{"id":28404,"date":"2024-09-15T00:44:51","date_gmt":"2024-09-15T00:44:51","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28404"},"modified":"2024-09-15T00:44:51","modified_gmt":"2024-09-15T00:44:51","slug":"algoritmo-de-forma-normal-e-aplicacoes","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/algoritmo-de-forma-normal-e-aplicacoes\/","title":{"rendered":"Algoritmo de Forma Normal e Aplica\u00e7\u00f5es"},"content":{"rendered":"<p><center><\/p>\n<h1>Algoritmo de Forma Normal e Aplica\u00e7\u00f5es<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>RESUMO<\/strong><br \/><em>Nesta aula, revisaremos o algoritmo FND\/FNC, que nos permitir\u00e1 encontrar, a partir de qualquer express\u00e3o na l\u00f3gica proposicional, sua express\u00e3o equivalente em forma normal conjuntiva ou disjuntiva. Come\u00e7aremos explicando os tr\u00eas passos que comp\u00f5em esse algoritmo, que consistem na elimina\u00e7\u00e3o de implica\u00e7\u00f5es e duplas implica\u00e7\u00f5es, na elimina\u00e7\u00e3o de duplas nega\u00e7\u00f5es e na aplica\u00e7\u00e3o de distribui\u00e7\u00e3o, conforme desejamos obter uma FNC ou uma FND. Al\u00e9m disso, apresentaremos exemplos de como aplicar esse algoritmo a express\u00f5es concretas. Posteriormente, abordaremos como obter a forma normal a partir das tabelas de verdade utilizando interruptores simples e compostos, e as caixas pretas. Para isso, ser\u00e3o utilizados conceitos como cabos, n\u00f3s, interruptores simples, interruptores compostos e caixas pretas. Finalmente, ser\u00e3o apresentados exerc\u00edcios de exemplo nos quais se dever\u00e1 resumir informa\u00e7\u00f5es em uma tabela de verdade e extrair a FND e FNC que reproduzam o funcionamento de um dispositivo, assim como projetar um interruptor composto que tenha o mesmo funcionamento que o dispositivo.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJETIVOS DE APRENDIZAGEM:<\/strong><br \/>\nAo final desta aula, o aluno ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Aplicar<\/strong> o algoritmo FND\/FNC a express\u00f5es concretas para encontrar suas formas normais conjuntiva e disjuntiva.<\/li>\n<li><strong>Compreender<\/strong> o uso de interruptores simples e compostos na l\u00f3gica proposicional.<\/li>\n<li><strong>Identificar<\/strong> a estrutura dos interruptores compostos e das caixas pretas.<\/li>\n<li><strong>Utilizar<\/strong> a tabela de verdade para resumir informa\u00e7\u00f5es sobre um dispositivo.<\/li>\n<li><strong>Extrair<\/strong> a FND e a FNC de um dispositivo a partir de sua tabela de verdade.<\/li>\n<li><strong>Projetar<\/strong> um interruptor composto que tenha o mesmo funcionamento que um dispositivo dado.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u00cdNDICE<\/strong><br \/>\n<a href=\"#1\">O ALGORITMO FND\/FNC<\/a><br \/>\n<a href=\"#2\">ALGORITMO PARA OBTER A FORMA NORMAL A PARTIR DAS TABELAS DE VERDADE: CAIXAS PRETAS E INTERRUPTORES COMPOSTOS<\/a><br \/>\n<a href=\"#3\">EXERC\u00cdCIOS DE EXEMPLO<\/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;\">Embora tenhamos provado que todas as express\u00f5es da l\u00f3gica proposicional s\u00e3o equivalentes a uma <strong>forma normal,<\/strong> nada dissemos sobre como encontrar tais formas normais. Para conseguir isso, revisaremos um algoritmo cujo objetivo \u00e9 gerar express\u00f5es em forma normal e, finalmente, revisaremos as aplica\u00e7\u00f5es que emergem desses t\u00f3picos.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>O Algoritmo FND\/FNC<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=107s\" target=\"_blank\" rel=\"noopener\"><strong>O algoritmo FND\/FNC \u00e9 uma s\u00e9rie de passos <\/strong><\/a>que permitir\u00e1 encontrar, a partir de qualquer express\u00e3o da l\u00f3gica proposicional, sua express\u00e3o equivalente em FND ou FNC (conforme o caso). Ele \u00e9 realizado da seguinte maneira:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>PASSO 1:<\/strong> Substitua todas as sub-express\u00f5es da forma <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\rightarrow G)<\/span><\/span> por <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\neg F\\vee G),<\/span><\/span> e da mesma forma com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\leftrightarrow G).<\/span><\/span> Repita este passo at\u00e9 eliminar todas as implica\u00e7\u00f5es e duplas implica\u00e7\u00f5es da express\u00e3o.<\/li>\n<li><strong>PASSO 2:<\/strong> Elimine as duplas nega\u00e7\u00f5es e aplique as leis de De Morgan onde for poss\u00edvel. Devem-se aplicar as seguintes substitui\u00e7\u00f5es\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;\">Quando n\u00e3o houver mais sub-express\u00f5es da forma <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> nem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(G \\vee H),<\/span><\/span> continue com o passo 3.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong>Passo 3:<\/strong> Este passo se divide em duas partes dependendo se voc\u00ea deseja chegar a uma FND ou a uma FNC\n<ul>\n<li><strong>Se voc\u00ea deseja chegar a uma FNC:<\/strong>\n<p style=\"text-align: justify;\">Utilize a <span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span>-distribui\u00e7\u00e3o onde for poss\u00edvel. Ou seja, devem-se aplicar as seguintes substitui\u00e7\u00f5es:<\/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;\">Quando n\u00e3o houver mais express\u00f5es da forma <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G\\vee(H\\wedge K)<\/span><\/span> ou <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(H\\wedge K)\\vee G<\/span><\/span>, voc\u00ea ter\u00e1 chegado a uma FNC.<\/p>\n<\/li>\n<li><strong>Se voc\u00ea deseja chegar a uma FND:<\/strong>\n<p style=\"text-align: justify;\">Utilize a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-distribui\u00e7\u00e3o onde for poss\u00edvel. Ou seja, devem-se aplicar as seguintes substitui\u00e7\u00f5es:<\/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;\">Quando n\u00e3o houver mais express\u00f5es da forma <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G\\wedge(H\\vee K)<\/span><\/span> ou <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(H\\vee K)\\wedge G<\/span><\/span>, voc\u00ea ter\u00e1 chegado a uma FND.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3>Exemplos<\/h3>\n<p style=\"text-align: justify;\">Utilize o Algoritmo FND\/FNC para as seguintes express\u00f5es em suas formas normais conjuntiva e disjuntiva.<\/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>[SOLU\u00c7\u00c3O]<\/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>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/li>\n<\/ol>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Algoritmo para Obter a Forma Normal a partir das Tabelas de Verdade: Caixas Pretas e Interruptores Compostos<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=915s\" target=\"_blank\" rel=\"noopener\"><strong>O algoritmo FND\/FNC nos permite encontrar,<\/strong><\/a> para qualquer express\u00e3o da l\u00f3gica proposicional, sua express\u00e3o equivalente em forma normal. Mas h\u00e1 situa\u00e7\u00f5es em que n\u00e3o temos uma express\u00e3o inicial com a qual trabalhar em primeiro lugar. Este \u00e9 o caso quando temos o resultado de uma tabela de verdade de alguma express\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> cuja estrutura proposicional desconhecemos. Explicar isso em palavras \u00e9 um processo longo; a t\u00e9cnica, no entanto, \u00e9 muito melhor compreendida mostrando exemplos de como ela se desenvolve, ent\u00e3o deixarei alguns exemplos que desenvolverei em v\u00eddeo, mas primeiro voc\u00ea deve revisar os seguintes conceitos:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Cabo:<\/strong> Meio pelo qual uma sinal circula<\/li>\n<li><strong>N\u00f3:<\/strong> Ponto onde 3 ou mais cabos se encontram.<\/li>\n<li><strong>Interruptor simples:<\/strong> Dispositivo que admite os estados de ligado (1) e desligado (0), estando sempre em um, e somente um, desses estados. O estado de ligado permite a passagem de um sinal e o estado de desligado o impede.<\/li>\n<li><strong>Interruptor composto:<\/strong> \u00c9 um dispositivo composto de interruptores simples e cabos.<\/li>\n<li><strong>Caixa Preta:<\/strong> \u00c9 qualquer dispositivo cuja estrutura interna n\u00e3o pode ser observada.<\/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>Os interruptores simples s\u00e3o modelados<\/strong><\/a> atrav\u00e9s de vari\u00e1veis proposicionais, e os compostos mediante express\u00f5es da l\u00f3gica proposicional. Os casos mais simples s\u00e3o os que se obt\u00eam dos conectores de disjun\u00e7\u00e3o e conjun\u00e7\u00e3o que se mostram a seguir<\/p>\n<h3>Esquema da Conjun\u00e7\u00e3o<\/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=\"Tabela 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=\"Tabela Conector Y\" class=\"alignnone size-full lazyload\" width=\"1041\" height=\"420\" \/><\/noscript><\/p>\n<h3>Esquema da Disjun\u00e7\u00e3o<\/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=\"Tabela do 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=\"Tabela do Conector O\" class=\"alignnone size-full lazyload\" width=\"1135\" height=\"598\" \/><\/noscript><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Exerc\u00edcios de Exemplo<\/h2>\n<ol style=\"text-align: justify;\">\n<li>Um dispositivo \u00e9 formado por uma caixa preta e 4 interruptores ordenados. A ativa\u00e7\u00e3o do dispositivo ocorre sob as seguintes condi\u00e7\u00f5es\n<ul>\n<li><strong>Condi\u00e7\u00e3o 1:<\/strong> \u00c9 ativado se houver dois interruptores consecutivos ligados. Esta condi\u00e7\u00e3o deixa de funcionar se houver tr\u00eas interruptores consecutivos ligados.<\/li>\n<li><strong>Condi\u00e7\u00e3o 2:<\/strong> \u00c9 ativado se todos os interruptores estiverem desligados.<\/li>\n<li><strong>Exce\u00e7\u00e3o:<\/strong> Se as condi\u00e7\u00f5es anteriores n\u00e3o forem atendidas, o dispositivo desliga.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">a) Resuma esta informa\u00e7\u00e3o em uma tabela de verdade <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=2158s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/p>\n<p style=\"text-align: justify;\">b) A partir da tabela de verdade, extraia a FND e a FNC que reproduzam o funcionamento da m\u00e1quina. <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=2283s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/p>\n<p style=\"text-align: justify;\">c) Utilize a FNC ou a FND obtida no passo anterior (a mais simples) para projetar um interruptor composto que tenha o mesmo funcionamento que o dispositivo. <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=2402s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/a><\/p>\n<\/li>\n<li>O mesmo que no exerc\u00edcio anterior, s\u00f3 que agora o dispositivo tem 5 interruptores. <strong>[DESAFIO PARA O LEITOR]<\/strong><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Algoritmo de Forma Normal e Aplica\u00e7\u00f5es RESUMONesta aula, revisaremos o algoritmo FND\/FNC, que nos permitir\u00e1 encontrar, a partir de qualquer express\u00e3o na l\u00f3gica proposicional, sua express\u00e3o equivalente em forma normal conjuntiva ou disjuntiva. Come\u00e7aremos explicando os tr\u00eas passos que comp\u00f5em esse algoritmo, que consistem na elimina\u00e7\u00e3o de implica\u00e7\u00f5es e duplas implica\u00e7\u00f5es, na elimina\u00e7\u00e3o de duplas [&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":34,"footnotes":""},"categories":[607,621,571],"tags":[],"class_list":["post-28404","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logica-matematica-pt","category-logica-proposicional-pt","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Algoritmo de Forma Normal e Aplica\u00e7\u00f5es - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Uma Forma Normal \u00e9 uma maneira padronizada de representar express\u00f5es l\u00f3gicas na l\u00f3gica proposicional. 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