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<span class="vector-toc-numb">2</span> <span>Infinite time Turing machines</span> </div> </a> <ul id="toc-Infinite_time_Turing_machines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computability" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computability"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Computability</span> </div> </a> <ul id="toc-Computability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div 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href="https://es.wikipedia.org/wiki/M%C3%A1quina_de_Zen%C3%B3n" title="Máquina de Zenón – Spanish" lang="es" hreflang="es" data-title="Máquina de Zenón" 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%85%D8%A7%D8%B4%DB%8C%D9%86_%D8%B2%D9%86%D9%88%D9%86" title="ماشین زنون – Persian" lang="fa" hreflang="fa" data-title="ماشین زنون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Zenonov_stroj" title="Zenonov stroj – Croatian" lang="hr" hreflang="hr" data-title="Zenonov stroj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%BC%E3%83%8E%E3%83%B3%E6%A9%9F%E6%A2%B0" title="ゼノン機械 – Japanese" lang="ja" hreflang="ja" data-title="ゼノン機械" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%88%D0%B8%D0%BD%D0%B0_%D0%97%D0%B5%D0%BD%D0%BE%D0%BD%D0%B0" title="Машина Зенона – Russian" lang="ru" hreflang="ru" data-title="Машина Зенона" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%97%D0%B5%D0%BD%D0%BE%D0%BD%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B0%D1%88%D0%B8%D0%BD%D0%B0" title="Зенонова машина – Serbian" lang="sr" hreflang="sr" data-title="Зенонова машина" data-language-autonym="Српски / srpski" 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0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a></th></tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Machine</th></tr><tr><td class="sidebar-content plainlist"> <ul><li><a href="/wiki/Turing_machine_equivalents" title="Turing machine equivalents">Turing machine equivalents</a></li> <li><a href="/wiki/Turing_machine_examples" title="Turing machine examples">Turing machine examples</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Variants</th></tr><tr><td class="sidebar-content plainlist"> <ul><li><a href="/wiki/Alternating_Turing_machine" title="Alternating Turing machine">Alternating Turing machine</a></li> <li><a href="/wiki/Neural_Turing_machine" title="Neural Turing machine">Neural Turing machine</a></li> <li><a href="/wiki/Nondeterministic_Turing_machine" title="Nondeterministic Turing machine">Nondeterministic Turing machine</a></li> <li><a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">Quantum Turing machine</a></li> <li><a href="/wiki/Post%E2%80%93Turing_machine" title="Post–Turing machine">Post–Turing machine</a></li> <li><a href="/wiki/Probabilistic_Turing_machine" title="Probabilistic Turing machine">Probabilistic Turing machine</a></li> <li><a href="/wiki/Multitape_Turing_machine" title="Multitape Turing machine">Multitape Turing machine</a></li> <li><a href="/wiki/Multi-track_Turing_machine" title="Multi-track Turing machine">Multi-track Turing machine</a></li> <li><a href="/wiki/Symmetric_Turing_machine" title="Symmetric Turing machine">Symmetric Turing machine</a></li> <li><a href="/wiki/Decider_(Turing_machine)" title="Decider (Turing machine)">Total Turing machine</a></li> <li><a href="/wiki/Unambiguous_Turing_machine" title="Unambiguous Turing machine">Unambiguous Turing machine</a></li> <li><a href="/wiki/Universal_Turing_machine" title="Universal Turing machine">Universal Turing machine</a></li> <li><a class="mw-selflink selflink">Zeno machine</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Science</th></tr><tr><td class="sidebar-content plainlist"> <ul><li><a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a></li> <li><a href="/wiki/Category:Turing_machine" title="Category:Turing machine">Category:Turing machine</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Turing" title="Template:Turing"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Turing" title="Template talk:Turing"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Turing" title="Special:EditPage/Template:Turing"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <b>Zeno machines</b> (abbreviated <b>ZM</b>, and also called <b>accelerated Turing machine</b>, <b>ATM</b>) are a hypothetical computational model related to <a href="/wiki/Turing_machines" class="mw-redirect" title="Turing machines">Turing machines</a> that are capable of carrying out computations involving a <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> number of algorithmic steps.<sup id="cite_ref-ITTM_1-0" class="reference"><a href="#cite_note-ITTM-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> These machines are ruled out in most models of computation. </p><p>The idea of Zeno machines was first discussed by <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> in 1927; the name refers to <a href="/wiki/Zeno%27s_paradoxes" title="Zeno's paradoxes">Zeno's paradoxes</a>, attributed to the ancient Greek philosopher <a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a>. Zeno machines play a crucial role in some theories. The theory of the <a href="/wiki/Frank_J._Tipler#The_Omega_Point" title="Frank J. Tipler">Omega Point</a> devised by physicist <a href="/wiki/Frank_J._Tipler" title="Frank J. Tipler">Frank J. Tipler</a>, for instance, can only be valid if Zeno machines are possible. </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=Zeno_machine&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A Zeno machine is a Turing machine that can take an infinite number of steps, and then continue take more steps. This can be thought of as a <a href="/wiki/Supertask" title="Supertask">supertask</a> 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 {\frac {1}{2^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98a4804516c6a111a316c88d88290a31951034d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.217ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{2^{n}}}}"></span> units of time are taken to perform the <span class="mwe-math-element"><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>-th step; thus, the first step takes 0.5 units of time, the second takes 0.25, the third 0.125 and so on, so that after one unit of time, a <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> number of steps will have been performed. </p> <div class="mw-heading mw-heading2"><h2 id="Infinite_time_Turing_machines">Infinite time Turing machines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno_machine&action=edit&section=2" title="Edit section: Infinite time Turing machines"><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:Animation_of_Thomson%27s_Lamp_Zeno_Machine.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Animation_of_Thomson%27s_Lamp_Zeno_Machine.gif/400px-Animation_of_Thomson%27s_Lamp_Zeno_Machine.gif" decoding="async" width="400" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Animation_of_Thomson%27s_Lamp_Zeno_Machine.gif/600px-Animation_of_Thomson%27s_Lamp_Zeno_Machine.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0f/Animation_of_Thomson%27s_Lamp_Zeno_Machine.gif 2x" data-file-width="749" data-file-height="434" /></a><figcaption>An animation of an infinite time Turing machine based on the <a href="/wiki/Thomson%27s_lamp" title="Thomson's lamp">Thomson's lamp</a> thought experiment. A cell alternates between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> for steps before <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>. The cell becomes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> since the sequence does not converge.</figcaption></figure> <p>A more formal model of the Zeno machine is the <b>infinite time Turing machine</b>. Defined first in unpublished work by Jeffrey Kidder and expanded upon by Joel Hamkins and Andy Lewis, in <i>Infinite Time Turing Machines</i>,<sup id="cite_ref-ITTM1_2-0" class="reference"><a href="#cite_note-ITTM1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> the infinite time Turing machine is an extension of the classical Turing machine model, to include <a href="/wiki/Transfinite_ordinal" class="mw-redirect" title="Transfinite ordinal">transfinite</a> time; that is time beyond all finite time.<sup id="cite_ref-ITTM1_2-1" class="reference"><a href="#cite_note-ITTM1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A classical Turing machine has a status at step <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> (in the start state, with an empty tape, read head at cell 0) and a procedure for getting from one status to the successive status. In this way the status of a Turing machine is defined for all steps corresponding to a natural number. An <abbr title="Infinite Time Turing Machine">ITTM</abbr> maintains these properties, but also defines the status of the machine at <a href="/wiki/Limit_ordinals" class="mw-redirect" title="Limit ordinals">limit ordinals</a>, that is ordinals that are neither <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> nor the successor of any ordinal. The status of a Turing machine consists of 3 parts: </p> <ol><li>The state</li> <li>The location of the read-write head</li> <li>The contents of the tape</li></ol> <p>Just as a classical Turing machine has a labeled start state, which is the state at the start of a program, an <abbr title="Infinite Time Turing Machine">ITTM</abbr> has a labeled <i>limit</i> state which is the state for the machine at any limit ordinal.<sup id="cite_ref-ITTM_1-1" class="reference"><a href="#cite_note-ITTM-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This is the case even if the machine has no other way to access this state, for example no node transitions to it. The location of the read-write head is set to zero for at any limit step.<sup id="cite_ref-ITTM_1-2" class="reference"><a href="#cite_note-ITTM-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ITTM1_2-2" class="reference"><a href="#cite_note-ITTM1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Lastly the state of the tape is determined by the <a href="/wiki/Limsup" class="mw-redirect" title="Limsup">limit supremum</a> of previous tape states. For some 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, a cell <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> and, a <a href="/wiki/Limit_ordinal" title="Limit ordinal">limit ordinal</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\lambda )_{k}=\limsup _{n\rightarrow \lambda }T(n)_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi>λ</mi> </mrow> </munder> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\lambda )_{k}=\limsup _{n\rightarrow \lambda }T(n)_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb98de348a2b5350e37dacde161d7c2d324fc70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.423ex; height:4.509ex;" alt="{\displaystyle T(\lambda )_{k}=\limsup _{n\rightarrow \lambda }T(n)_{k}}"></span> </p><p>That is the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>th cell at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the limit supremum of that same cell as the machine approaches <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>.<sup id="cite_ref-ITTM_1-3" class="reference"><a href="#cite_note-ITTM-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This can be thought of as the limit if it converges 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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> otherwise.<sup id="cite_ref-ITTM_1-4" class="reference"><a href="#cite_note-ITTM-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Computability">Computability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno_machine&action=edit&section=3" title="Edit section: Computability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zeno machines have been proposed as a model of computation more powerful than classical Turing machines, based on their ability to solve <a href="/wiki/The_halting_problem" class="mw-redirect" title="The halting problem">the halting problem</a> for classical Turing machines.<sup id="cite_ref-Shagrir_3-0" class="reference"><a href="#cite_note-Shagrir-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Cristian Calude and Ludwig Staiger present the following <a href="/wiki/Pseudocode" title="Pseudocode">pseudocode</a> algorithm as a solution to the halting problem when run on a Zeno machine.<sup id="cite_ref-CDMTCS_4-0" class="reference"><a href="#cite_note-CDMTCS-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <pre><b>begin program</b> write 0 on the first position of the output tape; <b>begin loop</b> simulate 1 successive step of the given Turing machine on the given input; <b>if</b> the Turing machine has halted <b>then</b> write 1 on the first position of the output tape and break out of loop; <b>end loop</b> <b>end program</b> </pre> <p>By inspecting the first position of the output tape after <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> unit of time has elapsed we can determine whether the given Turing machine halts.<sup id="cite_ref-CDMTCS_4-1" class="reference"><a href="#cite_note-CDMTCS-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In contrast Oron Shagrir argues that the state of a Zeno machine is only defined on the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f99b30b4451167959e97802252ad13b87af5505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.91ex; height:2.843ex;" alt="{\displaystyle [0,1)}"></span>, and so it is impossible to inspect the tape at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>. Furthermore since classical Turing machines don't have any timing information, the addition of timing information whether accelerating or not does not itself add any computational power.<sup id="cite_ref-Shagrir_3-1" class="reference"><a href="#cite_note-Shagrir-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Infinite time Turing machines however, are capable of implementing the given algorithm, halting at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> with the correct solution,<sup id="cite_ref-ITTM1_2-3" class="reference"><a href="#cite_note-ITTM1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> since they do define their state for transfinite steps.<sup id="cite_ref-Shagrir_3-2" class="reference"><a href="#cite_note-Shagrir-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> All <a href="/wiki/Projective_hierarchy" title="Projective hierarchy"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd076fb78a7a5ee340920a3d846ddf67455f941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{1}}"></span> sets</a> are <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a> by infinite time Turing machines, 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 \Delta _{2}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{2}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bb95191e7af7bd30330e983e82620aa5b2a1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.99ex; height:3.176ex;" alt="{\displaystyle \Delta _{2}^{1}}"></span> sets are <a href="/wiki/Decidability_(logic)#Semidecidability" title="Decidability (logic)">semidecidable</a>.<sup id="cite_ref-ITTM1_2-4" class="reference"><a href="#cite_note-ITTM1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><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="The text near this tag may need clarification or removal of jargon. (May 2021)">clarification needed</span></a></i>]</sup> </p><p>Zeno machines cannot solve their own halting problem.<sup id="cite_ref-CDMTCS_4-2" class="reference"><a href="#cite_note-CDMTCS-4"><span class="cite-bracket">[</span>4<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=Zeno_machine&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Computation_in_the_limit" title="Computation in the limit">Computation in the limit</a></li> <li><a href="/wiki/Specker_sequence" title="Specker sequence">Specker sequence</a></li> <li><a href="/wiki/Ross%E2%80%93Littlewood_paradox" title="Ross–Littlewood paradox">Ross–Littlewood paradox</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno_machine&action=edit&section=5" title="Edit section: References"><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-ITTM-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-ITTM_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ITTM_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-ITTM_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-ITTM_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-ITTM_1-4"><sup><i><b>e</b></i></sup></a></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="CITEREFHamkins2002" class="citation arxiv cs1">Hamkins, Joel (2002-12-03). "Infinite time Turing machines". <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/math/0212047">math/0212047</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=Infinite+time+Turing+machines&rft.date=2002-12-03&rft_id=info%3Aarxiv%2Fmath%2F0212047&rft.aulast=Hamkins&rft.aufirst=Joel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno+machine" class="Z3988"></span></span> </li> <li id="cite_note-ITTM1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-ITTM1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ITTM1_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-ITTM1_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-ITTM1_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-ITTM1_2-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamkinsLewis1998" class="citation arxiv cs1">Hamkins, Joel; Lewis, Andy (1998-08-21). "Infinite Time Turing Machines". <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/math/9808093">math/9808093</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=Infinite+Time+Turing+Machines&rft.date=1998-08-21&rft_id=info%3Aarxiv%2Fmath%2F9808093&rft.aulast=Hamkins&rft.aufirst=Joel&rft.au=Lewis%2C+Andy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno+machine" class="Z3988"></span></span> </li> <li id="cite_note-Shagrir-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Shagrir_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Shagrir_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Shagrir_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShagrir" class="citation cs2"><a href="/wiki/Oron_Shagrir" title="Oron Shagrir">Shagrir, Oron</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070709100858/http://edelstein.huji.ac.il/staff/shagrir/papers/Supertasks_Accelerating_Turing_Machines_and_Uncomputability.pdf"><i>Super-Tasks, Accelerating Turing Machines and Uncomputability</i></a> <span class="cs1-format">(PDF)</span>, archived from <a rel="nofollow" class="external text" href="http://edelstein.huji.ac.il/staff/shagrir/papers/Supertasks_Accelerating_Turing_Machines_and_Uncomputability.pdf">the original</a> <span class="cs1-format">(PDF)</span> on July 9, 2007</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Super-Tasks%2C+Accelerating+Turing+Machines+and+Uncomputability&rft.aulast=Shagrir&rft.aufirst=Oron&rft_id=http%3A%2F%2Fedelstein.huji.ac.il%2Fstaff%2Fshagrir%2Fpapers%2FSupertasks_Accelerating_Turing_Machines_and_Uncomputability.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno+machine" class="Z3988"></span></span> </li> <li id="cite_note-CDMTCS-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-CDMTCS_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-CDMTCS_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-CDMTCS_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaludeStaiger" class="citation cs2">Calude, Cristian; Staiger, Ludwig, <a rel="nofollow" class="external text" href="https://www.cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/350cris.pdf"><i>A Note on Accelerated Turing Machines</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Note+on+Accelerated+Turing+Machines&rft.aulast=Calude&rft.aufirst=Cristian&rft.au=Staiger%2C+Ludwig&rft_id=https%3A%2F%2Fwww.cs.auckland.ac.nz%2Fresearch%2Fgroups%2FCDMTCS%2Fresearchreports%2F350cris.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno+machine" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Zeno_machine&oldid=1227154789">https://en.wikipedia.org/w/index.php?title=Zeno_machine&oldid=1227154789</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Models_of_computation" title="Category:Models of computation">Models of computation</a></li><li><a href="/wiki/Category:Turing_machine" title="Category:Turing machine">Turing machine</a></li><li><a href="/wiki/Category:Hypercomputation" title="Category:Hypercomputation">Hypercomputation</a></li><li><a href="/wiki/Category:Supertasks" title="Category:Supertasks">Supertasks</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_May_2021" title="Category:Articles needing additional references from May 2021">Articles needing additional references from May 2021</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_May_2021" title="Category:Wikipedia articles needing clarification from May 2021">Wikipedia articles needing clarification from May 2021</a></li><li><a href="/wiki/Category:Articles_with_example_pseudocode" title="Category:Articles with example pseudocode">Articles with example pseudocode</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 4 June 2024, at 01:32<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; 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