{"id":28286,"date":"2021-04-17T13:00:49","date_gmt":"2021-04-17T13:00:49","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28286"},"modified":"2024-09-04T17:46:24","modified_gmt":"2024-09-04T17:46:24","slug":"inducao-sobre-a-complexidade-das-formulas","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/inducao-sobre-a-complexidade-das-formulas\/","title":{"rendered":"Indu\u00e7\u00e3o sobre a Complexidade das F\u00f3rmulas"},"content":{"rendered":"<p><center><\/p>\n<h1>Indu\u00e7\u00e3o sobre a Complexidade das F\u00f3rmulas<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>RESUMO<\/strong><br \/><em>Nesta aula voc\u00ea aprender\u00e1 sobre uma variante da indu\u00e7\u00e3o matem\u00e1tica, conhecida como \u00abindu\u00e7\u00e3o sobre a complexidade das express\u00f5es\u00bb, que \u00e9 muito \u00fatil para demonstrar propriedades na l\u00f3gica proposicional. Atrav\u00e9s de um exemplo simples, o teorema de substitui\u00e7\u00e3o, voc\u00ea ver\u00e1 como essa t\u00e9cnica \u00e9 aplicada e como se pode demonstrar que uma propriedade se cumpre para todas as express\u00f5es da l\u00f3gica proposicional. Al\u00e9m disso, ser\u00e1 explicado como funciona a hip\u00f3tese de indu\u00e7\u00e3o e o passo indutivo, para que voc\u00ea possa aplicar essa t\u00e9cnica em suas pr\u00f3prias demonstra\u00e7\u00f5es.<\/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 estudante ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Compreender<\/strong> o conceito de indu\u00e7\u00e3o sobre a complexidade das express\u00f5es.<\/li>\n<li><strong>Aplicar<\/strong> a indu\u00e7\u00e3o matem\u00e1tica sobre a complexidade das f\u00f3rmulas na l\u00f3gica proposicional.<\/li>\n<li><strong>Compreender<\/strong> a demonstra\u00e7\u00e3o por indu\u00e7\u00e3o sobre a complexidade e como ela \u00e9 aplicada na l\u00f3gica proposicional.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u00cdNDICE<\/strong><br \/>\n<a href=\"#1\">INDU\u00c7\u00c3O SOBRE A COMPLEXIDADE<\/a><br \/>\n<a href=\"#2\">UM EXEMPLO SIMPLES: O TEOREMA DE SUBSTITUI\u00c7\u00c3O<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/wipHo3D2lQY\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Indu\u00e7\u00e3o sobre a complexidade<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=407s\" target=\"_blank\" rel=\"noopener\"><strong>Suponha que queremos provar que uma propriedade<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{P}<\/span> se cumpre para qualquer express\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>. Uma forma de demonstrar isso utilizando a variante da <strong>indu\u00e7\u00e3o matem\u00e1tica<\/strong> conhecida como \u00abindu\u00e7\u00e3o sobre a complexidade das express\u00f5es\u00bb. Isso \u00e9 feito atrav\u00e9s da seguinte s\u00e9rie de passos:<\/p>\n<ul style=\"text-align: justify;\">\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=432s\" target=\"_blank\" rel=\"noopener\"><strong>Primeiro mostramos que<\/strong><\/a> todas as express\u00f5es at\u00f4micas satisfazem essa propriedade (isso corresponde ao caso n=1 de uma indu\u00e7\u00e3o tradicional).<\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=461s\" target=\"_blank\" rel=\"noopener\"><strong>Depois, assumindo que<\/strong><\/a> se cumpre para express\u00f5es quaisquer <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, provamos que, consequentemente, se cumpre para express\u00f5es da forma <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\downarrow G<\/span><\/span>; ou, de forma equivalente, para <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F<\/span><\/span> e algumas das seguintes: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\wedge G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\vee G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\rightarrow G.<\/span><\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Se conseguirmos fazer isso, ent\u00e3o conclu\u00edmos que a propriedade <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{P}<\/span> se cumpre para todas as express\u00f5es da l\u00f3gica proposicional. Isso \u00e9 o que chamamos de \u00abindu\u00e7\u00e3o matem\u00e1tica sobre a complexidade das express\u00f5es\u00bb.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Um Exemplo Simples: O Teorema de Substitui\u00e7\u00e3o<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=wipHo3D2lQY&amp;t=681s\" target=\"_blank\" rel=\"noopener\"><strong>Para ter uma ideia melhor sobre como se realiza a indu\u00e7\u00e3o<\/strong><\/a> sobre a complexidade das f\u00f3rmulas, revisaremos o (meta)teorema de substitui\u00e7\u00e3o<\/p>\n<p style=\"text-align: justify;\">Suponha que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv G.<\/span><\/span> Seja <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> uma express\u00e3o que cont\u00e9m <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> como sub-express\u00e3o e seja <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> a express\u00e3o obtida ao substituir todas as ocorr\u00eancias de <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> por <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H^\\prime.<\/span><\/span><\/p>\n<h3>Demonstra\u00e7\u00e3o por Indu\u00e7\u00e3o sobre a Complexidade das F\u00f3rmulas<\/h3>\n<p style=\"text-align: justify;\">Demonstrar por indu\u00e7\u00e3o sobre a complexidade \u00e9 mostrar que ocorrem duas coisas: 1) um caso inicial (para f\u00f3rmulas at\u00f4micas) e 2) o passo indutivo (se funciona para express\u00f5es quaisquer <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, ent\u00e3o funciona para <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\downarrow G<\/span><\/span>, ou de uma forma mais simples: funciona para <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg F<\/span><\/span> e pelo menos algum dos seguintes: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\vee G,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\wedge G<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\rightarrow G<\/span><\/span> ou <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\leftrightarrow G<\/span><\/span>).<\/p>\n<p style=\"text-align: justify;\">Suponha que <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> seja uma express\u00e3o at\u00f4mica, <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> \u00e9 sub-express\u00e3o de <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\equiv G.<\/span><\/span> Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> \u00e9 o resultado de substituir todas as sub-express\u00f5es <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> de <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span>, como <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u00e9 at\u00f4mica, ser\u00e1 que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime \\equiv G<\/span><\/span>. Por outro lado, como <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> \u00e9 at\u00f4mica e <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> \u00e9 sub-express\u00e3o de <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv F<\/span><\/span>. Finalmente teremos que:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv F \\equiv G \\equiv H^\\prime<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">ficando demonstrado o caso inicial para express\u00f5es at\u00f4micas.<\/p>\n<p style=\"text-align: justify;\">Agora vejamos o caso indutivo.<\/p>\n<h4>A hip\u00f3tese de indu\u00e7\u00e3o<\/h4>\n<p style=\"text-align: justify;\">Suponha que o teorema funcione para duas express\u00f5es quaisquer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2<\/span><\/span>, cada uma contendo <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> como sub-express\u00e3o com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F \\equiv G.<\/span><\/span> Ent\u00e3o, se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1^\\prime<\/span><\/span> \u00e9 o que se obt\u00e9m ao substituir todas as <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> por <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2^\\prime<\/span><\/span> \u00e9 o que se obt\u00e9m ao substituir todas as <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> por <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2<\/span><\/span>, teremos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1\\equiv H_1^\\prime<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_2\\equiv H_2^\\prime<\/span><\/span>.<\/p>\n<h4>O passo indutivo<\/h4>\n<p style=\"text-align: justify;\">Aqui verificaremos se, como consequ\u00eancia da hip\u00f3tese de indu\u00e7\u00e3o, o teorema tamb\u00e9m se aplica para <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg H_1<\/span><\/span> (ou <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg H_2,<\/span><\/span> qualquer um dos dois) e pelo menos um dos seguintes: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1 \\wedge H_2,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1 \\vee H_2,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H_1 \\rightarrow H_2.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H:= \\neg H_1<\/span><\/span>, ent\u00e3o, pela hip\u00f3tese de indu\u00e7\u00e3o, ser\u00e1 que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv \\neg H_1^\\prime=: H^\\prime <\/span><\/span>, onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> \u00e9 o resultado de substituir todas as ocorr\u00eancias de <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> em <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> por <span class=\"katex-eq\" data-katex-display=\"false\">G.<\/span> Portanto, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H^\\prime.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">De forma semelhante, se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H:= H_1 \\wedge H_2<\/span><\/span>, ent\u00e3o, pela hip\u00f3tese de indu\u00e7\u00e3o, ser\u00e1 que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H_1^\\prime \\wedge H_2 \\equiv H_1^\\prime \\wedge H_2^\\prime =: H^\\prime <\/span><\/span>, onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H^\\prime<\/span><\/span> \u00e9 o resultado de substituir todas as ocorr\u00eancias de <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> em <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> por <span class=\"katex-eq\" data-katex-display=\"false\">G.<\/span> Portanto, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">H\\equiv H^\\prime.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Portanto, a indu\u00e7\u00e3o est\u00e1 completa e o teorema de substitui\u00e7\u00e3o vale para todas as express\u00f5es da l\u00f3gica proposicional.<\/p>\n<p style=\"text-align: justify;\">\nAo aplicar esta forma de indu\u00e7\u00e3o, \u00e9 poss\u00edvel garantir que uma propriedade se mantenha para todas as express\u00f5es de um sistema l\u00f3gico, o que \u00e9 especialmente \u00fatil na l\u00f3gica proposicional para estruturar demonstra\u00e7\u00f5es rigorosas. Al\u00e9m disso, sua utilidade se estende a campos como a intelig\u00eancia artificial e o desenvolvimento de software, onde a verifica\u00e7\u00e3o de sistemas l\u00f3gicos \u00e9 essencial. Atrav\u00e9s dessa t\u00e9cnica, \u00e9 poss\u00edvel automatizar demonstra\u00e7\u00f5es e garantir a consist\u00eancia de express\u00f5es complexas, o que reduz o risco de erros e melhora a precis\u00e3o em ambientes que dependem da validade matem\u00e1tica e l\u00f3gica.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Indu\u00e7\u00e3o sobre a Complexidade das F\u00f3rmulas RESUMONesta aula voc\u00ea aprender\u00e1 sobre uma variante da indu\u00e7\u00e3o matem\u00e1tica, conhecida como \u00abindu\u00e7\u00e3o sobre a complexidade das express\u00f5es\u00bb, que \u00e9 muito \u00fatil para demonstrar propriedades na l\u00f3gica proposicional. Atrav\u00e9s de um exemplo simples, o teorema de substitui\u00e7\u00e3o, voc\u00ea ver\u00e1 como essa t\u00e9cnica \u00e9 aplicada e como se pode demonstrar [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28280,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":35,"footnotes":""},"categories":[607,621,571],"tags":[],"class_list":["post-28286","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>Indu\u00e7\u00e3o sobre a Complexidade das F\u00f3rmulas - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"A indu\u00e7\u00e3o sobre a complexidade \u00e9 uma t\u00e9cnica matem\u00e1tica utilizada para demonstrar 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