{"id":6522,"date":"2018-07-09T10:53:19","date_gmt":"2018-07-09T09:53:19","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=6522"},"modified":"2022-12-03T23:00:49","modified_gmt":"2022-12-03T22:00:49","slug":"resolution-par-le-lemme-darden","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/language-theory\/resolution-by-lemma-darden\/","title":{"rendered":"Solving by Arden&#039;s lemma"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"6522\" class=\"elementor elementor-6522\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a2d20e0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a2d20e0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-2e8d19f\" data-id=\"2e8d19f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d1da51d elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"d1da51d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/language-theory\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Language theory<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div 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class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0ab57e5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0ab57e5\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e07f0d8\" data-id=\"e07f0d8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-31aae94 elementor-widget 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elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8f478e1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-55d8d62\" data-id=\"55d8d62\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5af68af7 elementor-widget elementor-widget-text-editor\" data-id=\"5af68af7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/language-theory\/resolution-by-lemma-darden\/#Lemme-dArden\" >Lemma of Arden<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/language-theory\/resolution-by-lemma-darden\/#Version-de-droite\" >Right version<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/en\/language-theory\/resolution-by-lemma-darden\/#Version-de-gauche\" >Left version<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Lemme-dArden\"><\/span>Lemma of Arden<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>It is important to note that there are two symmetrical versions of Arden&#039;s lemma. Depending on the version you use, the method to generate the equations is slightly different.<\/p>\n\n<figure class=\"wp-block-image\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-6523\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage32.png\" alt=\"Arden lemma\" width=\"603\" height=\"100\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage32.png 603w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage32-300x50.png 300w\" sizes=\"(max-width: 603px) 100vw, 603px\" \/><\/figure>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Version-de-droite\"><\/span>Right version<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Consider the following automaton:<\/p>\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" class=\"alignnone wp-image-6524\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage33.png\" alt=\"Arden lemma\" width=\"291\" height=\"214\" title=\"\"><\/figure>\n\n<p>The construction of the equations is done as follows: the language recognized by a state is equal to the language followed by the transition symbol of its predecessors. For example, the language L<sub>2<\/sub> accepted by state 2 is equal to: L<sub>2<\/sub>\u00a0= L<sub>1<\/sub>To. A word w \u2208 L<sub>i<\/sub> if and only if he<br \/>exists a calculation of the automaton on w starting from the initial state and arriving in the state i (attention the version is not the same one on a left version).<\/p>\n\n<p>We have the following system of equation:<\/p>\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" class=\"alignnone wp-image-6525\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage34.png\" alt=\"Arden lemma\" width=\"292\" height=\"127\" title=\"\"><\/figure>\n\n<p>By substituting L<sub>1<\/sub>\u00a0in the second equation by its value then L<sub>2<\/sub>\u00a0in the third and fourth equations we get:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6526\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage35.png\" alt=\"Arden lemma\" width=\"259\" height=\"116\" title=\"\"><\/figure>\n\n<p>We apply the right Arden lemma to the last two equations:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6527\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage36.png\" alt=\"Arden lemma\" width=\"223\" height=\"131\" title=\"\"><\/figure>\n\n<p>The language recognized by the automaton is the set of languages recognized by its terminal states, which in this example gives: ab (a + b) * + (b + aa) a *<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Version-de-gauche\"><\/span>Left version<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Consider the following automaton:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6528\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage37.png\" alt=\"Arden lemma\" width=\"233\" height=\"185\" title=\"\"><\/figure>\n\n<p>The construction of the equations is done as follows: the language recognized by a state is equal to the language preceded by the transition symbol of its successors. For example, the language L<sub>2<\/sub> accepted by state 2 is equal to: L<sub>2<\/sub>\u00a0= aL<sub>1<\/sub>+ aL<sub>3<\/sub>+ \u03b5 (because it is a terminal state). A word w \u2208 L<sub>i<\/sub>\u00a0if and only if he<br \/>exists a computation of the automaton on w starting from state i and arriving in a final state (be careful if we used the other version of the lemma, the definition would be different).<\/p>\n\n<p>We have the following system of equations:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6529\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage38.png\" alt=\"Arden lemma\" width=\"253\" height=\"122\" title=\"\"><\/figure>\n\n<p>By substituting L<sub>3<\/sub>\u00a0and L<sub>4<\/sub>\u00a0in the first two equations then L<sub>2<\/sub>\u00a0in the first equation we get:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6530\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage39.png\" alt=\"Arden lemma\" width=\"386\" height=\"118\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage39.png 386w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/07\/langage39-300x92.png 300w\" sizes=\"(max-width: 386px) 100vw, 386px\" \/><\/figure>\n\n<p>Finally, by applying Arden&#039;s lemma (left version) to the first equation we<br \/>obtains: L1 = (a + bba + bbab) \u2217 (bb + \u03b5). Here we have the language recognized from the initial state, therefore from the automaton.<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Language Theory Wiki Home Difficulty Hard 80% Arden&#039;s Lemma It is important to note that there are two symmetrical versions of Arden&#039;s Lemma. Next \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":5028,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-6522","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6522","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=6522"}],"version-history":[{"count":9,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6522\/revisions"}],"predecessor-version":[{"id":18643,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6522\/revisions\/18643"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/5028"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=6522"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}