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id="toc-Prefix-free_Kolmogorov_complexity_K-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Invariance_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Invariance_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Invariance theorem</span> </div> </a> <button aria-controls="toc-Invariance_theorem-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 Invariance theorem subsection</span> </button> <ul id="toc-Invariance_theorem-sublist" class="vector-toc-list"> <li id="toc-Informal_treatment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Informal_treatment"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Informal treatment</span> </div> </a> <ul id="toc-Informal_treatment-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_more_formal_treatment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_more_formal_treatment"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>A more formal treatment</span> </div> </a> <ul id="toc-A_more_formal_treatment-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History_and_context" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History_and_context"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History and context</span> </div> </a> <ul id="toc-History_and_context-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_results" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Basic_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Basic results</span> </div> </a> <button aria-controls="toc-Basic_results-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 Basic results subsection</span> </button> <ul id="toc-Basic_results-sublist" class="vector-toc-list"> <li id="toc-Inequalities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inequalities"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Inequalities</span> </div> </a> <ul id="toc-Inequalities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uncomputability_of_Kolmogorov_complexity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uncomputability_of_Kolmogorov_complexity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Uncomputability of Kolmogorov complexity</span> </div> </a> <ul id="toc-Uncomputability_of_Kolmogorov_complexity-sublist" class="vector-toc-list"> <li id="toc-A_naive_attempt_at_a_program_to_compute_K" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#A_naive_attempt_at_a_program_to_compute_K"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>A naive attempt at a program to compute <i>K</i></span> </div> </a> <ul id="toc-A_naive_attempt_at_a_program_to_compute_K-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_proof_of_uncomputability_of_K" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Formal_proof_of_uncomputability_of_K"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.2</span> <span>Formal proof of uncomputability of <i>K</i></span> </div> </a> <ul id="toc-Formal_proof_of_uncomputability_of_K-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Chain_rule_for_Kolmogorov_complexity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chain_rule_for_Kolmogorov_complexity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Chain rule for Kolmogorov complexity</span> </div> </a> <ul id="toc-Chain_rule_for_Kolmogorov_complexity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Compression" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Compression"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Compression</span> </div> </a> <ul id="toc-Compression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chaitin's_incompleteness_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Chaitin's_incompleteness_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Chaitin's incompleteness theorem</span> </div> </a> <ul id="toc-Chaitin's_incompleteness_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minimum_message_length" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Minimum_message_length"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Minimum message length</span> </div> </a> <ul id="toc-Minimum_message_length-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kolmogorov_randomness" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Kolmogorov_randomness"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Kolmogorov randomness</span> </div> </a> <ul id="toc-Kolmogorov_randomness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_entropy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_to_entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Relation to entropy</span> </div> </a> <ul id="toc-Relation_to_entropy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Halting_problem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Halting_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Halting problem</span> </div> </a> <ul id="toc-Halting_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Universal_probability" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Universal_probability"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Universal probability</span> </div> </a> <ul id="toc-Universal_probability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditional_versions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Conditional_versions"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Conditional versions</span> </div> </a> <ul id="toc-Conditional_versions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time-bounded_complexity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Time-bounded_complexity"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Time-bounded complexity</span> </div> </a> <ul id="toc-Time-bounded_complexity-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">14</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">16</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </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">18</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 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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Kolmogorov complexity</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 23 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-23" 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">23 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/%D8%AA%D8%B9%D9%82%D9%8A%D8%AF_%D9%83%D9%88%D9%84%D9%85%D9%88%D8%BA%D8%B1%D9%88%D9%81" title="تعقيد كولموغروف – Arabic" lang="ar" hreflang="ar" data-title="تعقيد كولموغروف" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Complexitat_de_Kolmog%C3%B3rov" title="Complexitat de Kolmogórov – Catalan" lang="ca" hreflang="ca" data-title="Complexitat de Kolmogórov" 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/Kolmogorovsk%C3%A1_slo%C5%BEitost" title="Kolmogorovská složitost – Czech" lang="cs" hreflang="cs" data-title="Kolmogorovská složitost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kolmogorow-Komplexit%C3%A4t" title="Kolmogorow-Komplexität – German" lang="de" hreflang="de" data-title="Kolmogorow-Komplexität" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kolmogorovi_keerukus" title="Kolmogorovi keerukus – Estonian" lang="et" hreflang="et" data-title="Kolmogorovi keerukus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Complejidad_de_Kolmog%C3%B3rov" title="Complejidad de Kolmogórov – Spanish" lang="es" hreflang="es" data-title="Complejidad de Kolmogórov" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%DB%8C%DA%86%DB%8C%D8%AF%DA%AF%DB%8C_%DA%A9%D9%88%D9%84%D9%85%D9%88%DA%AF%D8%B1%D9%88%D9%81" title="پیچیدگی کولموگروف – Persian" lang="fa" hreflang="fa" data-title="پیچیدگی کولموگروف" 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/Complexit%C3%A9_de_Kolmogorov" title="Complexité de Kolmogorov – French" lang="fr" hreflang="fr" data-title="Complexité de Kolmogorov" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Complexidade_de_Kolmogorov" title="Complexidade de Kolmogorov – Galician" lang="gl" hreflang="gl" data-title="Complexidade de Kolmogorov" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%BD%9C%EB%AA%A8%EA%B3%A0%EB%A1%9C%ED%94%84_%EB%B3%B5%EC%9E%A1%EB%8F%84" title="콜모고로프 복잡도 – Korean" lang="ko" hreflang="ko" data-title="콜모고로프 복잡도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kompleksitas_Kolmogorov" title="Kompleksitas Kolmogorov – Indonesian" lang="id" hreflang="id" data-title="Kompleksitas Kolmogorov" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Complessit%C3%A0_di_Kolmogorov" title="Complessità di Kolmogorov – Italian" lang="it" hreflang="it" data-title="Complessità di Kolmogorov" 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%A1%D7%99%D7%91%D7%95%D7%9B%D7%99%D7%95%D7%AA_%D7%A7%D7%95%D7%9C%D7%9E%D7%95%D7%92%D7%95%D7%A8%D7%95%D7%91" title="סיבוכיות קולמוגורוב – Hebrew" lang="he" hreflang="he" data-title="סיבוכיות קולמוגורוב" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kolmogorov-complexiteit" title="Kolmogorov-complexiteit – Dutch" lang="nl" hreflang="nl" data-title="Kolmogorov-complexiteit" 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/%E3%82%B3%E3%83%AB%E3%83%A2%E3%82%B4%E3%83%AD%E3%83%95%E8%A4%87%E9%9B%91%E6%80%A7" title="コルモゴロフ複雑性 – Japanese" lang="ja" hreflang="ja" data-title="コルモゴロフ複雑性" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Z%C5%82o%C5%BCono%C5%9B%C4%87_Ko%C5%82mogorowa" title="Złożoność Kołmogorowa – Polish" lang="pl" hreflang="pl" data-title="Złożoność Kołmogorowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Complexidade_de_Kolmogorov" title="Complexidade de Kolmogorov – Portuguese" lang="pt" hreflang="pt" data-title="Complexidade de Kolmogorov" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D0%BC%D0%BE%D0%B3%D0%BE%D1%80%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D1%81%D0%BB%D0%BE%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Колмогоровская сложность – Russian" lang="ru" hreflang="ru" data-title="Колмогоровская сложность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kolmogorovkomplexitet" title="Kolmogorovkomplexitet – Swedish" lang="sv" hreflang="sv" data-title="Kolmogorovkomplexitet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kolmogorov_karma%C5%9F%C4%B1kl%C4%B1%C4%9F%C4%B1" title="Kolmogorov karmaşıklığı – Turkish" lang="tr" hreflang="tr" data-title="Kolmogorov karmaşıklığı" 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%9A%D0%BE%D0%BB%D0%BC%D0%BE%D0%B3%D0%BE%D1%80%D0%BE%D0%B2%D1%81%D1%8C%D0%BA%D0%B0_%D1%81%D0%BA%D0%BB%D0%B0%D0%B4%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Колмогоровська складність – Ukrainian" lang="uk" hreflang="uk" data-title="Колмогоровська складність" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%BC%94%E7%AE%97%E6%B3%95%E7%86%B5" title="演算法熵 – Cantonese" lang="yue" hreflang="yue" 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class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Mandelpart2_red.png/310px-Mandelpart2_red.png" decoding="async" width="310" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Mandelpart2_red.png/465px-Mandelpart2_red.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Mandelpart2_red.png/620px-Mandelpart2_red.png 2x" data-file-width="3200" data-file-height="2400" /></a><figcaption>This image illustrates part of the <a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a> <a href="/wiki/Fractal" title="Fractal">fractal</a>. Simply storing the 24-bit color of each pixel in this image would require 23 million bytes, but a small computer program can reproduce these 23 MB using the definition of the Mandelbrot set and the corner coordinates of the image. Thus, the Kolmogorov complexity of this image is much less than 23 MB in any pragmatic <a href="/wiki/Model_of_computation" title="Model of computation">model of computation</a>. <a href="/wiki/Portable_Network_Graphics" class="mw-redirect" title="Portable Network Graphics">PNG</a>'s general-purpose image compression only reduces it to 1.6 MB, smaller than the raw data but much larger than the Kolmogorov complexity.</figcaption></figure> <p>In <a href="/wiki/Algorithmic_information_theory" title="Algorithmic information theory">algorithmic information theory</a> (a subfield of <a href="/wiki/Computer_science" title="Computer science">computer science</a> and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>), the <b>Kolmogorov complexity</b> of an object, such as a piece of text, is the length of a shortest <a href="/wiki/Computer_program" title="Computer program">computer program</a> (in a predetermined <a href="/wiki/Programming_language" title="Programming language">programming language</a>) that produces the object as output. It is a measure of the <a href="/wiki/Computation" title="Computation">computational</a> resources needed to specify the object, and is also known as <b>algorithmic complexity</b>, <b>Solomonoff–Kolmogorov–Chaitin complexity</b>, <b>program-size complexity</b>, <b>descriptive complexity</b>, or <b>algorithmic entropy</b>. It is named after <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a>, who first published on the subject in 1963<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><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> and is a generalization of classical information theory. </p><p>The notion of Kolmogorov complexity can be used to state and <a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">prove impossibility</a> results akin to <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Cantor's diagonal argument</a>, <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's incompleteness theorem</a>, and <a href="/wiki/Halting_problem" title="Halting problem">Turing's halting problem</a>. In particular, no program <i>P</i> computing a <a href="/wiki/Lower_bound" class="mw-redirect" title="Lower bound">lower bound</a> for each text's Kolmogorov complexity can return a value essentially larger than <i>P</i>'s own length (see section <a href="#Chaitin's_incompleteness_theorem">§ Chaitin's incompleteness theorem</a>); hence no single program can compute the exact Kolmogorov complexity for infinitely many texts. Kolmogorov complexity is the length of the ultimately compressed version of a file (i.e., anything which can be put in a computer). Formally, it is the length of a shortest program from which the file can be reconstructed. While Kolmogorov complexity is uncomputable, various approaches have been proposed and reviewed.<sup id="cite_ref-zenil20202_3-0" class="reference"><a href="#cite_note-zenil20202-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Intuition">Intuition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=2" title="Edit section: Intuition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the following two <a href="/wiki/String_(computer_science)" title="String (computer science)">strings</a> of 32 lowercase letters and digits: </p> <dl><dd><code>abababababababababababababababab</code> , and</dd> <dd><code>4c1j5b2p0cv4w1x8rx2y39umgw5q85s7</code></dd></dl> <p>The first string has a short English-language description, namely "write ab 16 times", which consists of <b>17</b> characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, i.e., "write 4c1j5b2p0cv4w1x8rx2y39umgw5q85s7" which has <b>38</b> characters. Hence the operation of writing the first string can be said to have "less complexity" than writing the second. </p><p>More formally, the <a href="/wiki/Complexity" title="Complexity">complexity</a> of a string is the length of the shortest possible description of the string in some fixed <a href="/wiki/Turing_complete" class="mw-redirect" title="Turing complete">universal</a> description language (the sensitivity of complexity relative to the choice of description language is discussed below). It can be shown that the Kolmogorov complexity of any string cannot be more than a few bytes larger than the length of the string itself. Strings like the <i>abab</i> example above, whose Kolmogorov complexity is small relative to the string's size, are not considered to be complex. </p><p>The Kolmogorov complexity can be defined for any mathematical object, but for simplicity the scope of this article is restricted to strings. We must first specify a description language for strings. Such a description language can be based on any computer programming language, such as <a href="/wiki/Lisp_programming_language" class="mw-redirect" title="Lisp programming language">Lisp</a>, <a href="/wiki/Pascal_(programming_language)" title="Pascal (programming language)">Pascal</a>, or <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a>. If <b>P</b> is a program which outputs a string <i>x</i>, then <b>P</b> is a description of <i>x</i>. The length of the description is just the length of <b>P</b> as a character string, multiplied by the number of bits in a character (e.g., 7 for <a href="/wiki/ASCII" title="ASCII">ASCII</a>). </p><p>We could, alternatively, choose an encoding for <a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a>, where an <i>encoding</i> is a function which associates to each Turing Machine <b>M</b> a bitstring <<b>M</b>>. If <b>M</b> is a Turing Machine which, on input <i>w</i>, outputs string <i>x</i>, then the concatenated string <<b>M</b>> <i>w</i> is a description of <i>x</i>. For theoretical analysis, this approach is more suited for constructing detailed formal proofs and is generally preferred in the research literature. In this article, an informal approach is discussed. </p><p>Any string <i>s</i> has at least one description. For example, the second string above is output by the <a href="/wiki/Pseudo-code" class="mw-redirect" title="Pseudo-code">pseudo-code</a>: </p> <pre><b>function</b> GenerateString2() <b>return</b> "4c1j5b2p0cv4w1x8rx2y39umgw5q85s7" </pre> <p>whereas the first string is output by the (much shorter) pseudo-code: </p> <pre><b>function</b> GenerateString1() <b>return</b> "ab" × 16 </pre> <p>If a description <i>d</i>(<i>s</i>) of a string <i>s</i> is of minimal length (i.e., using the fewest bits), it is called a <b>minimal description</b> of <i>s</i>, and the length of <i>d</i>(<i>s</i>) (i.e. the number of bits in the minimal description) is the <b>Kolmogorov complexity</b> of <i>s</i>, written <i>K</i>(<i>s</i>). Symbolically, </p> <dl><dd><i>K</i>(<i>s</i>) = |<i>d</i>(<i>s</i>)|.</dd></dl> <p>The length of the shortest description will depend on the choice of description language; but the effect of changing languages is bounded (a result called the <i>invariance theorem</i>). </p> <div class="mw-heading mw-heading3"><h3 id="Plain_Kolmogorov_complexity_C">Plain Kolmogorov complexity <i>C</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=3" title="Edit section: Plain Kolmogorov complexity C"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are two definitions of Kolmogorov complexity: <i>plain</i> and <i>prefix-free</i>. The plain complexity is the minimal description length of any program, and denoted <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6b53682fddbe3028ffd75bd75771b86c6c7bd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.905ex; height:2.843ex;" alt="{\displaystyle C(x)}"></span> while the prefix-free complexity is the minimal description length of any program encoded in a prefix-free code, and denoted <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a621a6fbcc971649157a42ea42a599c22eee94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.205ex; height:2.843ex;" alt="{\displaystyle K(x)}"></span>. The plain complexity is more intuitive, but the prefix-free complexity is easier to study. </p><p>By default, all equations hold only up to an additive constant. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d0bb8fc04dfc0d04d53ee9ada38c794e51ca78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle f(x)=g(x)}"></span> really 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 f(x)=g(x)+O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=g(x)+O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80c21ce019319258761daaf31782d64b7cf230f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.356ex; height:2.843ex;" alt="{\displaystyle f(x)=g(x)+O(1)}"></span>, that 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 \exists c,\forall x,|f(x)-g(x)|\leq c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>c</mi> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists c,\forall x,|f(x)-g(x)|\leq c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4eb32bee0ea8e1b5305dac761162c0f7ee6573d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.901ex; height:2.843ex;" alt="{\displaystyle \exists c,\forall x,|f(x)-g(x)|\leq c}"></span>. </p><p>Let <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 U:2^{*}\to 2^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>:</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U:2^{*}\to 2^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eec0237626a8db2243b600cc689e1d553e64fbe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.767ex; height:2.343ex;" alt="{\displaystyle U:2^{*}\to 2^{*}}"></span> be a computable function mapping finite binary strings to binary strings. It is a universal function if, and only if, for any computable <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:2^{*}\to 2^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:2^{*}\to 2^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6219afe6ddda1092c072a18d91b961218e72471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.263ex; height:2.676ex;" alt="{\displaystyle f:2^{*}\to 2^{*}}"></span>, we can encode the function in a "program" <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 s_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861eb1d6654456882863ba0b53ca9ab551ad903c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.227ex; height:2.343ex;" alt="{\displaystyle s_{f}}"></span>, such 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 \forall x\in 2^{*},U(s_{f}x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in 2^{*},U(s_{f}x)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54645b5c6e64a33d5b2b8cdf9e9eca7fd92bdf04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.378ex; height:3.009ex;" alt="{\displaystyle \forall x\in 2^{*},U(s_{f}x)=f(x)}"></span>. We can think 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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> as a program interpreter, which takes in an initial segment describing the program, followed by data that the program should process. </p><p>One problem with plain complexity is 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 C(xy)\not <C(x)+C(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≮</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(xy)\not <C(x)+C(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d17428d41589b1c70b96b8d57403d7f1a70dc0ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.636ex; height:2.843ex;" alt="{\displaystyle C(xy)\not <C(x)+C(y)}"></span>, because intuitively speaking, there is no general way to tell where to divide an output string just by looking at the concatenated string. We can divide it by specifying the length 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 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> 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 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/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, but that would take <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(\min(\ln x,\ln y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> <mi>ln</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\min(\ln x,\ln y))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e78f589550e718ab91199d26e5ba9a010541972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.439ex; height:2.843ex;" alt="{\displaystyle O(\min(\ln x,\ln y))}"></span> extra symbols. Indeed, for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba126f626d61752f62eaacaf11761a54de4dc84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}"></span> there exists <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,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span> such 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 C(xy)\geq C(x)+C(y)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(xy)\geq C(x)+C(y)+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee0500bb5396e48c094b4914a23d157db4aaa03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.483ex; height:2.843ex;" alt="{\displaystyle C(xy)\geq C(x)+C(y)+c}"></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Typically, inequalities with plain complexity have a term like <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(\min(\ln x,\ln y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> <mi>ln</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\min(\ln x,\ln y))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e78f589550e718ab91199d26e5ba9a010541972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.439ex; height:2.843ex;" alt="{\displaystyle O(\min(\ln x,\ln y))}"></span> on one side, whereas the same inequalities with prefix-free complexity have only <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(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span>. </p><p>The main problem with plain complexity is that there is something extra sneaked into a program. A program not only represents for something with its code, but also represents its own length. In particular, a program <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> may represent a binary number up 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 \log _{2}|x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}|x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2820188ad3fea6ff5cd7387da1784a28942300b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.036ex; height:2.843ex;" alt="{\displaystyle \log _{2}|x|}"></span>, simply by its own length. Stated in another way, it is as if we are using a termination symbol to denote where a word ends, and so we are not using 2 symbols, but 3. To fix this defect, we introduce the prefix-free Kolmogorov complexity.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Prefix-free_Kolmogorov_complexity_K">Prefix-free Kolmogorov complexity <i>K</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=4" title="Edit section: Prefix-free Kolmogorov complexity K"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A prefix-free code is a subset 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 2^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f47b346be3d5aca04951ee13c4fbf15065496b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle 2^{*}}"></span> such that given any two different words <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,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span> in the set, neither is a prefix of the other. The benefit of a prefix-free code is that we can build a machine that reads words from the code forward in one direction, and as soon as it reads the last symbol of the word, it <i>knows</i> that the word is finished, and does not need to backtrack or a termination symbol. </p><p>Define a <b>prefix-free Turing machine</b> to be a Turing machine that comes with a prefix-free code, such that the Turing machine can read any string from the code in one direction, and stop reading as soon as it reads the last symbol. Afterwards, it may compute on a work tape and write to a write tape, but it cannot move its read-head anymore. </p><p>This gives us the following formal way to describe <i>K</i>.<sup id="cite_ref-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Fix a prefix-free universal Turing machine, with three tapes: a read tape infinite in one direction, a work tape infinite in two directions, and a write tape infinite in one direction.</li> <li>The machine can read from the read tape in one direction only (no backtracking), and write to the write tape in one direction only. It can read and write the work tape in both directions.</li> <li>The work tape and write tape start with all zeros. The read tape starts with an input prefix code, followed by all zeros.</li> <li>Let <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> be the prefix-free code on <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^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f47b346be3d5aca04951ee13c4fbf15065496b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle 2^{*}}"></span>, used by the universal Turing machine.</li></ul> <p>Note that some universal Turing machines may not be programmable with prefix codes. We must pick only a prefix-free universal Turing machine. </p><p>The prefix-free complexity of a string <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 shortest prefix code that makes the machine output <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>:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x):=\min\{|c|:c\in S,U(c)=x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mi>c</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x):=\min\{|c|:c\in S,U(c)=x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5868f25de180d0e055306ff23c68c15f389f244a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.796ex; height:2.843ex;" alt="{\displaystyle K(x):=\min\{|c|:c\in S,U(c)=x\}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Invariance_theorem">Invariance theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=5" title="Edit section: Invariance theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Informal_treatment">Informal treatment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=6" title="Edit section: Informal treatment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are some description languages which are optimal, in the following sense: given any description of an object in a description language, said description may be used in the optimal description language with a constant overhead. The constant depends only on the languages involved, not on the description of the object, nor the object being described. </p><p>Here is an example of an optimal description language. A description will have two parts: </p> <ul><li>The first part describes another description language.</li> <li>The second part is a description of the object in that language.</li></ul> <p>In more technical terms, the first part of a description is a computer program (specifically: a compiler for the object's language, written in the description language), with the second part being the input to that computer program which produces the object as output. </p><p><b>The invariance theorem follows:</b> Given any description language <i>L</i>, the optimal description language is at least as efficient as <i>L</i>, with some constant overhead. </p><p><b>Proof:</b> Any description <i>D</i> in <i>L</i> can be converted into a description in the optimal language by first describing <i>L</i> as a computer program <i>P</i> (part 1), and then using the original description <i>D</i> as input to that program (part 2). The total length of this new description <i>D′</i> is (approximately): </p> <dl><dd>|<i>D′</i> | = |<i>P</i>| + |<i>D</i>|</dd></dl> <p>The length of <i>P</i> is a constant that doesn't depend on <i>D</i>. So, there is at most a constant overhead, regardless of the object described. Therefore, the optimal language is universal <a href="/wiki/Up_to" title="Up to">up to</a> this additive constant. </p> <div class="mw-heading mw-heading3"><h3 id="A_more_formal_treatment">A more formal treatment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=7" title="Edit section: A more formal treatment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Theorem</b>: If <i>K</i><sub>1</sub> and <i>K</i><sub>2</sub> are the complexity functions relative to <a href="/wiki/Turing_complete" class="mw-redirect" title="Turing complete">Turing complete</a> description languages <i>L</i><sub>1</sub> and <i>L</i><sub>2</sub>, then there is a constant <i>c</i> – which depends only on the languages <i>L</i><sub>1</sub> and <i>L</i><sub>2</sub> chosen – such that </p> <dl><dd>∀<i>s</i>. −<i>c</i> ≤ <i>K</i><sub>1</sub>(<i>s</i>) − <i>K</i><sub>2</sub>(<i>s</i>) ≤ <i>c</i>.</dd></dl> <p><b>Proof</b>: By symmetry, it suffices to prove that there is some constant <i>c</i> such that for all strings <i>s</i> </p> <dl><dd><i>K</i><sub>1</sub>(<i>s</i>) ≤ <i>K</i><sub>2</sub>(<i>s</i>) + <i>c</i>.</dd></dl> <p>Now, suppose there is a program in the language <i>L</i><sub>1</sub> which acts as an <a href="/wiki/Interpreter_(computing)" title="Interpreter (computing)">interpreter</a> for <i>L</i><sub>2</sub>: </p> <pre><b>function</b> InterpretLanguage(<b>string</b> <i>p</i>) </pre> <p>where <i>p</i> is a program in <i>L</i><sub>2</sub>. The interpreter is characterized by the following property: </p> <dl><dd>Running <code>InterpretLanguage</code> on input <i>p</i> returns the result of running <i>p</i>.</dd></dl> <p>Thus, if <b>P</b> is a program in <i>L</i><sub>2</sub> which is a minimal description of <i>s</i>, then <code>InterpretLanguage</code>(<b>P</b>) returns the string <i>s</i>. The length of this description of <i>s</i> is the sum of </p> <ol><li>The length of the program <code>InterpretLanguage</code>, which we can take to be the constant <i>c</i>.</li> <li>The length of <b>P</b> which by definition is <i>K</i><sub>2</sub>(<i>s</i>).</li></ol> <p>This proves the desired upper bound. </p> <div class="mw-heading mw-heading2"><h2 id="History_and_context">History and context</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=8" title="Edit section: History and context"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Algorithmic_information_theory" title="Algorithmic information theory">Algorithmic information theory</a> is the area of computer science that studies Kolmogorov complexity and other complexity measures on strings (or other <a href="/wiki/Data_structure" title="Data structure">data structures</a>). </p><p>The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by <a href="/wiki/Ray_Solomonoff" title="Ray Solomonoff">Ray Solomonoff</a>, who published it in 1960, describing it in "A Preliminary Report on a General Theory of Inductive Inference"<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> as part of his invention of <a href="/wiki/Algorithmic_probability" title="Algorithmic probability">algorithmic probability</a>. He gave a more complete description in his 1964 publications, "A Formal Theory of Inductive Inference," Part 1 and Part 2 in <i>Information and Control</i>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><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> </p><p>Andrey Kolmogorov later <a href="/wiki/Multiple_discovery" title="Multiple discovery">independently published</a> this theorem in <i>Problems Inform. Transmission</i><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> in 1965. <a href="/wiki/Gregory_Chaitin" title="Gregory Chaitin">Gregory Chaitin</a> also presents this theorem in <i>J. ACM</i> – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.<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> </p><p>The theorem says that, among algorithms that decode strings from their descriptions (codes), there exists an optimal one. This algorithm, for all strings, allows codes as short as allowed by any other algorithm up to an additive constant that depends on the algorithms, but not on the strings themselves. Solomonoff used this algorithm and the code lengths it allows to define a "universal probability" of a string on which inductive inference of the subsequent digits of the string can be based. Kolmogorov used this theorem to define several functions of strings, including complexity, randomness, and information. </p><p>When Kolmogorov became aware of Solomonoff's work, he acknowledged Solomonoff's priority.<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> For several years, Solomonoff's work was better known in the Soviet Union than in the Western World. The general consensus in the scientific community, however, was to associate this type of complexity with Kolmogorov, who was concerned with randomness of a sequence, while Algorithmic Probability became associated with Solomonoff, who focused on prediction using his invention of the universal prior probability distribution. The broader area encompassing descriptional complexity and probability is often called Kolmogorov complexity. The computer scientist <a href="/wiki/Ming_Li" title="Ming Li">Ming Li</a> considers this an example of the <a href="/wiki/Matthew_effect_(sociology)" class="mw-redirect" title="Matthew effect (sociology)">Matthew effect</a>: "...to everyone who has, more will be given..."<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> </p><p>There are several other variants of Kolmogorov complexity or algorithmic information. The most widely used one is based on <a href="/w/index.php?title=Self-delimiting_program&action=edit&redlink=1" class="new" title="Self-delimiting program (page does not exist)">self-delimiting programs</a>, and is mainly due to <a href="/wiki/Leonid_Levin" title="Leonid Levin">Leonid Levin</a> (1974). </p><p>An axiomatic approach to Kolmogorov complexity based on <a href="/wiki/Blum_axioms" title="Blum axioms">Blum axioms</a> (Blum 1967) was introduced by Mark Burgin in the paper presented for publication by Andrey Kolmogorov.<sup id="cite_ref-Burgin1982_14-0" class="reference"><a href="#cite_note-Burgin1982-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Basic_results">Basic results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=9" title="Edit section: Basic results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We 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 K(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b4554c1b79ecaae2584d99cc3959f2eed7e6d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.394ex; height:2.843ex;" alt="{\displaystyle K(x,y)}"></span> 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 K((x,y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K((x,y))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f92c334856676ce3c009d748f0cb579e10e61db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle K((x,y))}"></span>, where <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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span> means some fixed way to code for a tuple of strings x and y. </p> <div class="mw-heading mw-heading3"><h3 id="Inequalities">Inequalities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=10" title="Edit section: Inequalities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We omit additive factors 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 O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span>. This section is based on.<sup id="cite_ref-:0_6-1" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><b>Theorem.</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(x)\leq K(x)+2\log _{2}C(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(x)\leq K(x)+2\log _{2}C(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb8a9dc8684ac54a3f6c3943a430a56c1b3d7ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.917ex; height:2.843ex;" alt="{\displaystyle C(x)\leq K(x)+2\log _{2}C(x)}"></span> </p><p><b>Proof.</b> Take any program for the universal Turing machine used to define plain complexity, and convert it to a prefix-free program by first coding the length of the program in binary, then convert the length to prefix-free coding. For example, suppose the program has length 9, then we can convert it as follows:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9\mapsto 1001\mapsto 11-00-00-11-\color {red}{01}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo stretchy="false">↦</mo> <mn>1001</mn> <mo stretchy="false">↦</mo> <mn>11</mn> <mo>−</mo> <mn>00</mn> <mo>−</mo> <mn>00</mn> <mo>−</mo> <mn>11</mn> <mo>−</mo> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9\mapsto 1001\mapsto 11-00-00-11-\color {red}{01}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7093cc2eb9fecda41b07384b09f8b242b3b2d939" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:36.027ex; height:2.343ex;" alt="{\displaystyle 9\mapsto 1001\mapsto 11-00-00-11-\color {red}{01}}"></span>where we double each digit, then add a termination code. The prefix-free universal Turing machine can then read in any program for the other machine as follows:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\text{code for simulating the other machine}}][{\text{coded length of the program}}][{\text{the program}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>code for simulating the other machine</mtext> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>coded length of the program</mtext> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>the program</mtext> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\text{code for simulating the other machine}}][{\text{coded length of the program}}][{\text{the program}}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7fb908ae6de37e990e880d0a297db3601251fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:82.084ex; height:2.843ex;" alt="{\displaystyle [{\text{code for simulating the other machine}}][{\text{coded length of the program}}][{\text{the program}}]}"></span>The first part programs the machine to simulate the other machine, and is a constant overhead <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(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span>. The second part has 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 \leq 2\log _{2}C(x)+3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mn>2</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq 2\log _{2}C(x)+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aefa7b2063105071f36ad39f67d8d676cce215f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.324ex; height:2.843ex;" alt="{\displaystyle \leq 2\log _{2}C(x)+3}"></span>. The third part has 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 C(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6b53682fddbe3028ffd75bd75771b86c6c7bd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.905ex; height:2.843ex;" alt="{\displaystyle C(x)}"></span>. </p><p><b>Theorem</b>: There exists <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/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> such 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 \forall x,C(x)\leq |x|+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,C(x)\leq |x|+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c20c8b32d3c5e3d974f717cf97b02e1c969a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.131ex; height:2.843ex;" alt="{\displaystyle \forall x,C(x)\leq |x|+c}"></span>. More succinctly, <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(x)\leq |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(x)\leq |x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245421cbfe61712de9ed40ef695e0aebb6480dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.627ex; height:2.843ex;" alt="{\displaystyle C(x)\leq |x|}"></span>. Similarly, <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(x)\leq |x|+2\log _{2}|x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>2</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x)\leq |x|+2\log _{2}|x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94f5d0e08b7467647615ac400479bd134ad77d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.353ex; height:2.843ex;" alt="{\displaystyle K(x)\leq |x|+2\log _{2}|x|}"></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 K(x||x|)\leq |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x||x|)\leq |x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae95c2bb5cd0251613d9df89b62b41e74f1fed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.197ex; height:2.843ex;" alt="{\displaystyle K(x||x|)\leq |x|}"></span>.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="This appears to be the first use of conditional complexity notation. Its definition should occur before here. Moreover, notation could better be changed to avoid confusion with string length (maybe, string length is better denoted by '#x'). (January 2024)">clarification needed</span></a></i>]</sup> </p><p><b>Proof.</b> For the plain complexity, just write a program that simply copies the input to the output. For the prefix-free complexity, we need to first describe the length of the string, before writing out the string itself. </p><p><b>Theorem. (extra information bounds, subadditivity)</b> </p> <ul><li><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(x|y)\leq K(x)\leq K(x,y)\leq \max(K(x|y)+K(y),K(y|x)+K(x))\leq K(x)+K(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x|y)\leq K(x)\leq K(x,y)\leq \max(K(x|y)+K(y),K(y|x)+K(x))\leq K(x)+K(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759b8a60b87044b3f131fe64040b19b2d61996bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:82.176ex; height:2.843ex;" alt="{\displaystyle K(x|y)\leq K(x)\leq K(x,y)\leq \max(K(x|y)+K(y),K(y|x)+K(x))\leq K(x)+K(y)}"></span></li></ul> <ul><li><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(xy)\leq K(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(xy)\leq K(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95829709755988e98b4bece18ab5ccfdc3105167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.853ex; height:2.843ex;" alt="{\displaystyle K(xy)\leq K(x,y)}"></span></li></ul> <p>Note that there is no way to compare <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(xy)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(xy)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c27779d859cebeb2c82e042cc8005d85d32e25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.36ex; height:2.843ex;" alt="{\displaystyle K(xy)}"></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 K(x|y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x|y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93d70e70ae35d7640bbc98a6042641e75b49f507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.007ex; height:2.843ex;" alt="{\displaystyle K(x|y)}"></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 K(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a621a6fbcc971649157a42ea42a599c22eee94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.205ex; height:2.843ex;" alt="{\displaystyle K(x)}"></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 K(y|x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(y|x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/701aeff166e07145d3794d1965684eb961d1910d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.007ex; height:2.843ex;" alt="{\displaystyle K(y|x)}"></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 K(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc34d1b0e7ceebda1e774fba6e33b8b3afbc236" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.031ex; height:2.843ex;" alt="{\displaystyle K(y)}"></span>. There are strings such that the whole string <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 xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span> is easy to describe, but its substrings are very hard to describe. </p><p><b>Theorem. (symmetry of information)</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x,y)=K(x|y,K(y))+K(y)=K(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x,y)=K(x|y,K(y))+K(y)=K(y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce64094e0356ea949b964ea063cbe328935cb542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.929ex; height:2.843ex;" alt="{\displaystyle K(x,y)=K(x|y,K(y))+K(y)=K(y,x)}"></span>. </p><p><b>Proof.</b> One side is simple. For the other side 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 K(x,y)\geq K(x|y,K(y))+K(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x,y)\geq K(x|y,K(y))+K(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852e63e17955e5446b8eee7f210224cef06d2c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.436ex; height:2.843ex;" alt="{\displaystyle K(x,y)\geq K(x|y,K(y))+K(y)}"></span>, we need to use a counting argument (page 38 <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>). </p><p><b>Theorem. (information non-increase)</b> For any computable 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, we 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 K(f(x))\leq K(x)+K(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(f(x))\leq K(x)+K(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b199484b74d2a88cb68edb4dae8a75175183fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.59ex; height:2.843ex;" alt="{\displaystyle K(f(x))\leq K(x)+K(f)}"></span>. </p><p><b>Proof.</b> Program the Turing machine to read two subsequent programs, one describing the function and one describing the string. Then run both programs on the work tape to produce <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span>, and write it out. </p> <div class="mw-heading mw-heading3"><h3 id="Uncomputability_of_Kolmogorov_complexity">Uncomputability of Kolmogorov complexity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=11" title="Edit section: Uncomputability of Kolmogorov complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="A_naive_attempt_at_a_program_to_compute_K">A naive attempt at a program to compute <i>K</i></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=12" title="Edit section: A naive attempt at a program to compute K"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At first glance it might seem trivial to write a program which can compute <i>K</i>(<i>s</i>) for any <i>s</i>, such as the following: </p> <pre><b>function</b> KolmogorovComplexity(<b>string</b> s) <b>for</b> i = 1 <b>to</b> infinity: <b>for each</b> string p <b>of</b> length exactly i <b>if</b> isValidProgram(p) <b>and</b> evaluate(p) == s <b>return</b> i </pre> <p>This program iterates through all possible programs (by iterating through all possible strings and only considering those which are valid programs), starting with the shortest. Each program is executed to find the result produced by that program, comparing it to the input <i>s</i>. If the result matches then the length of the program is returned. </p><p>However this will not work because some of the programs <i>p</i> tested will not terminate, e.g. if they contain infinite loops. There is no way to avoid all of these programs by testing them in some way before executing them due to the non-computability of the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a>. </p><p>What is more, no program at all can compute the function <i>K</i>, be it ever so sophisticated. This is proven in the following. </p> <div class="mw-heading mw-heading4"><h4 id="Formal_proof_of_uncomputability_of_K">Formal proof of uncomputability of <i>K</i></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=13" title="Edit section: Formal proof of uncomputability of K"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Theorem</b>: There exist strings of arbitrarily large Kolmogorov complexity. Formally: for each natural number <i>n</i>, there is a string <i>s</i> with <i>K</i>(<i>s</i>) ≥ <i>n</i>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> </p><p><b>Proof:</b> Otherwise all of the infinitely many possible finite strings could be generated by the finitely many<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> programs with a complexity below <i>n</i> bits. </p><p><b>Theorem</b>: <i>K</i> is not a <a href="/wiki/Computable_function" title="Computable function">computable function</a>. In other words, there is no program which takes any string <i>s</i> as input and produces the integer <i>K</i>(<i>s</i>) as output. </p><p>The following <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction"><b>proof</b> by contradiction</a> uses a simple <a href="/wiki/Pascal_(programming_language)" title="Pascal (programming language)">Pascal</a>-like language to denote programs; for sake of proof simplicity assume its description (i.e. an <a href="/wiki/Interpreter_(computing)" title="Interpreter (computing)">interpreter</a>) to have a length of <span class="nowrap"><span data-sort-value="7006140000000000000♠"></span>1<span style="margin-left:.25em;">400</span><span style="margin-left:.25em;">000</span></span> bits. Assume for contradiction there is a program </p> <pre><b>function</b> KolmogorovComplexity(<b>string</b> s) </pre> <p>which takes as input a string <i>s</i> and returns <i>K</i>(<i>s</i>). All programs are of finite length so, for sake of proof simplicity, assume it to be <span class="nowrap"><span data-sort-value="7009700000000000000♠"></span>7<span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">000</span></span> bits. Now, consider the following program of length <span class="nowrap"><span data-sort-value="7003128800000000000♠"></span>1288</span> bits: </p> <pre><b>function</b> GenerateComplexString() <b>for</b> i = 1 <b>to</b> infinity: <b>for each</b> string s <b>of</b> length exactly i <b>if</b> KolmogorovComplexity(s) ≥ 8000000000 <b>return</b> s </pre> <p>Using <code>KolmogorovComplexity</code> as a subroutine, the program tries every string, starting with the shortest, until it returns a string with Kolmogorov complexity at least <span class="nowrap"><span data-sort-value="7009800000000000000♠"></span>8<span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">000</span></span> bits,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> i.e. a string that cannot be produced by any program shorter than <span class="nowrap"><span data-sort-value="7009800000000000000♠"></span>8<span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">000</span></span> bits. However, the overall length of the above program that produced <i>s</i> is only <span class="nowrap"><span data-sort-value="7009700140128800000♠"></span>7<span style="margin-left:.25em;">001</span><span style="margin-left:.25em;">401</span><span style="margin-left:.25em;">288</span></span> bits,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>note 4<span class="cite-bracket">]</span></a></sup> which is a contradiction. (If the code of <code>KolmogorovComplexity</code> is shorter, the contradiction remains. If it is longer, the constant used in <code>GenerateComplexString</code> can always be changed appropriately.)<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>note 5<span class="cite-bracket">]</span></a></sup> </p><p>The above proof uses a contradiction similar to that of the <a href="/wiki/Berry_paradox" title="Berry paradox">Berry paradox</a>: "<sub><style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style><span class="tmp-color" style="color:#8080ff">1</span></sub>The <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">2</span></sub>smallest <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">3</span></sub>positive <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">4</span></sub>integer <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">5</span></sub>that <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">6</span></sub>cannot <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">7</span></sub>be <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">8</span></sub>defined <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">9</span></sub>in <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">10</span></sub>fewer <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">11</span></sub>than <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">12</span></sub>twenty <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">13</span></sub>English <sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#8080ff">14</span></sub>words". It is also possible to show the non-computability of <i>K</i> by reduction from the non-computability of the halting problem <i>H</i>, since <i>K</i> and <i>H</i> are <a href="/wiki/Turing_degree#Turing_equivalence" title="Turing degree">Turing-equivalent</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>There is a corollary, humorously called the "<a href="/wiki/Full_employment_theorem" class="mw-redirect" title="Full employment theorem">full employment theorem</a>" in the programming language community, stating that there is no perfect size-optimizing compiler. </p> <div class="mw-heading mw-heading3"><h3 id="Chain_rule_for_Kolmogorov_complexity">Chain rule for Kolmogorov complexity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=14" title="Edit section: Chain rule for Kolmogorov complexity"><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/Chain_rule_for_Kolmogorov_complexity" title="Chain rule for Kolmogorov complexity">Chain rule for Kolmogorov complexity</a></div> <p>The chain rule<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> for Kolmogorov complexity states that there exists a constant <i>c</i> such that for all <i>X</i> and <i>Y</i>: </p> <dl><dd><i>K</i>(<i>X</i>,<i>Y</i>) = <i>K</i>(<i>X</i>) + <i>K</i>(<i>Y</i>|<i>X</i>) + c*max(1,log(<i>K</i>(<i>X</i>,<i>Y</i>))).</dd></dl> <p>It states that the shortest program that reproduces <i>X</i> and <i>Y</i> is <a href="/wiki/Big-O_notation" class="mw-redirect" title="Big-O notation">no more</a> than a logarithmic term larger than a program to reproduce <i>X</i> and a program to reproduce <i>Y</i> given <i>X</i>. Using this statement, one can define <a href="/wiki/Mutual_information#Absolute_mutual_information" title="Mutual information">an analogue of mutual information for Kolmogorov complexity</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Compression">Compression</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=15" title="Edit section: Compression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is straightforward to compute upper bounds for <i>K</i>(<i>s</i>) – simply <a href="/wiki/Data_compression" title="Data compression">compress</a> the string <i>s</i> with some method, implement the corresponding decompressor in the chosen language, concatenate the decompressor to the compressed string, and measure the length of the resulting string – concretely, the size of a <a href="/wiki/Self-extracting_archive" title="Self-extracting archive">self-extracting archive</a> in the given language. </p><p>A string <i>s</i> is compressible by a number <i>c</i> if it has a description whose length does not exceed |<i>s</i>| − <i>c</i> bits. This is equivalent to saying that <i>K</i>(<i>s</i>) ≤ |<i>s</i>| − <i>c</i>. Otherwise, <i>s</i> is incompressible by <i>c</i>. A string incompressible by 1 is said to be simply <i>incompressible</i> – by the <a href="/wiki/Pigeonhole_principle" title="Pigeonhole principle">pigeonhole principle</a>, which applies because every compressed string maps to only one uncompressed string, <a href="/wiki/Incompressible_string" title="Incompressible string">incompressible strings</a> must exist, since there are 2<sup><i>n</i></sup> bit strings of length <i>n</i>, but only 2<sup><i>n</i></sup> − 1 shorter strings, that is, strings of length less than <i>n</i>, (i.e. with length 0, 1, ..., <i>n − 1).<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>note 6<span class="cite-bracket">]</span></a></sup></i> </p><p>For the same reason, most strings are complex in the sense that they cannot be significantly compressed – their <i>K</i>(<i>s</i>) is not much smaller than |<i>s</i>|, the length of <i>s</i> in bits. To make this precise, fix a value of <i>n</i>. There are 2<sup><i>n</i></sup> bitstrings of length <i>n</i>. The <a href="/wiki/Uniform_distribution_(discrete)" class="mw-redirect" title="Uniform distribution (discrete)">uniform</a> <a href="/wiki/Probability" title="Probability">probability</a> distribution on the space of these bitstrings assigns exactly equal weight 2<sup>−<i>n</i></sup> to each string of length <i>n</i>. </p><p><b>Theorem</b>: With the uniform probability distribution on the space of bitstrings of length <i>n</i>, the probability that a string is incompressible by <i>c</i> is at least 1 − 2<sup>−<i>c</i>+1</sup> + 2<sup>−<i>n</i></sup>. </p><p>To prove the theorem, note that the number of descriptions of length not exceeding <i>n</i> − <i>c</i> is given by the geometric series: </p> <dl><dd>1 + 2 + 2<sup>2</sup> + ... + 2<sup><i>n</i> − <i>c</i></sup> = 2<sup><i>n</i>−<i>c</i>+1</sup> − 1.</dd></dl> <p>There remain at least </p> <dl><dd>2<sup><i>n</i></sup> − 2<sup><i>n</i>−<i>c</i>+1</sup> + 1</dd></dl> <p>bitstrings of length <i>n</i> that are incompressible by <i>c</i>. To determine the probability, divide by 2<sup><i>n</i></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Chaitin's_incompleteness_theorem"><span id="Chaitin.27s_incompleteness_theorem"></span>Chaitin's incompleteness theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=16" title="Edit section: Chaitin's incompleteness theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Kolmogorov_complexity_and_computable_lower_bounds_svg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Kolmogorov_complexity_and_computable_lower_bounds_svg.svg/600px-Kolmogorov_complexity_and_computable_lower_bounds_svg.svg.png" decoding="async" width="600" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Kolmogorov_complexity_and_computable_lower_bounds_svg.svg/900px-Kolmogorov_complexity_and_computable_lower_bounds_svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Kolmogorov_complexity_and_computable_lower_bounds_svg.svg/1200px-Kolmogorov_complexity_and_computable_lower_bounds_svg.svg.png 2x" data-file-width="744" data-file-height="195" /></a><figcaption>Kolmogorov complexity <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#ff0000"><i>K</i>(<i>s</i>)</span>, and two computable lower bound functions <code><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#00e000">prog1(s)</span></code>, <code><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#0000ff">prog2(s)</span></code>. The horizontal axis (<a href="/wiki/Logarithmic_scale" title="Logarithmic scale">logarithmic scale</a>) enumerates all <a href="/wiki/String_(computer_science)" title="String (computer science)">strings</a> <i>s</i>, ordered by length; the vertical axis (<a href="/wiki/Linear_scale" title="Linear scale">linear scale</a>) measures Kolmogorov complexity in <a href="/wiki/Bit" title="Bit">bits</a>. Most strings are incompressible, i.e. their Kolmogorov complexity exceeds their length by a constant amount. 9 compressible strings are shown in the picture, appearing as almost vertical slopes. Due to Chaitin's incompleteness theorem (1974), the output of any program computing a lower bound of the Kolmogorov complexity cannot exceed some fixed limit, which is independent of the input string <i>s</i>.</figcaption></figure> <p>By the above theorem (<a href="#Compression">§ Compression</a>), most strings are complex in the sense that they cannot be described in any significantly "compressed" way. However, it turns out that the fact that a specific string is complex cannot be formally proven, if the complexity of the string is above a certain threshold. The precise formalization is as follows. First, fix a particular <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic system</a> <b>S</b> for the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. The axiomatic system has to be powerful enough so that, to certain assertions <b>A</b> about complexity of strings, one can associate a formula <b>F</b><sub><b>A</b></sub> in <b>S</b>. This association must have the following property: </p><p>If <b>F</b><sub><b>A</b></sub> is provable from the axioms of <b>S</b>, then the corresponding assertion <b>A</b> must be true. This "formalization" can be achieved based on a <a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a>. </p><p><b>Theorem</b>: There exists a constant <i>L</i> (which only depends on <b>S</b> and on the choice of description language) such that there does not exist a string <i>s</i> for which the statement </p> <dl><dd><i>K</i>(<i>s</i>) ≥ <i>L</i>       (as formalized in <b>S</b>)</dd></dl> <p>can be proven within <b>S</b>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p><b>Proof Idea</b>: The proof of this result is modeled on a self-referential construction used in <a href="/wiki/Berry%27s_paradox" class="mw-redirect" title="Berry's paradox">Berry's paradox</a>. We firstly obtain a program which enumerates the proofs within <b>S</b> and we specify a procedure <i>P</i> which takes as an input an integer <i>L</i> and prints the strings <i>x</i> which are within proofs within <b>S</b> of the statement <i>K</i>(<i>x</i>) ≥ <i>L</i>. By then setting <i>L</i> to greater than the length of this procedure <i>P</i>, we have that the required length of a program to print <i>x</i> as stated in <i>K</i>(<i>x</i>) ≥ <i>L</i> as being at least <i>L</i> is then less than the amount <i>L</i> since the string <i>x</i> was printed by the procedure <i>P</i>. This is a contradiction. So it is not possible for the proof system <b>S</b> to prove <i>K</i>(<i>x</i>) ≥ <i>L</i> for <i>L</i> arbitrarily large, in particular, for <i>L</i> larger than the length of the procedure <i>P</i>, (which is finite). </p><p><b>Proof</b>: </p><p>We can find an effective enumeration of all the formal proofs in <b>S</b> by some procedure </p> <pre><b>function</b> NthProof(<b>int</b> <i>n</i>) </pre> <p>which takes as input <i>n</i> and outputs some proof. This function enumerates all proofs. Some of these are proofs for formulas we do not care about here, since every possible proof in the language of <b>S</b> is produced for some <i>n</i>. Some of these are complexity formulas of the form <i>K</i>(<i>s</i>) ≥ <i>n</i> where <i>s</i> and <i>n</i> are constants in the language of <b>S</b>. There is a procedure </p> <pre><b>function</b> NthProofProvesComplexityFormula(<b>int</b> <i>n</i>) </pre> <p>which determines whether the <i>n</i>th proof actually proves a complexity formula <i>K</i>(<i>s</i>) ≥ <i>L</i>. The strings <i>s</i>, and the integer <i>L</i> in turn, are computable by procedure: </p> <pre><b>function</b> StringNthProof(<b>int</b> <i>n</i>) </pre> <pre><b>function</b> ComplexityLowerBoundNthProof(<b>int</b> <i>n</i>) </pre> <p>Consider the following procedure: </p> <pre><b>function</b> GenerateProvablyComplexString(<b>int</b> <i>n</i>) <b>for</b> i = 1 to infinity: <b>if</b> NthProofProvesComplexityFormula(i) <b>and</b> ComplexityLowerBoundNthProof(i) ≥ <i>n</i> <b>return</b> StringNthProof(<i>i</i>) </pre> <p>Given an <i>n</i>, this procedure tries every proof until it finds a string and a proof in the formal system <b>S</b> of the formula <i>K</i>(<i>s</i>) ≥ <i>L</i> for some <i>L</i> ≥ <i>n</i>; if no such proof exists, it loops forever. </p><p>Finally, consider the program consisting of all these procedure definitions, and a main call: </p> <pre>GenerateProvablyComplexString(<i>n</i><sub>0</sub>) </pre> <p>where the constant <i>n</i><sub>0</sub> will be determined later on. The overall program length can be expressed as <i>U</i>+log<sub>2</sub>(<i>n</i><sub>0</sub>), where <i>U</i> is some constant and log<sub>2</sub>(<i>n</i><sub>0</sub>) represents the length of the integer value <i>n</i><sub>0</sub>, under the reasonable assumption that it is encoded in binary digits. We will choose <i>n</i><sub>0</sub> to be greater than the program length, that is, such that <i>n</i><sub>0</sub> > <i>U</i>+log<sub>2</sub>(<i>n</i><sub>0</sub>). This is clearly true for <i>n</i><sub>0</sub> sufficiently large, because the left hand side grows linearly in <i>n</i><sub>0</sub> whilst the right hand side grows logarithmically in <i>n</i><sub>0</sub> up to the fixed constant <i>U</i>. </p><p>Then no proof of the form "<i>K</i>(<i>s</i>)≥<i>L</i>" with <i>L</i>≥<i>n</i><sub>0</sub> can be obtained in <b>S</b>, as can be seen by an <a href="/wiki/Indirect_argument" class="mw-redirect" title="Indirect argument">indirect argument</a>: If <code>ComplexityLowerBoundNthProof(i)</code> could return a value ≥<i>n</i><sub>0</sub>, then the loop inside <code>GenerateProvablyComplexString</code> would eventually terminate, and that procedure would return a string <i>s</i> such that </p> <table style="border: 1px solid darkgray;"> <tbody><tr> <td></td> <td><i>K</i>(<i>s</i>) </td></tr> <tr> <td>≥</td> <td><i>n</i><sub>0</sub></td> <td>         </td> <td>by construction of <code>GenerateProvablyComplexString</code> </td></tr> <tr> <td>></td> <td><i>U</i>+log<sub>2</sub>(<i>n</i><sub>0</sub>)</td> <td></td> <td>by the choice of <i>n</i><sub>0</sub> </td></tr> <tr> <td>≥</td> <td><i>K</i>(<i>s</i>)</td> <td></td> <td>since <i>s</i> was described by the program with that length </td></tr></tbody></table> <p>This is a contradiction, <a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a> </p><p>As a consequence, the above program, with the chosen value of <i>n</i><sub>0</sub>, must loop forever. </p><p>Similar ideas are used to prove the properties of <a href="/wiki/Chaitin%27s_constant" title="Chaitin's constant">Chaitin's constant</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Minimum_message_length">Minimum message length</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=17" title="Edit section: Minimum message length"><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/Minimum_message_length" title="Minimum message length">Minimum message length</a></div> <p>The minimum message length principle of statistical and inductive inference and machine learning was developed by <a href="/wiki/Chris_Wallace_(computer_scientist)" title="Chris Wallace (computer scientist)">C.S. Wallace</a> and D.M. Boulton in 1968. MML is <a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian</a> (i.e. it incorporates prior beliefs) and information-theoretic. It has the desirable properties of statistical invariance (i.e. the inference transforms with a re-parametrisation, such as from polar coordinates to Cartesian coordinates), statistical consistency (i.e. even for very hard problems, MML will converge to any underlying model) and efficiency (i.e. the MML model will converge to any true underlying model about as quickly as is possible). C.S. Wallace and D.L. Dowe (1999) showed a formal connection between MML and algorithmic information theory (or Kolmogorov complexity).<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Kolmogorov_randomness">Kolmogorov randomness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=18" title="Edit section: Kolmogorov randomness"><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/Algorithmically_random_sequence" title="Algorithmically random sequence">Algorithmically random sequence</a></div> <p><i>Kolmogorov randomness</i> defines a string (usually of <a href="/wiki/Bit" title="Bit">bits</a>) as being <a href="/wiki/Randomness" title="Randomness">random</a> if and only if every <a href="/wiki/Computer_program" title="Computer program">computer program</a> that can produce that string is at least as long as the string itself. To make this precise, a universal computer (or <a href="/wiki/Universal_Turing_machine" title="Universal Turing machine">universal Turing machine</a>) must be specified, so that "program" means a program for this universal machine. A random string in this sense is "incompressible" in that it is impossible to "compress" the string into a program that is shorter than the string itself. For every universal computer, there is at least one algorithmically random string of each length.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Whether a particular string is random, however, depends on the specific universal computer that is chosen. This is because a universal computer can have a particular string hard-coded in itself, and a program running on this universal computer can then simply refer to this hard-coded string using a short sequence of bits (i.e. much shorter than the string itself). </p><p>This definition can be extended to define a notion of randomness for <i>infinite</i> sequences from a finite alphabet. These <a href="/wiki/Algorithmically_random_sequence" title="Algorithmically random sequence">algorithmically random sequences</a> can be defined in three equivalent ways. One way uses an effective analogue of <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>; another uses effective <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingales</a>. The third way defines an infinite sequence to be random if the prefix-free Kolmogorov complexity of its initial segments grows quickly enough — there must be a constant <i>c</i> such that the complexity of an initial segment of length <i>n</i> is always at least <i>n</i>−<i>c</i>. This definition, unlike the definition of randomness for a finite string, is not affected by which universal machine is used to define prefix-free Kolmogorov complexity.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_entropy">Relation to entropy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=19" title="Edit section: Relation to entropy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For dynamical systems, entropy rate and algorithmic complexity of the trajectories are related by a theorem of Brudno, that the equality <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(x;T)=h(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x;T)=h(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e702d8fc2b76af6709b8df186cbee7839baf15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.758ex; height:2.843ex;" alt="{\displaystyle K(x;T)=h(T)}"></span> holds for almost all <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>.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>It can be shown<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> that for the output of <a href="/wiki/Markov_information_source" title="Markov information source">Markov information sources</a>, Kolmogorov complexity is related to the <a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">entropy</a> of the information source. More precisely, the Kolmogorov complexity of the output of a Markov information source, normalized by the length of the output, converges almost surely (as the length of the output goes to infinity) to the <a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">entropy</a> of the source. </p><p><b>Theorem.</b> (Theorem 14.2.5 <sup id="cite_ref-:1_31-0" class="reference"><a href="#cite_note-:1-31"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>) The conditional Kolmogorov complexity of a binary string <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_{1:n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>:</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1:n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22243edba296d9059dcd64505ca9252581a90ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.828ex; height:2.009ex;" alt="{\displaystyle x_{1:n}}"></span> satisfies<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n}}K(x_{1:n}|n)\leq H_{b}\left({\frac {1}{n}}\sum _{i}x_{i}\right)+{\frac {\log n}{2n}}+O(1/n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}K(x_{1:n}|n)\leq H_{b}\left({\frac {1}{n}}\sum _{i}x_{i}\right)+{\frac {\log n}{2n}}+O(1/n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/400d28435b9fc24158b61cd6c0b625b591488044" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.073ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{n}}K(x_{1:n}|n)\leq H_{b}\left({\frac {1}{n}}\sum _{i}x_{i}\right)+{\frac {\log n}{2n}}+O(1/n)}"></span>where <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 H_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574b50e73ed6ce90345d40bda58dd780bce30f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle H_{b}}"></span> is the <a href="/wiki/Binary_entropy_function" title="Binary entropy function">binary entropy function</a> (not to be confused with the entropy rate). </p> <div class="mw-heading mw-heading2"><h2 id="Halting_problem">Halting problem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=20" title="Edit section: Halting problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Kolmogorov complexity function is equivalent to deciding the halting problem. </p><p>If we have a halting oracle, then the Kolmogorov complexity of a string can be computed by simply trying every halting program, in lexicographic order, until one of them outputs the string. </p><p>The other direction is much more involved.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> It shows that given a Kolmogorov complexity function, we can construct a 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 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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, such 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(n)\geq BB(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>B</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)\geq BB(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ba6ffab48b4e97b6a53b177f154d533bf17634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.293ex; height:2.843ex;" alt="{\displaystyle p(n)\geq BB(n)}"></span> for all large <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>, where <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 BB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58e28c8465f8b93b870d374f0b92986e5a93934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.528ex; height:2.176ex;" alt="{\displaystyle BB}"></span> is the <a href="/wiki/Busy_beaver" title="Busy beaver">Busy Beaver</a> shift function (also denoted 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 S(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896aa95cb194816f48a89fecd7e304d4b7f1af71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.703ex; height:2.843ex;" alt="{\displaystyle S(n)}"></span>). By modifying the function at lower values 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> we get an upper bound on <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 BB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58e28c8465f8b93b870d374f0b92986e5a93934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.528ex; height:2.176ex;" alt="{\displaystyle BB}"></span>, which solves the halting problem. </p><p>Consider this program <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="{\textstyle p_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p_{K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c10cad3d27e93a9bd7c5313ad10a60b3aea5fb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.952ex; height:2.009ex;" alt="{\textstyle p_{K}}"></span>, which takes input 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="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" 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="{\textstyle n}"></span>, and uses <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="{\textstyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985dcc2532a2d4d91b9a9610139216c63cf832d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\textstyle K}"></span>. </p> <ul><li>List all strings 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="{\textstyle \leq 2n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>≤</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \leq 2n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c37911ff937539e7822453891fdd9c319d672990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.013ex; height:2.343ex;" alt="{\textstyle \leq 2n+1}"></span>.</li></ul> <ul><li>For each such string <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="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" 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="{\textstyle x}"></span>, enumerate all (prefix-free) programs 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 K(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a621a6fbcc971649157a42ea42a599c22eee94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.205ex; height:2.843ex;" alt="{\displaystyle K(x)}"></span> until one of them does output <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="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" 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="{\textstyle x}"></span>. Record its runtime <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="{\textstyle n_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbce3a8583d922f51a31639d306ed3ff254c98e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.009ex;" alt="{\textstyle n_{x}}"></span>.</li> <li>Output the largest <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="{\textstyle n_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbce3a8583d922f51a31639d306ed3ff254c98e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.009ex;" alt="{\textstyle n_{x}}"></span>.</li></ul> <p>We prove by contradiction 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="{\textstyle p_{K}(n)\geq BB(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>B</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p_{K}(n)\geq BB(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901cd87b63a68fd909b6ad9ee6bb4bc355ba032f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:15.986ex; height:2.843ex;" alt="{\textstyle p_{K}(n)\geq BB(n)}"></span> for all large <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="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" 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="{\textstyle n}"></span>. </p><p>Let <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="{\textstyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7608ce99e3bb9154aa761b6c456f013eb156ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\textstyle p_{n}}"></span> be a Busy Beaver 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>. Consider this (prefix-free) program, which takes no input: </p> <ul><li>Run the program <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="{\textstyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7608ce99e3bb9154aa761b6c456f013eb156ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\textstyle p_{n}}"></span>, and record its runtime 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="{\textstyle BB(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>B</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle BB(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7ccf789e71326861a89ac6485a4f3377c5027d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.732ex; height:2.843ex;" alt="{\textstyle BB(n)}"></span>.</li></ul> <ul><li>Generate all programs with 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="{\textstyle \leq 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>≤</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \leq 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62474deb86d518b8beebe4caeefba64f49a54185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.01ex; height:2.343ex;" alt="{\textstyle \leq 2n}"></span>. Run every one of them for up 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="{\textstyle BB(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>B</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle BB(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7ccf789e71326861a89ac6485a4f3377c5027d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.732ex; height:2.843ex;" alt="{\textstyle BB(n)}"></span> steps. Note the outputs of those that have halted.</li></ul> <ul><li>Output the string with the lowest lexicographic order that has not been output by any of those.</li></ul> <p>Let the string output by the program 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="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" 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="{\textstyle x}"></span>. </p><p>The program has 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="{\textstyle \leq n+2\log _{2}n+O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>≤</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \leq n+2\log _{2}n+O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec0a2cb6a67bc787526eb79643fe41547c49fd71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.631ex; height:2.843ex;" alt="{\textstyle \leq n+2\log _{2}n+O(1)}"></span>, where <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> comes from the length of the Busy Beaver <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="{\textstyle p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7608ce99e3bb9154aa761b6c456f013eb156ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.009ex;" alt="{\textstyle p_{n}}"></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 2\log _{2}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\log _{2}n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/832e6fc50367578fee4c8b0a8a2fdd35e20c3044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.357ex; height:2.676ex;" alt="{\displaystyle 2\log _{2}n}"></span> comes from using the (prefix-free) <a href="/wiki/Elias_delta_code" class="mw-redirect" title="Elias delta code">Elias delta code</a> for the number <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>, 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 O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span> comes from the rest of the program. Therefore,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x)\leq n+2\log _{2}n+O(1)\leq 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x)\leq n+2\log _{2}n+O(1)\leq 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac39159e0b8b60d62f73e8117aa884a473b69d7a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.137ex; height:2.843ex;" alt="{\displaystyle K(x)\leq n+2\log _{2}n+O(1)\leq 2n}"></span>for all big <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="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" 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="{\textstyle n}"></span>. Further, since there are only so many possible programs with 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="{\textstyle \leq 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>≤</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \leq 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62474deb86d518b8beebe4caeefba64f49a54185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.01ex; height:2.343ex;" alt="{\textstyle \leq 2n}"></span>, we 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="{\textstyle l(x)\leq 2n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>l</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle l(x)\leq 2n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3f2d3de6c35e98befbdf98f1bd62a1e35c31e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.491ex; height:2.843ex;" alt="{\textstyle l(x)\leq 2n+1}"></span> by <a href="/wiki/Pigeonhole_principle" title="Pigeonhole principle">pigeonhole principle</a>. By assumption, <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="{\textstyle p_{K}(n)<BB(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>B</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p_{K}(n)<BB(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b0e8306f5c17341fed9caae5e29ee2eebf8044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:15.986ex; height:2.843ex;" alt="{\textstyle p_{K}(n)<BB(n)}"></span>, so every string 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="{\textstyle \leq 2n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>≤</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \leq 2n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c37911ff937539e7822453891fdd9c319d672990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.013ex; height:2.343ex;" alt="{\textstyle \leq 2n+1}"></span> has a minimal program with runtime <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="{\textstyle <BB(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo><</mo> <mi>B</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle <BB(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94b759abb849a8496a1c6648d3e5cfaf28ba0931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.185ex; height:2.843ex;" alt="{\textstyle <BB(n)}"></span>. Thus, the string <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="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" 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="{\textstyle x}"></span> has a minimal program with runtime <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="{\textstyle <BB(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo><</mo> <mi>B</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle <BB(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94b759abb849a8496a1c6648d3e5cfaf28ba0931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.185ex; height:2.843ex;" alt="{\textstyle <BB(n)}"></span>. Further, that program has 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="{\textstyle K(x)\leq 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle K(x)\leq 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc357c7469214e26a7c7aaf47c2b473fab338a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.861ex; height:2.843ex;" alt="{\textstyle K(x)\leq 2n}"></span>. This contradicts how <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="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" 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="{\textstyle x}"></span> was constructed. </p> <div class="mw-heading mw-heading2"><h2 id="Universal_probability">Universal probability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=21" title="Edit section: Universal probability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fix a universal 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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>, the same one used to define the (prefix-free) Kolmogorov complexity. Define the (prefix-free) universal probability of a string <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> to be<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=\sum _{U(p)=x}2^{-l(p)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </munder> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>l</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=\sum _{U(p)=x}2^{-l(p)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/670b9b2561e84cb4bbe56d56799f8506283345ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:19.225ex; height:6.009ex;" alt="{\displaystyle P(x)=\sum _{U(p)=x}2^{-l(p)}}"></span>In other words, it is the probability that, given a uniformly random binary stream as input, the universal Turing machine would halt after reading a certain prefix of the stream, and output <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>. </p><p>Note. <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 U(p)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(p)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b16ecc59c08beb639bd674ea5a15a632f4d86d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.189ex; height:2.843ex;" alt="{\displaystyle U(p)=x}"></span> does not mean that the input stream 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 p000\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mn>000</mn> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p000\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb1e4faa44c12862489e919816a42e24ef90951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.857ex; height:2.509ex;" alt="{\displaystyle p000\cdots }"></span>, but that the universal Turing machine would halt at some point after reading the initial segment <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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, without reading any further input, and that, when it halts, its has written <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> to the output tape. </p><p><b>Theorem.</b> (Theorem 14.11.1<sup id="cite_ref-:1_31-1" class="reference"><a href="#cite_note-:1-31"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>) <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 \log {\frac {1}{P(x)}}=K(x)+O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log {\frac {1}{P(x)}}=K(x)+O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bbc4559747dceda679ffc305987ec43f33320f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.968ex; height:6.009ex;" alt="{\displaystyle \log {\frac {1}{P(x)}}=K(x)+O(1)}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Conditional_versions">Conditional versions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=22" title="Edit section: Conditional versions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Kolmogorov_complexity&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">July 2014</span>)</i></span></div></td></tr></tbody></table> <p>The conditional Kolmogorov complexity of two strings <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(x|y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x|y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93d70e70ae35d7640bbc98a6042641e75b49f507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.007ex; height:2.843ex;" alt="{\displaystyle K(x|y)}"></span> is, roughly speaking, defined as the Kolmogorov complexity of <i>x</i> given <i>y</i> as an auxiliary input to the procedure.<sup id="cite_ref-Rissanen2007_34-0" class="reference"><a href="#cite_note-Rissanen2007-34"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>There is also a length-conditional 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 K(x|L(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x|L(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12a0a546f451972d5e5127746d5574de65e1db9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.574ex; height:2.843ex;" alt="{\displaystyle K(x|L(x))}"></span>, which is the complexity of <i>x</i> given the length of <i>x</i> as known/input.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Time-bounded_complexity">Time-bounded complexity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=23" title="Edit section: Time-bounded complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Time-bounded Kolmogorov complexity is a modified version of Kolmogorov complexity where the space of programs to be searched for a solution is confined to only programs that can run within some pre-defined number of steps.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> It is hypothesised that the possibility of the existence of an efficient algorithm for determining approximate time-bounded Kolmogorov complexity is related to the question of whether true <a href="/wiki/One-way_function" title="One-way function">one-way functions</a> exist.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>34<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=Kolmogorov_complexity&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Berry_paradox" title="Berry paradox">Berry paradox</a></li> <li><a href="/wiki/Code_golf" title="Code golf">Code golf</a></li> <li><a href="/wiki/Data_compression" title="Data compression">Data compression</a></li> <li><a href="/wiki/Descriptive_complexity_theory" title="Descriptive complexity theory">Descriptive complexity theory</a></li> <li><a href="/wiki/Grammar_induction" title="Grammar induction">Grammar induction</a></li> <li><a href="/wiki/Inductive_reasoning" title="Inductive reasoning">Inductive reasoning</a></li> <li><a href="/wiki/Kolmogorov_structure_function" title="Kolmogorov structure function">Kolmogorov structure function</a></li> <li><a href="/wiki/Levenshtein_distance" title="Levenshtein distance">Levenshtein distance</a></li> <li><a href="/wiki/Solomonoff%27s_theory_of_inductive_inference" title="Solomonoff's theory of inductive inference">Solomonoff's theory of inductive inference</a></li> <li><a href="/wiki/Sample_entropy" title="Sample entropy">Sample entropy</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=25" title="Edit section: Notes"><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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">However, an <i>s</i> with <i>K</i>(<i>s</i>) = <i>n</i> need not exist for every <i>n</i>. For example, if <i>n</i> is not a multiple of 7, no <a href="/wiki/ASCII" title="ASCII">ASCII</a> program can have a length of exactly <i>n</i> bits.</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">There are 1 + 2 + 2<sup>2</sup> + 2<sup>3</sup> + ... + 2<sup><i>n</i></sup> = 2<sup><i>n</i>+1</sup> − 1 different program texts of length up to <i>n</i> bits; cf. <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a>. If program lengths are to be multiples of 7 bits, even fewer program texts exist.</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">By the previous theorem, such a string exists, hence the <code>for</code> loop will eventually terminate.</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">including the language interpreter and the subroutine code for <code>KolmogorovComplexity</code></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">If <code>KolmogorovComplexity</code> has length <i>n</i> bits, the constant <i>m</i> used in <code>GenerateComplexString</code> needs to be adapted to satisfy <span class="nowrap"><i>n</i> + <span class="nowrap"><span data-sort-value="7006140000000000000♠"></span>1<span style="margin-left:.25em;">400</span><span style="margin-left:.25em;">000</span></span> + <span class="nowrap"><span data-sort-value="7003121800000000000♠"></span>1218</span> + 7·log<sub>10</sub>(<i>m</i>) < <i>m</i></span>, which is always possible since <i>m</i> grows faster than log<sub>10</sub>(<i>m</i>).</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">As there are <span class="nowrap"><i>N</i><sub><i>L</i></sub> = 2<sup><i>L</i></sup></span> strings of length <i>L</i>, the number of strings of lengths <span class="nowrap"><i>L</i> = 0, 1, ..., <i>n</i> − 1</span> is <span class="nowrap"><i>N</i><sub>0</sub> + <i>N</i><sub>1</sub> + ... + <i>N</i><sub><i>n</i>−1</sub></span> = <span class="nowrap">2<sup>0</sup> + 2<sup>1</sup> + ... + 2<sup><i>n</i>−1</sup></span>, which is a finite <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a> with sum <span class="nowrap">2<sup>0</sup> + 2<sup>1</sup> + ... + 2<sup><i>n</i>−1</sup></span> = <span class="nowrap">2<sup>0</sup> × (1 − 2<sup><i>n</i></sup>) / (1 − 2) = 2<sup><i>n</i></sup> − 1</span></span> </li> </ol></div></div> <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=Kolmogorov_complexity&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><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"><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="CITEREFKolmogorov1963" class="citation journal cs1"><a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Kolmogorov, Andrey</a> (Dec 1963). "On Tables of Random Numbers". <i>Sankhyā: The Indian Journal of Statistics, Series A (1961-2002)</i>. <b>25</b> (4): 369–375. <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/0581-572X">0581-572X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/25049284">25049284</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=0178484">0178484</a>.</cite><span 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text" href="https://doi.org/10.1016%2FS0304-3975%2898%2900075-9">"On Tables of Random Numbers"</a>. <i>Theoretical Computer Science</i>. <b>207</b> (2): 387–395. <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/10.1016%2FS0304-3975%2898%2900075-9">10.1016/S0304-3975(98)00075-9</a></span>. <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=1643414">1643414</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=On+Tables+of+Random+Numbers&rft.volume=207&rft.issue=2&rft.pages=387-395&rft.date=1998&rft_id=info%3Adoi%2F10.1016%2FS0304-3975%2898%2900075-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1643414%23id-name%3DMR&rft.aulast=Kolmogorov&rft.aufirst=Andrey&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0304-3975%252898%252900075-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-zenil20202-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-zenil20202_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZenil2020" class="citation journal cs1">Zenil, Hector (2020). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7517143">"A Review of Methods for Estimating Algorithmic Complexity: Options, Challenges, and New Directions"</a>. <i>Entropy</i>. <b>22</b> (6): 612. <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/10.3390%2Fe22060612">10.3390/e22060612</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7517143">7517143</a></span>. <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/33286384">33286384</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Entropy&rft.atitle=A+Review+of+Methods+for+Estimating+Algorithmic+Complexity%3A+Options%2C+Challenges%2C+and+New+Directions&rft.volume=22&rft.issue=6&rft.pages=612&rft.date=2020&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7517143%23id-name%3DPMC&rft_id=info%3Apmid%2F33286384&rft_id=info%3Adoi%2F10.3390%2Fe22060612&rft.aulast=Zenil&rft.aufirst=Hector&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7517143&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">(Downey and Hirschfeldt, 2010), Theorem 3.1.4 </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">(Downey and Hirschfeldt, 2010), Section 3.5</span> </li> <li id="cite_note-:0-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_6-1"><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="CITEREFHutter2007" class="citation journal cs1"><a href="/wiki/Marcus_Hutter" title="Marcus Hutter">Hutter, Marcus</a> (2007-03-06). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.2519">"Algorithmic information theory"</a>. <i>Scholarpedia</i>. <b>2</b> (3): 2519. <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/2007SchpJ...2.2519H">2007SchpJ...2.2519H</a>. <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/10.4249%2Fscholarpedia.2519">10.4249/scholarpedia.2519</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1885%2F15015">1885/15015</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/1941-6016">1941-6016</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Algorithmic+information+theory&rft.volume=2&rft.issue=3&rft.pages=2519&rft.date=2007-03-06&rft_id=info%3Ahdl%2F1885%2F15015&rft.issn=1941-6016&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.2519&rft_id=info%3Abibcode%2F2007SchpJ...2.2519H&rft.aulast=Hutter&rft.aufirst=Marcus&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.2519&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" 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="CITEREFSolomonoff1960" class="citation report cs1"><a href="/wiki/Ray_Solomonoff" title="Ray Solomonoff">Solomonoff, Ray</a> (February 4, 1960). <a rel="nofollow" class="external text" href="http://world.std.com/~rjs/rayfeb60.pdf">A Preliminary Report on a General Theory of Inductive Inference</a> <span class="cs1-format">(PDF)</span>. <i>Report V-131</i> (Report). <a rel="nofollow" class="external text" href="http://world.std.com/~rjs/z138.pdf">Revision</a> published November 1960. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://world.std.com/~rjs/rayfeb60.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=A+Preliminary+Report+on+a+General+Theory+of+Inductive+Inference&rft.date=1960-02-04&rft.aulast=Solomonoff&rft.aufirst=Ray&rft_id=http%3A%2F%2Fworld.std.com%2F~rjs%2Frayfeb60.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolomonoff1964" class="citation journal cs1">Solomonoff, Ray (March 1964). <a rel="nofollow" class="external text" href="http://world.std.com/~rjs/1964pt1.pdf">"A Formal Theory of Inductive Inference Part I"</a> <span class="cs1-format">(PDF)</span>. <i>Information and Control</i>. <b>7</b> (1): 1–22. <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/10.1016%2FS0019-9958%2864%2990223-2">10.1016/S0019-9958(64)90223-2</a></span>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://world.std.com/~rjs/1964pt1.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+and+Control&rft.atitle=A+Formal+Theory+of+Inductive+Inference+Part+I&rft.volume=7&rft.issue=1&rft.pages=1-22&rft.date=1964-03&rft_id=info%3Adoi%2F10.1016%2FS0019-9958%2864%2990223-2&rft.aulast=Solomonoff&rft.aufirst=Ray&rft_id=http%3A%2F%2Fworld.std.com%2F~rjs%2F1964pt1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" 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="CITEREFSolomonoff1964" class="citation journal cs1">Solomonoff, Ray (June 1964). <a rel="nofollow" class="external text" href="http://world.std.com/~rjs/1964pt2.pdf">"A Formal Theory of Inductive Inference Part II"</a> <span class="cs1-format">(PDF)</span>. <i>Information and Control</i>. <b>7</b> (2): 224–254. <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/10.1016%2FS0019-9958%2864%2990131-7">10.1016/S0019-9958(64)90131-7</a></span>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://world.std.com/~rjs/1964pt2.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+and+Control&rft.atitle=A+Formal+Theory+of+Inductive+Inference+Part+II&rft.volume=7&rft.issue=2&rft.pages=224-254&rft.date=1964-06&rft_id=info%3Adoi%2F10.1016%2FS0019-9958%2864%2990131-7&rft.aulast=Solomonoff&rft.aufirst=Ray&rft_id=http%3A%2F%2Fworld.std.com%2F~rjs%2F1964pt2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" 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="CITEREFKolmogorov1965" class="citation journal cs1">Kolmogorov, A.N. (1965). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110928032821/http://www.ece.umd.edu/~abarg/ppi/contents/1-65-abstracts.html">"Three Approaches to the Quantitative Definition of Information"</a>. <i>Problems Inform. Transmission</i>. <b>1</b> (1): 1–7. Archived from <a rel="nofollow" class="external text" href="http://www.ece.umd.edu/~abarg/ppi/contents/1-65-abstracts.html#1-65.2">the original</a> on September 28, 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Problems+Inform.+Transmission&rft.atitle=Three+Approaches+to+the+Quantitative+Definition+of+Information&rft.volume=1&rft.issue=1&rft.pages=1-7&rft.date=1965&rft.aulast=Kolmogorov&rft.aufirst=A.N.&rft_id=http%3A%2F%2Fwww.ece.umd.edu%2F~abarg%2Fppi%2Fcontents%2F1-65-abstracts.html%231-65.2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" 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="CITEREFChaitin1969" class="citation journal cs1">Chaitin, Gregory J. (1969). "On the Simplicity and Speed of Programs for Computing Infinite Sets of Natural Numbers". <i>Journal of the ACM</i>. <b>16</b> (3): 407–422. <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=10.1.1.15.3821">10.1.1.15.3821</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/10.1145%2F321526.321530">10.1145/321526.321530</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:12584692">12584692</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+Simplicity+and+Speed+of+Programs+for+Computing+Infinite+Sets+of+Natural+Numbers&rft.volume=16&rft.issue=3&rft.pages=407-422&rft.date=1969&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.15.3821%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12584692%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F321526.321530&rft.aulast=Chaitin&rft.aufirst=Gregory+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKolmogorov1968" class="citation journal cs1">Kolmogorov, A. (1968). "Logical basis for information theory and probability theory". <i>IEEE Transactions on Information Theory</i>. <b>14</b> (5): 662–664. <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%2FTIT.1968.1054210">10.1109/TIT.1968.1054210</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:11402549">11402549</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+Information+Theory&rft.atitle=Logical+basis+for+information+theory+and+probability+theory&rft.volume=14&rft.issue=5&rft.pages=662-664&rft.date=1968&rft_id=info%3Adoi%2F10.1109%2FTIT.1968.1054210&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11402549%23id-name%3DS2CID&rft.aulast=Kolmogorov&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiVitányi2008" class="citation book cs1">Li, Ming; Vitányi, Paul (2008). "Preliminaries". <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoko00limi_695"><i>An Introduction to Kolmogorov Complexity and its Applications</i></a></span>. Texts in Computer Science. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoko00limi_695/page/n23">1</a>–99. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-49820-1_1">10.1007/978-0-387-49820-1_1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-33998-6" title="Special:BookSources/978-0-387-33998-6"><bdi>978-0-387-33998-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Preliminaries&rft.btitle=An+Introduction+to+Kolmogorov+Complexity+and+its+Applications&rft.series=Texts+in+Computer+Science&rft.pages=1-99&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-0-387-49820-1_1&rft.isbn=978-0-387-33998-6&rft.aulast=Li&rft.aufirst=Ming&rft.au=Vit%C3%A1nyi%2C+Paul&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoko00limi_695&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-Burgin1982-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Burgin1982_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgin1982" class="citation journal cs1">Burgin, M. (1982). <a rel="nofollow" class="external text" href="https://www.mathnet.ru/eng/dan45265">"Generalized Kolmogorov complexity and duality in theory of computations"</a>. <i>Notices of the Russian Academy of Sciences</i>. <b>25</b> (3): 19–23.</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+Russian+Academy+of+Sciences&rft.atitle=Generalized+Kolmogorov+complexity+and+duality+in+theory+of+computations&rft.volume=25&rft.issue=3&rft.pages=19-23&rft.date=1982&rft.aulast=Burgin&rft.aufirst=M.&rft_id=https%3A%2F%2Fwww.mathnet.ru%2Feng%2Fdan45265&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></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="CITEREFHutter2005" class="citation book cs1">Hutter, Marcus (2005). <i>Universal artificial intelligence: sequential decisions based on algorithmic probability</i>. Texts in theoretical computer science. Berlin New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-26877-2" title="Special:BookSources/978-3-540-26877-2"><bdi>978-3-540-26877-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Universal+artificial+intelligence%3A+sequential+decisions+based+on+algorithmic+probability&rft.place=Berlin+New+York&rft.series=Texts+in+theoretical+computer+science&rft.pub=Springer&rft.date=2005&rft.isbn=978-3-540-26877-2&rft.aulast=Hutter&rft.aufirst=Marcus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></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">Stated without proof in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP._B._Miltersen2005" class="citation web cs1">P. B. Miltersen (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090909132048/http://www.daimi.au.dk/~bromille/DC05/Kolmogorov.pdf">"Course notes for Data Compression - Kolmogorov complexity"</a> <span class="cs1-format">(PDF)</span>. p. 7. Archived from <a rel="nofollow" class="external text" href="http://www.daimi.au.dk/~bromille/DC05/Kolmogorov.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2009-09-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Course+notes+for+Data+Compression+-+Kolmogorov+complexity&rft.pages=7&rft.date=2005&rft.au=P.+B.+Miltersen&rft_id=http%3A%2F%2Fwww.daimi.au.dk%2F~bromille%2FDC05%2FKolmogorov.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></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="CITEREFZvonkinL._Levin1970" class="citation journal cs1">Zvonkin, A.; L. Levin (1970). <a rel="nofollow" class="external text" href="http://alexander.shen.free.fr/library/Zvonkin_Levin_70.pdf">"The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms"</a> <span class="cs1-format">(PDF)</span>. <i>Russian Mathematical Surveys</i>. <b>25</b> (6): 83–124. <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/1970RuMaS..25...83Z">1970RuMaS..25...83Z</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1070%2FRM1970v025n06ABEH001269">10.1070/RM1970v025n06ABEH001269</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:250850390">250850390</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Russian+Mathematical+Surveys&rft.atitle=The+complexity+of+finite+objects+and+the+development+of+the+concepts+of+information+and+randomness+by+means+of+the+theory+of+algorithms.&rft.volume=25&rft.issue=6&rft.pages=83-124&rft.date=1970&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250850390%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1070%2FRM1970v025n06ABEH001269&rft_id=info%3Abibcode%2F1970RuMaS..25...83Z&rft.aulast=Zvonkin&rft.aufirst=A.&rft.au=L.+Levin&rft_id=http%3A%2F%2Falexander.shen.free.fr%2Flibrary%2FZvonkin_Levin_70.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGregory_J._Chaitin1974" class="citation journal cs1">Gregory J. Chaitin (Jul 1974). <a rel="nofollow" class="external text" href="http://www.cas.mcmaster.ca/~sancheg/EE_UCU2006_thesis/biblio/Information-theoretic%20limitations%20of%20Formal%20Systems%20(acm74).pdf">"Information-theoretic limitations of formal systems"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of the ACM</i>. <b>21</b> (3): 403–434. <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%2F321832.321839">10.1145/321832.321839</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:2142553">2142553</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=Information-theoretic+limitations+of+formal+systems&rft.volume=21&rft.issue=3&rft.pages=403-434&rft.date=1974-07&rft_id=info%3Adoi%2F10.1145%2F321832.321839&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2142553%23id-name%3DS2CID&rft.au=Gregory+J.+Chaitin&rft_id=http%3A%2F%2Fwww.cas.mcmaster.ca%2F~sancheg%2FEE_UCU2006_thesis%2Fbiblio%2FInformation-theoretic%2520limitations%2520of%2520Formal%2520Systems%2520%28acm74%29.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span> Here: Thm.4.1b</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="CITEREFCalude2002" class="citation book cs1">Calude, Cristian S. (12 September 2002). <a rel="nofollow" class="external text" href="https://www.springer.com/br/book/9783540434665"><i>Information and Randomness: an algorithmic perspective</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540434665" title="Special:BookSources/9783540434665"><bdi>9783540434665</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+and+Randomness%3A+an+algorithmic+perspective&rft.pub=Springer&rft.date=2002-09-12&rft.isbn=9783540434665&rft.aulast=Calude&rft.aufirst=Cristian+S.&rft_id=https%3A%2F%2Fwww.springer.com%2Fbr%2Fbook%2F9783540434665&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWallaceDowe1999" class="citation journal cs1">Wallace, C. S.; Dowe, D. L. (1999). "Minimum Message Length and Kolmogorov Complexity". <i>Computer Journal</i>. <b>42</b> (4): 270–283. <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=10.1.1.17.321">10.1.1.17.321</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/10.1093%2Fcomjnl%2F42.4.270">10.1093/comjnl/42.4.270</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Journal&rft.atitle=Minimum+Message+Length+and+Kolmogorov+Complexity&rft.volume=42&rft.issue=4&rft.pages=270-283&rft.date=1999&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.17.321%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1093%2Fcomjnl%2F42.4.270&rft.aulast=Wallace&rft.aufirst=C.+S.&rft.au=Dowe%2C+D.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">There are 2<sup><i>n</i></sup> bit strings of length <i>n</i> but only 2<sup><i>n</i></sup>-1 shorter bit strings, hence at most that much compression results.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin-Löf1966" class="citation journal cs1">Martin-Löf, Per (1966). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0019-9958%2866%2980018-9">"The definition of random sequences"</a>. <i>Information and Control</i>. <b>9</b> (6): 602–619. <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/10.1016%2Fs0019-9958%2866%2980018-9">10.1016/s0019-9958(66)80018-9</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+and+Control&rft.atitle=The+definition+of+random+sequences&rft.volume=9&rft.issue=6&rft.pages=602-619&rft.date=1966&rft_id=info%3Adoi%2F10.1016%2Fs0019-9958%2866%2980018-9&rft.aulast=Martin-L%C3%B6f&rft.aufirst=Per&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fs0019-9958%252866%252980018-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalatoloHoyrupRojas2010" class="citation journal cs1">Galatolo, Stefano; Hoyrup, Mathieu; Rojas, Cristóbal (2010). <a rel="nofollow" class="external text" href="http://www.loria.fr/~hoyrup/random_ergodic.pdf">"Effective symbolic dynamics, random points, statistical behavior, complexity and entropy"</a> <span class="cs1-format">(PDF)</span>. <i>Information and Computation</i>. <b>208</b>: 23–41. <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/0801.0209">0801.0209</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/10.1016%2Fj.ic.2009.05.001">10.1016/j.ic.2009.05.001</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:5555443">5555443</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.loria.fr/~hoyrup/random_ergodic.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</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=Effective+symbolic+dynamics%2C+random+points%2C+statistical+behavior%2C+complexity+and+entropy&rft.volume=208&rft.pages=23-41&rft.date=2010&rft_id=info%3Aarxiv%2F0801.0209&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5555443%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.ic.2009.05.001&rft.aulast=Galatolo&rft.aufirst=Stefano&rft.au=Hoyrup%2C+Mathieu&rft.au=Rojas%2C+Crist%C3%B3bal&rft_id=http%3A%2F%2Fwww.loria.fr%2F~hoyrup%2Frandom_ergodic.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexei_Kaltchenko2004" class="citation arxiv cs1">Alexei Kaltchenko (2004). "Algorithms for Estimating Information Distance with Application to Bioinformatics and Linguistics". <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/cs.CC/0404039">cs.CC/0404039</a></span>.</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=Algorithms+for+Estimating+Information+Distance+with+Application+to+Bioinformatics+and+Linguistics&rft.date=2004&rft_id=info%3Aarxiv%2Fcs.CC%2F0404039&rft.au=Alexei+Kaltchenko&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-:1-31"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_31-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_31-1"><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="CITEREFCoverThomas2006" class="citation book cs1">Cover, Thomas M.; Thomas, Joy A. (2006). <i>Elements of information theory</i> (2nd ed.). Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-24195-4" title="Special:BookSources/0-471-24195-4"><bdi>0-471-24195-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+information+theory&rft.edition=2nd&rft.pub=Wiley-Interscience&rft.date=2006&rft.isbn=0-471-24195-4&rft.aulast=Cover&rft.aufirst=Thomas+M.&rft.au=Thomas%2C+Joy+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChaitinArslanovCalude1995" class="citation journal cs1">Chaitin, G.; Arslanov, A.; Calude, Cristian S. (1995-09-01). "Program-size Complexity Computes the Halting Problem". <i>Bull. EATCS</i>. <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:39718973">39718973</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+EATCS&rft.atitle=Program-size+Complexity+Computes+the+Halting+Problem&rft.date=1995-09-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A39718973%23id-name%3DS2CID&rft.aulast=Chaitin&rft.aufirst=G.&rft.au=Arslanov%2C+A.&rft.au=Calude%2C+Cristian+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiVitányi2008" class="citation book cs1">Li, Ming; Vitányi, Paul (2008). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-0-387-49820-1"><i>An Introduction to Kolmogorov Complexity and Its Applications</i></a>. Exercise 2.7.7. <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/2008ikca.book.....L">2008ikca.book.....L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-49820-1">10.1007/978-0-387-49820-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-33998-6" title="Special:BookSources/978-0-387-33998-6"><bdi>978-0-387-33998-6</bdi></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/1868-0941">1868-0941</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Kolmogorov+Complexity+and+Its+Applications&rft.pages=Exercise+2.7.7.&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-0-387-49820-1&rft.issn=1868-0941&rft_id=info%3Abibcode%2F2008ikca.book.....L&rft.isbn=978-0-387-33998-6&rft.aulast=Li&rft.aufirst=Ming&rft.au=Vit%C3%A1nyi%2C+Paul&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-0-387-49820-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|journal=</code> ignored (<a href="/wiki/Help:CS1_errors#periodical_ignored" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-Rissanen2007-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rissanen2007_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJorma_Rissanen2007" class="citation book cs1">Jorma Rissanen (2007). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/informationcompl00riss_364"><i>Information and Complexity in Statistical Modeling</i></a></span>. Springer S. p. <a rel="nofollow" class="external text" href="https://archive.org/details/informationcompl00riss_364/page/n59">53</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-68812-1" title="Special:BookSources/978-0-387-68812-1"><bdi>978-0-387-68812-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+and+Complexity+in+Statistical+Modeling&rft.pages=53&rft.pub=Springer+S&rft.date=2007&rft.isbn=978-0-387-68812-1&rft.au=Jorma+Rissanen&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Finformationcompl00riss_364&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMing_LiPaul_M.B._Vitányi2009" class="citation book cs1">Ming Li; Paul M.B. Vitányi (2009). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoko00limi_695"><i>An Introduction to Kolmogorov Complexity and Its Applications</i></a></span>. Springer. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoko00limi_695/page/n127">105</a>–106. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-49820-1" title="Special:BookSources/978-0-387-49820-1"><bdi>978-0-387-49820-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Kolmogorov+Complexity+and+Its+Applications&rft.pages=105-106&rft.pub=Springer&rft.date=2009&rft.isbn=978-0-387-49820-1&rft.au=Ming+Li&rft.au=Paul+M.B.+Vit%C3%A1nyi&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoko00limi_695&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMing_LiPaul_M.B._Vitányi2009" class="citation book cs1">Ming Li; Paul M.B. Vitányi (2009). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoko00limi_695"><i>An Introduction to Kolmogorov Complexity and Its Applications</i></a></span>. Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoko00limi_695/page/n141">119</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-49820-1" title="Special:BookSources/978-0-387-49820-1"><bdi>978-0-387-49820-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Kolmogorov+Complexity+and+Its+Applications&rft.pages=119&rft.pub=Springer&rft.date=2009&rft.isbn=978-0-387-49820-1&rft.au=Ming+Li&rft.au=Paul+M.B.+Vit%C3%A1nyi&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoko00limi_695&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVitányi2013" class="citation journal cs1">Vitányi, Paul M.B. (2013). <a rel="nofollow" class="external text" href="https://ir.cwi.nl/pub/26818">"Conditional Kolmogorov complexity and universal probability"</a>. <i>Theoretical Computer Science</i>. <b>501</b>: 93–100. <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/1206.0983">1206.0983</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/10.1016%2Fj.tcs.2013.07.009">10.1016/j.tcs.2013.07.009</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:12085503">12085503</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Conditional+Kolmogorov+complexity+and+universal+probability&rft.volume=501&rft.pages=93-100&rft.date=2013&rft_id=info%3Aarxiv%2F1206.0983&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12085503%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.tcs.2013.07.009&rft.aulast=Vit%C3%A1nyi&rft.aufirst=Paul+M.B.&rft_id=https%3A%2F%2Fir.cwi.nl%2Fpub%2F26818&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHiraharaKabanetsLuOliveira2024" class="citation journal cs1">Hirahara, Shuichi; Kabanets, Valentine; Lu, Zhenjian; Oliveira, Igor C. (2024). <a rel="nofollow" class="external text" href="https://doi.org/10.4230%2FLIPIcs.CCC.2024.29">"Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity"</a>. <i>39th Computational Complexity Conference (CCC 2024)</i>. Schloss Dagstuhl – Leibniz-Zentrum für Informatik: 29:1–29:56. <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/10.4230%2FLIPIcs.CCC.2024.29">10.4230/LIPIcs.CCC.2024.29</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=39th+Computational+Complexity+Conference+%28CCC+2024%29&rft.atitle=Exact+Search-To-Decision+Reductions+for+Time-Bounded+Kolmogorov+Complexity&rft.pages=29%3A1-29%3A56&rft.date=2024&rft_id=info%3Adoi%2F10.4230%2FLIPIcs.CCC.2024.29&rft.aulast=Hirahara&rft.aufirst=Shuichi&rft.au=Kabanets%2C+Valentine&rft.au=Lu%2C+Zhenjian&rft.au=Oliveira%2C+Igor+C.&rft_id=https%3A%2F%2Fdoi.org%2F10.4230%252FLIPIcs.CCC.2024.29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlarreich2022" class="citation web cs1">Klarreich, Erica (2022-04-06). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/researchers-identify-master-problem-underlying-all-cryptography-20220406/">"Researchers Identify 'Master Problem' Underlying All Cryptography"</a>. <i>Quanta Magazine</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quanta+Magazine&rft.atitle=Researchers+Identify+%27Master+Problem%27+Underlying+All+Cryptography&rft.date=2022-04-06&rft.aulast=Klarreich&rft.aufirst=Erica&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fresearchers-identify-master-problem-underlying-all-cryptography-20220406%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiuPass2020" class="citation cs2">Liu, Yanyi; Pass, Rafael (2020-09-24), <i>On One-way Functions and Kolmogorov Complexity</i>, <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/2009.11514">2009.11514</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+One-way+Functions+and+Kolmogorov+Complexity&rft.date=2020-09-24&rft_id=info%3Aarxiv%2F2009.11514&rft.aulast=Liu&rft.aufirst=Yanyi&rft.au=Pass%2C+Rafael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kolmogorov_complexity&action=edit&section=27" title="Edit section: Further reading"><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="CITEREFBlum1967" class="citation journal cs1"><a href="/wiki/Manuel_Blum" title="Manuel Blum">Blum, M.</a> (1967). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0019-9958%2867%2990546-3">"On the size of machines"</a>. <i>Information and Control</i>. <b>11</b> (3): 257. <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/10.1016%2FS0019-9958%2867%2990546-3">10.1016/S0019-9958(67)90546-3</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+and+Control&rft.atitle=On+the+size+of+machines&rft.volume=11&rft.issue=3&rft.pages=257&rft.date=1967&rft_id=info%3Adoi%2F10.1016%2FS0019-9958%2867%2990546-3&rft.aulast=Blum&rft.aufirst=M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0019-9958%252867%252990546-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrudno1983" class="citation journal cs1">Brudno, A. (1983). "Entropy and the complexity of the trajectories of a dynamical system". <i>Transactions of the Moscow Mathematical Society</i>. <b>2</b>: 127–151.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Moscow+Mathematical+Society&rft.atitle=Entropy+and+the+complexity+of+the+trajectories+of+a+dynamical+system&rft.volume=2&rft.pages=127-151&rft.date=1983&rft.aulast=Brudno&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoverThomas2006" class="citation book cs1">Cover, Thomas M.; Thomas, Joy A. (2006). <i>Elements of information theory</i> (2nd ed.). Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-24195-4" title="Special:BookSources/0-471-24195-4"><bdi>0-471-24195-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+information+theory&rft.edition=2nd&rft.pub=Wiley-Interscience&rft.date=2006&rft.isbn=0-471-24195-4&rft.aulast=Cover&rft.aufirst=Thomas+M.&rft.au=Thomas%2C+Joy+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLajosGáborRéka1999" class="citation book cs1">Lajos, Rónyai; Gábor, Ivanyos; Réka, Szabó (1999). <i>Algoritmusok</i>. TypoTeX. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/963-279-014-6" title="Special:BookSources/963-279-014-6"><bdi>963-279-014-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algoritmusok&rft.pub=TypoTeX&rft.date=1999&rft.isbn=963-279-014-6&rft.aulast=Lajos&rft.aufirst=R%C3%B3nyai&rft.au=G%C3%A1bor%2C+Ivanyos&rft.au=R%C3%A9ka%2C+Szab%C3%B3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiVitányi1997" class="citation book cs1">Li, Ming; Vitányi, Paul (1997). <i>An Introduction to Kolmogorov Complexity and Its Applications</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0387339986" title="Special:BookSources/978-0387339986"><bdi>978-0387339986</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Kolmogorov+Complexity+and+Its+Applications&rft.pub=Springer&rft.date=1997&rft.isbn=978-0387339986&rft.aulast=Li&rft.aufirst=Ming&rft.au=Vit%C3%A1nyi%2C+Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYu1977" class="citation book cs1">Yu, Manin (1977). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/courseinmathemat0000bell"><i>A Course in Mathematical Logic</i></a></span>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7204-2844-5" title="Special:BookSources/978-0-7204-2844-5"><bdi>978-0-7204-2844-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Mathematical+Logic&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=978-0-7204-2844-5&rft.aulast=Yu&rft.aufirst=Manin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcourseinmathemat0000bell&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSipser1997" class="citation book cs1">Sipser, Michael (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00sips"><i>Introduction to the Theory of Computation</i></a></span>. PWS. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-95097-3" title="Special:BookSources/0-534-95097-3"><bdi>0-534-95097-3</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.pub=PWS&rft.date=1997&rft.isbn=0-534-95097-3&rft.aulast=Sipser&rft.aufirst=Michael&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoth00sips&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDowneyHirschfeldt2010" class="citation journal cs1">Downey, Rodney G.; Hirschfeldt, Denis R. (2010). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-0-387-68441-3">"Algorithmic Randomness and Complexity"</a>. <i>Theory and Applications of Computability</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-68441-3">10.1007/978-0-387-68441-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95567-4" title="Special:BookSources/978-0-387-95567-4"><bdi>978-0-387-95567-4</bdi></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/2190-619X">2190-619X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theory+and+Applications+of+Computability&rft.atitle=Algorithmic+Randomness+and+Complexity&rft.date=2010&rft.issn=2190-619X&rft_id=info%3Adoi%2F10.1007%2F978-0-387-68441-3&rft.isbn=978-0-387-95567-4&rft.aulast=Downey&rft.aufirst=Rodney+G.&rft.au=Hirschfeldt%2C+Denis+R.&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-0-387-68441-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" 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=Kolmogorov_complexity&action=edit&section=28" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180321163508/http://kolmogorov.com/">The Legacy of Andrei Nikolaevich Kolmogorov</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150215210504/http://www.cs.umaine.edu/~chaitin/">Chaitin's online publications</a></li> <li><a rel="nofollow" class="external text" href="http://www.idsia.ch/~juergen/ray.html">Solomonoff's IDSIA page</a></li> <li><a rel="nofollow" class="external text" href="http://www.idsia.ch/~juergen/kolmogorov.html">Generalizations of algorithmic information</a> by <a href="/wiki/Juergen_Schmidhuber" class="mw-redirect" title="Juergen Schmidhuber">J. Schmidhuber</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://homepages.cwi.nl/~paulv/kolmogorov.html">"Review of Li Vitányi 1997"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Review+of+Li+Vit%C3%A1nyi+1997&rft_id=http%3A%2F%2Fhomepages.cwi.nl%2F~paulv%2Fkolmogorov.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTromp" class="citation web cs1">Tromp, John. <a rel="nofollow" class="external text" href="https://tromp.github.io/cl/cl.html">"John's Lambda Calculus and Combinatory Logic Playground"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=John%27s+Lambda+Calculus+and+Combinatory+Logic+Playground&rft.aulast=Tromp&rft.aufirst=John&rft_id=https%3A%2F%2Ftromp.github.io%2Fcl%2Fcl.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span> Tromp's lambda calculus computer model offers a concrete definition of K()]</li> <li>Universal AI based on Kolmogorov Complexity <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-22139-5" title="Special:BookSources/3-540-22139-5">3-540-22139-5</a> by <a href="/wiki/Marcus_Hutter" title="Marcus Hutter">M. Hutter</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-22139-5" title="Special:BookSources/3-540-22139-5">3-540-22139-5</a></li> <li><a rel="nofollow" class="external text" href="http://www.csse.monash.edu.au/~dld">David Dowe</a>'s <a rel="nofollow" class="external text" href="http://www.csse.monash.edu.au/~dld/MML.html">Minimum Message Length (MML)</a> and <a rel="nofollow" class="external text" href="http://www.csse.monash.edu.au/~dld/Occam.html">Occam's razor</a> pages.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrunwaldPitt2005" class="citation book cs1">Grunwald, P.; Pitt, M.A. (2005). Myung, I. J. (ed.). <i>Advances in Minimum Description Length: Theory and Applications</i>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-07262-9" title="Special:BookSources/0-262-07262-9"><bdi>0-262-07262-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advances+in+Minimum+Description+Length%3A+Theory+and+Applications&rft.pub=MIT+Press&rft.date=2005&rft.isbn=0-262-07262-9&rft.aulast=Grunwald&rft.aufirst=P.&rft.au=Pitt%2C+M.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKolmogorov+complexity" class="Z3988"></span></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 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href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</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/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</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/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</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/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</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/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</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/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</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/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</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/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></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>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a class="mw-selflink selflink">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/Logicism" title="Logicism">Logicism</a></li> <li><a href="/wiki/Mathematical_object" title="Mathematical object">Mathematical object</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Supertask" title="Supertask">Supertask</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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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:Compression_methods" title="Template:Compression methods"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Compression_methods" title="Template talk:Compression methods"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Compression_methods" title="Special:EditPage/Template:Compression methods"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_compression_methods" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_compression" title="Data compression">Data compression</a> methods</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lossless_compression" title="Lossless compression">Lossless</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Entropy_coding" title="Entropy coding">Entropy type</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_coding" title="Adaptive coding">Adaptive coding</a></li> <li><a href="/wiki/Arithmetic_coding" title="Arithmetic coding">Arithmetic</a></li> <li><a href="/wiki/Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric numeral systems</a></li> <li><a href="/wiki/Golomb_coding" title="Golomb coding">Golomb</a></li> <li><a href="/wiki/Huffman_coding" title="Huffman coding">Huffman</a> <ul><li><a href="/wiki/Adaptive_Huffman_coding" title="Adaptive Huffman coding">Adaptive</a></li> <li><a href="/wiki/Canonical_Huffman_code" title="Canonical Huffman code">Canonical</a></li> <li><a href="/wiki/Modified_Huffman_coding" title="Modified Huffman coding">Modified</a></li></ul></li> <li><a href="/wiki/Range_coding" title="Range coding">Range</a></li> <li><a href="/wiki/Shannon_coding" title="Shannon coding">Shannon</a></li> <li><a href="/wiki/Shannon%E2%80%93Fano_coding" title="Shannon–Fano coding">Shannon–Fano</a></li> <li><a href="/wiki/Shannon%E2%80%93Fano%E2%80%93Elias_coding" title="Shannon–Fano–Elias coding">Shannon–Fano–Elias</a></li> <li><a href="/wiki/Tunstall_coding" title="Tunstall coding">Tunstall</a></li> <li><a href="/wiki/Unary_coding" title="Unary coding">Unary</a></li> <li><a href="/wiki/Universal_code_(data_compression)" title="Universal code (data compression)">Universal</a> <ul><li><a href="/wiki/Exponential-Golomb_coding" title="Exponential-Golomb coding">Exp-Golomb</a></li> <li><a href="/wiki/Fibonacci_coding" title="Fibonacci coding">Fibonacci</a></li> <li><a href="/wiki/Elias_gamma_coding" title="Elias gamma coding">Gamma</a></li> <li><a href="/wiki/Levenshtein_coding" title="Levenshtein coding">Levenshtein</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Dictionary_coder" title="Dictionary coder">Dictionary type</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Byte_pair_encoding" title="Byte pair encoding">Byte pair encoding</a></li> <li><a href="/wiki/LZ77_and_LZ78" title="LZ77 and LZ78">Lempel–Ziv</a> <ul><li><a href="/wiki/842_(compression_algorithm)" title="842 (compression algorithm)">842</a></li> <li><a href="/wiki/LZ4_(compression_algorithm)" title="LZ4 (compression algorithm)">LZ4</a></li> <li><a href="/wiki/LZJB" class="mw-redirect" title="LZJB">LZJB</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Oberhumer" title="Lempel–Ziv–Oberhumer">LZO</a></li> <li><a href="/wiki/LZRW" title="LZRW">LZRW</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Storer%E2%80%93Szymanski" title="Lempel–Ziv–Storer–Szymanski">LZSS</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Welch" title="Lempel–Ziv–Welch">LZW</a></li> <li><a href="/wiki/LZWL" title="LZWL">LZWL</a></li> <li><a href="/wiki/Snappy_(compression)" title="Snappy (compression)">Snappy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Other types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burrows%E2%80%93Wheeler_transform" title="Burrows–Wheeler transform">BWT</a></li> <li><a href="/wiki/Context_tree_weighting" title="Context tree weighting">CTW</a></li> <li><a href="/wiki/Context_mixing" title="Context mixing">CM</a></li> <li><a href="/wiki/Delta_encoding" title="Delta encoding">Delta</a> <ul><li><a href="/wiki/Incremental_encoding" title="Incremental encoding">Incremental</a></li></ul></li> <li><a href="/wiki/Dynamic_Markov_compression" title="Dynamic Markov compression">DMC</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a></li> <li><a href="/wiki/Grammar-based_code" title="Grammar-based code">Grammar</a> <ul><li><a href="/wiki/Re-Pair" title="Re-Pair">Re-Pair</a></li> <li><a href="/wiki/Sequitur_algorithm" title="Sequitur algorithm">Sequitur</a></li></ul></li> <li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">LDCT</a></li> <li><a href="/wiki/Move-to-front_transform" title="Move-to-front transform">MTF</a></li> <li><a href="/wiki/PAQ" title="PAQ">PAQ</a></li> <li><a href="/wiki/Prediction_by_partial_matching" title="Prediction by partial matching">PPM</a></li> <li><a href="/wiki/Run-length_encoding" title="Run-length encoding">RLE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Hybrid</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li>LZ77 + Huffman <ul><li><a href="/wiki/Deflate" title="Deflate">Deflate</a></li> <li><a href="/wiki/LZX" title="LZX">LZX</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Stac" title="Lempel–Ziv–Stac">LZS</a></li></ul></li> <li>LZ77 + ANS <ul><li><a href="/wiki/LZFSE" title="LZFSE">LZFSE</a></li></ul></li> <li>LZ77 + Huffman + ANS <ul><li><a href="/wiki/Zstd" title="Zstd">Zstandard</a></li></ul></li> <li>LZ77 + Huffman + context <ul><li><a href="/wiki/Brotli" title="Brotli">Brotli</a></li></ul></li> <li>LZSS + Huffman <ul><li><a href="/wiki/LHA_(file_format)" title="LHA (file format)">LHA/LZH</a></li></ul></li> <li>LZ77 + Range <ul><li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Markov_chain_algorithm" title="Lempel–Ziv–Markov chain algorithm">LZMA</a></li> <li>LZHAM</li></ul></li> <li>RLE + BWT + MTF + Huffman <ul><li><a href="/wiki/Bzip2" title="Bzip2">bzip2</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lossy_compression" title="Lossy compression">Lossy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Transform_coding" title="Transform coding">Transform type</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a> <ul><li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li></ul></li> <li><a href="/wiki/Discrete_sine_transform" title="Discrete sine transform">DST</a></li> <li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li> <li><a href="/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Predictive type</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a> <ul><li><a href="/wiki/Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li></ul></li> <li><a href="/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li> <li><a href="/wiki/Log_area_ratio" title="Log area ratio">LAR</a></li> <li><a href="/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a></li> <li><a href="/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li> <li>Motion <ul><li><a href="/wiki/Motion_compensation" title="Motion compensation">Compensation</a></li> <li><a href="/wiki/Motion_estimation" title="Motion estimation">Estimation</a></li> <li><a href="/wiki/Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li> <li><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Data_compression#Audio" title="Data compression">Audio</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_rate" title="Bit rate">Bit rate</a> <ul><li><a href="/wiki/Average_bitrate" title="Average bitrate">ABR</a></li> <li><a href="/wiki/Constant_bitrate" title="Constant bitrate">CBR</a></li> <li><a href="/wiki/Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li> <li><a href="/wiki/Companding" title="Companding">Companding</a></li> <li><a href="/wiki/Convolution" title="Convolution">Convolution</a></li> <li><a href="/wiki/Dynamic_range" title="Dynamic range">Dynamic range</a></li> <li><a href="/wiki/Latency_(audio)" title="Latency (audio)">Latency</a></li> <li><a href="/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon theorem</a></li> <li><a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></li> <li><a href="/wiki/Silence_compression" title="Silence compression">Silence compression</a></li> <li><a href="/wiki/Sound_quality" title="Sound quality">Sound quality</a></li> <li><a href="/wiki/Speech_coding" title="Speech coding">Speech coding</a></li> <li><a href="/wiki/Sub-band_coding" title="Sub-band coding">Sub-band coding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Audio_codec" title="Audio codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/A-law_algorithm" title="A-law algorithm">A-law</a></li> <li><a href="/wiki/%CE%9C-law_algorithm" title="Μ-law algorithm">μ-law</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a> <ul><li><a href="/wiki/Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li> <li><a href="/wiki/Delta_modulation" title="Delta modulation">DM</a></li></ul></li> <li><a href="/wiki/Fourier_transform" title="Fourier transform">FT</a> <ul><li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li></ul></li> <li><a href="/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li> <li><a href="/wiki/Log_area_ratio" title="Log area ratio">LAR</a></li> <li><a href="/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a></li> <li><a href="/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li> <li><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustic model</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Image_compression" title="Image compression">Image</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chroma_subsampling" title="Chroma subsampling">Chroma subsampling</a></li> <li><a href="/wiki/Coding_tree_unit" title="Coding tree unit">Coding tree unit</a></li> <li><a href="/wiki/Color_space" title="Color space">Color space</a></li> <li><a href="/wiki/Compression_artifact" title="Compression artifact">Compression artifact</a></li> <li><a href="/wiki/Image_resolution" title="Image resolution">Image resolution</a></li> <li><a href="/wiki/Macroblock" title="Macroblock">Macroblock</a></li> <li><a href="/wiki/Pixel" title="Pixel">Pixel</a></li> <li><a href="/wiki/Peak_signal-to-noise_ratio" title="Peak signal-to-noise ratio">PSNR</a></li> <li><a href="/wiki/Quantization_(image_processing)" title="Quantization (image processing)">Quantization</a></li> <li><a href="/wiki/Standard_test_image" title="Standard test image">Standard test image</a></li> <li><a href="/wiki/Texture_compression" title="Texture compression">Texture compression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chain_code" title="Chain code">Chain code</a></li> <li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li> <li><a href="/wiki/Deflate" title="Deflate">Deflate</a></li> <li><a href="/wiki/Fractal_compression" title="Fractal compression">Fractal</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">KLT</a></li> <li><a href="/wiki/Pyramid_(image_processing)" title="Pyramid (image processing)">LP</a></li> <li><a href="/wiki/Run-length_encoding" title="Run-length encoding">RLE</a></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li> <li><a href="/wiki/Embedded_zerotrees_of_wavelet_transforms" title="Embedded zerotrees of wavelet transforms">EZW</a></li> <li><a href="/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Data_compression#Video" title="Data compression">Video</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_rate" title="Bit rate">Bit rate</a> <ul><li><a href="/wiki/Average_bitrate" title="Average bitrate">ABR</a></li> <li><a href="/wiki/Constant_bitrate" title="Constant bitrate">CBR</a></li> <li><a href="/wiki/Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li> <li><a href="/wiki/Display_resolution" title="Display resolution">Display resolution</a></li> <li><a href="/wiki/Film_frame" title="Film frame">Frame</a></li> <li><a href="/wiki/Frame_rate" title="Frame rate">Frame rate</a></li> <li><a href="/wiki/Video_compression_picture_types" title="Video compression picture types">Frame types</a></li> <li><a href="/wiki/Interlaced_video" title="Interlaced video">Interlace</a></li> <li><a href="/wiki/Video#Characteristics_of_video_streams" title="Video">Video characteristics</a></li> <li><a href="/wiki/Video_quality" title="Video quality">Video quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Video_codec" title="Video codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a></li> <li><a href="/wiki/Deblocking_filter" title="Deblocking filter">Deblocking filter</a></li> <li><a href="/wiki/Lapped_transform" title="Lapped transform">Lapped transform</a></li> <li>Motion <ul><li><a href="/wiki/Motion_compensation" title="Motion compensation">Compensation</a></li> <li><a href="/wiki/Motion_estimation" title="Motion estimation">Estimation</a></li> <li><a href="/wiki/Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_theory" title="Information theory">Theory</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/Compressed_data_structure" title="Compressed data structure">Compressed data structures</a> <ul><li><a href="/wiki/Compressed_suffix_array" title="Compressed suffix array">Compressed suffix array</a></li> <li><a href="/wiki/FM-index" title="FM-index">FM-index</a></li></ul></li> <li><a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a> <ul><li><a href="/wiki/Timeline_of_information_theory" title="Timeline of information theory">Timeline</a></li></ul></li> <li><a class="mw-selflink selflink">Kolmogorov complexity</a></li> <li><a href="/wiki/Prefix_code" title="Prefix code">Prefix code</a></li> <li><a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li> <li><a href="/wiki/Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion</a></li> <li><a href="/wiki/Redundancy_(information_theory)" title="Redundancy (information theory)">Redundancy</a></li> <li><a href="/wiki/Data_compression_symmetry" title="Data compression symmetry">Symmetry</a></li> <li><a href="/wiki/Smallest_grammar_problem" title="Smallest grammar problem">Smallest grammar problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Community</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/Hutter_Prize" title="Hutter Prize">Hutter Prize</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</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/Mark_Adler" title="Mark Adler">Mark Adler</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Template"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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