CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Computational complexity theory - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"f1ba9d1c-bb01-4f2a-87dc-3dec47ef190d","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Computational_complexity_theory","wgTitle":"Computational complexity theory","wgCurRevisionId":1273212307,"wgRevisionId":1273212307,"wgArticleId":7543,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1: long volume value","Articles with short description","Short description is different from Wikidata","Use mdy dates from September 2017","Commons category link from Wikidata","Computational complexity theory","Computational fields of study"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Computational_complexity_theory","wgRelevantArticleId":7543,"wgIsProbablyEditable":true, "wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":50000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q205084","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false, "wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.tablesorter.styles":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.tablesorter","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher", "ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cjquery.tablesorter.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.18"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Computational complexity theory - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Computational_complexity_theory"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Computational_complexity_theory&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 (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Computational_complexity_theory"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&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-Computational_complexity_theory rootpage-Computational_complexity_theory skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</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="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <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="Main menu" > <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">Main menu</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">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" 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="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </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/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</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="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search 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="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <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="Appearance"> <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="Appearance" > <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">Appearance</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="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</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=Special:CreateAccount&returnto=Computational+complexity+theory" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</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=Special:UserLogin&returnto=Computational+complexity+theory" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</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="Log in and more options" > <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="Personal tools" > <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">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <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="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Computational+complexity+theory" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Computational+complexity+theory" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</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"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</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/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</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="Site"> <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="Contents" 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">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</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">(Top)</div> </a> </li> <li id="toc-Computational_problems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Computational_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Computational problems</span> </div> </a> <button aria-controls="toc-Computational_problems-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>Toggle Computational problems subsection</span> </button> <ul id="toc-Computational_problems-sublist" class="vector-toc-list"> <li id="toc-Problem_instances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problem_instances"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Problem instances</span> </div> </a> <ul id="toc-Problem_instances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representing_problem_instances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representing_problem_instances"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Representing problem instances</span> </div> </a> <ul id="toc-Representing_problem_instances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decision_problems_as_formal_languages" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decision_problems_as_formal_languages"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Decision problems as formal languages</span> </div> </a> <ul id="toc-Decision_problems_as_formal_languages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Function_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Function_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Function problems</span> </div> </a> <ul id="toc-Function_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measuring_the_size_of_an_instance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measuring_the_size_of_an_instance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Measuring the size of an instance</span> </div> </a> <ul id="toc-Measuring_the_size_of_an_instance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Machine_models_and_complexity_measures" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Machine_models_and_complexity_measures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Machine models and complexity measures</span> </div> </a> <button aria-controls="toc-Machine_models_and_complexity_measures-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>Toggle Machine models and complexity measures subsection</span> </button> <ul id="toc-Machine_models_and_complexity_measures-sublist" class="vector-toc-list"> <li id="toc-Turing_machine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Turing_machine"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Turing machine</span> </div> </a> <ul id="toc-Turing_machine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_machine_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_machine_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Other machine models</span> </div> </a> <ul id="toc-Other_machine_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complexity_measures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complexity_measures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Complexity measures</span> </div> </a> <ul id="toc-Complexity_measures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Best,_worst_and_average_case_complexity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Best,_worst_and_average_case_complexity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Best, worst and average case complexity</span> </div> </a> <ul id="toc-Best,_worst_and_average_case_complexity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Upper_and_lower_bounds_on_the_complexity_of_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Upper_and_lower_bounds_on_the_complexity_of_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Upper and lower bounds on the complexity of problems</span> </div> </a> <ul id="toc-Upper_and_lower_bounds_on_the_complexity_of_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complexity_classes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Complexity_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Complexity classes</span> </div> </a> <button aria-controls="toc-Complexity_classes-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>Toggle Complexity classes subsection</span> </button> <ul id="toc-Complexity_classes-sublist" class="vector-toc-list"> <li id="toc-Defining_complexity_classes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Defining_complexity_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Defining complexity classes</span> </div> </a> <ul id="toc-Defining_complexity_classes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Important_complexity_classes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Important_complexity_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Important complexity classes</span> </div> </a> <ul id="toc-Important_complexity_classes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hierarchy_theorems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hierarchy_theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Hierarchy theorems</span> </div> </a> <ul id="toc-Hierarchy_theorems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reduction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Reduction</span> </div> </a> <ul id="toc-Reduction-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Important_open_problems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Important_open_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Important open problems</span> </div> </a> <button aria-controls="toc-Important_open_problems-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>Toggle Important open problems subsection</span> </button> <ul id="toc-Important_open_problems-sublist" class="vector-toc-list"> <li id="toc-P_versus_NP_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#P_versus_NP_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>P versus NP problem</span> </div> </a> <ul id="toc-P_versus_NP_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_in_NP_not_known_to_be_in_P_or_NP-complete" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_in_NP_not_known_to_be_in_P_or_NP-complete"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Problems in NP not known to be in P or NP-complete</span> </div> </a> <ul id="toc-Problems_in_NP_not_known_to_be_in_P_or_NP-complete-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Separations_between_other_complexity_classes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Separations_between_other_complexity_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Separations between other complexity classes</span> </div> </a> <ul id="toc-Separations_between_other_complexity_classes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Intractability" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Intractability"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Intractability</span> </div> </a> <ul id="toc-Intractability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuous_complexity_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Continuous_complexity_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Continuous complexity theory</span> </div> </a> <ul id="toc-Continuous_complexity_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Works_on_complexity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Works_on_complexity"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Works on complexity</span> </div> </a> <ul id="toc-Works_on_complexity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Textbooks"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Textbooks</span> </div> </a> <ul id="toc-Textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surveys" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surveys"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.3</span> <span>Surveys</span> </div> </a> <ul id="toc-Surveys-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <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="Toggle the table of contents" > <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">Toggle the table of contents</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">Computational complexity theory</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="Go to an article in another language. Available in 35 languages" > <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-35" 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">35 languages</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/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%AA%D8%B9%D9%82%D9%8A%D8%AF_%D8%A7%D9%84%D8%AD%D8%B3%D8%A7%D8%A8%D9%8A" title="نظرية التعقيد الحسابي – Arabic" lang="ar" hreflang="ar" data-title="نظرية التعقيد الحسابي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_la_complexid%C3%A1_computacional" title="Teoría de la complexidá computacional – Asturian" lang="ast" hreflang="ast" data-title="Teoría de la complexidá computacional" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Hesablama_m%C3%BCr%C9%99kk%C9%99bliyi_n%C9%99z%C9%99riyy%C9%99si" title="Hesablama mürəkkəbliyi nəzəriyyəsi – Azerbaijani" lang="az" hreflang="az" data-title="Hesablama mürəkkəbliyi nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%B0%E0%A6%BF%E0%A6%97%E0%A6%A3%E0%A6%A8%E0%A6%BE%E0%A6%AE%E0%A7%82%E0%A6%B2%E0%A6%95_%E0%A6%9C%E0%A6%9F%E0%A6%BF%E0%A6%B2%E0%A6%A4%E0%A6%BE_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="পরিগণনামূলক জটিলতা তত্ত্ব – Bangla" lang="bn" hreflang="bn" data-title="পরিগণনামূলক জটিলতা তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%B8%D0%B7%D1%87%D0%B8%D1%81%D0%BB%D0%B8%D1%82%D0%B5%D0%BB%D0%BD%D0%B0%D1%82%D0%B0_%D1%81%D0%BB%D0%BE%D0%B6%D0%BD%D0%BE%D1%81%D1%82" title="Теория на изчислителната сложност – Bulgarian" lang="bg" hreflang="bg" data-title="Теория на изчислителната сложност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_la_complexitat_computacional" title="Teoria de la complexitat computacional – Catalan" lang="ca" hreflang="ca" data-title="Teoria de la complexitat computacional" data-language-autonym="Català" data-language-local-name="Catalan" 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/Teorie_slo%C5%BEitosti" title="Teorie složitosti – Czech" lang="cs" hreflang="cs" data-title="Teorie složitosti" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://de.wikipedia.org/wiki/Komplexit%C3%A4tstheorie" title="Komplexitätstheorie – German" lang="de" hreflang="de" data-title="Komplexitätstheorie" data-language-autonym="Deutsch" data-language-local-name="German" 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%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CF%80%CE%BF%CE%BB%CF%85%CF%80%CE%BB%CE%BF%CE%BA%CF%8C%CF%84%CE%B7%CF%84%CE%B1%CF%82" title="Θεωρία πολυπλοκότητας – Greek" lang="el" hreflang="el" data-title="Θεωρία πολυπλοκότητας" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_la_complejidad_computacional" title="Teoría de la complejidad computacional – Spanish" lang="es" hreflang="es" data-title="Teoría de la complejidad computacional" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Konplexutasun_konputazionalaren_teoria" title="Konplexutasun konputazionalaren teoria – Basque" lang="eu" hreflang="eu" data-title="Konplexutasun konputazionalaren teoria" 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/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D9%BE%DB%8C%DA%86%DB%8C%D8%AF%DA%AF%DB%8C_%D9%85%D8%AD%D8%A7%D8%B3%D8%A8%D8%A7%D8%AA%DB%8C" title="نظریه پیچیدگی محاسباتی – Persian" lang="fa" hreflang="fa" data-title="نظریه پیچیدگی محاسباتی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_de_la_complexit%C3%A9_(informatique_th%C3%A9orique)" title="Théorie de la complexité (informatique théorique) – French" lang="fr" hreflang="fr" data-title="Théorie de la complexité (informatique théorique)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%84%EC%82%B0_%EB%B3%B5%EC%9E%A1%EB%8F%84_%EC%9D%B4%EB%A1%A0" title="계산 복잡도 이론 – Korean" lang="ko" hreflang="ko" data-title="계산 복잡도 이론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ra%C4%8Dunska_teorija_slo%C5%BEenosti" title="Računska teorija složenosti – Croatian" lang="hr" hreflang="hr" data-title="Računska teorija složenosti" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_della_complessit%C3%A0_computazionale" title="Teoria della complessità computazionale – Italian" lang="it" hreflang="it" data-title="Teoria della complessità computazionale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A1%D7%99%D7%91%D7%95%D7%9B%D7%99%D7%95%D7%AA" title="תורת הסיבוכיות – Hebrew" lang="he" hreflang="he" data-title="תורת הסיבוכיות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_kekompleksan_pengiraan" title="Teori kekompleksan pengiraan – Malay" lang="ms" hreflang="ms" data-title="Teori kekompleksan pengiraan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Computationele_complexiteitstheorie" title="Computationele complexiteitstheorie – Dutch" lang="nl" hreflang="nl" data-title="Computationele complexiteitstheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A8%88%E7%AE%97%E8%A4%87%E9%9B%91%E6%80%A7%E7%90%86%E8%AB%96" title="計算複雑性理論 – Japanese" lang="ja" hreflang="ja" data-title="計算複雑性理論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kompleksitetsteori" title="Kompleksitetsteori – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kompleksitetsteori" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_complexit%C4%83%C8%9Bii" title="Teoria complexității – Romanian" lang="ro" hreflang="ro" data-title="Teoria complexității" data-language-autonym="Română" data-language-local-name="Romanian" 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%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D1%81%D0%BB%D0%BE%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D0%B8_%D0%B2%D1%8B%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%B8%D0%B9" title="Теория сложности вычислений – Russian" lang="ru" hreflang="ru" data-title="Теория сложности вычислений" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Computational_complexity_theory" title="Computational complexity theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Computational complexity theory" 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/Te%C3%B3ria_zlo%C5%BEitosti" title="Teória zložitosti – Slovak" lang="sk" hreflang="sk" data-title="Teória zložitosti" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Теорија комплексности – Serbian" lang="sr" hreflang="sr" data-title="Теорија комплексности" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ra%C4%8Dunska_teorija_slo%C5%BEenosti" title="Računska teorija složenosti – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Računska teorija složenosti" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Laskennallisen_kompleksisuuden_teoria" title="Laskennallisen kompleksisuuden teoria – Finnish" lang="fi" hreflang="fi" data-title="Laskennallisen kompleksisuuden teoria" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorya_ng_komputasyonal_na_komplehidad" title="Teorya ng komputasyonal na komplehidad – Tagalog" lang="tl" hreflang="tl" data-title="Teorya ng komputasyonal na komplehidad" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%8B%E0%B8%B1%E0%B8%9A%E0%B8%8B%E0%B9%89%E0%B8%AD%E0%B8%99%E0%B9%83%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%84%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%93" title="ทฤษฎีความซับซ้อนในการคำนวณ – Thai" lang="th" hreflang="th" data-title="ทฤษฎีความซับซ้อนในการคำนวณ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hesaplamal%C4%B1_karma%C5%9F%C4%B1kl%C4%B1k_teorisi" title="Hesaplamalı karmaşıklık teorisi – Turkish" lang="tr" hreflang="tr" data-title="Hesaplamalı karmaşıklık teorisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D1%81%D0%BA%D0%BB%D0%B0%D0%B4%D0%BD%D0%BE%D1%81%D1%82%D1%96_%D0%BE%D0%B1%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D1%8C" title="Теорія складності обчислень – Ukrainian" lang="uk" hreflang="uk" data-title="Теорія складності обчислень" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_%C4%91%E1%BB%99_ph%E1%BB%A9c_t%E1%BA%A1p_t%C3%ADnh_to%C3%A1n" title="Lý thuyết độ phức tạp tính toán – Vietnamese" lang="vi" hreflang="vi" data-title="Lý thuyết độ phức tạp tính toán" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%81%8B%E7%AE%97%E8%A4%87%E9%9B%9C%E5%BA%A6%E7%90%86%E8%AB%96" title="運算複雜度理論 – Cantonese" lang="yue" hreflang="yue" data-title="運算複雜度理論" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%A8%88%E7%AE%97%E8%A4%87%E9%9B%9C%E6%80%A7%E7%90%86%E8%AB%96" title="計算複雜性理論 – Chinese" lang="zh" hreflang="zh" data-title="計算複雜性理論" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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/Q205084#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> <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/Computational_complexity_theory" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Computational_complexity_theory" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</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="Change language variant" > <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">English</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="Views"> <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/Computational_complexity_theory"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Computational_complexity_theory&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Computational_complexity_theory&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <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="Tools" > <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">Tools</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">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </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/Computational_complexity_theory"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Computational_complexity_theory&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Computational_complexity_theory&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Computational_complexity_theory" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Computational_complexity_theory" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Computational_complexity_theory&oldid=1273212307" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Computational_complexity_theory&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Computational_complexity_theory&id=1273212307&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FComputational_complexity_theory"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FComputational_complexity_theory"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Computational_complexity_theory&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Computational_complexity_theory&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Computational_complexity_theory" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikiversity mw-list-item"><a href="https://en.wikiversity.org/wiki/Introduction_to_Complexity_Theory" hreflang="en"><span>Wikiversity</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q205084" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Inherent difficulty of computational problems</div> <p> In <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a> and mathematics, <b>computational complexity theory</b> focuses on classifying <a href="/wiki/Computational_problem" title="Computational problem">computational problems</a> according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a>. </p><p>A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical <a href="/wiki/Models_of_computation" class="mw-redirect" title="Models of computation">models of computation</a> to study these problems and quantifying their <a href="/wiki/Computational_complexity" title="Computational complexity">computational complexity</a>, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in <a href="/wiki/Communication_complexity" title="Communication complexity">communication complexity</a>), the number of <a href="/wiki/Logic_gate" title="Logic gate">gates</a> in a circuit (used in <a href="/wiki/Circuit_complexity" title="Circuit complexity">circuit complexity</a>) and the number of processors (used in <a href="/wiki/Parallel_computing" title="Parallel computing">parallel computing</a>). One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do. The <a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a>, one of the seven <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>,<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> is part of the field of computational complexity. </p><p>Closely related fields in <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a> are <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">analysis of algorithms</a> and <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>. A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem. More precisely, computational complexity theory tries to classify problems that can or cannot be solved with appropriately restricted resources. In turn, imposing restrictions on the available resources is what distinguishes computational complexity from computability theory: the latter theory asks what kinds of problems can, in principle, be solved algorithmically. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Computational_problems">Computational problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=1" title="Edit section: Computational problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TSP_Deutschland_3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c4/TSP_Deutschland_3.png" decoding="async" width="326" height="350" class="mw-file-element" data-file-width="326" data-file-height="350" /></a><figcaption>A traveling salesman tour through 14 German cities</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Problem_instances">Problem instances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=2" title="Edit section: Problem instances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Computational_problem" title="Computational problem">computational problem</a> can be viewed as an infinite collection of <i>instances</i> together with a set (possibly empty) of <i>solutions</i> for every instance. The input string for a computational problem is referred to as a problem instance, and should not be confused with the problem itself. In computational complexity theory, a problem refers to the abstract question to be solved. In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a decision problem. For example, consider the problem of <a href="/wiki/Primality_testing" class="mw-redirect" title="Primality testing">primality testing</a>. The instance is a number (e.g., 15) and the solution is "yes" if the number is prime and "no" otherwise (in this case, 15 is not prime and the answer is "no"). Stated another way, the <i>instance</i> is a particular input to the problem, and the <i>solution</i> is the output corresponding to the given input. </p><p>To further highlight the difference between a problem and an instance, consider the following instance of the decision version of the <a href="/wiki/Travelling_salesman_problem" title="Travelling salesman problem">travelling salesman problem</a>: Is there a route of at most 2000 kilometres passing through all of Germany's 15 largest cities? The quantitative answer to this particular problem instance is of little use for solving other instances of the problem, such as asking for a round trip through all sites in <a href="/wiki/Milan" title="Milan">Milan</a> whose total length is at most 10 km. For this reason, complexity theory addresses computational problems and not particular problem instances. </p> <div class="mw-heading mw-heading3"><h3 id="Representing_problem_instances">Representing problem instances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=3" title="Edit section: Representing problem instances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When considering computational problems, a problem instance is a <a href="/wiki/String_(computer_science)" title="String (computer science)">string</a> over an <a href="/wiki/Alphabet_(computer_science)" class="mw-redirect" title="Alphabet (computer science)">alphabet</a>. Usually, the alphabet is taken to be the binary alphabet (i.e., the set {0,1}), and thus the strings are <a href="/wiki/Bitstring" class="mw-redirect" title="Bitstring">bitstrings</a>. As in a real-world <a href="/wiki/Computer" title="Computer">computer</a>, mathematical objects other than bitstrings must be suitably encoded. For example, <a href="/wiki/Integer" title="Integer">integers</a> can be represented in <a href="/wiki/Binary_notation" class="mw-redirect" title="Binary notation">binary notation</a>, and <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> can be encoded directly via their <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrices</a>, or by encoding their <a href="/wiki/Adjacency_list" title="Adjacency list">adjacency lists</a> in binary. </p><p>Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently. </p> <div class="mw-heading mw-heading3"><h3 id="Decision_problems_as_formal_languages">Decision problems as formal languages</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=4" title="Edit section: Decision problems as formal languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Decision_Problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Decision_Problem.svg/220px-Decision_Problem.svg.png" decoding="async" width="220" height="274" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Decision_Problem.svg/330px-Decision_Problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Decision_Problem.svg/440px-Decision_Problem.svg.png 2x" data-file-width="247" data-file-height="308" /></a><figcaption>A <a href="/wiki/Decision_problem" title="Decision problem">decision problem</a> has only two possible outputs, <i>yes</i> or <i>no</i> (or alternately 1 or 0) on any input.</figcaption></figure> <p><a href="/wiki/Decision_problem" title="Decision problem">Decision problems</a> are one of the central objects of study in computational complexity theory. A decision problem is a type of computational problem where the answer is either <i>yes</i> or <i>no</i> (alternatively, 1 or 0). A decision problem can be viewed as a <a href="/wiki/Formal_language" title="Formal language">formal language</a>, where the members of the language are instances whose output is yes, and the non-members are those instances whose output is no. The objective is to decide, with the aid of an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a>, whether a given input string is a member of the formal language under consideration. If the algorithm deciding this problem returns the answer <i>yes</i>, the algorithm is said to accept the input string, otherwise it is said to reject the input. </p><p>An example of a decision problem is the following. The input is an arbitrary <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>. The problem consists in deciding whether the given graph is <a href="/wiki/Connectivity_(graph_theory)" title="Connectivity (graph theory)">connected</a> or not. The formal language associated with this decision problem is then the set of all connected graphs — to obtain a precise definition of this language, one has to decide how graphs are encoded as binary strings. </p> <div class="mw-heading mw-heading3"><h3 id="Function_problems">Function problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=5" title="Edit section: Function problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Function_problem" title="Function problem">function problem</a> is a computational problem where a single output (of a <a href="/wiki/Total_function" class="mw-redirect" title="Total function">total function</a>) is expected for every input, but the output is more complex than that of a <a href="/wiki/Decision_problem" title="Decision problem">decision problem</a>—that is, the output is not just yes or no. Notable examples include the <a href="/wiki/Traveling_salesman_problem" class="mw-redirect" title="Traveling salesman problem">traveling salesman problem</a> and the <a href="/wiki/Integer_factorization_problem" class="mw-redirect" title="Integer factorization problem">integer factorization problem</a>. </p><p>It is tempting to think that the notion of function problems is much richer than the notion of decision problems. However, this is not really the case, since function problems can be recast as decision problems. For example, the multiplication of two integers can be expressed as the set of triples <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 (a,b,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae973a762a92b9cd3eafe7f283890ccfa9b887e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}" /></span> such that the relation <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 a\times b=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times b=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1272c2667a4b31fc639ab3c998d30fb011f89b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.173ex; height:2.176ex;" alt="{\displaystyle a\times b=c}" /></span> holds. Deciding whether a given triple is a member of this set corresponds to solving the problem of multiplying two numbers. </p> <div class="mw-heading mw-heading3"><h3 id="Measuring_the_size_of_an_instance">Measuring the size of an instance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=6" title="Edit section: Measuring the size of an instance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To measure the difficulty of solving a computational problem, one may wish to see how much time the best algorithm requires to solve the problem. However, the running time may, in general, depend on the instance. In particular, larger instances will require more time to solve. Thus the time required to solve a problem (or the space required, or any measure of complexity) is calculated as a function of the size of the instance. The input size is typically measured in bits. Complexity theory studies how algorithms scale as input size increases. For instance, in the problem of finding whether a graph is connected, how much more time does it take to solve a problem for a graph with <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 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" /></span> vertices compared to the time taken for a graph with <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> vertices? </p><p>If the input size is <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>, the time taken can be expressed as a function of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>. Since the time taken on different inputs of the same size can be different, the worst-case time complexity <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 T(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be5a46684e1279c27414b285fa995f30407d002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.84ex; height:2.843ex;" alt="{\displaystyle T(n)}" /></span> is defined to be the maximum time taken over all inputs of size <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>. If <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 T(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be5a46684e1279c27414b285fa995f30407d002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.84ex; height:2.843ex;" alt="{\displaystyle T(n)}" /></span> is a polynomial in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>, then the algorithm is said to be a <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a> algorithm. <a href="/wiki/Cobham%27s_thesis" title="Cobham's thesis">Cobham's thesis</a> argues that a problem can be solved with a feasible amount of resources if it admits a polynomial-time algorithm. </p> <div class="mw-heading mw-heading2"><h2 id="Machine_models_and_complexity_measures">Machine models and complexity measures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=7" title="Edit section: Machine models and complexity measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Turing_machine">Turing machine</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=8" title="Edit section: Turing machine"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output 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}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Turing_machine_2b.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Turing_machine_2b.svg/220px-Turing_machine_2b.svg.png" decoding="async" width="220" height="52" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Turing_machine_2b.svg/330px-Turing_machine_2b.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Turing_machine_2b.svg/440px-Turing_machine_2b.svg.png 2x" data-file-width="550" data-file-height="130" /></a><figcaption>An illustration of a Turing machine</figcaption></figure> <p>A Turing machine is a mathematical model of a general computing machine. It is a theoretical device that manipulates symbols contained on a strip of tape. Turing machines are not intended as a practical computing technology, but rather as a general model of a computing machine—anything from an advanced supercomputer to a mathematician with a pencil and paper. It is believed that if a problem can be solved by an algorithm, there exists a Turing machine that solves the problem. Indeed, this is the statement of the <a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a>. Furthermore, it is known that everything that can be computed on other models of computation known to us today, such as a <a href="/wiki/RAM_machine" class="mw-redirect" title="RAM machine">RAM machine</a>, <a href="/wiki/Conway%27s_Game_of_Life" title="Conway's Game of Life">Conway's Game of Life</a>, <a href="/wiki/Cellular_automata" class="mw-redirect" title="Cellular automata">cellular automata</a>, <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a> or any programming language can be computed on a Turing machine. Since Turing machines are easy to analyze mathematically, and are believed to be as powerful as any other model of computation, the Turing machine is the most commonly used model in complexity theory. </p><p>Many types of Turing machines are used to define complexity classes, such as <a href="/wiki/Deterministic_Turing_machine" class="mw-redirect" title="Deterministic Turing machine">deterministic Turing machines</a>, <a href="/wiki/Probabilistic_Turing_machine" title="Probabilistic Turing machine">probabilistic Turing machines</a>, <a href="/wiki/Non-deterministic_Turing_machine" class="mw-redirect" title="Non-deterministic Turing machine">non-deterministic Turing machines</a>, <a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">quantum Turing machines</a>, <a href="/wiki/Symmetric_Turing_machine" title="Symmetric Turing machine">symmetric Turing machines</a> and <a href="/wiki/Alternating_Turing_machine" title="Alternating Turing machine">alternating Turing machines</a>. They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others. </p><p>A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called <a href="/wiki/Randomized_algorithm" title="Randomized algorithm">randomized algorithms</a>. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see <a href="/wiki/Non-deterministic_algorithm" class="mw-redirect" title="Non-deterministic algorithm">non-deterministic algorithm</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Other_machine_models">Other machine models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=9" title="Edit section: Other machine models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many machine models different from the standard <a href="/wiki/Turing_machine_equivalents#Multi-tape_Turing_machines" title="Turing machine equivalents">multi-tape Turing machines</a> have been proposed in the literature, for example <a href="/wiki/Random-access_machine" title="Random-access machine">random-access machines</a>. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> What all these models have in common is that the machines operate <a href="/wiki/Deterministic_algorithm" title="Deterministic algorithm">deterministically</a>. </p><p>However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that <a href="/wiki/Non-deterministic_time" class="mw-redirect" title="Non-deterministic time">non-deterministic time</a> is a very important resource in analyzing computational problems. </p> <div class="mw-heading mw-heading3"><h3 id="Complexity_measures">Complexity measures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=10" title="Edit section: Complexity measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the <a href="/wiki/Deterministic_Turing_machine" class="mw-redirect" title="Deterministic Turing machine">deterministic Turing machine</a> is used. The time required by a deterministic Turing machine <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> on input <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> is said to operate within time <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span> if the time required by <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /></span> on each input of length <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> is at most <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span>. A decision problem <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> can be solved in time <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span> if there exists a Turing machine operating in time <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span> that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span> on a deterministic Turing machine is then denoted by <a href="/wiki/DTIME" title="DTIME">DTIME</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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span>). </p><p>Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any <a href="/wiki/Complexity" title="Complexity">complexity measure</a> can be viewed as a computational resource. Complexity measures are very generally defined by the <a href="/wiki/Blum_complexity_axioms" class="mw-redirect" title="Blum complexity axioms">Blum complexity axioms</a>. Other complexity measures used in complexity theory include <a href="/wiki/Communication_complexity" title="Communication complexity">communication complexity</a>, <a href="/wiki/Circuit_complexity" title="Circuit complexity">circuit complexity</a>, and <a href="/wiki/Decision_tree_complexity" class="mw-redirect" title="Decision tree complexity">decision tree complexity</a>. </p><p>The complexity of an algorithm is often expressed using <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Best,_worst_and_average_case_complexity"><span id="Best.2C_worst_and_average_case_complexity"></span>Best, worst and average case complexity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=11" title="Edit section: Best, worst and average case complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sorting_quicksort_anim.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Sorting_quicksort_anim.gif/220px-Sorting_quicksort_anim.gif" decoding="async" width="220" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/6a/Sorting_quicksort_anim.gif 1.5x" data-file-width="280" data-file-height="214" /></a><figcaption>Visualization of the <a href="/wiki/Quicksort" title="Quicksort">quicksort</a> <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> that has <a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">average case performance</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 {\mathcal {O}}(n\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(n\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9981ede263cbf28215d3a70bf30f55db41a6e692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.195ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(n\log n)}" /></span></figcaption></figure> <p>The <a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">best, worst and average case</a> complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> may be faster to solve than others, we define the following complexities: </p> <ol><li>Best-case complexity: This is the complexity of solving the problem for the best input of size <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>.</li> <li>Average-case complexity: This is the complexity of solving the problem on an average. This complexity is only defined with respect to a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> over the inputs. For instance, if all inputs of the same size are assumed to be equally likely to appear, the average case complexity can be defined with respect to the uniform distribution over all inputs of size <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>.</li> <li><a href="/wiki/Amortized_analysis" title="Amortized analysis">Amortized analysis</a>: Amortized analysis considers both the costly and less costly operations together over the whole series of operations of the algorithm.</li> <li>Worst-case complexity: This is the complexity of solving the problem for the worst input of size <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>.</li></ol> <p>The order from cheap to costly is: Best, average (of <a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">discrete uniform distribution</a>), amortized, worst. </p><p>For example, the deterministic sorting algorithm <a href="/wiki/Quicksort" title="Quicksort">quicksort</a> addresses the problem of sorting a list of integers. The worst-case is when the pivot is always the largest or smallest value in the list (so the list is never divided). In this case, the algorithm takes time <a href="/wiki/Big_O_notation" title="Big O notation">O</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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}" /></span>). If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is <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 O(n\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2320768fb54880ca4356e61f60eb02a3f9d9f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="{\displaystyle O(n\log n)}" /></span>. The best case occurs when each pivoting divides the list in half, also needing <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 O(n\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2320768fb54880ca4356e61f60eb02a3f9d9f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="{\displaystyle O(n\log n)}" /></span> time. </p> <div class="mw-heading mw-heading3"><h3 id="Upper_and_lower_bounds_on_the_complexity_of_problems">Upper and lower bounds on the complexity of problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=12" title="Edit section: Upper and lower bounds on the complexity of problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To classify the computation time (or similar resources, such as space consumption), it is helpful to demonstrate upper and lower bounds on the maximum amount of time required by the most efficient algorithm to solve a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity unless specified otherwise. Analyzing a particular algorithm falls under the field of <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">analysis of algorithms</a>. To show an upper bound <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 T(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be5a46684e1279c27414b285fa995f30407d002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.84ex; height:2.843ex;" alt="{\displaystyle T(n)}" /></span> on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most <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 T(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be5a46684e1279c27414b285fa995f30407d002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.84ex; height:2.843ex;" alt="{\displaystyle T(n)}" /></span>. However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of <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 T(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be5a46684e1279c27414b285fa995f30407d002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.84ex; height:2.843ex;" alt="{\displaystyle T(n)}" /></span> for a problem requires showing that no algorithm can have time complexity lower than <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 T(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be5a46684e1279c27414b285fa995f30407d002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.84ex; height:2.843ex;" alt="{\displaystyle T(n)}" /></span>. </p><p>Upper and lower bounds are usually stated using the <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a>, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if <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 T(n)=7n^{2}+15n+40}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>7</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>15</mn> <mi>n</mi> <mo>+</mo> <mn>40</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)=7n^{2}+15n+40}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/290239c7be0f10b25e09862f2cbb6429ff503ce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.275ex; height:3.176ex;" alt="{\displaystyle T(n)=7n^{2}+15n+40}" /></span>, in big O notation one would write <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 T(n)\in O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)\in O(n^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc66b3cf4747029fad4ae4d3a3c565bcf22ed237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.712ex; height:3.176ex;" alt="{\displaystyle T(n)\in O(n^{2})}" /></span>. </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Complexity_classes">Complexity classes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=13" title="Edit section: Complexity classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Complexity_class" title="Complexity class">Complexity class</a></div> <div class="mw-heading mw-heading3"><h3 id="Defining_complexity_classes">Defining complexity classes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=14" title="Edit section: Defining complexity classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>complexity class</b> is a set of problems of related complexity. Simpler complexity classes are defined by the following factors: </p> <ul><li>The type of computational problem: The most commonly used problems are decision problems. However, complexity classes can be defined based on <a href="/wiki/Function_problem" title="Function problem">function problems</a>, <a href="/wiki/Counting_problem_(complexity)" title="Counting problem (complexity)">counting problems</a>, <a href="/wiki/Optimization_problem" title="Optimization problem">optimization problems</a>, <a href="/wiki/Promise_problem" title="Promise problem">promise problems</a>, etc.</li> <li>The model of computation: The most common model of computation is the deterministic Turing machine, but many complexity classes are based on non-deterministic Turing machines, <a href="/wiki/Boolean_circuit" title="Boolean circuit">Boolean circuits</a>, <a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">quantum Turing machines</a>, <a href="/wiki/Monotone_circuit" class="mw-redirect" title="Monotone circuit">monotone circuits</a>, etc.</li> <li>The resource (or resources) that is being bounded and the bound: These two properties are usually stated together, such as "polynomial time", "logarithmic space", "constant depth", etc.</li></ul> <p>Some complexity classes have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following: </p> <dl><dd>The set of decision problems solvable by a deterministic Turing machine within time <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span>. (This complexity class is known as DTIME(<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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span>).)</dd></dl> <p>But bounding the computation time above by some concrete function <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span> often yields complexity classes that depend on the chosen machine model. For instance, the language <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 \{xx\mid x{\text{ is any binary string}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mi>x</mi> <mo>∣</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is any binary string</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{xx\mid x{\text{ is any binary string}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18769151cf5f5bfcc9b996d4beebb23bcfc32dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.187ex; height:2.843ex;" alt="{\displaystyle \{xx\mid x{\text{ is any binary string}}\}}" /></span> can be solved in <a href="/wiki/Linear_time" class="mw-redirect" title="Linear time">linear time</a> on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, <a href="/wiki/Cobham%27s_thesis" title="Cobham's thesis">Cobham-Edmonds thesis</a> states that "the time complexities in any two reasonable and general models of computation are polynomially related" (<a href="#CITEREFGoldreich2008">Goldreich 2008</a>, Chapter 1.2). This forms the basis for the complexity class <a href="/wiki/P_(complexity)" title="P (complexity)">P</a>, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is <a href="/wiki/FP_(complexity)" title="FP (complexity)">FP</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Important_complexity_classes">Important complexity classes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=15" title="Edit section: Important complexity classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complexity_subsets_pspace.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Complexity_subsets_pspace.svg/220px-Complexity_subsets_pspace.svg.png" decoding="async" width="220" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Complexity_subsets_pspace.svg/330px-Complexity_subsets_pspace.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Complexity_subsets_pspace.svg/440px-Complexity_subsets_pspace.svg.png 2x" data-file-width="485" data-file-height="441" /></a><figcaption>A representation of the relation among complexity classes; L would be another step "inside" NL</figcaption></figure> <p>Many important complexity classes can be defined by bounding the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following: </p> <table class="wikitable sortable"> <tbody><tr> <th scope="col">Resource </th> <th scope="col"><abbr title="of Turing Machine used to Model Computation">Determinism</abbr> </th> <th scope="col">Complexity class </th> <th scope="col">Resource constraint </th></tr> <tr> <th scope="rowgroup" rowspan="8" style="text-align:center;">Space </th> <th scope="rowgroup" rowspan="4" style="text-align:center;">Non-Deterministic </th> <td><a href="/wiki/NSPACE" title="NSPACE">NSPACE</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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span>) </td> <td data-sort-value="0"><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 O(f(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</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 O(f(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/756b4d8648334719f65bf5e6269c7a2b3a502f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.065ex; height:2.843ex;" alt="{\displaystyle O(f(n))}" /></span> </td></tr> <tr> <td><a href="/wiki/NL_(complexity)" title="NL (complexity)">NL</a> </td> <td data-sort-value="1"><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 O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}" /></span> </td></tr> <tr> <td><a href="/wiki/NPSPACE" class="mw-redirect" title="NPSPACE">NPSPACE</a> </td> <td data-sort-value="2"><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 O({\text{poly}}(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>poly</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({\text{poly}}(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668c1881ecd020e37ec301b68377cf2b38dc3d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.116ex; height:2.843ex;" alt="{\displaystyle O({\text{poly}}(n))}" /></span> </td></tr> <tr> <td><a href="/wiki/NEXPSPACE" class="mw-redirect" title="NEXPSPACE">NEXPSPACE</a> </td> <td data-sort-value="4"><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 O(2^{{\text{poly}}(n)})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>poly</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(2^{{\text{poly}}(n)})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa41c21ee3cdbe36df6dc6ffed8a44ba63b5034f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.304ex; height:3.343ex;" alt="{\displaystyle O(2^{{\text{poly}}(n)})}" /></span> </td></tr> <tr> <th scope="rowgroup" rowspan="4" style="text-align:center;">Deterministic </th> <td><a href="/wiki/DSPACE" title="DSPACE">DSPACE</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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span>) </td> <td data-sort-value="0"><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 O(f(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</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 O(f(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/756b4d8648334719f65bf5e6269c7a2b3a502f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.065ex; height:2.843ex;" alt="{\displaystyle O(f(n))}" /></span> </td></tr> <tr> <td><a href="/wiki/L_(complexity)" title="L (complexity)">L</a> </td> <td data-sort-value="1"><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 O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}" /></span> </td></tr> <tr> <td><a href="/wiki/PSPACE" title="PSPACE">PSPACE</a> </td> <td data-sort-value="2"><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 O({\text{poly}}(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>poly</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({\text{poly}}(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668c1881ecd020e37ec301b68377cf2b38dc3d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.116ex; height:2.843ex;" alt="{\displaystyle O({\text{poly}}(n))}" /></span> </td></tr> <tr> <td><a href="/wiki/EXPSPACE" title="EXPSPACE">EXPSPACE</a> </td> <td data-sort-value="4"><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 O(2^{{\text{poly}}(n)})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>poly</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(2^{{\text{poly}}(n)})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa41c21ee3cdbe36df6dc6ffed8a44ba63b5034f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.304ex; height:3.343ex;" alt="{\displaystyle O(2^{{\text{poly}}(n)})}" /></span> </td></tr> <tr> <th scope="rowgroup" rowspan="6" style="text-align:center;">Time </th> <th scope="rowgroup" rowspan="3" style="text-align:center;">Non-Deterministic </th> <td><a href="/wiki/NTIME" title="NTIME">NTIME</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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span>) </td> <td data-sort-value="0"><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 O(f(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</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 O(f(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/756b4d8648334719f65bf5e6269c7a2b3a502f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.065ex; height:2.843ex;" alt="{\displaystyle O(f(n))}" /></span> </td></tr> <tr> <td><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a> </td> <td data-sort-value="2"><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 O({\text{poly}}(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>poly</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({\text{poly}}(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668c1881ecd020e37ec301b68377cf2b38dc3d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.116ex; height:2.843ex;" alt="{\displaystyle O({\text{poly}}(n))}" /></span> </td></tr> <tr> <td><a href="/wiki/NEXPTIME" title="NEXPTIME">NEXPTIME</a> </td> <td data-sort-value="4"><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 O(2^{{\text{poly}}(n)})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>poly</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(2^{{\text{poly}}(n)})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa41c21ee3cdbe36df6dc6ffed8a44ba63b5034f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.304ex; height:3.343ex;" alt="{\displaystyle O(2^{{\text{poly}}(n)})}" /></span> </td></tr> <tr> <th scope="rowgroup" rowspan="3" style="text-align:center;">Deterministic </th> <td><a href="/wiki/DTIME" title="DTIME">DTIME</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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" /></span>) </td> <td data-sort-value="0"><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 O(f(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</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 O(f(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/756b4d8648334719f65bf5e6269c7a2b3a502f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.065ex; height:2.843ex;" alt="{\displaystyle O(f(n))}" /></span> </td></tr> <tr> <td><a href="/wiki/P_(complexity)" title="P (complexity)">P</a> </td> <td data-sort-value="2"><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 O({\text{poly}}(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>poly</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({\text{poly}}(n))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668c1881ecd020e37ec301b68377cf2b38dc3d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.116ex; height:2.843ex;" alt="{\displaystyle O({\text{poly}}(n))}" /></span> </td></tr> <tr> <td><a href="/wiki/EXPTIME" title="EXPTIME">EXPTIME</a> </td> <td data-sort-value="4"><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 O(2^{{\text{poly}}(n)})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>poly</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(2^{{\text{poly}}(n)})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa41c21ee3cdbe36df6dc6ffed8a44ba63b5034f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.304ex; height:3.343ex;" alt="{\displaystyle O(2^{{\text{poly}}(n)})}" /></span> </td></tr></tbody></table> <p>Logarithmic-space classes do not account for the space required to represent the problem. </p><p>It turns out that PSPACE = NPSPACE and EXPSPACE = NEXPSPACE by <a href="/wiki/Savitch%27s_theorem" title="Savitch's theorem">Savitch's theorem</a>. </p><p>Other important complexity classes include <a href="/wiki/BPP_(complexity)" title="BPP (complexity)">BPP</a>, <a href="/wiki/ZPP_(complexity)" title="ZPP (complexity)">ZPP</a> and <a href="/wiki/RP_(complexity)" title="RP (complexity)">RP</a>, which are defined using <a href="/wiki/Probabilistic_Turing_machine" title="Probabilistic Turing machine">probabilistic Turing machines</a>; <a href="/wiki/AC_(complexity)" title="AC (complexity)">AC</a> and <a href="/wiki/NC_(complexity)" title="NC (complexity)">NC</a>, which are defined using Boolean circuits; and <a href="/wiki/BQP" title="BQP">BQP</a> and <a href="/wiki/QMA" title="QMA">QMA</a>, which are defined using quantum Turing machines. <a href="/wiki/Sharp-P" class="mw-redirect" title="Sharp-P">#P</a> is an important complexity class of counting problems (not decision problems). Classes like <a href="/wiki/IP_(complexity)" title="IP (complexity)">IP</a> and <a href="/wiki/AM_(complexity)" class="mw-redirect" title="AM (complexity)">AM</a> are defined using <a href="/wiki/Interactive_proof_system" title="Interactive proof system">Interactive proof systems</a>. <a href="/wiki/ALL_(complexity)" title="ALL (complexity)">ALL</a> is the class of all decision problems. </p> <div class="mw-heading mw-heading3"><h3 id="Hierarchy_theorems">Hierarchy theorems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=16" title="Edit section: Hierarchy theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Time_hierarchy_theorem" title="Time hierarchy theorem">time hierarchy theorem</a> and <a href="/wiki/Space_hierarchy_theorem" title="Space hierarchy theorem">space hierarchy theorem</a></div> <p>For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>) is contained in DTIME(<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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}" /></span>), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved. </p><p>More precisely, the <a href="/wiki/Time_hierarchy_theorem" title="Time hierarchy theorem">time hierarchy theorem</a> states that <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 {\mathsf {DTIME}}{\big (}o(f(n)){\big )}\subsetneq {\mathsf {DTIME}}{\big (}f(n)\cdot \log(f(n)){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>o</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>⊊</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">I</mi> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {DTIME}}{\big (}o(f(n)){\big )}\subsetneq {\mathsf {DTIME}}{\big (}f(n)\cdot \log(f(n)){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3132fd5c216316bd45568d19d23eb750c12a0c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.861ex; height:3.176ex;" alt="{\displaystyle {\mathsf {DTIME}}{\big (}o(f(n)){\big )}\subsetneq {\mathsf {DTIME}}{\big (}f(n)\cdot \log(f(n)){\big )}}" /></span>. </p><p>The <a href="/wiki/Space_hierarchy_theorem" title="Space hierarchy theorem">space hierarchy theorem</a> states that <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 {\mathsf {DSPACE}}{\big (}o(f(n)){\big )}\subsetneq {\mathsf {DSPACE}}{\big (}f(n){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>o</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>⊊</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">D</mi> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {DSPACE}}{\big (}o(f(n)){\big )}\subsetneq {\mathsf {DSPACE}}{\big (}f(n){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/988ed215d1eb420de36722d6440842177aefd288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.019ex; height:3.176ex;" alt="{\displaystyle {\mathsf {DSPACE}}{\big (}o(f(n)){\big )}\subsetneq {\mathsf {DSPACE}}{\big (}f(n){\big )}}" /></span>. </p><p>The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in PSPACE. </p> <div class="mw-heading mw-heading3"><h3 id="Reduction">Reduction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=17" title="Edit section: Reduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Reduction_(complexity)" title="Reduction (complexity)">Reduction (complexity)</a></div> <p>Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at most as difficult as another problem. For instance, if a problem <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> can be solved using an algorithm for <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> is no more difficult than <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></span>, and we say that <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> <i>reduces</i> to <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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></span>. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as <a href="/wiki/Polynomial-time_reduction" title="Polynomial-time reduction">polynomial-time reductions</a> or <a href="/wiki/Log-space_reduction" title="Log-space reduction">log-space reductions</a>. </p><p>The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication. </p><p>This motivates the concept of a problem being hard for a complexity class. A problem <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> is <i>hard</i> for a class of problems <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> if every problem in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> can be reduced to <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span>. Thus no problem in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> is harder than <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span>, since an algorithm for <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> allows us to solve any problem in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>. The notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of <a href="/wiki/NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a> problems. </p><p>If a problem <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> is in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> and hard for <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>, then <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> is said to be <i><a href="/wiki/Complete_(complexity)" title="Complete (complexity)">complete</a></i> for <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>. This means that <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> is the hardest problem in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>. (Since many problems could be equally hard, one might say that <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span> is one of the hardest problems in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>.) Thus the class of <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, <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 \Pi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10e0f32ab9da5560199913701cfdb210e7b32736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \Pi _{2}}" /></span>, to another problem, <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 \Pi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aab5a28da997de9084eef3e569bd1e072efc1aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \Pi _{1}}" /></span>, would indicate that there is no known polynomial-time solution for <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 \Pi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aab5a28da997de9084eef3e569bd1e072efc1aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \Pi _{1}}" /></span>. This is because a polynomial-time solution to <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 \Pi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aab5a28da997de9084eef3e569bd1e072efc1aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \Pi _{1}}" /></span> would yield a polynomial-time solution to <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 \Pi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10e0f32ab9da5560199913701cfdb210e7b32736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \Pi _{2}}" /></span>. Similarly, because all NP problems can be reduced to the set, finding an <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> problem that can be solved in polynomial time would mean that P = NP.<sup id="cite_ref-Sipser2006_3-0" class="reference"><a href="#cite_note-Sipser2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Important_open_problems">Important open problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=18" title="Edit section: Important open problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Complexity_classes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Complexity_classes.svg/220px-Complexity_classes.svg.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Complexity_classes.svg/330px-Complexity_classes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Complexity_classes.svg/440px-Complexity_classes.svg.png 2x" data-file-width="414" data-file-height="255" /></a><figcaption>Diagram of complexity classes provided that P ≠ NP. The existence of problems in NP outside both P and NP-complete in this case was established by Ladner.<sup id="cite_ref-Ladner75_4-0" class="reference"><a href="#cite_note-Ladner75-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="P_versus_NP_problem">P versus NP problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=19" title="Edit section: P versus NP problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></div> <p>The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the <a href="/wiki/Cobham%E2%80%93Edmonds_thesis" class="mw-redirect" title="Cobham–Edmonds thesis">Cobham–Edmonds thesis</a>. The complexity class <a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a>, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the <a href="/wiki/Boolean_satisfiability_problem" title="Boolean satisfiability problem">Boolean satisfiability problem</a>, the <a href="/wiki/Hamiltonian_path_problem" title="Hamiltonian path problem">Hamiltonian path problem</a> and the <a href="/wiki/Vertex_cover_problem" class="mw-redirect" title="Vertex cover problem">vertex cover problem</a>. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP. </p><p>The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution.<sup id="cite_ref-Sipser2006_3-1" class="reference"><a href="#cite_note-Sipser2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of <a href="/wiki/Integer_programming" title="Integer programming">integer programming</a> problems in <a href="/wiki/Operations_research" title="Operations research">operations research</a>, many problems in <a href="/wiki/Logistics" title="Logistics">logistics</a>, <a href="/wiki/Protein_structure_prediction" title="Protein structure prediction">protein structure prediction</a> in <a href="/wiki/Biology" title="Biology">biology</a>,<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> and the ability to find formal proofs of <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a> theorems.<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> The P versus NP problem is one of the <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a> proposed by the <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a>. There is a US$1,000,000 prize for resolving the problem.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Problems_in_NP_not_known_to_be_in_P_or_NP-complete">Problems in NP not known to be in P or NP-complete</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=20" title="Edit section: Problems in NP not known to be in P or NP-complete"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It was shown by Ladner that if <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 P\neq NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≠</mo> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\neq NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91730efe5e62816c7c61cdc6eafc8e77e924540e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.653ex; height:2.676ex;" alt="{\displaystyle P\neq NP}" /></span> then there exist problems in <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span> that are neither in <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> nor <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span>-complete.<sup id="cite_ref-Ladner75_4-1" class="reference"><a href="#cite_note-Ladner75-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Such problems are called <a href="/wiki/NP-intermediate" title="NP-intermediate">NP-intermediate</a> problems. The <a href="/wiki/Graph_isomorphism_problem" title="Graph isomorphism problem">graph isomorphism problem</a>, the <a href="/wiki/Discrete_logarithm_problem" class="mw-redirect" title="Discrete logarithm problem">discrete logarithm problem</a> and the <a href="/wiki/Integer_factorization_problem" class="mw-redirect" title="Integer factorization problem">integer factorization problem</a> are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> or to be <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span>-complete. </p><p>The <a href="/wiki/Graph_isomorphism_problem" title="Graph isomorphism problem">graph isomorphism problem</a> is the computational problem of determining whether two finite <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> are <a href="/wiki/Graph_isomorphism" title="Graph isomorphism">isomorphic</a>. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span>-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete.<sup id="cite_ref-AK06_8-0" class="reference"><a href="#cite_note-AK06-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> If graph isomorphism is NP-complete, the <a href="/wiki/Polynomial_time_hierarchy" class="mw-redirect" title="Polynomial time hierarchy">polynomial time hierarchy</a> collapses to its second level.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to <a href="/wiki/L%C3%A1szl%C3%B3_Babai" title="László Babai">László Babai</a> and <a href="/wiki/Eugene_Luks" class="mw-redirect" title="Eugene Luks">Eugene Luks</a> has run time <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 O(2^{\sqrt {n\log n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </msqrt> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(2^{\sqrt {n\log n}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4186036dc1b46116f1fbd2d2ca92d9c4bf3d040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.468ex; height:3.509ex;" alt="{\displaystyle O(2^{\sqrt {n\log n}})}" /></span> for graphs with <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> vertices, although some recent work by Babai offers some potentially new perspectives on this.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Integer_factorization_problem" class="mw-redirect" title="Integer factorization problem">integer factorization problem</a> is the computational problem of determining the <a href="/wiki/Prime_factorization" class="mw-redirect" title="Prime factorization">prime factorization</a> of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a prime factor less than <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span>. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the <a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a> algorithm. The integer factorization problem is in <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span> and in <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 co{\text{-}}NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> </mrow> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle co{\text{-}}NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c3aec36706d41c2817026fd07e1bda1b251093d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.718ex; height:2.176ex;" alt="{\displaystyle co{\text{-}}NP}" /></span> (and even in UP and co-UP<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>). If the problem is <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span>-complete, the polynomial time hierarchy will collapse to its first level (i.e., <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span> will equal <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 co{\text{-}}NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> </mrow> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle co{\text{-}}NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c3aec36706d41c2817026fd07e1bda1b251093d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.718ex; height:2.176ex;" alt="{\displaystyle co{\text{-}}NP}" /></span>). The best known algorithm for integer factorization is the <a href="/wiki/General_number_field_sieve" title="General number field sieve">general number field sieve</a>, which takes time <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 O(e^{\left({\sqrt[{3}]{\frac {64}{9}}}\right){\sqrt[{3}]{(\log n)}}{\sqrt[{3}]{(\log \log n)^{2}}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mn>64</mn> <mn>9</mn> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(e^{\left({\sqrt[{3}]{\frac {64}{9}}}\right){\sqrt[{3}]{(\log n)}}{\sqrt[{3}]{(\log \log n)^{2}}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d50912fbf223d0bbadd2a028262414397fbee1a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.246ex; height:5.343ex;" alt="{\displaystyle O(e^{\left({\sqrt[{3}]{\frac {64}{9}}}\right){\sqrt[{3}]{(\log n)}}{\sqrt[{3}]{(\log \log n)^{2}}}})}" /></span><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> to factor an odd integer <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>. However, the best known <a href="/wiki/Quantum_algorithm" title="Quantum algorithm">quantum algorithm</a> for this problem, <a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's algorithm</a>, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. </p> <div class="mw-heading mw-heading3"><h3 id="Separations_between_other_complexity_classes">Separations between other complexity classes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=21" title="Edit section: Separations between other complexity classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many known complexity classes are suspected to be unequal, but this has not been proved. For instance <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 P\subseteq NP\subseteq PP\subseteq PSPACE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>⊆</mo> <mi>N</mi> <mi>P</mi> <mo>⊆</mo> <mi>P</mi> <mi>P</mi> <mo>⊆</mo> <mi>P</mi> <mi>S</mi> <mi>P</mi> <mi>A</mi> <mi>C</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\subseteq NP\subseteq PP\subseteq PSPACE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0617bae24485207f43afe3a906af897f71d201f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.616ex; height:2.343ex;" alt="{\displaystyle P\subseteq NP\subseteq PP\subseteq PSPACE}" /></span>, but it is possible that <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 P=PSPACE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>P</mi> <mi>S</mi> <mi>P</mi> <mi>A</mi> <mi>C</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=PSPACE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34004fd89f5ee33f15ee2128dd397d0054a0b316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.119ex; height:2.176ex;" alt="{\displaystyle P=PSPACE}" /></span>. If <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> is not equal to <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span>, then <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> is not equal to <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 PSPACE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>S</mi> <mi>P</mi> <mi>A</mi> <mi>C</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PSPACE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39a7c3cf8fd055a634ae6bed85e7bf680c913ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.275ex; height:2.176ex;" alt="{\displaystyle PSPACE}" /></span> either. Since there are many known complexity classes between <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> and <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 PSPACE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>S</mi> <mi>P</mi> <mi>A</mi> <mi>C</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PSPACE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39a7c3cf8fd055a634ae6bed85e7bf680c913ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.275ex; height:2.176ex;" alt="{\displaystyle PSPACE}" /></span>, such as <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 RP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6ee2dec8276007eca18596d7bf5a7aeac64e7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.509ex; height:2.176ex;" alt="{\displaystyle RP}" /></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 BPP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>P</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BPP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b80e7fbd490e008c7b4c7b903d0f4e1059124401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.255ex; height:2.176ex;" alt="{\displaystyle BPP}" /></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 PP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811106f33a07c68d0569ab65da38994d96533099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.491ex; height:2.176ex;" alt="{\displaystyle PP}" /></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 BQP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>Q</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BQP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4911ab75d35efb67fb52e0908479b7ad94a322d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.348ex; height:2.509ex;" alt="{\displaystyle BQP}" /></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 MA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6250feb34ae98d9b710a8b069d15811718a9c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.185ex; height:2.176ex;" alt="{\displaystyle MA}" /></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 PH}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PH}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aeedfd422ea2359bafabc7c3dd1d5beff4b5833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle PH}" /></span>, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory. </p><p>Along the same lines, <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 co{\text{-}}NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> </mrow> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle co{\text{-}}NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c3aec36706d41c2817026fd07e1bda1b251093d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.718ex; height:2.176ex;" alt="{\displaystyle co{\text{-}}NP}" /></span> is the class containing the <a href="/wiki/Complement_(complexity)" title="Complement (complexity)">complement</a> problems (i.e. problems with the <i>yes</i>/<i>no</i> answers reversed) of <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span> problems. It is believed<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> that <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span> is not equal to <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 co{\text{-}}NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> </mrow> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle co{\text{-}}NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c3aec36706d41c2817026fd07e1bda1b251093d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.718ex; height:2.176ex;" alt="{\displaystyle co{\text{-}}NP}" /></span>; however, it has not yet been proven. It is clear that if these two complexity classes are not equal then <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> is not equal to <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span>, since <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 P=co{\text{-}}P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> </mrow> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=co{\text{-}}P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a838cc47bad6da7542476e48533edc5695a6bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.498ex; height:2.176ex;" alt="{\displaystyle P=co{\text{-}}P}" /></span>. Thus if <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 P=NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04c4c7e4303f3c01c40c865c6a1a09c4632bb7e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.653ex; height:2.176ex;" alt="{\displaystyle P=NP}" /></span> we would have <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 co{\text{-}}P=co{\text{-}}NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> </mrow> <mi>P</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> </mrow> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle co{\text{-}}P=co{\text{-}}NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2be5b0889f16c2e33584ac006c7577796f7af71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.471ex; height:2.176ex;" alt="{\displaystyle co{\text{-}}P=co{\text{-}}NP}" /></span> whence <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 NP=P=co{\text{-}}P=co{\text{-}}NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> <mo>=</mo> <mi>P</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> </mrow> <mi>P</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> </mrow> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP=P=co{\text{-}}P=co{\text{-}}NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9acdef3c4feeca7776bd8bb215707319a9f40b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.222ex; height:2.176ex;" alt="{\displaystyle NP=P=co{\text{-}}P=co{\text{-}}NP}" /></span>. </p><p>Similarly, it is not known if <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" /></span> (the set of all problems that can be solved in logarithmic space) is strictly contained in <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> or equal to <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span>. Again, there are many complexity classes between the two, such as <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 NL}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NL}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cac9c46b1af6aada8ab8f9799fddbba8fc451f90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.646ex; height:2.176ex;" alt="{\displaystyle NL}" /></span> and <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 NC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d01d5d050f3f5f55ffc77dc9a4bb8bb606d3abc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.83ex; height:2.176ex;" alt="{\displaystyle NC}" /></span>, and it is not known if they are distinct or equal classes. </p><p>It is suspected that <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> and <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 BPP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>P</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BPP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b80e7fbd490e008c7b4c7b903d0f4e1059124401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.255ex; height:2.176ex;" alt="{\displaystyle BPP}" /></span> are equal. However, it is currently open if <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 BPP=NEXP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>P</mi> <mi>P</mi> <mo>=</mo> <mi>N</mi> <mi>E</mi> <mi>X</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BPP=NEXP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc756f901663e753360ce8baabf116d5d97eb43d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.918ex; height:2.176ex;" alt="{\displaystyle BPP=NEXP}" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Intractability">Intractability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=22" title="Edit section: Intractability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Combinatorial_explosion" title="Combinatorial explosion">Combinatorial explosion</a></div> <p>A problem that can theoretically be solved, but requires impractical and finite resources (e.g., time) to do so, is known as an <i><b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="intractable_problem"></span><span class="vanchor-text">intractable problem</span></span></b></i>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Conversely, a problem that can be solved in practice is called a <i><b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509" /><span class="vanchor"><span id="tractable_problem"></span><span class="vanchor-text">tractable problem</span></span></b></i>, literally "a problem that can be handled". The term <i><a href="https://en.wiktionary.org/wiki/infeasible" class="extiw" title="wikt:infeasible">infeasible</a></i> (literally "cannot be done") is sometimes used interchangeably with <i><a href="https://en.wiktionary.org/wiki/intractable" class="extiw" title="wikt:intractable">intractable</a></i>,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> though this risks confusion with a <a href="/wiki/Feasible_solution" class="mw-redirect" title="Feasible solution">feasible solution</a> in <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Tractable problems are frequently identified with problems that have polynomial-time solutions (<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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></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 PTIME}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>T</mi> <mi>I</mi> <mi>M</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PTIME}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6269acfa6002633c5c4eff68e66cbd8e9863a1e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.771ex; height:2.176ex;" alt="{\displaystyle PTIME}" /></span>); this is known as the <a href="/wiki/Cobham%E2%80%93Edmonds_thesis" class="mw-redirect" title="Cobham–Edmonds thesis">Cobham–Edmonds thesis</a>. Problems that are known to be intractable in this sense include those that are <a href="/wiki/EXPTIME" title="EXPTIME">EXPTIME</a>-hard. If <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 NP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb96b0c7cdeba71278d234f3478d41964dfa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle NP}" /></span> is not the same as <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span>, then <a href="/wiki/NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a> problems are also intractable in this sense. </p><p>However, this identification is inexact: a polynomial-time solution with large degree or large leading coefficient grows quickly, and may be impractical for practical size problems; conversely, an exponential-time solution that grows slowly may be practical on realistic input, or a solution that takes a long time in the worst case may take a short time in most cases or the average case, and thus still be practical. Saying that a problem is not in <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in <a href="/wiki/Presburger_arithmetic" title="Presburger arithmetic">Presburger arithmetic</a> has been shown not to be in <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span>, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete <a href="/wiki/Knapsack_problem" title="Knapsack problem">knapsack problem</a> over a wide range of sizes in less than quadratic time and <a href="/wiki/SAT_solver" title="SAT solver">SAT solvers</a> routinely handle large instances of the NP-complete <a href="/wiki/Boolean_satisfiability_problem" title="Boolean satisfiability problem">Boolean satisfiability problem</a>. </p><p>To see why exponential-time algorithms are generally unusable in practice, consider a program that makes <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 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}" /></span> operations before halting. For small <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>, say 100, and assuming for the sake of example that the computer does <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 10^{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c90fc8874dafeb075a01985c74c1d31f51e7c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.201ex; height:2.676ex;" alt="{\displaystyle 10^{12}}" /></span> operations each second, the program would run for about <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 4\times 10^{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 10^{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5c76e9b40d690e1411fc8c799d63492a11678b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.204ex; height:2.676ex;" alt="{\displaystyle 4\times 10^{10}}" /></span> years, which is the same order of magnitude as the <a href="/wiki/Age_of_the_universe" title="Age of the universe">age of the universe</a>. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. However, an exponential-time algorithm that takes <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 1.0001^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1.0001</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.0001^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/586ed7ba1079e3822a1cb4252e24ea5d90092187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.678ex; height:2.343ex;" alt="{\displaystyle 1.0001^{n}}" /></span> operations is practical until <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> gets relatively large. </p><p>Similarly, a polynomial time algorithm is not always practical. If its running time is, say, <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^{15}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{15}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc2e63ff8edb996fc3e92714dd8f6a1f3561f6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.271ex; height:2.676ex;" alt="{\displaystyle n^{15}}" /></span>, it is unreasonable to consider it efficient and it is still useless except on small instances. Indeed, in practice even <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^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e9d1a52e455a7a5272a345b2697e35f1579b681" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{3}}" /></span> or <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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}" /></span> algorithms are often impractical on realistic sizes of problems. </p> <div class="mw-heading mw-heading2"><h2 id="Continuous_complexity_theory">Continuous complexity theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=23" title="Edit section: Continuous complexity theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Continuous complexity theory can refer to complexity theory of problems that involve continuous functions that are approximated by discretizations, as studied in <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>. One approach to complexity theory of numerical analysis<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> is <a href="/wiki/Information_based_complexity" class="mw-redirect" title="Information based complexity">information based complexity</a>. </p><p>Continuous complexity theory can also refer to complexity theory of the use of <a href="/wiki/Analog_computation" class="mw-redirect" title="Analog computation">analog computation</a>, which uses continuous <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a> and <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Control_theory" title="Control theory">Control theory</a> can be considered a form of computation and differential equations are used in the modelling of continuous-time and hybrid discrete-continuous-time systems.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=24" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An early example of algorithm complexity analysis is the running time analysis of the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> done by <a href="/wiki/Gabriel_Lam%C3%A9" title="Gabriel Lamé">Gabriel Lamé</a> in 1844. </p><p>Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> in 1936, which turned out to be a very robust and flexible simplification of a computer. </p><p>The beginning of systematic studies in computational complexity is attributed to the seminal 1965 paper "On the Computational Complexity of Algorithms" by <a href="/wiki/Juris_Hartmanis" title="Juris Hartmanis">Juris Hartmanis</a> and <a href="/wiki/Richard_E._Stearns" title="Richard E. Stearns">Richard E. Stearns</a>, which laid out the definitions of <a href="/wiki/Time_complexity" title="Time complexity">time complexity</a> and <a href="/wiki/Space_complexity" title="Space complexity">space complexity</a>, and proved the hierarchy theorems.<sup id="cite_ref-Fortnow_2003_20-0" class="reference"><a href="#cite_note-Fortnow_2003-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> In addition, in 1965 <a href="/wiki/Jack_Edmonds" title="Jack Edmonds">Edmonds</a> suggested to consider a "good" algorithm to be one with running time bounded by a polynomial of the input size.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Earlier papers studying problems solvable by Turing machines with specific bounded resources include<sup id="cite_ref-Fortnow_2003_20-1" class="reference"><a href="#cite_note-Fortnow_2003-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <a href="/wiki/John_Myhill" title="John Myhill">John Myhill</a>'s definition of <a href="/wiki/Linear_bounded_automata" class="mw-redirect" title="Linear bounded automata">linear bounded automata</a> (Myhill 1960), <a href="/wiki/Raymond_Smullyan" title="Raymond Smullyan">Raymond Smullyan</a>'s study of rudimentary sets (1961), as well as <a href="/wiki/Hisao_Yamada" title="Hisao Yamada">Hisao Yamada</a>'s paper<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> on real-time computations (1962). Somewhat earlier, <a href="/wiki/Boris_Trakhtenbrot" title="Boris Trakhtenbrot">Boris Trakhtenbrot</a> (1956), a pioneer in the field from the USSR, studied another specific complexity measure.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> As he remembers: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>However, [my] initial interest [in automata theory] was increasingly set aside in favor of computational complexity, an exciting fusion of combinatorial methods, inherited from <a href="/wiki/Switching_theory" class="mw-redirect" title="Switching theory">switching theory</a>, with the conceptual arsenal of the theory of algorithms. These ideas had occurred to me earlier in 1955 when I coined the term "signalizing function", which is nowadays commonly known as "complexity measure".<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>In 1967, <a href="/wiki/Manuel_Blum" title="Manuel Blum">Manuel Blum</a> formulated a set of <a href="/wiki/Axiom" title="Axiom">axioms</a> (now known as <a href="/wiki/Blum_axioms" title="Blum axioms">Blum axioms</a>) specifying desirable properties of complexity measures on the set of computable functions and proved an important result, the so-called <a href="/wiki/Blum%27s_speedup_theorem" title="Blum's speedup theorem">speed-up theorem</a>. The field began to flourish in 1971 when <a href="/wiki/Stephen_Cook" title="Stephen Cook">Stephen Cook</a> and <a href="/wiki/Leonid_Levin" title="Leonid Levin">Leonid Levin</a> <a href="/wiki/Cook%E2%80%93Levin_theorem" title="Cook–Levin theorem">proved</a> the existence of practically relevant problems that are <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a>. In 1972, <a href="/wiki/Richard_Karp" class="mw-redirect" title="Richard Karp">Richard Karp</a> took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse <a href="/wiki/Combinatorics" title="Combinatorics">combinatorial</a> and <a href="/wiki/Graph_theory" title="Graph theory">graph theoretical</a> problems, each infamous for its computational intractability, are NP-complete.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 25em;"> <ul><li><a href="/wiki/Computational_complexity" title="Computational complexity">Computational complexity</a></li> <li><a href="/wiki/Descriptive_complexity_theory" title="Descriptive complexity theory">Descriptive complexity theory</a></li> <li><a href="/wiki/Game_complexity" title="Game complexity">Game complexity</a></li> <li><a href="/wiki/Leaf_language" title="Leaf language">Leaf language</a></li> <li><a href="/wiki/Limits_of_computation" title="Limits of computation">Limits of computation</a></li> <li><a href="/wiki/List_of_complexity_classes" title="List of complexity classes">List of complexity classes</a></li> <li><a href="/wiki/List_of_computability_and_complexity_topics" title="List of computability and complexity topics">List of computability and complexity topics</a></li> <li><a href="/wiki/List_of_unsolved_problems_in_computer_science" title="List of unsolved problems in computer science">List of unsolved problems in computer science</a></li> <li><a href="/wiki/Parameterized_complexity" title="Parameterized complexity">Parameterized complexity</a></li> <li><a href="/wiki/Proof_complexity" title="Proof complexity">Proof complexity</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Structural_complexity_theory" title="Structural complexity theory">Structural complexity theory</a></li> <li><a href="/wiki/Transcomputational_problem" title="Transcomputational problem">Transcomputational problem</a></li> <li><a href="/wiki/Computational_complexity_of_mathematical_operations" title="Computational complexity of mathematical operations">Computational complexity of mathematical operations</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Works_on_complexity">Works on complexity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=26" title="Edit section: Works on complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWuppuluriDoria2020" class="citation cs2">Wuppuluri, Shyam; Doria, Francisco A., eds. (2020), <a rel="nofollow" class="external text" href="https://www.worldscientific.com/worldscibooks/10.1142/11270"><i>Unravelling Complexity: The Life and Work of Gregory Chaitin</i></a>, World Scientific, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F11270">10.1142/11270</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-12-0006-9" title="Special:BookSources/978-981-12-0006-9"><bdi>978-981-12-0006-9</bdi></a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:198790362">198790362</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unravelling+Complexity%3A+The+Life+and+Work+of+Gregory+Chaitin&rft.pub=World+Scientific&rft.date=2020&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A198790362%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2F11270&rft.isbn=978-981-12-0006-9&rft_id=https%3A%2F%2Fwww.worldscientific.com%2Fworldscibooks%2F10.1142%2F11270&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=27" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=28" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180706075006/http://www.claymath.org/millennium-problems/p-vs-np-problem">"P vs NP Problem | Clay Mathematics Institute"</a>. <i>www.claymath.org</i>. Archived from <a rel="nofollow" class="external text" href="http://www.claymath.org/millennium-problems/p-vs-np-problem">the original</a> on July 6, 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">July 6,</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.claymath.org&rft.atitle=P+vs+NP+Problem+%7C+Clay+Mathematics+Institute&rft_id=http%3A%2F%2Fwww.claymath.org%2Fmillennium-problems%2Fp-vs-np-problem&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">See <a href="#CITEREFAroraBarak2009">Arora & Barak 2009</a>, Chapter 1: The computational model and why it doesn't matter</span> </li> <li id="cite_note-Sipser2006-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Sipser2006_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Sipser2006_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">See <a href="#CITEREFSipser2006">Sipser 2006</a>, Chapter 7: Time complexity</span> </li> <li id="cite_note-Ladner75-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ladner75_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ladner75_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLadner1975" class="citation cs2">Ladner, Richard E. (1975), "On the structure of polynomial time reducibility", <i><a href="/wiki/Journal_of_the_ACM" title="Journal of the ACM">Journal of the ACM</a></i>, <b>22</b> (1): <span class="nowrap">151–</span>171, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F321864.321877">10.1145/321864.321877</a></span>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14352974">14352974</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+ACM&rft.atitle=On+the+structure+of+polynomial+time+reducibility&rft.volume=22&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E151-%3C%2Fspan%3E171&rft.date=1975&rft_id=info%3Adoi%2F10.1145%2F321864.321877&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14352974%23id-name%3DS2CID&rft.aulast=Ladner&rft.aufirst=Richard+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBergerLeighton1998" class="citation cs2"><a href="/wiki/Bonnie_Berger" title="Bonnie Berger">Berger, Bonnie A.</a>; <a href="/wiki/F._Thomson_Leighton" title="F. Thomson Leighton">Leighton, T</a> (1998), "Protein folding in the hydrophobic-hydrophilic (HP) model is NP-complete", <i>Journal of Computational Biology</i>, <b>5</b> (1): <span class="nowrap">27–</span>40, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.139.5547">10.1.1.139.5547</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1089%2Fcmb.1998.5.27">10.1089/cmb.1998.5.27</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9541869">9541869</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Biology&rft.atitle=Protein+folding+in+the+hydrophobic-hydrophilic+%28HP%29+model+is+NP-complete&rft.volume=5&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E27-%3C%2Fspan%3E40&rft.date=1998&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.139.5547%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F9541869&rft_id=info%3Adoi%2F10.1089%2Fcmb.1998.5.27&rft.aulast=Berger&rft.aufirst=Bonnie+A.&rft.au=Leighton%2C+T&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCook2000" class="citation cs2"><a href="/wiki/Stephen_Cook" title="Stephen Cook">Cook, Stephen</a> (April 2000), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101212035424/http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf"><i>The P versus NP Problem</i></a> <span class="cs1-format">(PDF)</span>, <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a>, archived from <a rel="nofollow" class="external text" href="http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf">the original</a> <span class="cs1-format">(PDF)</span> on December 12, 2010<span class="reference-accessdate">, retrieved <span class="nowrap">October 18,</span> 2006</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+P+versus+NP+Problem&rft.pub=Clay+Mathematics+Institute&rft.date=2000-04&rft.aulast=Cook&rft.aufirst=Stephen&rft_id=http%3A%2F%2Fwww.claymath.org%2Fmillennium%2FP_vs_NP%2FOfficial_Problem_Description.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJaffe2006" class="citation cs2"><a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Jaffe, Arthur M.</a> (2006), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200606/fea-jaffe.pdf">"The Millennium Grand Challenge in Mathematics"</a> <span class="cs1-format">(PDF)</span>, <i>Notices of the AMS</i>, <b>53</b> (6), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060612225513/http://www.ams.org/notices/200606/fea-jaffe.pdf">archived</a> <span class="cs1-format">(PDF)</span> from the original on June 12, 2006<span class="reference-accessdate">, retrieved <span class="nowrap">October 18,</span> 2006</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+AMS&rft.atitle=The+Millennium+Grand+Challenge+in+Mathematics&rft.volume=53&rft.issue=6&rft.date=2006&rft.aulast=Jaffe&rft.aufirst=Arthur+M.&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200606%2Ffea-jaffe.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-AK06-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-AK06_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFArvindKurur2006" class="citation cs2">Arvind, Vikraman; Kurur, Piyush P. (2006), "Graph isomorphism is in SPP", <i>Information and Computation</i>, <b>204</b> (5): <span class="nowrap">835–</span>852, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ic.2006.02.002">10.1016/j.ic.2006.02.002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+and+Computation&rft.atitle=Graph+isomorphism+is+in+SPP&rft.volume=204&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E835-%3C%2Fspan%3E852&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.ic.2006.02.002&rft.aulast=Arvind&rft.aufirst=Vikraman&rft.au=Kurur%2C+Piyush+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchöning1988" class="citation cs2"><a href="/wiki/Uwe_Sch%C3%B6ning" title="Uwe Schöning">Schöning, Uwe</a> (1988), "Graph Isomorphism is in the Low Hierarchy", <i>Journal of Computer and System Sciences</i>, <b>37</b> (3): <span class="nowrap">312–</span>323, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-0000%2888%2990010-4">10.1016/0022-0000(88)90010-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computer+and+System+Sciences&rft.atitle=Graph+Isomorphism+is+in+the+Low+Hierarchy&rft.volume=37&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E312-%3C%2Fspan%3E323&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2F0022-0000%2888%2990010-4&rft.aulast=Sch%C3%B6ning&rft.aufirst=Uwe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBabai2016" class="citation arxiv cs1">Babai, László (2016). "Graph Isomorphism in Quasipolynomial Time". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1512.03547">1512.03547</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.DS">cs.DS</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Graph+Isomorphism+in+Quasipolynomial+Time&rft.date=2016&rft_id=info%3Aarxiv%2F1512.03547&rft.aulast=Babai&rft.aufirst=L%C3%A1szl%C3%B3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFortnow2002" class="citation web cs1"><a href="/wiki/Lance_Fortnow" title="Lance Fortnow">Fortnow, Lance</a> (September 13, 2002). <a rel="nofollow" class="external text" href="http://weblog.fortnow.com/2002/09/complexity-class-of-week-factoring.html">"Computational Complexity Blog: Factoring"</a>. <i>weblog.fortnow.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=weblog.fortnow.com&rft.atitle=Computational+Complexity+Blog%3A+Factoring&rft.date=2002-09-13&rft.aulast=Fortnow&rft.aufirst=Lance&rft_id=http%3A%2F%2Fweblog.fortnow.com%2F2002%2F09%2Fcomplexity-class-of-week-factoring.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Wolfram MathWorld: <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/NumberFieldSieve.html">Number Field Sieve</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/courses/archive/spr06/cos522/">Boaz Barak's course on Computational Complexity</a> <a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/courses/archive/spr06/cos522/lec2.pdf">Lecture 2</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Hopcroft, J.E., Motwani, R. and Ullman, J.D. (2007) <a href="/wiki/Introduction_to_Automata_Theory,_Languages,_and_Computation" title="Introduction to Automata Theory, Languages, and Computation">Introduction to Automata Theory, Languages, and Computation</a>, Addison Wesley, Boston/San Francisco/New York (page 368)</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMeurant2014" class="citation book cs1">Meurant, Gerard (2014). <i>Algorithms and Complexity</i>. Elsevier. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6WriBQAAQBAJ&pg=PA4">p. 4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08093391-7" title="Special:BookSources/978-0-08093391-7"><bdi>978-0-08093391-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithms+and+Complexity&rft.pages=p.+4&rft.pub=Elsevier&rft.date=2014&rft.isbn=978-0-08093391-7&rft.aulast=Meurant&rft.aufirst=Gerard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZobel2015" class="citation book cs1">Zobel, Justin (2015). <i>Writing for Computer Science</i>. Springer. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LWCYBgAAQBAJ&pg=PA132">132</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-44716639-9" title="Special:BookSources/978-1-44716639-9"><bdi>978-1-44716639-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Writing+for+Computer+Science&rft.pages=132&rft.pub=Springer&rft.date=2015&rft.isbn=978-1-44716639-9&rft.aulast=Zobel&rft.aufirst=Justin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmale1997" class="citation journal cs1">Smale, Steve (1997). "Complexity Theory and Numerical Analysis". <i>Acta Numerica</i>. <b>6</b>. Cambridge Univ Press: <span class="nowrap">523–</span>551. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997AcNum...6..523S">1997AcNum...6..523S</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.4678">10.1.1.33.4678</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0962492900002774">10.1017/s0962492900002774</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5949193">5949193</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Numerica&rft.atitle=Complexity+Theory+and+Numerical+Analysis&rft.volume=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E523-%3C%2Fspan%3E551&rft.date=1997&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.33.4678%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5949193%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2Fs0962492900002774&rft_id=info%3Abibcode%2F1997AcNum...6..523S&rft.aulast=Smale&rft.aufirst=Steve&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBabaiCampagnolo2009" class="citation arxiv cs1">Babai, László; Campagnolo, Manuel (2009). "A Survey on Continuous Time Computations". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0907.3117">0907.3117</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.CC">cs.CC</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+Survey+on+Continuous+Time+Computations&rft.date=2009&rft_id=info%3Aarxiv%2F0907.3117&rft.aulast=Babai&rft.aufirst=L%C3%A1szl%C3%B3&rft.au=Campagnolo%2C+Manuel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTomlinMitchellBayenOishi2003" class="citation journal cs1">Tomlin, Claire J.; Mitchell, Ian; Bayen, Alexandre M.; <a href="/wiki/Meeko_Oishi" title="Meeko Oishi">Oishi, Meeko</a> (July 2003). "Computational Techniques for the Verification of Hybrid Systems". <i>Proceedings of the IEEE</i>. <b>91</b> (7): <span class="nowrap">986–</span>1001. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.4296">10.1.1.70.4296</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fjproc.2003.814621">10.1109/jproc.2003.814621</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+IEEE&rft.atitle=Computational+Techniques+for+the+Verification+of+Hybrid+Systems&rft.volume=91&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E986-%3C%2Fspan%3E1001&rft.date=2003-07&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.70.4296%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1109%2Fjproc.2003.814621&rft.aulast=Tomlin&rft.aufirst=Claire+J.&rft.au=Mitchell%2C+Ian&rft.au=Bayen%2C+Alexandre+M.&rft.au=Oishi%2C+Meeko&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-Fortnow_2003-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fortnow_2003_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fortnow_2003_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFFortnowHomer2003">Fortnow & Homer (2003)</a></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Richard M. Karp, "<a rel="nofollow" class="external text" href="http://cecas.clemson.edu/~shierd/Shier/MthSc816/turing-karp.pdf">Combinatorics, Complexity, and Randomness</a>", 1985 Turing Award Lecture</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYamada1962" class="citation journal cs1 cs1-prop-long-vol">Yamada, H. (1962). "Real-Time Computation and Recursive Functions Not Real-Time Computable". <i>IEEE Transactions on Electronic Computers</i>. EC-11 (6): <span class="nowrap">753–</span>760. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTEC.1962.5219459">10.1109/TEC.1962.5219459</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Electronic+Computers&rft.atitle=Real-Time+Computation+and+Recursive+Functions+Not+Real-Time+Computable&rft.volume=EC-11&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E753-%3C%2Fspan%3E760&rft.date=1962&rft_id=info%3Adoi%2F10.1109%2FTEC.1962.5219459&rft.aulast=Yamada&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Trakhtenbrot, B.A.: Signalizing functions and tabular operators. Uchionnye Zapiski Penzenskogo Pedinstituta (Transactions of the Penza Pedagogoical Institute) 4, 75–87 (1956) (in Russian)</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Boris Trakhtenbrot, "<a rel="nofollow" class="external text" href="https://books.google.com/books?id=GFX2qiLuRAMC&pg=PA1">From Logic to Theoretical Computer Science – An Update</a>". In: <i>Pillars of Computer Science</i>, LNCS 4800, Springer 2008.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRichard_M._Karp1972" class="citation cs2">Richard M. Karp (1972), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110629023717/http://www.cs.berkeley.edu/~luca/cs172/karp.pdf">"Reducibility Among Combinatorial Problems"</a> <span class="cs1-format">(PDF)</span>, in R. E. Miller; J. W. Thatcher (eds.), <i>Complexity of Computer Computations</i>, New York: Plenum, pp. <span class="nowrap">85–</span>103, archived from <a rel="nofollow" class="external text" href="http://www.cs.berkeley.edu/~luca/cs172/karp.pdf">the original</a> <span class="cs1-format">(PDF)</span> on June 29, 2011<span class="reference-accessdate">, retrieved <span class="nowrap">September 28,</span> 2009</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reducibility+Among+Combinatorial+Problems&rft.btitle=Complexity+of+Computer+Computations&rft.pages=%3Cspan+class%3D%22nowrap%22%3E85-%3C%2Fspan%3E103&rft.pub=New+York%3A+Plenum&rft.date=1972&rft.au=Richard+M.+Karp&rft_id=http%3A%2F%2Fwww.cs.berkeley.edu%2F~luca%2Fcs172%2Fkarp.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Textbooks">Textbooks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=29" title="Edit section: Textbooks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAroraBarak2009" class="citation cs2"><a href="/wiki/Sanjeev_Arora" title="Sanjeev Arora">Arora, Sanjeev</a>; Barak, Boaz (2009), <a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/theory/complexity/"><i>Computational Complexity: A Modern Approach</i></a>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-42426-4" title="Special:BookSources/978-0-521-42426-4"><bdi>978-0-521-42426-4</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1193.68112">1193.68112</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Complexity%3A+A+Modern+Approach&rft.pub=Cambridge+University+Press&rft.date=2009&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1193.68112%23id-name%3DZbl&rft.isbn=978-0-521-42426-4&rft.aulast=Arora&rft.aufirst=Sanjeev&rft.au=Barak%2C+Boaz&rft_id=http%3A%2F%2Fwww.cs.princeton.edu%2Ftheory%2Fcomplexity%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDowneyFellows1999" class="citation cs2">Downey, Rod; <a href="/wiki/Michael_Fellows" title="Michael Fellows">Fellows, Michael</a> (1999), <i>Parameterized complexity</i>, Monographs in Computer Science, Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387948836" title="Special:BookSources/9780387948836"><bdi>9780387948836</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Parameterized+complexity&rft.place=Berlin%2C+New+York&rft.series=Monographs+in+Computer+Science&rft.pub=Springer-Verlag&rft.date=1999&rft.isbn=9780387948836&rft.aulast=Downey&rft.aufirst=Rod&rft.au=Fellows%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDuKo,_Ker-I2000" class="citation cs2">Du, Ding-Zhu; Ko, Ker-I (2000), <i>Theory of Computational Complexity</i>, John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-34506-0" title="Special:BookSources/978-0-471-34506-0"><bdi>978-0-471-34506-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Computational+Complexity&rft.pub=John+Wiley+%26+Sons&rft.date=2000&rft.isbn=978-0-471-34506-0&rft.aulast=Du&rft.aufirst=Ding-Zhu&rft.au=Ko%2C+Ker-I&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGareyJohnson1979" class="citation book cs1"><a href="/wiki/Michael_Garey" title="Michael Garey">Garey, Michael R.</a>; <a href="/wiki/David_S._Johnson" title="David S. Johnson">Johnson, David S.</a> (1979). <i><a href="/wiki/Computers_and_Intractability" title="Computers and Intractability">Computers and Intractability: A Guide to the Theory of NP-Completeness</a></i>. Series of Books in the Mathematical Sciences (1st ed.). New York: <a href="/wiki/W._H._Freeman_and_Company" title="W. H. Freeman and Company">W. H. Freeman and Company</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780716710455" title="Special:BookSources/9780716710455"><bdi>9780716710455</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0519066">0519066</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/247570676">247570676</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computers+and+Intractability%3A+A+Guide+to+the+Theory+of+NP-Completeness&rft.place=New+York&rft.series=Series+of+Books+in+the+Mathematical+Sciences&rft.edition=1st&rft.pub=W.+H.+Freeman+and+Company&rft.date=1979&rft_id=info%3Aoclcnum%2F247570676&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D519066%23id-name%3DMR&rft.isbn=9780716710455&rft.aulast=Garey&rft.aufirst=Michael+R.&rft.au=Johnson%2C+David+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGoldreich2008" class="citation cs2"><a href="/wiki/Oded_Goldreich" title="Oded Goldreich">Goldreich, Oded</a> (2008), <a rel="nofollow" class="external text" href="http://www.wisdom.weizmann.ac.il/~oded/cc-book.html"><i>Computational Complexity: A Conceptual Perspective</i></a>, Cambridge University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Complexity%3A+A+Conceptual+Perspective&rft.pub=Cambridge+University+Press&rft.date=2008&rft.aulast=Goldreich&rft.aufirst=Oded&rft_id=http%3A%2F%2Fwww.wisdom.weizmann.ac.il%2F~oded%2Fcc-book.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFvan_Leeuwen1990" class="citation cs2"><a href="/wiki/Jan_van_Leeuwen" title="Jan van Leeuwen">van Leeuwen, Jan</a>, ed. (1990), <i>Handbook of theoretical computer science (vol. A): algorithms and complexity</i>, MIT Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-88071-0" title="Special:BookSources/978-0-444-88071-0"><bdi>978-0-444-88071-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+theoretical+computer+science+%28vol.+A%29%3A+algorithms+and+complexity&rft.pub=MIT+Press&rft.date=1990&rft.isbn=978-0-444-88071-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPapadimitriou1994" class="citation cs2"><a href="/wiki/Christos_Papadimitriou" title="Christos Papadimitriou">Papadimitriou, Christos</a> (1994), <i>Computational Complexity</i> (1st ed.), Addison Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-53082-7" title="Special:BookSources/978-0-201-53082-7"><bdi>978-0-201-53082-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Complexity&rft.edition=1st&rft.pub=Addison+Wesley&rft.date=1994&rft.isbn=978-0-201-53082-7&rft.aulast=Papadimitriou&rft.aufirst=Christos&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSipser2006" class="citation cs2"><a href="/wiki/Michael_Sipser" title="Michael Sipser">Sipser, Michael</a> (2006), <a href="/wiki/Introduction_to_the_Theory_of_Computation" title="Introduction to the Theory of Computation"><i>Introduction to the Theory of Computation</i></a> (2nd ed.), USA: Thomson Course Technology, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-95097-2" title="Special:BookSources/978-0-534-95097-2"><bdi>978-0-534-95097-2</bdi></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+the+Theory+of+Computation&rft.place=USA&rft.edition=2nd&rft.pub=Thomson+Course+Technology&rft.date=2006&rft.isbn=978-0-534-95097-2&rft.aulast=Sipser&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Surveys">Surveys</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=30" title="Edit section: Surveys"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKhalilUlery1976" class="citation cs2">Khalil, Hatem; <a href="/wiki/Dana_Ulery" title="Dana Ulery">Ulery, Dana</a> (1976), "A review of current studies on complexity of algorithms for partial differential equations", <a rel="nofollow" class="external text" href="http://portal.acm.org/citation.cfm?id=800191.805573"><i>Proceedings of the annual conference on - ACM 76</i></a>, pp. <span class="nowrap">197–</span>201, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F800191.805573">10.1145/800191.805573</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781450374897" title="Special:BookSources/9781450374897"><bdi>9781450374897</bdi></a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15497394">15497394</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+review+of+current+studies+on+complexity+of+algorithms+for+partial+differential+equations&rft.btitle=Proceedings+of+the+annual+conference+on+-+ACM+76&rft.pages=%3Cspan+class%3D%22nowrap%22%3E197-%3C%2Fspan%3E201&rft.date=1976&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15497394%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F800191.805573&rft.isbn=9781450374897&rft.aulast=Khalil&rft.aufirst=Hatem&rft.au=Ulery%2C+Dana&rft_id=http%3A%2F%2Fportal.acm.org%2Fcitation.cfm%3Fid%3D800191.805573&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCook1983" class="citation cs2"><a href="/wiki/Stephen_Cook" title="Stephen Cook">Cook, Stephen</a> (1983), "An overview of computational complexity", <i>Communications of the ACM</i>, <b>26</b> (6): <span class="nowrap">400–</span>408, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F358141.358144">10.1145/358141.358144</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-0782">0001-0782</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14323396">14323396</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=An+overview+of+computational+complexity&rft.volume=26&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E400-%3C%2Fspan%3E408&rft.date=1983&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14323396%23id-name%3DS2CID&rft.issn=0001-0782&rft_id=info%3Adoi%2F10.1145%2F358141.358144&rft.aulast=Cook&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFortnowHomer2003" class="citation cs2">Fortnow, Lance; Homer, Steven (2003), <a rel="nofollow" class="external text" href="http://people.cs.uchicago.edu/~fortnow/papers/history.pdf">"A Short History of Computational Complexity"</a> <span class="cs1-format">(PDF)</span>, <i>Bulletin of the EATCS</i>, <b>80</b>: <span class="nowrap">95–</span>133</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+EATCS&rft.atitle=A+Short+History+of+Computational+Complexity&rft.volume=80&rft.pages=%3Cspan+class%3D%22nowrap%22%3E95-%3C%2Fspan%3E133&rft.date=2003&rft.aulast=Fortnow&rft.aufirst=Lance&rft.au=Homer%2C+Steven&rft_id=http%3A%2F%2Fpeople.cs.uchicago.edu%2F~fortnow%2Fpapers%2Fhistory.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMertens2002" class="citation cs2">Mertens, Stephan (2002), "Computational Complexity for Physicists", <i>Computing in Science & Engineering</i>, <b>4</b> (3): <span class="nowrap">31–</span>47, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0012185">cond-mat/0012185</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002CSE.....4c..31M">2002CSE.....4c..31M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F5992.998639">10.1109/5992.998639</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1521-9615">1521-9615</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:633346">633346</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computing+in+Science+%26+Engineering&rft.atitle=Computational+Complexity+for+Physicists&rft.volume=4&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E31-%3C%2Fspan%3E47&rft.date=2002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A633346%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2002CSE.....4c..31M&rft_id=info%3Aarxiv%2Fcond-mat%2F0012185&rft.issn=1521-9615&rft_id=info%3Adoi%2F10.1109%2F5992.998639&rft.aulast=Mertens&rft.aufirst=Stephan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_complexity_theory&action=edit&section=31" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.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:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.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;min-width:0}}@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><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.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 class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/tractable" class="extiw" title="wiktionary:tractable">tractable</a></b></i>, <i><b><a href="https://en.wiktionary.org/wiki/feasible" class="extiw" title="wiktionary:feasible">feasible</a></b></i>, <i><b><a href="https://en.wiktionary.org/wiki/intractability" class="extiw" title="wiktionary:intractability">intractability</a></b></i>, or <i><b><a href="https://en.wiktionary.org/wiki/infeasible" class="extiw" title="wiktionary:infeasible">infeasible</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735" /><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Computational_complexity_theory" class="extiw" title="commons:Category:Computational complexity theory">Computational complexity theory</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://complexityzoo.net/Complexity_Zoo">The Complexity Zoo</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Computational_complexity_classes">"Computational complexity classes"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Computational+complexity+classes&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DComputational_complexity_classes&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+complexity+theory" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.scottaaronson.com/papers/philos.pdf">Scott Aaronson: Why Philosophers Should Care About Computational Complexity</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Complexity_classes148" 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:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Complexity_classes" title="Template:Complexity classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Complexity_classes" title="Template talk:Complexity classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Complexity_classes" title="Special:EditPage/Template:Complexity classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Complexity_classes148" style="font-size:114%;margin:0 4em"><a href="/wiki/Complexity_class" title="Complexity class">Complexity classes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Considered feasible</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/DLOGTIME" title="DLOGTIME">DLOGTIME</a></li> <li><a href="/wiki/AC0" title="AC0">AC<sup>0</sup></a></li> <li><a href="/wiki/ACC0" title="ACC0">ACC<sup>0</sup></a></li> <li><a href="/wiki/TC0" title="TC0">TC<sup>0</sup></a></li> <li><a href="/wiki/L_(complexity)" title="L (complexity)">L</a></li> <li><a href="/wiki/SL_(complexity)" title="SL (complexity)">SL</a></li> <li><a href="/wiki/RL_(complexity)" title="RL (complexity)">RL</a></li> <li><a href="/wiki/FL_(complexity)" title="FL (complexity)">FL</a></li> <li><a href="/wiki/NL_(complexity)" title="NL (complexity)">NL</a> <ul><li><a href="/wiki/NL-complete" title="NL-complete">NL-complete</a></li></ul></li> <li><a href="/wiki/NC_(complexity)" title="NC (complexity)">NC</a></li> <li><a href="/wiki/SC_(complexity)" title="SC (complexity)">SC</a></li> <li><a href="/wiki/CC_(complexity)" title="CC (complexity)">CC</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a> <ul><li><a href="/wiki/P-complete" title="P-complete">P-complete</a></li></ul></li> <li><a href="/wiki/ZPP_(complexity)" title="ZPP (complexity)">ZPP</a></li> <li><a href="/wiki/RP_(complexity)" title="RP (complexity)">RP</a></li> <li><a href="/wiki/BPP_(complexity)" title="BPP (complexity)">BPP</a></li> <li><a href="/wiki/BQP" title="BQP">BQP</a></li> <li><a href="/wiki/APX" title="APX">APX</a></li> <li><a href="/wiki/FP_(complexity)" title="FP (complexity)">FP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Suspected infeasible</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="/wiki/UP_(complexity)" title="UP (complexity)">UP</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a> <ul><li><a href="/wiki/NP-completeness" title="NP-completeness">NP-complete</a></li> <li><a href="/wiki/NP-hardness" title="NP-hardness">NP-hard</a></li> <li><a href="/wiki/Co-NP" title="Co-NP">co-NP</a></li> <li><a href="/wiki/Co-NP-complete" title="Co-NP-complete">co-NP-complete</a></li></ul></li> <li><a href="/wiki/TFNP" title="TFNP">TFNP</a></li> <li><a href="/wiki/FNP_(complexity)" title="FNP (complexity)">FNP</a></li> <li><a href="/wiki/Arthur%E2%80%93Merlin_protocol" title="Arthur–Merlin protocol">AM</a></li> <li><a href="/wiki/QMA" title="QMA">QMA</a></li> <li><a href="/wiki/PH_(complexity)" title="PH (complexity)">PH</a></li> <li><a href="/wiki/Parity_P" title="Parity P">⊕P</a></li> <li><a href="/wiki/PP_(complexity)" title="PP (complexity)">PP</a></li> <li><a href="/wiki/%E2%99%AFP" title="♯P">#P</a> <ul><li><a href="/wiki/%E2%99%AFP-complete" title="♯P-complete">#P-complete</a></li></ul></li> <li><a href="/wiki/IP_(complexity)" title="IP (complexity)">IP</a></li> <li><a href="/wiki/PSPACE" title="PSPACE">PSPACE</a> <ul><li><a href="/wiki/PSPACE-complete" title="PSPACE-complete">PSPACE-complete</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Considered infeasible</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/EXPTIME" title="EXPTIME">EXPTIME</a></li> <li><a href="/wiki/NEXPTIME" title="NEXPTIME">NEXPTIME</a></li> <li><a href="/wiki/EXPSPACE" title="EXPSPACE">EXPSPACE</a></li> <li><a href="/wiki/2-EXPTIME" title="2-EXPTIME">2-EXPTIME</a></li> <li><a href="/wiki/ELEMENTARY" title="ELEMENTARY">ELEMENTARY</a></li> <li><a href="/wiki/PR_(complexity)" title="PR (complexity)">PR</a></li> <li><a href="/wiki/R_(complexity)" title="R (complexity)">R</a></li> <li><a href="/wiki/RE_(complexity)" title="RE (complexity)">RE</a></li> <li><a href="/wiki/ALL_(complexity)" title="ALL (complexity)">ALL</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Class hierarchies</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="/wiki/Polynomial_hierarchy" title="Polynomial hierarchy">Polynomial hierarchy</a></li> <li><a href="/wiki/Exponential_hierarchy" title="Exponential hierarchy">Exponential hierarchy</a></li> <li><a href="/wiki/Grzegorczyk_hierarchy" title="Grzegorczyk hierarchy">Grzegorczyk hierarchy</a></li> <li><a href="/wiki/Arithmetical_hierarchy" title="Arithmetical hierarchy">Arithmetical hierarchy</a></li> <li><a href="/wiki/Boolean_hierarchy" title="Boolean hierarchy">Boolean hierarchy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families of classes</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/DTIME" title="DTIME">DTIME</a></li> <li><a href="/wiki/NTIME" title="NTIME">NTIME</a></li> <li><a href="/wiki/DSPACE" title="DSPACE">DSPACE</a></li> <li><a href="/wiki/NSPACE" title="NSPACE">NSPACE</a></li> <li><a href="/wiki/Probabilistically_checkable_proof" title="Probabilistically checkable proof">Probabilistically checkable proof</a></li> <li><a href="/wiki/Interactive_proof_system" title="Interactive proof system">Interactive proof system</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="/wiki/List_of_complexity_classes" title="List of complexity classes">List of complexity classes</a></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Computer_science1028" style="padding:3px"><table class="nowraplinks hlist 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:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Computer_science" title="Template:Computer science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Computer_science" title="Template talk:Computer science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Computer_science" title="Special:EditPage/Template:Computer science"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Computer_science1028" style="font-size:114%;margin:0 4em"><a href="/wiki/Computer_science" title="Computer science">Computer science</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Note: This template roughly follows the 2012 <a href="/wiki/ACM_Computing_Classification_System" title="ACM Computing Classification System">ACM Computing Classification System</a>.</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_hardware" title="Computer hardware">Hardware</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Printed_circuit_board" title="Printed circuit board">Printed circuit board</a></li> <li><a href="/wiki/Peripheral" title="Peripheral">Peripheral</a></li> <li><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a></li> <li><a href="/wiki/Very_Large_Scale_Integration" class="mw-redirect" title="Very Large Scale Integration">Very Large Scale Integration</a></li> <li><a href="/wiki/System_on_a_chip" title="System on a chip">Systems on Chip (SoCs)</a></li> <li><a href="/wiki/Green_computing" title="Green computing">Energy consumption (Green computing)</a></li> <li><a href="/wiki/Electronic_design_automation" title="Electronic design automation">Electronic design automation</a></li> <li><a href="/wiki/Hardware_acceleration" title="Hardware acceleration">Hardware acceleration</a></li> <li><a href="/wiki/Processor_(computing)" title="Processor (computing)">Processor</a></li> <li><a href="/wiki/List_of_computer_size_categories" title="List of computer size categories">Size</a> / <a href="/wiki/Form_factor_(design)" title="Form factor (design)">Form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Computer systems organization</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_architecture" title="Computer architecture">Computer architecture</a></li> <li><a href="/wiki/Computational_complexity" title="Computational complexity">Computational complexity</a></li> <li><a href="/wiki/Dependability" title="Dependability">Dependability</a></li> <li><a href="/wiki/Embedded_system" title="Embedded system">Embedded system</a></li> <li><a href="/wiki/Real-time_computing" title="Real-time computing">Real-time computing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_network" title="Computer network">Networks</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Network_architecture" title="Network architecture">Network architecture</a></li> <li><a href="/wiki/Network_protocol" class="mw-redirect" title="Network protocol">Network protocol</a></li> <li><a href="/wiki/Networking_hardware" title="Networking hardware">Network components</a></li> <li><a href="/wiki/Network_scheduler" title="Network scheduler">Network scheduler</a></li> <li><a href="/wiki/Network_performance" title="Network performance">Network performance evaluation</a></li> <li><a href="/wiki/Network_service" title="Network service">Network service</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software organization</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpreter_(computing)" title="Interpreter (computing)">Interpreter</a></li> <li><a href="/wiki/Middleware" title="Middleware">Middleware</a></li> <li><a href="/wiki/Virtual_machine" title="Virtual machine">Virtual machine</a></li> <li><a href="/wiki/Operating_system" title="Operating system">Operating system</a></li> <li><a href="/wiki/Software_quality" title="Software quality">Software quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Programming_language_theory" title="Programming language theory">Software notations</a> and <a href="/wiki/Programming_tool" title="Programming tool">tools</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Programming_paradigm" title="Programming paradigm">Programming paradigm</a></li> <li><a href="/wiki/Programming_language" title="Programming language">Programming language</a></li> <li><a href="/wiki/Compiler_construction" class="mw-redirect" title="Compiler construction">Compiler</a></li> <li><a href="/wiki/Domain-specific_language" title="Domain-specific language">Domain-specific language</a></li> <li><a href="/wiki/Modeling_language" title="Modeling language">Modeling language</a></li> <li><a href="/wiki/Software_framework" title="Software framework">Software framework</a></li> <li><a href="/wiki/Integrated_development_environment" title="Integrated development environment">Integrated development environment</a></li> <li><a href="/wiki/Software_configuration_management" title="Software configuration management">Software configuration management</a></li> <li><a href="/wiki/Library_(computing)" title="Library (computing)">Software library</a></li> <li><a href="/wiki/Software_repository" title="Software repository">Software repository</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Software_development" title="Software development">Software development</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Control_variable_(programming)" class="mw-redirect" title="Control variable (programming)">Control variable</a></li> <li><a href="/wiki/Software_development_process" title="Software development process">Software development process</a></li> <li><a href="/wiki/Requirements_analysis" title="Requirements analysis">Requirements analysis</a></li> <li><a href="/wiki/Software_design" title="Software design">Software design</a></li> <li><a href="/wiki/Software_construction" title="Software construction">Software construction</a></li> <li><a href="/wiki/Software_deployment" title="Software deployment">Software deployment</a></li> <li><a href="/wiki/Software_engineering" title="Software engineering">Software engineering</a></li> <li><a href="/wiki/Software_maintenance" title="Software maintenance">Software maintenance</a></li> <li><a href="/wiki/Programming_team" title="Programming team">Programming team</a></li> <li><a href="/wiki/Open-source_software" title="Open-source software">Open-source model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Model_of_computation" title="Model of computation">Model of computation</a> <ul><li><a href="/wiki/Stochastic_computing" title="Stochastic computing">Stochastic</a></li></ul></li> <li><a href="/wiki/Formal_language" title="Formal language">Formal language</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a></li> <li><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></li> <li><a class="mw-selflink selflink">Computational complexity theory</a></li> <li><a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Logic</a></li> <li><a href="/wiki/Semantics_(computer_science)" title="Semantics (computer science)">Semantics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algorithm" title="Algorithm">Algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algorithm_design" class="mw-redirect" title="Algorithm design">Algorithm design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></li> <li><a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">Algorithmic efficiency</a></li> <li><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematics of <a href="/wiki/Computing" title="Computing">computing</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_system" title="Information system">Information systems</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Database" title="Database">Database management system</a></li> <li><a href="/wiki/Computer_data_storage" title="Computer data storage">Information storage systems</a></li> <li><a href="/wiki/Enterprise_information_system" title="Enterprise information system">Enterprise information system</a></li> <li><a href="/wiki/Social_software" title="Social software">Social information systems</a></li> <li><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a></li> <li><a href="/wiki/Decision_support_system" title="Decision support system">Decision support system</a></li> <li><a href="/wiki/Process_control" class="mw-redirect" title="Process control">Process control system</a></li> <li><a href="/wiki/Multimedia_database" title="Multimedia database">Multimedia information system</a></li> <li><a href="/wiki/Data_mining" title="Data mining">Data mining</a></li> <li><a href="/wiki/Digital_library" title="Digital library">Digital library</a></li> <li><a href="/wiki/Computing_platform" title="Computing platform">Computing platform</a></li> <li><a href="/wiki/Digital_marketing" title="Digital marketing">Digital marketing</a></li> <li><a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a></li> <li><a href="/wiki/Information_retrieval" title="Information retrieval">Information retrieval</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_security" title="Computer security">Security</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Formal_methods" title="Formal methods">Formal methods</a></li> <li><a href="/wiki/Security_hacker" title="Security hacker">Security hacker</a></li> <li><a href="/wiki/Security_service_(telecommunication)" title="Security service (telecommunication)">Security services</a></li> <li><a href="/wiki/Intrusion_detection_system" title="Intrusion detection system">Intrusion detection system</a></li> <li><a href="/wiki/Hardware_security" title="Hardware security">Hardware security</a></li> <li><a href="/wiki/Network_security" title="Network security">Network security</a></li> <li><a href="/wiki/Information_security" title="Information security">Information security</a></li> <li><a href="/wiki/Application_security" title="Application security">Application security</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Human%E2%80%93computer_interaction" title="Human–computer interaction">Human–computer interaction</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interaction_design" title="Interaction design">Interaction design</a></li> <li><a href="/wiki/Social_computing" title="Social computing">Social computing</a></li> <li><a href="/wiki/Ubiquitous_computing" title="Ubiquitous computing">Ubiquitous computing</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Visualization</a></li> <li><a href="/wiki/Computer_accessibility" title="Computer accessibility">Accessibility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Concurrency_(computer_science)" title="Concurrency (computer science)">Concurrency</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Concurrent_computing" title="Concurrent computing">Concurrent computing</a></li> <li><a href="/wiki/Parallel_computing" title="Parallel computing">Parallel computing</a></li> <li><a href="/wiki/Distributed_computing" title="Distributed computing">Distributed computing</a></li> <li><a href="/wiki/Multithreading_(computer_architecture)" title="Multithreading (computer architecture)">Multithreading</a></li> <li><a href="/wiki/Multiprocessing" title="Multiprocessing">Multiprocessing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_language_processing" title="Natural language processing">Natural language processing</a></li> <li><a href="/wiki/Knowledge_representation_and_reasoning" title="Knowledge representation and reasoning">Knowledge representation and reasoning</a></li> <li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Automated_planning_and_scheduling" title="Automated planning and scheduling">Automated planning and scheduling</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Search methodology</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control method</a></li> <li><a href="/wiki/Philosophy_of_artificial_intelligence" title="Philosophy of artificial intelligence">Philosophy of artificial intelligence</a></li> <li><a href="/wiki/Distributed_artificial_intelligence" title="Distributed artificial intelligence">Distributed artificial intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Multi-task_learning" title="Multi-task learning">Multi-task learning</a></li> <li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_graphics" title="Computer graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_animation" title="Computer animation">Animation</a></li> <li><a href="/wiki/Extended_reality" title="Extended reality">Extended reality</a> <ul><li><a href="/wiki/Augmented_reality" title="Augmented reality">Augmented</a></li> <li><a href="/wiki/Mixed_reality" title="Mixed reality">Mixed</a></li> <li><a href="/wiki/Virtual_reality" title="Virtual reality">Virtual</a></li></ul></li> <li><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a></li> <li><a href="/wiki/Photograph_manipulation" title="Photograph manipulation">Photograph manipulation</a></li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a></li> <li><a href="/wiki/Image_compression" title="Image compression">Image compression</a></li> <li><a href="/wiki/Solid_modeling" title="Solid modeling">Solid modeling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applied computing</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Computing" class="mw-redirect" title="Quantum Computing">Quantum Computing</a></li> <li><a href="/wiki/E-commerce" title="E-commerce">E-commerce</a></li> <li><a href="/wiki/Enterprise_software" title="Enterprise software">Enterprise software</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational mathematics</a></li> <li><a href="/wiki/Computational_physics" title="Computational physics">Computational physics</a></li> <li><a href="/wiki/Computational_chemistry" title="Computational chemistry">Computational chemistry</a></li> <li><a href="/wiki/Computational_biology" title="Computational biology">Computational biology</a></li> <li><a href="/wiki/Computational_social_science" title="Computational social science">Computational social science</a></li> <li><a href="/wiki/Computational_engineering" title="Computational engineering">Computational engineering</a></li> <li><a href="/wiki/Template:Differentiable_computing" title="Template:Differentiable computing">Differentiable computing</a></li> <li><a href="/wiki/Health_informatics" title="Health informatics">Computational healthcare</a></li> <li><a href="/wiki/Digital_art" title="Digital art">Digital art</a></li> <li><a href="/wiki/Electronic_publishing" title="Electronic publishing">Electronic publishing</a></li> <li><a href="/wiki/Cyberwarfare" title="Cyberwarfare">Cyberwarfare</a></li> <li><a href="/wiki/Electronic_voting" title="Electronic voting">Electronic voting</a></li> <li><a href="/wiki/Video_game" title="Video game">Video games</a></li> <li><a href="/wiki/Word_processor" title="Word processor">Word processing</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Educational_technology" title="Educational technology">Educational technology</a></li> <li><a href="/wiki/Document_management_system" title="Document management system">Document management</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Computer_science" title="Category:Computer science">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Outline"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/10px-Global_thinking.svg.png" decoding="async" width="10" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/15px-Global_thinking.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/21px-Global_thinking.svg.png 2x" data-file-width="130" data-file-height="200" /></span></span> <a href="/wiki/Outline_of_computer_science" title="Outline of computer science">Outline</a></li> <li><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/16px-Symbol_question.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/23px-Symbol_question.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/31px-Symbol_question.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Template:Glossaries_of_computers" title="Template:Glossaries of computers">Glossaries</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /></div><div role="navigation" class="navbox authority-control" aria-label="Navbox1013" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q205084#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4120591-1">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Computational complexity"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85029473">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="teorie složitosti"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph305884&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007545779105171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Computational_complexity_theory&oldid=1273212307">https://en.wikipedia.org/w/index.php?title=Computational_complexity_theory&oldid=1273212307</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Computational_complexity_theory" title="Category:Computational complexity theory">Computational complexity theory</a></li><li><a href="/wiki/Category:Computational_fields_of_study" title="Category:Computational fields of study">Computational fields of study</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_mdy_dates_from_September_2017" title="Category:Use mdy dates from September 2017">Use mdy dates from September 2017</a></li><li><a href="/wiki/Category:Commons_category_link_from_Wikidata" title="Category:Commons category link from Wikidata">Commons category link from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 1 February 2025, at 08:36<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Computational_complexity_theory&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div 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="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-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">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Computational complexity theory</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>35 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-64b5bb4b79-zt8sp","wgBackendResponseTime":239,"wgPageParseReport":{"limitreport":{"cputime":"0.807","walltime":"1.067","ppvisitednodes":{"value":4097,"limit":1000000},"postexpandincludesize":{"value":118332,"limit":2097152},"templateargumentsize":{"value":3086,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":9,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":128768,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 690.439 1 -total"," 24.58% 169.689 19 Template:Citation"," 18.17% 125.443 1 Template:Reflist"," 13.34% 92.137 2 Template:Navbox"," 11.23% 77.550 1 Template:ComplexityClasses"," 11.14% 76.888 1 Template:Short_description"," 7.96% 54.927 2 Template:Sister_project"," 7.72% 53.270 2 Template:Side_box"," 7.41% 51.196 2 Template:Pagetype"," 6.59% 45.531 5 Template:Main"]},"scribunto":{"limitreport-timeusage":{"value":"0.415","limit":"10.000"},"limitreport-memusage":{"value":7182255,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREF\"] = 2,\n [\"CITEREFAroraBarak2009\"] = 1,\n [\"CITEREFArvindKurur2006\"] = 1,\n [\"CITEREFBabai2016\"] = 1,\n [\"CITEREFBabaiCampagnolo2009\"] = 1,\n [\"CITEREFBergerLeighton1998\"] = 1,\n [\"CITEREFCook1983\"] = 1,\n [\"CITEREFCook2000\"] = 1,\n [\"CITEREFDowneyFellows1999\"] = 1,\n [\"CITEREFDuKo,_Ker-I2000\"] = 1,\n [\"CITEREFFortnow2002\"] = 1,\n [\"CITEREFFortnowHomer2003\"] = 1,\n [\"CITEREFGoldreich2008\"] = 1,\n [\"CITEREFJaffe2006\"] = 1,\n [\"CITEREFKhalilUlery1976\"] = 1,\n [\"CITEREFLadner1975\"] = 1,\n [\"CITEREFMertens2002\"] = 1,\n [\"CITEREFMeurant2014\"] = 1,\n [\"CITEREFPapadimitriou1994\"] = 1,\n [\"CITEREFRichard_M._Karp1972\"] = 1,\n [\"CITEREFSchöning1988\"] = 1,\n [\"CITEREFSipser2006\"] = 1,\n [\"CITEREFSmale1997\"] = 1,\n [\"CITEREFTomlinMitchellBayenOishi2003\"] = 1,\n [\"CITEREFWuppuluriDoria2020\"] = 1,\n [\"CITEREFYamada1962\"] = 1,\n [\"CITEREFZobel2015\"] = 1,\n [\"CITEREFvan_Leeuwen1990\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 1,\n [\"Authority control\"] = 1,\n [\"Blockquote\"] = 1,\n [\"Citation\"] = 19,\n [\"Cite arXiv\"] = 2,\n [\"Cite book\"] = 2,\n [\"Cite journal\"] = 3,\n [\"Cite web\"] = 2,\n [\"Commons category\"] = 1,\n [\"ComplexityClasses\"] = 1,\n [\"Computer science\"] = 1,\n [\"DEFAULTSORT:Computational Complexity Theory\"] = 1,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"Garey-Johnson\"] = 1,\n [\"Harv\"] = 1,\n [\"Harvnb\"] = 2,\n [\"Harvtxt\"] = 1,\n [\"Main\"] = 5,\n [\"Reflist\"] = 1,\n [\"See also\"] = 1,\n [\"Short description\"] = 1,\n [\"Springer\"] = 1,\n [\"Use mdy dates\"] = 1,\n [\"Visible anchor\"] = 2,\n [\"Wiktionary\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-api-int.codfw.main-5b65fffc7d-6bfbf","timestamp":"20250214040453","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Computational complexity theory","url":"https:\/\/en.wikipedia.org\/wiki\/Computational_complexity_theory","sameAs":"http:\/\/www.wikidata.org\/entity\/Q205084","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q205084","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-12-05T23:12:20Z","dateModified":"2025-02-01T08:36:56Z","headline":"theoretical computer science and mathematics theory that classifies problems according to their inherent difficulty, and relates those classes to each other"}</script> </body> </html>