.PNG\" alt=\"Conector Y\" class=\"alignnone size-full lazyload\" width=\"901\" height=\"407\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-xK_trhHRmzw\/YHeQhyDxUBI\/AAAAAAAAE48\/VUFuzNDMtBslDTcM7X2UlrmSzJ_evD4-ACLcBGAsYHQ\/s0\/conectorY.PNG\" alt=\"Conector Y\" class=\"alignnone size-full lazyload\" width=\"901\" height=\"407\" \/><\/noscript><\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-alLO75lFiw8\/YHeQiiHLF-I\/AAAAAAAAE5I\/yN2bqhtGxtIAzpLLAus5PLTMbse5uA2BwCLcBGAsYHQ\/s0\/tablaConectorY.PNG\" alt=\"Tabla Conector Y\" class=\"alignnone size-full lazyload\" width=\"1041\" height=\"420\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-alLO75lFiw8\/YHeQiiHLF-I\/AAAAAAAAE5I\/yN2bqhtGxtIAzpLLAus5PLTMbse5uA2BwCLcBGAsYHQ\/s0\/tablaConectorY.PNG\" alt=\"Tabla Conector Y\" class=\"alignnone size-full lazyload\" width=\"1041\" height=\"420\" \/><\/noscript><\/p>\n<h3>Disjunction Scheme<\/h3>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-xLVAp74wmRE\/YHeQh1p67rI\/AAAAAAAAE5A\/QRB4g9BENGgj4Jqw6Lee200YT02fUaaIQCLcBGAsYHQ\/s0\/conectorO.PNG\" alt=\"Conector O\" class=\"alignnone size-full lazyload\" width=\"1077\" height=\"661\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-xLVAp74wmRE\/YHeQh1p67rI\/AAAAAAAAE5A\/QRB4g9BENGgj4Jqw6Lee200YT02fUaaIQCLcBGAsYHQ\/s0\/conectorO.PNG\" alt=\"Conector O\" class=\"alignnone size-full lazyload\" width=\"1077\" height=\"661\" \/><\/noscript><\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-ZhxmnR7oxZU\/YHeQiDdLT_I\/AAAAAAAAE5E\/Gbh-I7c6yWwif685Z4GRLO_6iHMT3_yrQCLcBGAsYHQ\/s0\/tablaConectorO.PNG\" alt=\"Tabla del Conector O\" class=\"alignnone size-full lazyload\" width=\"1135\" height=\"598\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-ZhxmnR7oxZU\/YHeQiDdLT_I\/AAAAAAAAE5E\/Gbh-I7c6yWwif685Z4GRLO_6iHMT3_yrQCLcBGAsYHQ\/s0\/tablaConectorO.PNG\" alt=\"Tabla del Conector O\" class=\"alignnone size-full lazyload\" width=\"1135\" height=\"598\" \/><\/noscript><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Example Exercises<\/h2>\n<ol style=\"text-align: justify;\">\n<li>A device is formed by a black box and 4 ordered switches. The activation of the device occurs under the following conditions\n<ul>\n<li><strong>Condition 1:<\/strong> It is activated if there are two consecutive switches on. This condition stops working if there are three consecutive switches on.<\/li>\n<li><strong>Condition 2:<\/strong> It is activated if all the switches are off.<\/li>\n<li><strong>Exception:<\/strong> If the previous conditions are not met, then the device turns off.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">a) Summarize this information in a truth table <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=2158s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/p>\n<p style=\"text-align: justify;\">b) From the truth table, extract the DNF and CNF that reproduce the operation of the machine. <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=2283s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTION]<\/strong><\/a><\/p>\n<p style=\"text-align: justify;\">c) Use the CNF or DNF obtained in the previous step (the simplest one) to design a compound switch that has the same operation as the device. <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=2402s\" target=\"_blank\" rel=\"noopener\"><strong> [SOLUTION]<\/strong><\/a><\/p>\n<\/li>\n<li>The same as in the previous exercise, but now the device has 5 switches. <strong>[CHALLENGE FOR THE READER]<\/strong><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Normal Form Algorithm and Applications SUMMARYIn this class, we will review the DNF\/CNF algorithm, which will allow us to find, from any expression in propositional logic, its equivalent expression in conjunctive or disjunctive normal form. We will begin by explaining the three steps that make up this algorithm, which consist of eliminating implications and double [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28401,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":14,"footnotes":""},"categories":[605,567,619],"tags":[],"class_list":["post-28402","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematical-logic","category-mathematics","category-propositional-logic"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Normal Form Algorithm and Applications - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"A Normal Form is a standardized way of representing logical expressions in propositional logic. 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