{"id":33937,"date":"2021-04-21T13:00:48","date_gmt":"2021-04-21T13:00:48","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33937"},"modified":"2025-08-01T02:46:22","modified_gmt":"2025-08-01T02:46:22","slug":"algorithmus-formae-normalis-et-applicationes","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/algorithmus-formae-normalis-et-applicationes\/","title":{"rendered":"Algorithmus Formae Normalis et Applicationes"},"content":{"rendered":"<p><center><\/p>\n<h1>Algorithmus Formae Normalis et Applicationes<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>SUMMARIUM<\/strong><br \/><em>In hac lectione algorithmum FND\/FNC recognoscemus, qui nobis permittet, ex qualibet expressione logicae propositionis, eius expressionem aequivalentem in forma normali coniunctiva vel disiunctiva invenire. Incipiemus explicando tres gradus huius algoritmi componentes, qui constant in eliminatione implicationum et biimplicationum, eliminatione negationum duplicium atque applicatione distributionis, secundum utrum FNC an FND obtinere velimus. Praeterea exempla dabimus quomodo hic algorithmus ad expressiones concretas applicari possit. Postea tractabimus quomodo forma normalis ex tabulis veritatis obtineri possit, utens interruptoribus simplicibus et compositis, necnon capsis nigris. Ad hoc, conceptus tales uti filis, nodis, interruptoribus simplicibus, interruptoribus compositis et capsis nigris adhibebuntur. Denique exercitia exemplaria exhibebuntur, in quibus informatio in tabula veritatis summatim reddenda est et FND et FNC quae functionem cuiusdam machinae reddant extrahendae sunt, itemque interruptor compositus designandus qui eandem functionem ac machina habeat.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>METAE DISCENDI:<\/strong><br \/>\nPost hanc lectionem, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Applicare<\/strong> algorithmum FND\/FNC ad expressiones concretas ad formas normales coniunctivam et disiunctivam inveniendas.<\/li>\n<li><strong>Intellegere<\/strong> usum interruptorum simplicium et compositorum in logica propositionali.<\/li>\n<li><strong>Agnoscere<\/strong> structuram interruptorum compositorum et capsarum nigrarum.<\/li>\n<li><strong>Uti<\/strong> tabula veritatis ad informationem de machina summatim referendam.<\/li>\n<li><strong>Extrahere<\/strong> FND et FNC machinae ex eius tabula veritatis.<\/li>\n<li><strong>Designare<\/strong> interruptorem compositum qui eandem functionem ac machina proposita habeat.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">ALGORITHMUS FND\/FNC<\/a><br \/>\n<a href=\"#2\">ALGORITHMUS AD FORMAM NORMALIS EX TABULIS VERITATIS OBTINENDAM: CAPSAE NIGRAE ET INTERRUPTORES COMPOSITI<\/a><br \/>\n<a href=\"#3\">EXERCITIA EXEMPLARIA<\/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;\">Quamquam probavimus omnes expressiones logicae propositionis aequivalere <strong>formae normali,<\/strong> nihil diximus de modo ad has formas inveniendas. Ad hoc assequendum, algorithmum recognoscemus cuius finis est expressiones in forma normali generare atque denique applicationes quae ex his rebus emergunt examinabimus.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Algorithmus 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>Algorithmus FND\/FNC est series graduum <\/strong><\/a>qui tibi permittet, ex qualibet expressione logicae propositionis, eius expressionem aequivalentem in FND vel FNC (pro opportunitate) invenire. Hoc fit modo sequente:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>GRADUS 1:<\/strong> Omnes subexpressiones formae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\rightarrow G)<\/span><\/span> reponere per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\neg F\\vee G),<\/span><\/span> similiter cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(F\\leftrightarrow G).<\/span><\/span> Hic gradus iteretur donec omnes implicationes et biimplicationes ex expressione removeantur.<\/li>\n<li><strong>GRADUS 2:<\/strong> Elimina negationes duplices et lege De Morgan ubi fieri potest utere. Haec substitutiones applicandae sunt:\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;\">Cum iam nulla subexpressio formae <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> nec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(G \\vee H)<\/span><\/span> maneat, perge ad gradum 3.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong>GRADUS 3:<\/strong> Hic gradus in duas partes dividitur, secundum utrum ad FND an ad FNC pervenire cupias.\n<ul>\n<li><strong>Si ad FNC pervenire vis:<\/strong>\n<p style=\"text-align: justify;\">Utere <span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span>-distributione ubicumque fieri potest. Id est, sequentia substituenda sunt:<\/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;\">Cum nulla expressio formae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G\\vee(H\\wedge K)<\/span><\/span> aut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(H\\wedge K)\\vee G<\/span><\/span> maneat, ad FNC perventum erit.<\/p>\n<\/li>\n<li><strong>Si ad FND pervenire vis:<\/strong>\n<p style=\"text-align: justify;\">Utere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>-distributione ubicumque fieri potest. Id est, sequentia substituenda sunt:<\/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;\">Cum nulla expressio formae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">G\\wedge(H\\vee K)<\/span><\/span> aut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(H\\vee K)\\wedge G<\/span><\/span> maneat, ad FND perventum erit.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3>Exempla<\/h3>\n<p style=\"text-align: justify;\">Utere Algorithmo FND\/FNC ad sequentia enuntiata, ut in formis normalibus coniunctiva et disiunctiva exprimantur.<\/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>[SOLUTIO]<\/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>[SOLUTIO]<\/strong><\/a><\/li>\n<\/ol>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Algorithmus ad Formam Normalem ex Tabulis Veritatis Obtineundam: Capsae Nigrae et Interruptores Compositi<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=915s\" target=\"_blank\" rel=\"noopener\"><strong>Algorithmus FND\/FNC nobis permittit<\/strong><\/a> pro qualibet expressione logicae propositionis, eius expressionem aequivalentem in forma normali invenire. Sed sunt casus in quibus nulla expressio initialis praesto est. Tale exemplum est cum exitu tabulae veritatis cuidam expressioni <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> respondente laboramus, cuius structura propositionalis ignota est. Hoc verbis explicare processus longus est; tamen technica ipsa per exempla multo melius intellegitur. Itaque exempla videoclips explicanda proponam, sed prius conceptus sequentes recognoscendi sunt:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Funiculus:<\/strong> Medium per quod signum fluit.<\/li>\n<li><strong>Nodus:<\/strong> Punctum ubi tres vel plures funiculi conveniunt.<\/li>\n<li><strong>Interruptor simplex:<\/strong> Instrumentum quod status accensionis (1) et exstinctionis (0) recipit, semper in uno tantum ex istis statibus existens. Status accensionis signi transitum permittit, status exstinctionis impedit.<\/li>\n<li><strong>Interruptor compositus:<\/strong> Instrumentum ex interruptoribus simplicibus et funiculis compositum.<\/li>\n<li><strong>Capsa Nigra:<\/strong> Quodlibet instrumentum cuius structura interna observari non potest.<\/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>Interruptores simplices repraesentantur<\/strong><\/a> per variabiles propositionales, compositi vero per expressiones logicae propositionis. Causae faciliores ex connectoribus disiunctionis et coniunctionis oriuntur, ut infra monstratur:<\/p>\n<h3>Schema Coniunctionis<\/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>Schema Disiunctionis<\/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>Exercitia Exemplaria<\/h2>\n<ol style=\"text-align: justify;\">\n<li>Habetur instrumentum ex capsula nigra et quattuor interruptoribus ordinatis compositum. Activatio huius instrumenti fit sub sequentibus condicionibus:\n<ul>\n<li><strong>Conditio 1:<\/strong> Instrumentum activatur si duo interruptores continuati accensi sunt. Haec condicio desinit operari si tres interruptores continuati accensi sunt.<\/li>\n<li><strong>Conditio 2:<\/strong> Activatur si omnes interruptores exstincti sunt.<\/li>\n<li><strong>Exceptio:<\/strong> Si condiciones superiores non complentur, tunc instrumentum exstinguitur.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">a) Hanc informationem in tabula veritatis summatim repraesenta. <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=2158s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/p>\n<p style=\"text-align: justify;\">b) Ex tabula veritatis, extrahatur FND et FNC quae machinae functionem repraesentent. <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=2283s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/p>\n<p style=\"text-align: justify;\">c) Utere FNC vel FND in gradu superiore obtenta (simpliciore) ad designandum interruptorem compositum qui eandem functionem ac instrumentum habeat. <a href=\"https:\/\/www.youtube.com\/watch?v=aEQ2WFB4dls&amp;t=2402s\" target=\"_blank\" rel=\"noopener\"><strong>[SOLUTIO]<\/strong><\/a><\/p>\n<\/li>\n<li>Idem ac in superiore exercitatione, sed nunc instrumentum quinque interruptores habet. <strong>[PROVOCATIO LECTORI]<\/strong><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Algorithmus Formae Normalis et Applicationes SUMMARIUMIn hac lectione algorithmum FND\/FNC recognoscemus, qui nobis permittet, ex qualibet expressione logicae propositionis, eius expressionem aequivalentem in forma normali coniunctiva vel disiunctiva invenire. Incipiemus explicando tres gradus huius algoritmi componentes, qui constant in eliminatione implicationum et biimplicationum, eliminatione negationum duplicium atque applicatione distributionis, secundum utrum FNC an FND obtinere [&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":2,"footnotes":""},"categories":[1352,1358,1298],"tags":[],"class_list":["post-33937","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logica-mathematica","category-logica-propositionalis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Algorithmus Formae Normalis et Applicationes - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Forma Normalis est modus normatus expressiones logicas in logica propositionali repraesentandi. 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