{"id":160369,"date":"2023-09-26T14:14:49","date_gmt":"2023-09-26T08:44:49","guid":{"rendered":"https:\/\/www.gkseries.com\/blog\/?p=160369"},"modified":"2023-09-26T14:15:35","modified_gmt":"2023-09-26T08:45:35","slug":"given-a-language-%f0%9d%90%bf-define-%f0%9d%90%bf%f0%9d%91%96-as-follows","status":"publish","type":"post","link":"https:\/\/www.gkseries.com\/blog\/given-a-language-%f0%9d%90%bf-define-%f0%9d%90%bf%f0%9d%91%96-as-follows\/","title":{"rendered":"Given a language \ud835\udc3f, define \ud835\udc3f\ud835\udc56 as follows"},"content":{"rendered":"\n

Q. Given a language \ud835\udc3f, define \ud835\udc3f\ud835\udc56<\/sup> as follows:<\/p>\n\n\n\n

\ud835\udc3f0<\/sup> = {\ud835\udf00}<\/p>\n\n\n\n

\ud835\udc3f\ud835\udc56<\/sup> = \ud835\udc3f\ud835\udc56\u22121<\/sup> \u22c5 \ud835\udc3f \ud835\udc53\ud835\udc5c\ud835\udc5f \ud835\udc4e\ud835\udc59\ud835\udc59 \ud835\udc56 > 0<\/p>\n\n\n\n

The order of a language L <\/em>is defined as the smallest k <\/em>such that \ud835\udc3f\ud835\udc58<\/sup> =\u00a0\u00a0\u00a0 \ud835\udc3f\ud835\udc58+1<\/sup>. Consider the language L<\/em>1<\/sub> (over alphabet 0) accepted by the following automaton.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The order of L<\/em>1<\/sub> is<\/p>\n\n\n\n

Ans: 2<\/p>\n\n\n\n

Sol:<\/p>\n\n\n\n

L1<\/em> = \u03b5 + 0(00)*<\/strong>
L0<\/sup> = \u03b5
L1<\/sup> = \u03b5 . (\u03b5 + 0(00)*)
    = \u03b5 + 0(00)* = L1<\/p>\n\n\n\n

L2<\/sup> = L1<\/sup> . L1<\/em> = L1<\/em> . L1<\/em>
    = (\u03b5 + 0(00)*) (\u03b5 + 0(00)*)
    = \u03b5 + 0(00)* + 0(00)* + 0(00)* . 0(00)*
    = \u03b5 + 0(00)* + 0(00)* 0(00)* = 0*<\/strong><\/p>\n\n\n\n

Given automaton contains epsilon and even number of 0\u2019s and odd numbers of 0\u2019s.<\/p>\n\n\n\n

L3<\/sup> = L2<\/sup> . L1<\/em>
    = {0*} . {\u03b5 + 0(00)*} = 0* 0(00)* = 0*
So, L2<\/sup> = L3<\/sup> = L2+1<\/sup>
So, smallest value of k = 2<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Q. Given a language \ud835\udc3f, define \ud835\udc3f\ud835\udc56 as follows: \ud835\udc3f0 = {\ud835\udf00} \ud835\udc3f\ud835\udc56 = \ud835\udc3f\ud835\udc56\u22121 \u22c5 \ud835\udc3f \ud835\udc53\ud835\udc5c\ud835\udc5f \ud835\udc4e\ud835\udc59\ud835\udc59 \ud835\udc56 > 0 The order of a language L is defined as the smallest k such that \ud835\udc3f\ud835\udc58 =\u00a0\u00a0\u00a0 \ud835\udc3f\ud835\udc58+1. Consider the language L1 (over alphabet 0) accepted by the following automaton. The order of L1 […]<\/p>\n","protected":false},"author":419,"featured_media":160371,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[5141],"tags":[5140],"class_list":["post-160369","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-gate","tag-gate-questions"],"_links":{"self":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts\/160369","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/users\/419"}],"replies":[{"embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/comments?post=160369"}],"version-history":[{"count":1,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts\/160369\/revisions"}],"predecessor-version":[{"id":160372,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts\/160369\/revisions\/160372"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/media\/160371"}],"wp:attachment":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/media?parent=160369"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/categories?post=160369"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/tags?post=160369"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}