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content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//id.m.wikipedia.org/wiki/Rekursi"> <link rel="alternate" type="application/x-wiki" title="Sunting" href="/w/index.php?title=Rekursi&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (id)"> <link rel="EditURI" type="application/rsd+xml" href="//id.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://id.wikipedia.org/wiki/Rekursi"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.id"> <link rel="alternate" type="application/atom+xml" title="Umpan Atom Wikipedia" href="/w/index.php?title=Istimewa:Perubahan_terbaru&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Rekursi rootpage-Rekursi skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Lompat ke isi</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Situs"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menu utama" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menu utama</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menu utama</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">sembunyikan</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigasi </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage" class="mw-list-item"><a href="/wiki/Halaman_Utama" title="Kunjungi Halaman Utama [z]" accesskey="z"><span>Halaman Utama</span></a></li><li id="n-Daftar-isi" class="mw-list-item"><a href="/wiki/Wikipedia:Isi"><span>Daftar isi</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Istimewa:Perubahan_terbaru" title="Daftar perubahan terbaru dalam wiki. [r]" accesskey="r"><span>Perubahan terbaru</span></a></li><li id="n-Artikel-pilihan" class="mw-list-item"><a href="/wiki/Wikipedia:Artikel_pilihan/Topik"><span>Artikel pilihan</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Peristiwa_terkini" title="Temukan informasi tentang peristiwa terkini"><span>Peristiwa terkini</span></a></li><li id="n-newpage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_baru"><span>Halaman baru</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_sembarang" title="Tampilkan sembarang halaman [x]" accesskey="x"><span>Halaman sembarang</span></a></li> </ul> </div> </div> <div id="p-Komunitas" class="vector-menu mw-portlet mw-portlet-Komunitas" > <div class="vector-menu-heading"> Komunitas </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-Warung-Kopi" class="mw-list-item"><a href="/wiki/Wikipedia:Warung_Kopi"><span>Warung Kopi</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Portal:Komunitas" title="Tentang proyek, apa yang dapat Anda lakukan, di mana untuk mencari sesuatu"><span>Portal komunitas</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Bantuan:Isi" title="Tempat mencari bantuan."><span>Bantuan</span></a></li> </ul> </div> </div> <div id="p-Wikipedia" class="vector-menu mw-portlet mw-portlet-Wikipedia" > <div class="vector-menu-heading"> Wikipedia </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:Perihal"><span>Tentang Wikipedia</span></a></li><li id="n-Pancapilar" class="mw-list-item"><a href="/wiki/Wikipedia:Pancapilar"><span>Pancapilar</span></a></li><li id="n-Kebijakan" class="mw-list-item"><a href="/wiki/Wikipedia:Kebijakan_dan_pedoman"><span>Kebijakan</span></a></li><li id="n-Hubungi-kami" class="mw-list-item"><a href="/wiki/Wikipedia:Hubungi_kami"><span>Hubungi kami</span></a></li><li id="n-Bak-pasir" class="mw-list-item"><a href="/wiki/Wikipedia:Bak_pasir"><span>Bak pasir</span></a></li> </ul> </div> </div> <div id="p-Bagikan" class="vector-menu mw-portlet mw-portlet-Bagikan emptyPortlet" > <div class="vector-menu-heading"> Bagikan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Halaman_Utama" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="Ensiklopedia Bebas" src="/static/images/mobile/copyright/wikipedia-tagline-id.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Istimewa:Pencarian" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cari di Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Pencarian</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Telusuri Wikipedia" aria-label="Telusuri Wikipedia" autocapitalize="sentences" title="Cari di Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Istimewa:Pencarian"> </div> <button class="cdx-button cdx-search-input__end-button">Cari</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Perkakas pribadi"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Tampilan" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Tampilan</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_id.wikipedia.org&uselang=id" class=""><span>Menyumbang</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Istimewa:Buat_akun&returnto=Rekursi" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan" class=""><span>Buat akun baru</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Istimewa:Masuk_log&returnto=Rekursi" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o" class=""><span>Masuk log</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Opsi lainnya" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas pribadi" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Perkakas pribadi</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu pengguna" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_id.wikipedia.org&uselang=id"><span>Menyumbang</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Buat_akun&returnto=Rekursi" title="Anda dianjurkan untuk membuat akun dan masuk log; meskipun, hal itu tidak diwajibkan"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Buat akun baru</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Istimewa:Masuk_log&returnto=Rekursi" title="Anda disarankan untuk masuk log, meskipun hal itu tidak diwajibkan. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Masuk log</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Halaman penyunting yang telah keluar log <a href="/wiki/Bantuan:Pengantar" aria-label="Pelajari lebih lanjut tentang menyunting"><span>pelajari lebih lanjut</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Istimewa:Kontribusi_saya" title="Daftar suntingan yang dibuat dari alamat IP ini [y]" accesskey="y"><span>Kontribusi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Istimewa:Pembicaraan_saya" title="Pembicaraan tentang suntingan dari alamat IP ini [n]" accesskey="n"><span>Pembicaraan</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Situs"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Daftar isi" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Awal</div> </a> </li> <li id="toc-Definisi_formal_dari_rekursi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definisi_formal_dari_rekursi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definisi formal dari rekursi</span> </div> </a> <ul id="toc-Definisi_formal_dari_rekursi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definisi_informal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definisi_informal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definisi informal</span> </div> </a> <ul id="toc-Definisi_informal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rekursi_dalam_bahasa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rekursi_dalam_bahasa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Rekursi dalam bahasa</span> </div> </a> <button aria-controls="toc-Rekursi_dalam_bahasa-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Rekursi dalam bahasa</span> </button> <ul id="toc-Rekursi_dalam_bahasa-sublist" class="vector-toc-list"> <li id="toc-Humor_rekursif" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Humor_rekursif"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Humor rekursif</span> </div> </a> <ul id="toc-Humor_rekursif-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rekursi_dalam_matematika" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rekursi_dalam_matematika"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Rekursi dalam matematika</span> </div> </a> <button aria-controls="toc-Rekursi_dalam_matematika-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Rekursi dalam matematika</span> </button> <ul id="toc-Rekursi_dalam_matematika-sublist" class="vector-toc-list"> <li id="toc-Himpunan_yang_didefinisikan_secara_rekursif" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Himpunan_yang_didefinisikan_secara_rekursif"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Himpunan yang didefinisikan secara rekursif</span> </div> </a> <ul id="toc-Himpunan_yang_didefinisikan_secara_rekursif-sublist" class="vector-toc-list"> <li id="toc-Contoh:_bilangan_asli" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Contoh:_bilangan_asli"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Contoh: bilangan asli</span> </div> </a> <ul id="toc-Contoh:_bilangan_asli-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contoh:_himpunan_dari_proposisi_benar_terjangkau" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Contoh:_himpunan_dari_proposisi_benar_terjangkau"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>Contoh: himpunan dari proposisi benar terjangkau</span> </div> </a> <ul id="toc-Contoh:_himpunan_dari_proposisi_benar_terjangkau-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rekursi_fungsional" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rekursi_fungsional"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Rekursi fungsional</span> </div> </a> <ul id="toc-Rekursi_fungsional-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pembuktian_yang_mengikutkan_definisi_rekursif" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pembuktian_yang_mengikutkan_definisi_rekursif"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Pembuktian yang mengikutkan definisi rekursif</span> </div> </a> <ul id="toc-Pembuktian_yang_mengikutkan_definisi_rekursif-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optimisasi_rekursif" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimisasi_rekursif"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Optimisasi rekursif</span> </div> </a> <ul id="toc-Optimisasi_rekursif-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rekursi_dalam_ilmu_komputer" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rekursi_dalam_ilmu_komputer"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Rekursi dalam ilmu komputer</span> </div> </a> <ul id="toc-Rekursi_dalam_ilmu_komputer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rekursi_dalam_Seni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rekursi_dalam_Seni"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Rekursi dalam Seni</span> </div> </a> <ul id="toc-Rekursi_dalam_Seni-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_rekursi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Teorema_rekursi"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Teorema rekursi</span> </div> </a> <button aria-controls="toc-Teorema_rekursi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Teorema rekursi</span> </button> <ul id="toc-Teorema_rekursi-sublist" class="vector-toc-list"> <li id="toc-Pembuktian_keunikan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pembuktian_keunikan"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Pembuktian keunikan</span> </div> </a> <ul id="toc-Pembuktian_keunikan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contoh-contoh" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Contoh-contoh"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Contoh-contoh</span> </div> </a> <ul id="toc-Contoh-contoh-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bibliografi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografi"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliografi</span> </div> </a> <ul id="toc-Bibliografi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lihat_juga" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_juga"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Lihat juga</span> </div> </a> <ul id="toc-Lihat_juga-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Referensi</span> </div> </a> <ul id="toc-Referensi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pranala_luar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pranala_luar"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Pranala luar</span> </div> </a> <ul id="toc-Pranala_luar-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Daftar isi" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Gulingkan daftar isi" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Gulingkan daftar isi</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Rekursi</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Pergi ke artikel dalam bahasa lain. Terdapat 61 bahasa" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-61" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">61 bahasa</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D9%88%D8%AF%D9%8A%D8%A9" title="عودية – Arab" lang="ar" hreflang="ar" data-title="عودية" data-language-autonym="العربية" data-language-local-name="Arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Rekursiya" title="Rekursiya – Azerbaijani" lang="az" hreflang="az" data-title="Rekursiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BA%D1%83%D1%80%D1%81%D0%B8%D1%8F" title="Рекурсия – Bulgaria" lang="bg" hreflang="bg" data-title="Рекурсия" data-language-autonym="Български" data-language-local-name="Bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%81%E0%A6%A8%E0%A6%B0%E0%A6%BE%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A6%BF_(%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%B6%E0%A6%A8)" title="পুনরাবৃত্তি (রিকার্শন) – Bengali" lang="bn" hreflang="bn" data-title="পুনরাবৃত্তি (রিকার্শন)" data-language-autonym="বাংলা" data-language-local-name="Bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Recursivitat" title="Recursivitat – Katalan" lang="ca" hreflang="ca" data-title="Recursivitat" data-language-autonym="Català" data-language-local-name="Katalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rekurze" title="Rekurze – Cheska" lang="cs" hreflang="cs" data-title="Rekurze" data-language-autonym="Čeština" data-language-local-name="Cheska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Rekursion" title="Rekursion – Dansk" lang="da" hreflang="da" data-title="Rekursion" data-language-autonym="Dansk" data-language-local-name="Dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Rekursion" title="Rekursion – Jerman" lang="de" hreflang="de" data-title="Rekursion" data-language-autonym="Deutsch" data-language-local-name="Jerman" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CE%B1%CE%B4%CF%81%CE%BF%CE%BC%CE%AE" title="Αναδρομή – Yunani" lang="el" hreflang="el" data-title="Αναδρομή" data-language-autonym="Ελληνικά" data-language-local-name="Yunani" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Recursion" title="Recursion – Inggris" lang="en" hreflang="en" data-title="Recursion" data-language-autonym="English" data-language-local-name="Inggris" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rikuro" title="Rikuro – Esperanto" lang="eo" hreflang="eo" data-title="Rikuro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Recursi%C3%B3n" title="Recursión – Spanyol" lang="es" hreflang="es" data-title="Recursión" data-language-autonym="Español" data-language-local-name="Spanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Rekursioon" title="Rekursioon – Esti" lang="et" hreflang="et" data-title="Rekursioon" data-language-autonym="Eesti" data-language-local-name="Esti" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Errekurtsio" title="Errekurtsio – Basque" lang="eu" hreflang="eu" data-title="Errekurtsio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D8%A7%D8%B2%DA%AF%D8%B4%D8%AA" title="بازگشت – Persia" lang="fa" hreflang="fa" data-title="بازگشت" data-language-autonym="فارسی" data-language-local-name="Persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Rekursio" title="Rekursio – Suomi" lang="fi" hreflang="fi" data-title="Rekursio" data-language-autonym="Suomi" data-language-local-name="Suomi" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9cursivit%C3%A9" title="Récursivité – Prancis" lang="fr" hreflang="fr" data-title="Récursivité" data-language-autonym="Français" data-language-local-name="Prancis" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Recursividade" title="Recursividade – Galisia" lang="gl" hreflang="gl" data-title="Recursividade" data-language-autonym="Galego" data-language-local-name="Galisia" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A8%D7%A7%D7%95%D7%A8%D7%A1%D7%99%D7%94" title="רקורסיה – Ibrani" lang="he" hreflang="he" data-title="רקורסיה" data-language-autonym="עברית" data-language-local-name="Ibrani" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%A8" title="प्रतिवर्तन – Hindi" lang="hi" hreflang="hi" data-title="प्रतिवर्तन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Rekurzija" title="Rekurzija – Kroasia" lang="hr" hreflang="hr" data-title="Rekurzija" data-language-autonym="Hrvatski" data-language-local-name="Kroasia" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rekurzi%C3%B3" title="Rekurzió – Hungaria" lang="hu" hreflang="hu" data-title="Rekurzió" data-language-autonym="Magyar" data-language-local-name="Hungaria" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8C%D5%A5%D5%AF%D5%B8%D6%82%D6%80%D5%BD%D5%AB%D5%A1" title="Ռեկուրսիա – Armenia" lang="hy" hreflang="hy" data-title="Ռեկուրսիա" data-language-autonym="Հայերեն" data-language-local-name="Armenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Recursion" title="Recursion – Interlingua" lang="ia" hreflang="ia" data-title="Recursion" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Rekurso" title="Rekurso – Ido" lang="io" hreflang="io" data-title="Rekurso" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Endurkv%C3%A6mt_fall" title="Endurkvæmt fall – Islandia" lang="is" hreflang="is" data-title="Endurkvæmt fall" data-language-autonym="Íslenska" data-language-local-name="Islandia" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%8D%E5%B8%B0" title="再帰 – Jepang" lang="ja" hreflang="ja" data-title="再帰" data-language-autonym="日本語" data-language-local-name="Jepang" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BA%D1%83%D1%80%D1%81%D0%B8%D1%8F" title="Рекурсия – Kazakh" lang="kk" hreflang="kk" data-title="Рекурсия" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%AC%EA%B7%80" title="재귀 – Korea" lang="ko" hreflang="ko" data-title="재귀" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Recorsion" title="Recorsion – Lombard" lang="lmo" hreflang="lmo" data-title="Recorsion" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Rekursija" title="Rekursija – Lituavi" lang="lt" hreflang="lt" data-title="Rekursija" data-language-autonym="Lietuvių" data-language-local-name="Lituavi" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Rekursija" title="Rekursija – Latvi" lang="lv" hreflang="lv" data-title="Rekursija" data-language-autonym="Latviešu" data-language-local-name="Latvi" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%B5%E0%B5%BC%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%A8%E0%B4%82" title="സ്വാവർത്തനം – Malayalam" lang="ml" hreflang="ml" data-title="സ്വാവർത്തനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%A8" title="स्वावर्तन – Marathi" lang="mr" hreflang="mr" data-title="स्वावर्तन" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nds-nl mw-list-item"><a href="https://nds-nl.wikipedia.org/wiki/Rekursie" title="Rekursie – Low Saxon" lang="nds-NL" hreflang="nds-NL" data-title="Rekursie" data-language-autonym="Nedersaksies" data-language-local-name="Low Saxon" class="interlanguage-link-target"><span>Nedersaksies</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Recursie" title="Recursie – Belanda" lang="nl" hreflang="nl" data-title="Recursie" data-language-autonym="Nederlands" data-language-local-name="Belanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Rekursjon" title="Rekursjon – Nynorsk Norwegia" lang="nn" hreflang="nn" data-title="Rekursjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Nynorsk Norwegia" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Rekursjon" title="Rekursjon – Bokmål Norwegia" lang="nb" hreflang="nb" data-title="Rekursjon" data-language-autonym="Norsk bokmål" data-language-local-name="Bokmål Norwegia" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rekurencja" title="Rekurencja – Polski" lang="pl" hreflang="pl" data-title="Rekurencja" data-language-autonym="Polski" data-language-local-name="Polski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Recursividade" title="Recursividade – Portugis" lang="pt" hreflang="pt" data-title="Recursividade" data-language-autonym="Português" data-language-local-name="Portugis" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Recursivitate" title="Recursivitate – Rumania" lang="ro" hreflang="ro" data-title="Recursivitate" data-language-autonym="Română" data-language-local-name="Rumania" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BA%D1%83%D1%80%D1%81%D0%B8%D1%8F" title="Рекурсия – Rusia" lang="ru" hreflang="ru" data-title="Рекурсия" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BA%D1%83%D1%80%D0%B7%D1%96%D1%8F" title="Рекурзія – Rusyn" lang="rue" hreflang="rue" data-title="Рекурзія" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%AA%E0%A5%81%E0%A4%A8%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A4%A8%E0%A4%B5%E0%A4%BE%E0%A4%A6" title="पुनर्गमनवाद – Sanskerta" lang="sa" hreflang="sa" data-title="पुनर्गमनवाद" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskerta" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Rekurzija" title="Rekurzija – Serbo-Kroasia" lang="sh" hreflang="sh" data-title="Rekurzija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Kroasia" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Recursion" title="Recursion – Simple English" lang="en-simple" hreflang="en-simple" data-title="Recursion" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Rekurzia_(matematika)" title="Rekurzia (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Rekurzia (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Rekurzija" title="Rekurzija – Sloven" lang="sl" hreflang="sl" data-title="Rekurzija" data-language-autonym="Slovenščina" data-language-local-name="Sloven" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BA%D1%83%D1%80%D0%B7%D0%B8%D1%98%D0%B0" title="Рекурзија – Serbia" lang="sr" hreflang="sr" data-title="Рекурзија" data-language-autonym="Српски / srpski" data-language-local-name="Serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Rekursion" title="Rekursion – Swedia" lang="sv" hreflang="sv" data-title="Rekursion" data-language-autonym="Svenska" data-language-local-name="Swedia" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BA%D1%83%D1%80%D1%81%D0%B8%D1%8F" title="Рекурсия – Tajik" lang="tg" hreflang="tg" data-title="Рекурсия" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%A7%E0%B8%B5%E0%B8%A2%E0%B8%99%E0%B9%80%E0%B8%81%E0%B8%B4%E0%B8%94" title="การเวียนเกิด – Thai" lang="th" hreflang="th" data-title="การเวียนเกิด" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Rekursiyon" title="Rekursiyon – Tagalog" lang="tl" hreflang="tl" data-title="Rekursiyon" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96zyineleme" title="Özyineleme – Turki" lang="tr" hreflang="tr" data-title="Özyineleme" data-language-autonym="Türkçe" data-language-local-name="Turki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BA%D1%83%D1%80%D1%81%D1%96%D1%8F" title="Рекурсія – Ukraina" lang="uk" hreflang="uk" data-title="Рекурсія" data-language-autonym="Українська" data-language-local-name="Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Rekursiya" title="Rekursiya – Uzbek" lang="uz" hreflang="uz" data-title="Rekursiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%87_quy" title="Đệ quy – Vietnam" lang="vi" hreflang="vi" data-title="Đệ quy" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%80%92%E5%BD%92" title="递归 – Wu Tionghoa" lang="wuu" hreflang="wuu" data-title="递归" data-language-autonym="吴语" data-language-local-name="Wu Tionghoa" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%80%92%E5%BD%92" title="递归 – Tionghoa" lang="zh" hreflang="zh" data-title="递归" data-language-autonym="中文" data-language-local-name="Tionghoa" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C3%A0i-kui" title="Chài-kui – Minnan" lang="nan" hreflang="nan" data-title="Chài-kui" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%81%9E%E6%AD%B8" title="遞歸 – Kanton" lang="yue" hreflang="yue" data-title="遞歸" data-language-autonym="粵語" data-language-local-name="Kanton" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q179976#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ruang nama"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Rekursi" title="Lihat halaman isi [c]" accesskey="c"><span>Halaman</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Pembicaraan:Rekursi" rel="discussion" title="Pembicaraan halaman isi [t]" accesskey="t"><span>Pembicaraan</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Ubah varian bahasa" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Bahasa Indonesia</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Tampilan"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Rekursi"><span>Baca</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Rekursi&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Rekursi&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Rekursi&action=history" title="Revisi sebelumnya dari halaman ini. [h]" accesskey="h"><span>Lihat riwayat</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Perkakas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Perkakas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">sembunyikan</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Opsi lainnya" > <div class="vector-menu-heading"> Tindakan </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Rekursi"><span>Baca</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Rekursi&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Rekursi&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Rekursi&action=history"><span>Lihat riwayat</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Umum </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Istimewa:Pranala_balik/Rekursi" title="Daftar semua 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div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Untuk kegunaan lain, lihat <a href="/w/index.php?title=Rekursi_(disambiguasi)&action=edit&redlink=1" class="new" title="Rekursi (disambiguasi) (halaman belum tersedia)">Rekursi (disambiguasi)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Droste_cacao_100gr_blikje,_foto_02.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Droste_cacao_100gr_blikje%2C_foto_02.JPG/220px-Droste_cacao_100gr_blikje%2C_foto_02.JPG" decoding="async" width="220" height="334" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Droste_cacao_100gr_blikje%2C_foto_02.JPG/330px-Droste_cacao_100gr_blikje%2C_foto_02.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Droste_cacao_100gr_blikje%2C_foto_02.JPG/440px-Droste_cacao_100gr_blikje%2C_foto_02.JPG 2x" data-file-width="1768" data-file-height="2683" /></a><figcaption>Suatu bentuk rekursi yang dikenal dengan <i><a href="/wiki/Efek_Droste" title="Efek Droste">Efek Droste</a></i>. Wanita dalam gambar ini memegang suatu objek yang memiliki gambar kecil-nya yang memegang objek yang identik, yang juga memiliki gambar kecil dirinya sendiri yang memegang objek yang identik, dan seterusnya.</figcaption></figure> <p><b>Rekursi</b> adalah proses pengulangan sesuatu dengan cara <a href="/w/index.php?title=Kesamaan-diri&action=edit&redlink=1" class="new" title="Kesamaan-diri (halaman belum tersedia)">kesamaan-diri</a>. Sebagai contohnya, saat dua cermin berada paralel antara satu dengan yang lain, gambar yang tertangkap adalah suatu bentuk rekursi tak-terbatas. Istilah ini memiliki makna beragam bergantung kepada ragam disiplin mulai dari <a href="/wiki/Linguistik" title="Linguistik">linguistik</a> sampai <a href="/wiki/Logika" title="Logika">logika</a>. Penggunaan paling umum dari rekursi yaitu dalam <a href="/wiki/Matematika" title="Matematika">matematika</a> dan <a href="/wiki/Ilmu_komputer" title="Ilmu komputer">ilmu komputer</a>, yang mengacu kepada suatu metode mendefinisikan <a href="/wiki/Fungsi_(matematika)" title="Fungsi (matematika)">fungsi</a> yang mana fungsi tersebut menggunakan definisinya sendiri. Secara spesifik hal ini mendefinisikan suatu instansi tak-terbatas (nilai fungsi), menggunakan ekpresi terbatas dengan beberapa instansi bisa merujuk ke instansi lainnya, tetapi dengan suatu cara sehingga tidak ada perulangan atau keterkaitan tak-terbatas dapat terjadi. Istilah ini juga digunakan secara umum untuk menjelaskan suatu proses pengulangan objek dengan cara kesamaan-diri. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definisi_formal_dari_rekursi">Definisi formal dari rekursi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=1" title="Sunting bagian: Definisi formal dari rekursi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=1" title="Sunting kode sumber bagian: Definisi formal dari rekursi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Screenshot_Recursion_via_vlc.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Screenshot_Recursion_via_vlc.png/220px-Screenshot_Recursion_via_vlc.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Screenshot_Recursion_via_vlc.png/330px-Screenshot_Recursion_via_vlc.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Screenshot_Recursion_via_vlc.png/440px-Screenshot_Recursion_via_vlc.png 2x" data-file-width="1280" data-file-height="1024" /></a><figcaption>Rekursi dalam program perekaman layar, dengan suatu jendela paling kecil mengandung foto keseluruhan layar.</figcaption></figure> <p>Dalam <a href="/wiki/Matematika" title="Matematika">matematika</a> dan <a href="/wiki/Ilmu_komputer" title="Ilmu komputer">ilmu komputer</a>, kelas dari objek atau metode memperlihatkan perilaku rekursif bila mereka dapat didefinisikan oleh dua properti berikut: </p> <ol><li>Sebuah kasus (atau beberapa kasus) dasar sederhana</li> <li>Sejumlah aturan yang mengurangi kasus-kasus lainnya sampai ke kasus dasarnya.</li></ol> <p>Sebagai contoh, berikut ini adalah definisi rekursif dari leluhur seseorang: </p> <ul><li><a href="/wiki/Orang_tua" title="Orang tua">Orang tua</a> seseorang adalah <a href="/wiki/Leluhur" title="Leluhur">leluhur</a> seseorang (<i>kasus dasar</i>).</li> <li>Orang tua dari suatu leluhur juga merupakan leluhur-nya (<i>langkah rekursi</i>).</li></ul> <p><a href="/wiki/Bilangan_Fibonacci" class="mw-redirect" title="Bilangan Fibonacci">Bilangan Fibonacci</a> adalah contoh klasik dari rekursi: </p> <ul><li>Fib(0) adalah 0 [kasus dasar]</li> <li>Fib(1) adalah 1 [kasus dasar]</li> <li>Untuk semua integer n > 1: Fib(n) adalah (Fib(n-1) + Fib(n-2)) [definisi rekursif]</li></ul> <p>Banyak aksioma matematika berdasarkan aturan-aturan rekursif. Sebagai contohnya, definisi formal dari <a href="/wiki/Bilangan_asli" title="Bilangan asli">bilangan asli</a> pada <a href="/wiki/Aksioma_Peano" title="Aksioma Peano">Aksioma Peano</a> dapat dideskripsikan sebagai: <i>0 adalah bilangan asli, dan setiap bilangan asli memiliki sebuah suksesor, yang juga merupakan bilangan asli.</i> Dengan kasus dasar ini dan aturan rekursif, seseorang dapat membuat himpunan dari semua bilangan asli. </p><p>Gambaran humornya berbunyi: <i>"Untuk memahami rekursi, pertama anda harus memahami rekursi."</i> Atau mungkin yang lebih akurat, dari <a href="/w/index.php?title=Andrew_Plotkin&action=edit&redlink=1" class="new" title="Andrew Plotkin (halaman belum tersedia)">Andrew Plotkin</a>: <i>"Jika anda telah mengetahui apa itu rekursi, cukup ingat jawabannya. Kalau tidak, cari orang yang berdiri paling dekat dengan <a href="/w/index.php?title=Douglas_Hofstadter&action=edit&redlink=1" class="new" title="Douglas Hofstadter (halaman belum tersedia)">Douglas Hofstadter</a> selain anda; lalu tanya dia rekursi itu apa."</i> </p><p>Objek matematika yang didefinisikan secara rekursif termasuk <a href="/wiki/Fungsi_(matematika)" title="Fungsi (matematika)">fungsi</a>, <a href="/wiki/Himpunan_(matematika)" title="Himpunan (matematika)">himpunan</a>, dan terutama sekali <a href="/wiki/Fraktal" title="Fraktal">fraktal</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definisi_informal">Definisi informal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=2" title="Sunting bagian: Definisi informal" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=2" title="Sunting kode sumber bagian: Definisi informal"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rekursi adalah suatu proses dengan salah satu langkah dalam prosedur tersebut menjalankan prosedur itu sendiri. Prosedur yang melakukan rekursi disebut dengan 'rekursif'. </p><p>Untuk memahami rekursi, seseorang harus mengetahui perbedaan antara sebuah prosedur dan jalannya sebuah prosedur. Sebuah prosedur yaitu kumpulan langkah-langkah berdasarkan sekumpulan aturan. Jalannya sebuah prosedur mengikuti aturan-aturan dan melakukan langkah-langkah. Analoginya mungkin sebuah prosedur adalah seperti resep yang tertulis; menjalankan sebuah prosedur adalah seperti menyiapkan makanan. </p><p>Rekursi berhubungan dengan, tetapi tidak sama, suatu referensi dalam spesifikasi prosedur sampai pada eksekusi beberapa prosedur lainnya. Misalnya, suatu resep bisa mengacu pada memasak sayuran, yang merupakan prosedur yang kemudian membutuhkan memanaskan air, dan seterusnya. Namun, prosedur rekursif adalah spesial dengan (paling tidak) salah satu langkahnya memanggil instansi baru dari prosedur yang sama, seperti suatu resep <a href="/wiki/Gandum_hitam" title="Gandum hitam">gandum hitam</a> menggunakan beberapa sisa adonan dari resep yang sama yang telah dibuat. Hal ini tentu saja membuat suatu kemungkinan perulangan tanpa berakhir; rekursi hanya dapat digunakan secara tepat dalam definisi jika langkah yang bersangkutan dilewat pada beberapa kasus sehingga prosedur dapat selesai, seperti resep gandum-hitam yang memberitahu anda bagaimana membuat adonan awal seandainya anda belum pernah membuatnya sebelumnya. Bahkan jika didefinisikan secara tepat, prosedur rekursif tidak mudah dilakukan oleh manusia, karena ia membutuhkan membedakan pemanggilan prosedur yang baru dengan yang lama (yang telah dieksekusi sebagian); hal ini membutuhkan beberapa administrasi sejauh mana berbagai prosedur instan yang berjalan bersamaan telah berjalan. Karena hal ini definisi rekursif sangat jarang dalam keadaan harian. Contohnya dapat berupa prosedur untuk menemukan jalan melewati sebuah <a href="/wiki/Labirin" title="Labirin">labirin</a>. Terus ke depan sampai menemui jalan keluar atau titik percabangan (sebuah titik mati dianggap sebagai sebuah titik percabangan dengan 0 cabang). Jika titik yang ditemui adalah suatu jalan keluar,berhenti. Kalau tidak coba setiap cabang bergantian, menggunakan prosedur secara rekursif; jika setiap percobaan gagal karena mencapai titik mati, kembali ke jalur yang menyebabkan titik percabangan dan laporkan kegagalan. Apakah ini tepat mendefinisikan suatu prosedur pemberhentian bergantung kepada bentuk labirinnya: ia tidak membolehkan perulangan. Dalam semua kasus, mengeksekusi prosedur membutuhkan pencatatan teliti semua titik percabangan yang telah dieksplorasi, dan cabang-cabang mana yang telah dicoba.. </p> <div class="mw-heading mw-heading2"><h2 id="Rekursi_dalam_bahasa">Rekursi dalam bahasa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=3" title="Sunting bagian: Rekursi dalam bahasa" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=3" title="Sunting kode sumber bagian: Rekursi dalam bahasa"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ahli linguistik <a href="/wiki/Noam_Chomsky" title="Noam Chomsky">Noam Chomsky</a> memberikan teori bahwa ekstensi tak-terhingga dari setiap <a href="/wiki/Bahasa_alami" title="Bahasa alami">bahasa alami</a> adalah memungkinkan menggunakan perangkat rekursif dengan menanamkan klausa dalam kalimat (Aspects of the Theory of Syntax. 1965). Sebagai contoh, dua kalimat sederhana -- "<i>Dorothy bertemu dengan Penyihir Jahat dari Barat di Munchkin Land</i>" dan "<i>Saudara perempuan Penyihir Jahat dibunuh di Munchkin Land</i>"—dapat disisipkan dalam kalimat ketiga, "<i>Dorothy menyiram Penyihir Jahat dengan seember air</i>", untuk mendapatkan kalimat rekursif: "<i>Dorothy, yang bertemu dengan Penyihir Jahat dari Barat di Munchkin Land tempat saudara perempuannya dibunuh, menyiramnya dengan seember air.</i>" </p><p>Ide bahwa rekursi adalah suatu properti esensi dari bahasa manusia (seperti yang Chomsky ajukan) dibantah oleh <a href="/wiki/Linguis" class="mw-redirect" title="Linguis">linguis</a> <a href="/w/index.php?title=Daniel_Everett&action=edit&redlink=1" class="new" title="Daniel Everett (halaman belum tersedia)">Daniel Everett</a> dalam karyanya <i>Cultural Constraint on Grammar and Cognition in Pirahã: Another Look at the Design Features of Human Language</i>, yang di sana dia berhipotesis bahwa faktor kultur membuat rekursi tidak dibutuhkan dalam perkembangan <a href="/w/index.php?title=Bahasa_Piraha&action=edit&redlink=1" class="new" title="Bahasa Piraha (halaman belum tersedia)">Bahasa Piraha</a>. Konsep ini, yang menantang ide Chomsky bahwa rekursi satu-satunya sifat yang membedakan komunikasi manusia dan hewan, sekarang sedang diperdebatkan. Andrew Nevins, David Pesetsky dan Cilene Rodrigues berdebat melawan proposal tersebut. <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Rekursi dalam linguistik membolehkan 'diskrit tak-terbatas' dengan menanamkan frasa dalam tipe frasa yang sama dalam suatu struktur hierarki. Tanpa rekursi, bahasa tidak memiliki 'diskrit tak-terbatas' dan tidak dapat menanamkan kalimat menjadi tak-terbatas (dengan suatu efek '<a href="/wiki/Boneka_Matryoshka" class="mw-redirect" title="Boneka Matryoshka">Sarang boneka Rusia</a>'). Everett membantah bahwa bahasa harus memiliki diskrit tak-terbatas, dan menegaskan bahwa bahasa Piraha—yang diklaimnya tidak memiliki rekursi—pada kenyataannya terbatas. Dia menyamakannya dengan permainan terbatas <a href="/wiki/Catur" title="Catur">catur</a>, yang memiliki sejumlah pergerakan terbatas tetapi sangat produktif, dengan gerakan-gerakan baru diciptakan lewat sejarah. </p> <div class="mw-heading mw-heading3"><h3 id="Humor_rekursif">Humor rekursif</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=4" title="Sunting bagian: Humor rekursif" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=4" title="Sunting kode sumber bagian: Humor rekursif"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rekursi terkadang digunakan secara humor dalam buku ilmu komputer, pemrograman, filsafat, atau matematika. Adalah hal yang biasa bagi buku-buku tersebut untuk memasukan lelucon dalam <a href="/wiki/Glosarium" title="Glosarium">glosarium</a>-nya di antara barisan: </p> <dl><dd>Rekursi, <i>lihat Rekursi</i>.<sup id="cite_ref-Hunter_2-0" class="reference"><a href="#cite_note-Hunter-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Sebuah variasi ditemukan di halaman 269 dalam <a href="/w/index.php?title=Belakang-buku_indeks&action=edit&redlink=1" class="new" title="Belakang-buku indeks (halaman belum tersedia)">indeks</a> dari beberapa edisi buku Kernighan dan Ritchie <i><a href="/w/index.php?title=The_C_Programming_Language_(book)&action=edit&redlink=1" class="new" title="The C Programming Language (book) (halaman belum tersedia)">The C Programming Language</a></i>; isi indeks secara rekursif mengacu kepada dirinya sendiri ("rekursi 86, 139, 141, 182, 202, 269"). Versi awal dari lelucon ini ada di dalam "Software Tools" oleh Kernighan dan Plauger, dan juga muncul dalam "The UNIX Programming Environment" oleh Kernighan dan Pike. Ia tidak ada di edisi pertama dari <i>The C Programming Language</i>. </p><p>Lelucon yang lain adalah "Untuk memahami rekursi, anda harus memahami rekursi."<sup id="cite_ref-Hunter_2-1" class="reference"><a href="#cite_note-Hunter-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Dalam versi bahasa Inggris dari mesin pencari web <a href="/wiki/Google" title="Google">Google</a>, saat mencari kata untuk rekursi dalam bahasa Inggris "recursion" dilakukan, situs tersebut menyarankan "Did you mean: <i>recursion</i>." </p><p><a href="/wiki/Akronim_berulang" class="mw-redirect" title="Akronim berulang">Akronim berulang</a> juga dapat sebagai contoh dari humor rekursif. <a href="/wiki/PHP" title="PHP">PHP</a>, contohnya, singkatan dari "PHP Hypertext Preprocessor", <a href="/wiki/Wine_(perangkat_lunak)" title="Wine (perangkat lunak)">WINE</a> singkatan dari "Wine Is Not an Emulator", <a href="/wiki/Proyek_GNU" title="Proyek GNU">GNU</a> singkatan dari "GNU's not Unix". </p> <div class="mw-heading mw-heading2"><h2 id="Rekursi_dalam_matematika">Rekursi dalam matematika</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=5" title="Sunting bagian: Rekursi dalam matematika" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=5" title="Sunting kode sumber bagian: Rekursi dalam matematika"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Sierpinski_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Sierpinski_triangle.svg/250px-Sierpinski_triangle.svg.png" decoding="async" width="250" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Sierpinski_triangle.svg/375px-Sierpinski_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Sierpinski_triangle.svg/500px-Sierpinski_triangle.svg.png 2x" data-file-width="1024" data-file-height="887" /></a><figcaption><a href="/w/index.php?title=Segitiga_Sierpinski&action=edit&redlink=1" class="new" title="Segitiga Sierpinski (halaman belum tersedia)">Segitiga Sierpinski</a>—sebuah rekursi terbatas dari segitiga membentuk suatu <a href="/w/index.php?title=Jeruji&action=edit&redlink=1" class="new" title="Jeruji (halaman belum tersedia)">jeruji</a> geometris.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Himpunan_yang_didefinisikan_secara_rekursif">Himpunan yang didefinisikan secara rekursif</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=6" title="Sunting bagian: Himpunan yang didefinisikan secara rekursif" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=6" title="Sunting kode sumber bagian: Himpunan yang didefinisikan secara rekursif"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Definisi_rekursif&action=edit&redlink=1" class="new" title="Definisi rekursif (halaman belum tersedia)">Definisi rekursif</a></div> <div class="mw-heading mw-heading4"><h4 id="Contoh:_bilangan_asli">Contoh: bilangan asli</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=7" title="Sunting bagian: Contoh: bilangan asli" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=7" title="Sunting kode sumber bagian: Contoh: bilangan asli"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Lihat pula: <a href="/w/index.php?title=Closure_(matematika)&action=edit&redlink=1" class="new" title="Closure (matematika) (halaman belum tersedia)">Closure (matematika)</a></div> <p>Contoh kanonikal dari himpunan yang didefinisikan secara rekursif yaitu diberikan oleh <a href="/wiki/Bilangan_asli" title="Bilangan asli">bilangan asli</a>: </p> <dl><dd>0 ada dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></dd> <dd>jika <i>n</i> ada dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>, maka <i>n</i> + 1 ada dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></dd> <dd>Himpunan dari bilangan asli adalah himpunan terkecil yang memenuhi dua properti sebelumnya.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Contoh:_himpunan_dari_proposisi_benar_terjangkau">Contoh: himpunan dari proposisi benar terjangkau</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=8" title="Sunting bagian: Contoh: himpunan dari proposisi benar terjangkau" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=8" title="Sunting kode sumber bagian: Contoh: himpunan dari proposisi benar terjangkau"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Contoh menarik lainnya adalah himpunan dari semua proposisi "benar terjangkau" dalam suatu <a href="/wiki/Sistem_aksioma" title="Sistem aksioma">sistem aksioma</a>. </p> <ul><li>Jika suatu proposisi adalah sebuah aksioma, maka ia adalah suatu proposisi benar terjangkau.</li> <li>Jika suatu proposisi dapat dihasilkan dari proposisi benar terjangkau dengan menggunakan aturan-aturan inferensi, maka ia adalah proposisi benar terjangkau.</li> <li>Himpunan dari proposisi benar-terjangkau adalah himpunan terkecil dari proposisi yang memenuhi kondisi tersebut.</li></ul> <p>Himpunan ini disebut 'proposisi benar terjangkau' karena dalam pendekatan non-konstruktif terhadap fondasi matematika, himpunan dari proposisi benar bisa lebih besar daripada himpunan yang dibangun secara rekursif dari aksioma-aksioma dan aturan-aturan inferensi. Lihat juga <a href="/w/index.php?title=Teorema_ketaklengkapan_Godel&action=edit&redlink=1" class="new" title="Teorema ketaklengkapan Godel (halaman belum tersedia)">Teorema ketaklengkapan Godel</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Rekursi_fungsional">Rekursi fungsional</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=9" title="Sunting bagian: Rekursi fungsional" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=9" title="Sunting kode sumber bagian: Rekursi fungsional"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sebuah <a href="/wiki/Fungsi_(matematika)" title="Fungsi (matematika)">fungsi</a> bisa didefinisikan sebagai bagian dari dirinya sendiri. Contoh yang terkenal adalah urutan <a href="/wiki/Bilangan_Fibonacci" class="mw-redirect" title="Bilangan Fibonacci">bilangan Fibonacci</a>: <i>F</i>(<i>n</i>) = <i>F</i>(<i>n</i> − 1) + <i>F</i>(<i>n</i> − 2). Supaya definisi tersebut dapat berguna, ia harus mengarah pada nilai yang terdefinisi secara tak-rekursif, dalam kasus ini <i>F</i>(0) = 0 dan <i>F</i>(1) = 1. </p><p>Fungsi rekursif terkenal yaitu <a href="/w/index.php?title=Fungsi_Ackermann&action=edit&redlink=1" class="new" title="Fungsi Ackermann (halaman belum tersedia)">fungsi Ackermann</a> yang mana—tidak seperti urutan Fibonacci—tidak dapat dengan mudah diekspresikan tanpa rekursi. </p> <div class="mw-heading mw-heading3"><h3 id="Pembuktian_yang_mengikutkan_definisi_rekursif">Pembuktian yang mengikutkan definisi rekursif</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=10" title="Sunting bagian: Pembuktian yang mengikutkan definisi rekursif" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=10" title="Sunting kode sumber bagian: Pembuktian yang mengikutkan definisi rekursif"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Menerapkan teknik standar dari <a href="/w/index.php?title=Pembuktian_dengan_kasus&action=edit&redlink=1" class="new" title="Pembuktian dengan kasus (halaman belum tersedia)">pembuktian dengan kasus</a> untuk mendefinisikan secara rekursif suatu himpunan atau fungsi, seperti bagian sebelumnya, menghasilkan <a href="/w/index.php?title=Induksi_struktural&action=edit&redlink=1" class="new" title="Induksi struktural (halaman belum tersedia)">induksi struktural</a>, generalisasi ampuh dari <a href="/wiki/Induksi_matematika" title="Induksi matematika">induksi matematika</a> secara luas digunakan untuk menurunkan pembuktian dalam <a href="/wiki/Logika_matematika" title="Logika matematika">logika matematika</a> dan <a href="/wiki/Ilmu_komputer" title="Ilmu komputer">ilmu komputer</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Optimisasi_rekursif">Optimisasi rekursif</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=11" title="Sunting bagian: Optimisasi rekursif" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=11" title="Sunting kode sumber bagian: Optimisasi rekursif"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Pemrograman_dinamis" title="Pemrograman dinamis">Pemrograman dinamis</a> adalah suatu pendekatan terhadap <a href="/wiki/Optimisasi_(matematika)" class="mw-redirect" title="Optimisasi (matematika)">optimisasi</a> yang menempatkan ulang suatu permasalahan multiperiode atau tahapan dalam bentuk rekursif. Kunci jawaban dari pemrograman dinamis adalah <a href="/w/index.php?title=Persamaan_Bellman&action=edit&redlink=1" class="new" title="Persamaan Bellman (halaman belum tersedia)">persamaan Bellman</a>, yang menuliskan nilai dari permasalahan optimisasi pada waktu awal (atau langkah awal) berkenaan dengan nilainya pada waktu kemudian (atau langkah selanjutnya). </p> <div class="mw-heading mw-heading2"><h2 id="Rekursi_dalam_ilmu_komputer">Rekursi dalam ilmu komputer</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=12" title="Sunting bagian: Rekursi dalam ilmu komputer" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=12" title="Sunting kode sumber bagian: Rekursi dalam ilmu komputer"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Rekursi_(ilmu_komputer)&action=edit&redlink=1" class="new" title="Rekursi (ilmu komputer) (halaman belum tersedia)">Rekursi (ilmu komputer)</a></div> <p>Metode umum dari penyederhanaan adalah dengan membagi suatu permasalah menjadi beberapa sub-permasalahan dengan tipe yang sama. Sebagai sebuah teknik dalam <a href="/wiki/Pemrograman_komputer" class="mw-redirect" title="Pemrograman komputer">pemrograman komputer</a>, hal ini disebut dengan <i><a href="/wiki/Divide_and_conquer" class="mw-redirect" title="Divide and conquer">divide and conquer</a></i> dan merupakan kunci dari perancangan berbagai algoritme penting. <i>Divide and conquer</i> menyediakan pendekatan atas-bawah dalam pemecahan masalah, dengan permasalahan diselesaikan dengan menyelesaikan instansi yang lebih kecil. Pendekatan sebaliknya yaitu <a href="/wiki/Pemrograman_dinamis" title="Pemrograman dinamis">pemrograman dinamis</a>. Pendekatan ini menyelesaikannya secara bawah-atas, dengan permasalahan diselesaikan dengan menyelesaikan instansi yang lebih besar, sampai ukuran yang diinginkan dicapai. </p><p>Contoh klasik dari rekursi adalah definisi dari fungsi <a href="/wiki/Faktorial" title="Faktorial">faktorial</a>, diberikan dalam kode C: </p> <div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre><span></span><span class="kt">unsigned</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">factorial</span><span class="p">(</span><span class="kt">unsigned</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span> <span class="p">{</span> <span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span> <span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="mi">-1</span><span class="p">);</span> <span class="w"> </span><span class="p">}</span> <span class="p">}</span> </pre></div> <p>Fungsi tersebut memanggil dirinya sendiri secara rekursif terhadap versi input yang lebih kecil (n-1) dan mengkalikan hasil dari pemanggilan rekursif dengan n, sampai pada <a href="/w/index.php?title=Kasus_dasar&action=edit&redlink=1" class="new" title="Kasus dasar (halaman belum tersedia)">kasus dasar</a>, sama analoginya dengan definisi matematika dari faktorial. </p><p>Rekursi dalam pemrograman komputer dicontohkan saat sebuah fungsi didefinisikan dalam bentuk sederhana, bahkan versi terkecil dari dirinya. Solusi dari permasalahan kemudian dirancang dengan menggabungkan solusi-solusi yang didapat dari versi sederhana dari permasalahan. Salah satu contoh aplikasi rekursi yaitu dalam <i><a href="/wiki/Parsing" title="Parsing">parsing</a></i> untuk bahasa pemrograman. Keuntungan utama dari rekursi adalah suatu himpunan tak-terbatas dari kalimat yang memungkinkan, perancangan atau data lainnya dapat didefinisikan, diurai atau dihasilkan dengan suatu program komputer yang terbatas. </p><p><a href="/wiki/Relasi_perulangan" title="Relasi perulangan">Relasi perulangan</a> adalah persamaan-persamaan untuk menentukan satu atau lebih urutan-urutan secara rekursif. Beberapa relasi perulangan tertentu dapat "diselesaikan" untuk mendapatkan definisi bukan-rekursif. </p><p>Penggunaan rekursi dalam suatu algoritme memiliki kelebihan dan kekurangan. Kelebihan utamanya adalah biasanya kesederhanaan. Kekurangan utamanya adalah terkadang algoritme tersebut membutuhkan memori yang sangat banyak jika kedalaman rekursi sangat besar. </p> <div class="mw-heading mw-heading2"><h2 id="Rekursi_dalam_Seni">Rekursi dalam Seni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=13" title="Sunting bagian: Rekursi dalam Seni" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=13" title="Sunting kode sumber bagian: Rekursi dalam Seni"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:First_matryoshka_museum_doll_open.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/First_matryoshka_museum_doll_open.jpg/220px-First_matryoshka_museum_doll_open.jpg" decoding="async" width="220" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/First_matryoshka_museum_doll_open.jpg/330px-First_matryoshka_museum_doll_open.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/First_matryoshka_museum_doll_open.jpg/440px-First_matryoshka_museum_doll_open.jpg 2x" data-file-width="500" data-file-height="352" /></a><figcaption>Boneka rekursif: kumpulan orisinil dari '<a href="/w/index.php?title=Matryoshka_doll&action=edit&redlink=1" class="new" title="Matryoshka doll (halaman belum tersedia)">Matryoshka dolls</a>' yang dibuat oleh <a href="/w/index.php?title=Vasily_Zvyozdochkin&action=edit&redlink=1" class="new" title="Vasily Zvyozdochkin (halaman belum tersedia)">Zvyozdochkin</a> dan <a href="/w/index.php?title=Sergey_Malyutin&action=edit&redlink=1" class="new" title="Sergey Malyutin (halaman belum tersedia)">Malyutin</a>, pada 1892</figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Polittico_stefaneschi,_verso.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Polittico_stefaneschi%2C_verso.jpg/220px-Polittico_stefaneschi%2C_verso.jpg" decoding="async" width="220" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Polittico_stefaneschi%2C_verso.jpg/330px-Polittico_stefaneschi%2C_verso.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Polittico_stefaneschi%2C_verso.jpg/440px-Polittico_stefaneschi%2C_verso.jpg 2x" data-file-width="600" data-file-height="529" /></a><figcaption> Wajah depan <a href="/w/index.php?title=Stefaneschi_Triptych&action=edit&redlink=1" class="new" title="Stefaneschi Triptych (halaman belum tersedia)">Stefaneschi Triptych</a> karya <a href="/wiki/Giotto" class="mw-redirect" title="Giotto">Giotto</a>, 1320, secara rekursif berisi gambar dirinya sendiri (dipegang oleh figur yang berlutut di panel tengah).</figcaption></figure> <p>Boneka Matryoshka adalah contoh artistik fisik dari konsep rekursif.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><br /> Rekursi telah digunakan dalam lukisan sejak <a href="/w/index.php?title=Stefaneschi_Triptych&action=edit&redlink=1" class="new" title="Stefaneschi Triptych (halaman belum tersedia)">Stefaneschi Triptych</a> karya <a href="/wiki/Giotto" class="mw-redirect" title="Giotto">Giotto</a>, yang dibuat pada tahun 1320. Panel tengahnya berisi figur Kardinal Stefaneschi yang sedang berlutut, mengangkat triptych itu sendiri sebagai persembahan.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Praktik ini secara umum dikenal sebagai efek Droste, sebuah contoh teknik Mise en abyme. </p><p>M. C. Escher's Print Gallery (1956) adalah sebuah cetakan yang menggambarkan sebuah kota yang terdistorsi yang berisi sebuah galeri yang secara berulang-ulang berisi gambar tersebut, dan seterusnya secara <i><span lang="la" dir="ltr">ad infinitum</span></i>(tak terhingga).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Teorema_rekursi">Teorema rekursi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=14" title="Sunting bagian: Teorema rekursi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=14" title="Sunting kode sumber bagian: Teorema rekursi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dalam <a href="/wiki/Teori_himpunan" title="Teori himpunan">teori himpunan</a>, ini adalah teorema yang menjamin bahwa fungsi yang terdefinisi secara rekursif itu ada. Diberikan suatu himpunan <i>X</i>, sebuah elemen <i>a</i> dari <i>X</i> dan sebuah fungsi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\rightarrow X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\rightarrow X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245918678040123f009e350000a38b663951fff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.79ex; height:2.509ex;" alt="{\displaystyle f:X\rightarrow X}"></span>, teorema menyatakan bahwa ada fungsi unik <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:\mathbb {N} \rightarrow X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:\mathbb {N} \rightarrow X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9b2e4d4c898d2319d2cea22660c03c406f9761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.95ex; height:2.176ex;" alt="{\displaystyle F:\mathbb {N} \rightarrow X}"></span> (dengan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span> menunjukkan himpunan dari bilangan asli termasuk nol) sehingga </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(0)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(0)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13d054d8df6f9f2770997b66344c4f390bae2d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.041ex; height:2.843ex;" alt="{\displaystyle F(0)=a}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(n+1)=f(F(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(n+1)=f(F(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/872fa88af5e4b3ca934ba4abee41be187e3d3f94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.079ex; height:2.843ex;" alt="{\displaystyle F(n+1)=f(F(n))}"></span></dd></dl> <p>untuk setiap bilangan asli <i>n</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Pembuktian_keunikan">Pembuktian keunikan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=15" title="Sunting bagian: Pembuktian keunikan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=15" title="Sunting kode sumber bagian: Pembuktian keunikan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ambil dua fungsi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:\mathbb {N} \rightarrow X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:\mathbb {N} \rightarrow X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9b2e4d4c898d2319d2cea22660c03c406f9761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.95ex; height:2.176ex;" alt="{\displaystyle F:\mathbb {N} \rightarrow X}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:\mathbb {N} \rightarrow X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G:\mathbb {N} \rightarrow X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ad6ce89f36f7242cda0de98f9645cf7f975b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.036ex; height:2.176ex;" alt="{\displaystyle G:\mathbb {N} \rightarrow X}"></span> sehingga: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(0)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(0)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13d054d8df6f9f2770997b66344c4f390bae2d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.041ex; height:2.843ex;" alt="{\displaystyle F(0)=a}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(0)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(0)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4144e1a9d47560e0b5930da11b91807bde69f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.127ex; height:2.843ex;" alt="{\displaystyle G(0)=a}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(n+1)=f(F(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(n+1)=f(F(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/872fa88af5e4b3ca934ba4abee41be187e3d3f94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.079ex; height:2.843ex;" alt="{\displaystyle F(n+1)=f(F(n))}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n+1)=f(G(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n+1)=f(G(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0c178891ef17320110f41eaeb21ab5307f996a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.251ex; height:2.843ex;" alt="{\displaystyle G(n+1)=f(G(n))}"></span></dd></dl> <p>dengan <i>a</i> adalah elemen dari <i>X</i>. </p><p>Ia dapat dibuktikan dengan <a href="/wiki/Induksi_matematika" title="Induksi matematika">induksi matematika</a> bahwa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(n)=G(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(n)=G(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/333fb070c367ee695f65924c6c2e6fb63d5edcb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.074ex; height:2.843ex;" alt="{\displaystyle F(n)=G(n)}"></span> untuk semua bilangan asli <i>n</i>: </p> <dl><dd><b>Kasus dasar</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(0)=a=G(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(0)=a=G(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2dbf2a4023102987e8a67d1405f7a47678d43e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.938ex; height:2.843ex;" alt="{\displaystyle F(0)=a=G(0)}"></span> sehingga persamaan memenuhi untuk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}"></span>.</dd></dl> <dl><dd><b>Langkah Induktif</b>: Misalkan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k)=G(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k)=G(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e36aed00803ca0ee521a5727da0926f78cd10c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.707ex; height:2.843ex;" alt="{\displaystyle F(k)=G(k)}"></span> untuk beberapa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }"></span>. Maka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k+1)=f(F(k))=f(G(k))=G(k+1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k+1)=f(F(k))=f(G(k))=G(k+1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ebed4c94ea428c026075006a3bf76e722cf54f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.341ex; height:2.843ex;" alt="{\displaystyle F(k+1)=f(F(k))=f(G(k))=G(k+1).}"></span> <dl><dd>Karenanya F(k) = G(k) menyiratkan F(k+1) = G(k+1).</dd></dl></dd></dl> <p>Dengan induksi, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(n)=G(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(n)=G(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/333fb070c367ee695f65924c6c2e6fb63d5edcb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.074ex; height:2.843ex;" alt="{\displaystyle F(n)=G(n)}"></span> untuk semua <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Contoh-contoh">Contoh-contoh</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=16" title="Sunting bagian: Contoh-contoh" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=16" title="Sunting kode sumber bagian: Contoh-contoh"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Beberapa relasi perulangan umum yaitu: </p> <div> <p class="mw-empty-elt"></p><table class="multicol" role="presentation" style="border-collapse: collapse; padding: 0; border: 0; background:transparent; width:100%;"> <tbody><tr> <td width="" align="left" valign="top" style="padding-left:;"> <ul><li><a href="/w/index.php?title=Rasio_Golden&action=edit&redlink=1" class="new" title="Rasio Golden (halaman belum tersedia)">Rasio Golden</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =1+(1/\phi )=1+(1/(1+(1/(1+1/...))))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =1+(1/\phi )=1+(1/(1+(1/(1+1/...))))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69d4690170b13ddecd687d239e7ccb8a8437eda6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.427ex; height:2.843ex;" alt="{\displaystyle \phi =1+(1/\phi )=1+(1/(1+(1/(1+1/...))))}"></span></li> <li><a href="/wiki/Faktorial" title="Faktorial">Faktorial</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!=n(n-1)!=n(n-1)\cdots 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!=n(n-1)!=n(n-1)\cdots 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e8856ae0afe9a15fe8ed2141e7f6a25bb6530c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.748ex; height:2.843ex;" alt="{\displaystyle n!=n(n-1)!=n(n-1)\cdots 1}"></span></li> <li><a href="/wiki/Bilangan_Fibonacci" class="mw-redirect" title="Bilangan Fibonacci">Bilangan Fibonacci</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=f(n-1)+f(n-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=f(n-1)+f(n-2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5292c0129a4ebd0f560bf6b1b3647dc5ac5eda6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.392ex; height:2.843ex;" alt="{\displaystyle f(n)=f(n-1)+f(n-2)}"></span></li> <li><a href="/w/index.php?title=Bilangan_Catalan&action=edit&redlink=1" class="new" title="Bilangan Catalan (halaman belum tersedia)">Bilangan Catalan</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f16d74f9b1af01a229f5b576167e4f1d7969c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.977ex; height:2.509ex;" alt="{\displaystyle C_{0}=1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n+1}=(4n+2)C_{n}/(n+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n+1}=(4n+2)C_{n}/(n+2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0927f514faab4d5636046f0371a7839081d1b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.698ex; height:2.843ex;" alt="{\displaystyle C_{n+1}=(4n+2)C_{n}/(n+2)}"></span></li> <li>Perhitungan <a href="/wiki/Suku_bunga" title="Suku bunga">suku bunga</a></li> <li><a href="/wiki/Menara_Hanoi" title="Menara Hanoi">Menara Hanoi</a></li> <li><a href="/w/index.php?title=Fungsi_Ackermann&action=edit&redlink=1" class="new" title="Fungsi Ackermann (halaman belum tersedia)">Fungsi Ackermann</a></li></ul> <p class="mw-empty-elt"></p> </td></tr></tbody></table></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografi">Bibliografi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=17" title="Sunting bagian: Bibliografi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=17" title="Sunting kode sumber bagian: Bibliografi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation journal"><a href="/wiki/Edsger_W._Dijkstra" class="mw-redirect" title="Edsger W. Dijkstra">Dijkstra, Edsger W.</a> (1960). "Recursive Programming". <i>Numerische Mathematik</i>. <b>2</b> (1): 312–318. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01386232">10.1007/BF01386232</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Numerische+Mathematik&rft.atitle=Recursive+Programming&rft.volume=2&rft.issue=1&rft.pages=312-318&rft.date=1960&rft_id=info%3Adoi%2F10.1007%2FBF01386232&rft.aulast=Dijkstra&rft.aufirst=Edsger+W.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Johnsonbaugh, Richard (2004). <i>Discrete Mathematics</i>. Prentice Hall. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-13-117686-2" title="Istimewa:Sumber buku/0-13-117686-2">0-13-117686-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics&rft.pub=Prentice+Hall&rft.date=2004&rft.isbn=0-13-117686-2&rft.au=Johnsonbaugh%2C+Richard&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Hofstadter, Douglas (1999). <i>Gödel, Escher, Bach: an Eternal Golden Braid</i>. Basic Books. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-465-02656-7" title="Istimewa:Sumber buku/0-465-02656-7">0-465-02656-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=G%C3%B6del%2C+Escher%2C+Bach%3A+an+Eternal+Golden+Braid&rft.pub=Basic+Books&rft.date=1999&rft.isbn=0-465-02656-7&rft.au=Hofstadter%2C+Douglas&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Shoenfield, Joseph R. (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/recursiontheory0000shoe"><i>Recursion Theory</i></a>. A K Peters Ltd. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/1-56881-149-7" title="Istimewa:Sumber buku/1-56881-149-7">1-56881-149-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recursion+Theory&rft.pub=A+K+Peters+Ltd&rft.date=2000&rft.isbn=1-56881-149-7&rft.au=Shoenfield%2C+Joseph+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Frecursiontheory0000shoe&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Causey, Robert L. (2001). <a rel="nofollow" class="external text" href="https://archive.org/details/logicsetsrecursi0000caus"><i>Logic, Sets, and Recursion</i></a>. Jones & Bartlett. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-7637-1695-2" title="Istimewa:Sumber buku/0-7637-1695-2">0-7637-1695-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic%2C+Sets%2C+and+Recursion&rft.pub=Jones+%26+Bartlett&rft.date=2001&rft.isbn=0-7637-1695-2&rft.au=Causey%2C+Robert+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flogicsetsrecursi0000caus&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Cori, Rene; Lascar, Daniel; Pelletier, Donald H. (2001). <i>Recursion Theory, Gödel's Theorems, Set Theory, Model Theory</i>. Oxford University Press. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-19-850050-5" title="Istimewa:Sumber buku/0-19-850050-5">0-19-850050-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recursion+Theory%2C+G%C3%B6del%27s+Theorems%2C+Set+Theory%2C+Model+Theory&rft.pub=Oxford+University+Press&rft.date=2001&rft.isbn=0-19-850050-5&rft.au=Cori%2C+Rene%3B+Lascar%2C+Daniel%3B+Pelletier%2C+Donald+H.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33; margin-left:0.3em">Pemeliharaan CS1: Banyak nama: authors list (<a href="/wiki/Kategori:Pemeliharaan_CS1:_Banyak_nama:_authors_list" title="Kategori:Pemeliharaan CS1: Banyak nama: authors list">link</a>) </span></li> <li><cite class="citation book">Barwise, Jon; Moss, Lawrence S. (1996). <i>Vicious Circles</i>. Stanford Univ Center for the Study of Language and Information. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-19-850050-5" title="Istimewa:Sumber buku/0-19-850050-5">0-19-850050-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vicious+Circles&rft.pub=Stanford+Univ+Center+for+the+Study+of+Language+and+Information&rft.date=1996&rft.isbn=0-19-850050-5&rft.au=Barwise%2C+Jon%3B+Moss%2C+Lawrence+S.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33; margin-left:0.3em">Pemeliharaan CS1: Banyak nama: authors list (<a href="/wiki/Kategori:Pemeliharaan_CS1:_Banyak_nama:_authors_list" title="Kategori:Pemeliharaan CS1: Banyak nama: authors list">link</a>) </span> - menjelaskan pengerjaan dari <a href="/w/index.php?title=Corecursion&action=edit&redlink=1" class="new" title="Corecursion (halaman belum tersedia)">corecursion</a>.</li> <li><cite class="citation book">Rosen, Kenneth H. (2002). <i>Discrete Mathematics and Its Applications</i>. McGraw-Hill College. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-07-293033-0" title="Istimewa:Sumber buku/0-07-293033-0">0-07-293033-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+and+Its+Applications&rft.pub=McGraw-Hill+College&rft.date=2002&rft.isbn=0-07-293033-0&rft.au=Rosen%2C+Kenneth+H.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Cormen, Thomas H., Charles E. Leiserson, Ronald L. Rivest, Clifford Stein (2001). <i>Introduction to Algorithms</i>. Mit Pr. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-262-03293-7" title="Istimewa:Sumber buku/0-262-03293-7">0-262-03293-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Algorithms&rft.pub=Mit+Pr&rft.date=2001&rft.isbn=0-262-03293-7&rft.au=Cormen%2C+Thomas+H.%2C+Charles+E.+Leiserson%2C+Ronald+L.+Rivest%2C+Clifford+Stein&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33; margin-left:0.3em">Pemeliharaan CS1: Banyak nama: authors list (<a href="/wiki/Kategori:Pemeliharaan_CS1:_Banyak_nama:_authors_list" title="Kategori:Pemeliharaan CS1: Banyak nama: authors list">link</a>) </span></li> <li><cite class="citation book">Kernighan, B.; Ritchie, D. (1988). <i>The C programming Language</i>. Prentice Hall. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-13-110362-8" title="Istimewa:Sumber buku/0-13-110362-8">0-13-110362-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+C+programming+Language&rft.pub=Prentice+Hall&rft.date=1988&rft.isbn=0-13-110362-8&rft.au=Kernighan%2C+B.%3B+Ritchie%2C+D.&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33; margin-left:0.3em">Pemeliharaan CS1: Banyak nama: authors list (<a href="/wiki/Kategori:Pemeliharaan_CS1:_Banyak_nama:_authors_list" title="Kategori:Pemeliharaan CS1: Banyak nama: authors list">link</a>) </span></li> <li><cite class="citation book">Stokey, Nancy,; Robert Lucas; Edward Prescott (1989). <i>Recursive Methods in Economic Dynamics</i>. Harvard University Press. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-674-75096-9" title="Istimewa:Sumber buku/0-674-75096-9">0-674-75096-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recursive+Methods+in+Economic+Dynamics&rft.pub=Harvard+University+Press&rft.date=1989&rft.isbn=0-674-75096-9&rft.au=Stokey%2C+Nancy%2C%3B+Robert+Lucas%3B+Edward+Prescott&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33; margin-left:0.3em">Pemeliharaan CS1: Banyak nama: authors list (<a href="/wiki/Kategori:Pemeliharaan_CS1:_Banyak_nama:_authors_list" title="Kategori:Pemeliharaan CS1: Banyak nama: authors list">link</a>) </span></li> <li><cite class="citation book">Hungerford (1980). <i>Algebra</i>. Springer. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-387-90518-1" title="Istimewa:Sumber buku/978-0-387-90518-1">978-0-387-90518-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pub=Springer&rft.date=1980&rft.isbn=978-0-387-90518-1&rft.au=Hungerford&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span>, bagian pertama dari teori himpunan.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Lihat_juga">Lihat juga</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=18" title="Sunting bagian: Lihat juga" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=18" title="Sunting kode sumber bagian: Lihat juga"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="-moz-column-count:3; column-count:3;"> <ul><li><a href="/w/index.php?title=Korekursi&action=edit&redlink=1" class="new" title="Korekursi (halaman belum tersedia)">Korekursi</a></li> <li><a href="/w/index.php?title=Rekursi_Course-of-values&action=edit&redlink=1" class="new" title="Rekursi Course-of-values (halaman belum tersedia)">Rekursi Course-of-values</a></li> <li><a href="/w/index.php?title=Ketakterbatasan_digital&action=edit&redlink=1" class="new" title="Ketakterbatasan digital (halaman belum tersedia)">Ketakterbatasan digital</a></li> <li><a href="/w/index.php?title=Kombinator_titik_tetap&action=edit&redlink=1" class="new" title="Kombinator titik tetap (halaman belum tersedia)">Kombinator titik tetap</a></li> <li><a href="/w/index.php?title=Perulangan_Takterbatas&action=edit&redlink=1" class="new" title="Perulangan Takterbatas (halaman belum tersedia)">Perulangan Takterbatas</a></li> <li><a href="/w/index.php?title=Infinitisme&action=edit&redlink=1" class="new" title="Infinitisme (halaman belum tersedia)">Infinitisme</a></li> <li><a href="/w/index.php?title=Fungsi_Iterasi&action=edit&redlink=1" class="new" title="Fungsi Iterasi (halaman belum tersedia)">Fungsi Iterasi</a></li> <li><a href="/w/index.php?title=Mise_en_abyme&action=edit&redlink=1" class="new" title="Mise en abyme (halaman belum tersedia)">Mise en abyme</a></li> <li><a href="/w/index.php?title=Reentrant_(subroutine)&action=edit&redlink=1" class="new" title="Reentrant (subroutine) (halaman belum tersedia)">Reentrant (subroutine)</a></li> <li><a href="/w/index.php?title=Referensi-diri&action=edit&redlink=1" class="new" title="Referensi-diri (halaman belum tersedia)">Referensi-diri</a></li> <li><a href="/w/index.php?title=Perulangan_ganjil&action=edit&redlink=1" class="new" title="Perulangan ganjil (halaman belum tersedia)">Perulangan ganjil</a></li> <li><a href="/w/index.php?title=Rekursi_buntut&action=edit&redlink=1" class="new" title="Rekursi buntut (halaman belum tersedia)">Rekursi buntut</a></li> <li><a href="/w/index.php?title=Formula_Tupper&action=edit&redlink=1" class="new" title="Formula Tupper (halaman belum tersedia)">Formula Tupper</a></li> <li><a href="/w/index.php?title=Turtles_all_the_way_down&action=edit&redlink=1" class="new" title="Turtles all the way down (halaman belum tersedia)">Turtles all the way down</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Referensi">Referensi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=19" title="Sunting bagian: Referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=19" title="Sunting kode sumber bagian: Referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation journal">Nevins, Andrew; Pesetsky, David; Rodrigues, Cilene (2009). <a rel="nofollow" class="external text" href="http://web.mit.edu/linguistics/people/faculty/pesetsky/Nevins_Pesetsky_Rodrigues_2_Evidence_and_Argumentation_Reply_to_Everett.pdf">"Evidence and argumentation: A reply to Everett (2009)"</a> <span style="font-size:85%;">(PDF)</span>. <i>Language</i>. <b>85</b> (3): 671–681. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1353%2Flan.0.0140">10.1353/lan.0.0140</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Language&rft.atitle=Evidence+and+argumentation%3A+A+reply+to+Everett+%282009%29&rft.volume=85&rft.issue=3&rft.pages=671-681&rft.date=2009&rft_id=info%3Adoi%2F10.1353%2Flan.0.0140&rft.aulast=Nevins&rft.aufirst=Andrew&rft.au=Pesetsky%2C+David&rft.au=Rodrigues%2C+Cilene&rft_id=http%3A%2F%2Fweb.mit.edu%2Flinguistics%2Fpeople%2Ffaculty%2Fpesetsky%2FNevins_Pesetsky_Rodrigues_2_Evidence_and_Argumentation_Reply_to_Everett.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Hunter-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hunter_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hunter_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <cite class="citation book">Hunter, David (2011). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=kuwhTxCVovQC&dq=recursion+joke&source=gbs_navlinks_s"><i>Essentials of Discrete Mathematics</i></a>. Jones and Bartlett. hlm. 494.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essentials+of+Discrete+Mathematics&rft.pages=494&rft.pub=Jones+and+Bartlett&rft.date=2011&rft.aulast=Hunter&rft.aufirst=David&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkuwhTxCVovQC%26dq%3Drecursion%2Bjoke%26source%3Dgbs_navlinks_s&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web">Tang, Daisy. <a rel="nofollow" class="external text" href="http://www.cpp.edu/~ftang/courses/CS240/lectures/recursion.htm">"Recursion"</a><span class="reference-accessdate">. Diakses tanggal <span class="nowrap">24 September</span> 2015</span>. <q>More examples of recursion: Russian Matryoshka dolls. Each doll is made of solid wood or is hollow and contains another Matryoshka doll inside it.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Recursion&rft.aulast=Tang&rft.aufirst=Daisy&rft_id=http%3A%2F%2Fwww.cpp.edu%2F~ftang%2Fcourses%2FCS240%2Flectures%2Frecursion.htm&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://mv.vatican.va/3_EN/pages/PIN/PIN_Sala02_03.html">"Giotto di Bondone and assistants: Stefaneschi triptych"</a>. The Vatican<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">16 September</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Giotto+di+Bondone+and+assistants%3A+Stefaneschi+triptych&rft.pub=The+Vatican&rft_id=http%3A%2F%2Fmv.vatican.va%2F3_EN%2Fpages%2FPIN%2FPIN_Sala02_03.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation book">Svozil, Karl (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gxBMDwAAQBAJ&pg=PA12"><i>Physical (A)Causality: Determinism, Randomness and Uncaused Events</i></a>. Springer. hlm. 12. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/9783319708157" title="Istimewa:Sumber buku/9783319708157">9783319708157</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+%28A%29Causality%3A+Determinism%2C+Randomness+and+Uncaused+Events&rft.pages=12&rft.pub=Springer&rft.date=2018&rft.isbn=9783319708157&rft.aulast=Svozil&rft.aufirst=Karl&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgxBMDwAAQBAJ%26pg%3DPA12&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation web">Cooper, Jonathan (5 September 2007). <a rel="nofollow" class="external text" href="https://unwrappingart.com/art/art-and-mathematics/">"Art and Mathematics"</a><span class="reference-accessdate">. Diakses tanggal <span class="nowrap">5 July</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Art+and+Mathematics&rft.date=2007-09-05&rft.aulast=Cooper&rft.aufirst=Jonathan&rft_id=https%3A%2F%2Funwrappingart.com%2Fart%2Fart-and-mathematics%2F&rfr_id=info%3Asid%2Fid.wikipedia.org%3ARekursi" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol> <div class="mw-heading mw-heading2"><h2 id="Pranala_luar">Pranala luar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rekursi&veaction=edit&section=20" title="Sunting bagian: Pranala luar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rekursi&action=edit&section=20" title="Sunting kode sumber bagian: Pranala luar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r23035139">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r23782729">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiktionary-logo-en.svg/37px-Wiktionary-logo-en.svg.png" decoding="async" width="37" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiktionary-logo-en.svg/55px-Wiktionary-logo-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiktionary-logo-en.svg/73px-Wiktionary-logo-en.svg.png 2x" data-file-width="1000" data-file-height="1089" /></span></span></div> <div class="side-box-text plainlist">Lihat entri <i><b><a href="https://en.wiktionary.org/wiki/recursion" class="extiw" title="wiktionary:recursion">recursion</a></b></i> atau <i><b><a href="https://en.wiktionary.org/wiki/recursivity" class="extiw" title="wiktionary:recursivity">recursivity</a></b></i> di kamus bebas Wiktionary.</div></div> </div> <ul><li><span style="font-size:0.95em; font-weight:bold; color:#777; cursor:help;" title="Bahasa Inggris" lang="Inggris">(Inggris)</span> <a rel="nofollow" class="external text" href="http://www.freenetpages.co.uk/hp/alan.gauld/tutrecur.htm">Recursion</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050206051223/http://www.freenetpages.co.uk/hp/alan.gauld/tutrecur.htm">Diarsipkan</a> 2005-02-06 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. - tutorial oleh Alan Gauld</li> <li><span style="font-size:0.95em; font-weight:bold; color:#777; cursor:help;" title="Bahasa Inggris" lang="Inggris">(Inggris)</span> <a rel="nofollow" class="external text" href="http://amitksaha.files.wordpress.com/2009/05/recursion-primer.pdf">A Primer on Recursion</a>- mengandung petunjuk untuk rekursi dalam Bahasa Formal, Linguistik, Matematika dan Ilmu Komputer</li> <li><span style="font-size:0.95em; font-weight:bold; color:#777; cursor:help;" title="Bahasa Inggris" lang="Inggris">(Inggris)</span> <a rel="nofollow" class="external text" href="http://research.swtch.com/2010/03/zip-files-all-way-down.html">Zip Files All The Way Down</a></li> <li><span style="font-size:0.95em; font-weight:bold; color:#777; cursor:help;" title="Bahasa Inggris" lang="Inggris">(Inggris)</span> <a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/psychlangsci/staff/linguistics-staff/nevins-publications/npr09b">Nevins, Andrew and David Pesetsky and Cilene Rodrigues. Evidence and Argumentation: A Reply to Everett (2009). 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href="/wiki/Templat:Fraktal" title="Templat:Fraktal"><abbr title="Lihat templat ini">l</abbr></a></li><li class="nv-bicara"><a href="/w/index.php?title=Pembicaraan_Templat:Fraktal&action=edit&redlink=1" class="new" title="Pembicaraan Templat:Fraktal (halaman belum tersedia)"><abbr title="Diskusikan templat ini">b</abbr></a></li><li class="nv-sunting"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Templat:Fraktal&action=edit"><abbr title="Sunting templat ini">s</abbr></a></li></ul></div><div id="Fraktal" style="font-size:114%;margin:0 4em"><a href="/wiki/Fraktal" title="Fraktal">Fraktal</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Karakteristik</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dimensi_fraktal" title="Dimensi fraktal">Dimensi fraktal</a> (<a href="/wiki/Dimensi_Hausdorff" title="Dimensi Hausdorff">Dimensi Hausdorff</a> & <a href="/w/index.php?title=Dimensi_penutup_Lebesgue&action=edit&redlink=1" class="new" title="Dimensi penutup Lebesgue (halaman belum tersedia)">Dimensi Topologis</a>)</li> <li><a href="/w/index.php?title=Kesamaan-diri&action=edit&redlink=1" class="new" title="Kesamaan-diri (halaman belum tersedia)">Kesamaan-diri</a></li> <li><a class="mw-selflink selflink">Rekursi</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Barnsley_fern" title="Barnsley fern"><img alt="Sebuah gambar daun pakis yang memperlihatkan kesamaan-diri affine." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Fractal_fern_explained.png/80px-Fractal_fern_explained.png" decoding="async" width="80" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Fractal_fern_explained.png/120px-Fractal_fern_explained.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Fractal_fern_explained.png/160px-Fractal_fern_explained.png 2x" data-file-width="333" data-file-height="465" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=Sistem_fungsi_iterasi&action=edit&redlink=1" class="new" title="Sistem fungsi iterasi (halaman belum tersedia)">Sistem fungsi iterasi</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Bunga_salju_Koch&action=edit&redlink=1" class="new" title="Bunga salju Koch (halaman belum tersedia)">Bunga salju Koch</a></li> <li><a href="/w/index.php?title=Himpunan_Cantor&action=edit&redlink=1" class="new" title="Himpunan Cantor (halaman belum tersedia)">Himpunan Cantor</a></li> <li><a href="/w/index.php?title=Permadani_Sierpinski&action=edit&redlink=1" class="new" title="Permadani Sierpinski (halaman belum tersedia)">Permadani Sierpinski</a></li> <li><a href="/w/index.php?title=Segitiga_Sierpinski&action=edit&redlink=1" class="new" title="Segitiga Sierpinski (halaman belum tersedia)">Segitiga Sierpinski</a></li> <li><a href="/w/index.php?title=Kurva_pengisi-ruang&action=edit&redlink=1" class="new" title="Kurva pengisi-ruang (halaman belum tersedia)">Kurva pengisi-ruang</a></li> <li><a href="/wiki/Kurva_naga" title="Kurva naga">Kurva naga</a></li> <li><a href="/w/index.php?title=Persegi-T_(fraktal)&action=edit&redlink=1" class="new" title="Persegi-T (fraktal) (halaman belum tersedia)">Persegi-T</a></li> <li><a href="/wiki/Spons_Menger" title="Spons Menger">Spons Menger</a></li> <li><a href="/w/index.php?title=Pakis_Barnsley&action=edit&redlink=1" class="new" title="Pakis Barnsley (halaman belum tersedia)">Pakis Barnsley</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=Penarik_ganjil&action=edit&redlink=1" class="new" title="Penarik ganjil (halaman belum tersedia)">Penarik ganjil</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Sistem_multifraktal&action=edit&redlink=1" class="new" title="Sistem multifraktal (halaman belum tersedia)">Sistem multifraktal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Sistem-L" class="mw-redirect" title="Sistem-L">sistem-L</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Kurva_pengisi-ruang&action=edit&redlink=1" class="new" title="Kurva pengisi-ruang (halaman belum tersedia)">Kurva pengisi-ruang</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fraktal diluar-waktu</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Himpunan_Mandelbrot" title="Himpunan Mandelbrot">Himpunan Mandelbrot</a></li> <li><a href="/wiki/Himpunan_Julia" title="Himpunan Julia">Himpunan Julia</a></li> <li><a href="/w/index.php?title=Fraktal_Burning_Ship&action=edit&redlink=1" class="new" title="Fraktal Burning Ship (halaman belum tersedia)">Fraktal Burning Ship</a></li> <li><a href="/w/index.php?title=Fraktal_Nova&action=edit&redlink=1" class="new" title="Fraktal Nova (halaman belum tersedia)">Fraktal Nova</a></li> <li><a href="/w/index.php?title=Fraktal_Lyapunov&action=edit&redlink=1" class="new" title="Fraktal Lyapunov (halaman belum tersedia)">Fraktal Lyapunov</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fraktal acak</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Penerbangan_L%C3%A9vy&action=edit&redlink=1" class="new" title="Penerbangan Lévy (halaman belum tersedia)">Penerbangan Lévy</a></li> <li><a href="/w/index.php?title=Teori_Percolation&action=edit&redlink=1" class="new" title="Teori Percolation (halaman belum tersedia)">Teori Percolation</a></li> <li><a href="/w/index.php?title=Perjalanan_menghindari-diri&action=edit&redlink=1" class="new" title="Perjalanan menghindari-diri (halaman belum tersedia)">Perjalanan menghindari-diri</a></li> <li><a href="/w/index.php?title=Lanskap_Fraktal&action=edit&redlink=1" class="new" title="Lanskap Fraktal (halaman belum tersedia)">Lanskap Fraktal</a></li> <li><a href="/w/index.php?title=Pergerakan_Brownian&action=edit&redlink=1" class="new" title="Pergerakan Brownian (halaman belum tersedia)">Pergerakan Brownian</a></li> <li><a href="/w/index.php?title=Pohon_Brownian&action=edit&redlink=1" class="new" title="Pohon Brownian (halaman belum tersedia)">Pohon Brownian</a></li> <li><a href="/w/index.php?title=Agregasi_Difusi-terbatas&action=edit&redlink=1" class="new" title="Agregasi Difusi-terbatas (halaman belum tersedia)">Agregasi Difusi-terbatas</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Orang</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a></li> <li><a href="/w/index.php?title=Gaston_Julia&action=edit&redlink=1" class="new" title="Gaston Julia (halaman belum tersedia)">Gaston Julia</a></li> <li><a href="/w/index.php?title=Paul_Pierre_L%C3%A9vy&action=edit&redlink=1" class="new" title="Paul Pierre Lévy (halaman belum tersedia)">Paul Pierre Lévy</a></li> <li><a href="/w/index.php?title=Aleksandr_Lyapunov&action=edit&redlink=1" class="new" title="Aleksandr Lyapunov (halaman belum tersedia)">Aleksandr Lyapunov</a></li> <li><a href="/wiki/Benoit_Mandelbrot" class="mw-redirect" title="Benoit Mandelbrot">Benoît Mandelbrot</a></li> <li><a href="/w/index.php?title=Lewis_Fry_Richardson&action=edit&redlink=1" class="new" title="Lewis Fry Richardson (halaman belum tersedia)">Lewis Fry Richardson</a></li> <li><a href="/w/index.php?title=Wac%C5%82aw_Sierpi%C5%84ski&action=edit&redlink=1" class="new" title="Wacław Sierpiński (halaman belum tersedia)">Wacław Sierpiński</a></li> <li><a href="/w/index.php?title=Helge_von_Koch&action=edit&redlink=1" class="new" title="Helge von Koch (halaman belum tersedia)">Helge von Koch</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lainnya</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Daftar_fraktal_oleh_dimensi_Hausdorff&action=edit&redlink=1" class="new" title="Daftar fraktal oleh dimensi Hausdorff (halaman belum tersedia)">Daftar fraktal oleh dimensi Hausdorff</a></li> <li>"<a href="/w/index.php?title=Berapakah_panjang_pantai_Britania%3F_Dimensi_Fraksional_dan_Statistik_Kesamaan-Diri&action=edit&redlink=1" class="new" title="Berapakah panjang pantai Britania? Dimensi Fraksional dan Statistik Kesamaan-Diri (halaman belum tersedia)">Berapakah panjang pantai Britania? Dimensi Fraksional dan Statistik Kesamaan-Diri</a>"</li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r25847331"></div><div role="navigation" class="navbox" aria-labelledby="Logika_matematika" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18590415"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-lihat"><a href="/wiki/Templat:Logika_matematika" title="Templat:Logika matematika"><abbr title="Lihat templat ini">l</abbr></a></li><li class="nv-bicara"><a href="/w/index.php?title=Pembicaraan_Templat:Logika_matematika&action=edit&redlink=1" class="new" 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