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.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 class="mw-selflink selflink">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 href="/wiki/Zeno_machine" title="Zeno machine">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/Computability_theory" title="Computability theory">computability theory</a>, a <b>decider</b> is a <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> that halts for every input.<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> A decider is also called a <b>total Turing machine</b><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> as it represents a <a href="/wiki/Total_function" class="mw-redirect" title="Total function">total function</a>. </p><p>Because it always halts, such a machine is able to decide whether a given string is a member of a <a href="/wiki/Formal_language" title="Formal language">formal language</a>. The class of languages which can be decided by such machines is the set of <a href="/wiki/Recursive_language" title="Recursive language">recursive languages</a>. </p><p>Given an arbitrary Turing machine, determining whether it is a decider is an <a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable problem</a>. This is a variant of the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a>, which asks for whether a Turing machine halts on a specific input. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Functions_computable_by_total_Turing_machines">Functions computable by total Turing machines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decider_(Turing_machine)&action=edit&section=1" title="Edit section: Functions computable by total Turing machines"><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/Computable_function" title="Computable function">Computable function</a></div> <p>In practice, many functions of interest are computable by machines that always halt. A machine that uses only finite memory on any particular input can be forced to halt for every input by restricting its <a href="/wiki/Control_flow" title="Control flow">flow control</a> capabilities so that no input will ever cause the machine to enter an <a href="/wiki/Infinite_loop" title="Infinite loop">infinite loop</a>. As a trivial example, a machine implementing a finitary <a href="/wiki/Decision_tree" title="Decision tree">decision tree</a> will always halt. </p><p>It is not required that the machine be entirely free of looping capabilities, however, to guarantee halting. If we restrict loops to be of a predictably finite size (like the FOR loop in <a href="/wiki/BASIC" title="BASIC">BASIC</a>), we can express all of the <a href="/wiki/Primitive_recursive_functions" class="mw-redirect" title="Primitive recursive functions">primitive recursive functions</a> (Meyer and Ritchie, 1967). An example of such a machine is provided by the <a href="/wiki/Toy_programming_language" class="mw-redirect" title="Toy programming language">toy programming language</a> PL-{GOTO} of Brainerd and Landweber (1974). </p><p>We can further define a programming language in which we can ensure that even more sophisticated functions always halt. For example, the <a href="/wiki/Ackermann_function" title="Ackermann function">Ackermann function</a>, which is not primitive recursive, nevertheless is a total computable function computable by a <a href="/wiki/Term_rewriting" class="mw-redirect" title="Term rewriting">term rewriting</a> system with a <a href="/wiki/Well-order" title="Well-order">reduction ordering</a> on its arguments (Ohlebusch, 2002, pp. 67). </p><p>Despite the above examples of programming languages which guarantee termination of the programs, there exists no programming language which captures exactly the <a href="/wiki/Total_recursive_function" class="mw-redirect" title="Total recursive function">total recursive functions</a>, i.e. the functions which can be computed by a Turing machine that always halts. This is because existence of such a programming language would be a contradiction to the non-semi-decidability of the problem whether a Turing machine halts on every input. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_partial_Turing_machines">Relationship to partial Turing machines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decider_(Turing_machine)&action=edit&section=2" title="Edit section: Relationship to partial Turing machines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A general Turing machine will compute a partial function. Two questions can be asked about the relationship between partial Turing machines and total Turing machines: </p> <ol><li>Can every partial function computable by a partial Turing machine be extended (that is, have its domain enlarged) to become a total computable function?</li> <li>Is it possible to change the definition of a Turing machine so that a particular class of total Turing machines, computing all the total computable functions, can be found?</li></ol> <p>The answer to each of these questions is no. </p><p>The following theorem shows that the functions computable by machines that always halt do not include extensions of all partial computable functions, which implies the first question above has a negative answer. This fact is closely related to the algorithmic unsolvability of the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a>. </p> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>There are Turing computable <a href="/wiki/Partial_functions" class="mw-redirect" title="Partial functions">partial functions</a> that have no extension to a total Turing computable function. In particular, the partial function <i>f</i> defined so that <i>f</i>(<i>n</i>) = <i>m</i> if and only if the Turing machine with index <i>n</i> halts on input <span class="nowrap"><span data-sort-value="5000000000000000000♠"></span>0</span> with output <i>m</i> has no extension to a total computable function. </p> </div> <p>Indeed, if <i>g</i> were a total computable function extending <i>f</i> then <i>g</i> would be computable by some Turing machine; fix <i>e</i> as the index of such a machine. Build a Turing machine <i>M</i>, using <a href="/wiki/Kleene%27s_recursion_theorem" title="Kleene's recursion theorem">Kleene's recursion theorem</a>, which on input <span class="nowrap"><span data-sort-value="5000000000000000000♠"></span>0</span> simulates the machine with index <i>e</i> running on an index <i>n<sub>M</sub></i> for <i>M</i> (thus the machine <i>M</i> can produce an index of itself; this is the role of the recursion theorem). By assumption, this simulation will eventually return an answer. <span class="clarify-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">Define <i>M</i></span><sup class="noprint Inline-Template Template-Clarify" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The machine M has just been constructed, so it can't be defined again here. (July 2021)">clarify</span></a></i>]</sup> so that if <i>g</i>(<i>n<sub>M</sub></i>) = <i>m</i> then the return value of <i>M</i> is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f7ed29a2b4a62d3b6af05cd91a58ffc6094201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m+1}"></span>⁠</span>. Thus <i>f</i>(<i>n<sub>M</sub></i>), the true return value of <i>M</i> on input <span class="nowrap"><span data-sort-value="5000000000000000000♠"></span>0</span>, will not equal <i>g</i>(<i>n<sub>M</sub></i>). Hence <i>g</i> does not extend <i>f</i>. </p><p>The second question asks, in essence, whether there is another reasonable model of computation which computes only total functions and computes all the total computable functions. Informally, if such a model existed then each of its computers could be simulated by a Turing machine. Thus if this new model of computation consisted of a sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1},M_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1},M_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87b864fa03c5877ec46f9b9f279f117fa41b9f66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.408ex; height:2.509ex;" alt="{\displaystyle M_{1},M_{2},\ldots }"></span> of machines, there would be a <a href="/wiki/Recursively_enumerable" class="mw-redirect" title="Recursively enumerable">recursively enumerable</a> sequence <span class="mwe-math-element"><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_{1},\ldots T_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1},\ldots T_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ffe332bb807113cfed5a3e1b480ff55f88b12c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.725ex; height:2.509ex;" alt="{\displaystyle T_{1},\ldots T_{2},\ldots }"></span> of Turing machines that compute total functions and so that every total computable function is computable by one of the machines <i>T<sub>i</sub></i>. This is impossible, because a machine <span class="texhtml mvar" style="font-style:italic;">T</span> could be constructed such that on input <i>i</i> the machine <i>T</i> returns <span class="mwe-math-element"><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_{i}(i)+1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{i}(i)+1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a05942793d10ba4416095cd547765b63f9a5a911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.159ex; height:2.843ex;" alt="{\displaystyle T_{i}(i)+1\,}"></span>. This machine cannot be equivalent to any machine <span class="texhtml mvar" style="font-style:italic;">T</span> on the list: suppose it were on the list at index <i>j</i>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{j}(j)=T_{j}(j)+1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{j}(j)=T_{j}(j)+1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd2de585ffb805ffb8b49c7823f53c8472f44232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.558ex; height:3.009ex;" alt="{\displaystyle T_{j}(j)=T_{j}(j)+1\,}"></span>, which does not return an integer result. Therefore, it cannot be total, but the function by construction must be total (if total functions are recursively enumerable, then this function can be constructed), which is a contradiction. This shows that the second question has a negative answer. </p> <div class="mw-heading mw-heading2"><h2 id="The_set_of_indices_of_total_Turing_machines">The set of indices of total Turing machines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decider_(Turing_machine)&action=edit&section=3" title="Edit section: The set of indices of total Turing machines"><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/Termination_analysis" title="Termination analysis">Termination analysis</a></div> <p>The <a href="/wiki/Decision_problem" title="Decision problem">decision problem</a> of whether the Turing machine with index <i>e</i> will halt on every input is not decidable. In fact, this problem is at level <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{2}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{2}^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d58b0d35851996c3e8fa2ed4b5f4c583a3337df" 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 _{2}^{0}}"></span> of the <a href="/wiki/Arithmetical_hierarchy" title="Arithmetical hierarchy">arithmetical hierarchy</a>. Thus this problem is strictly more difficult than the <a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a>, which asks whether the machine with index <i>e</i> halts on input <i>0</i>. Intuitively, this difference in unsolvability is because each instance of the "total machine" problem represents infinitely many instances of the Halting problem. </p> <div class="mw-heading mw-heading2"><h2 id="Provability">Provability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Decider_(Turing_machine)&action=edit&section=4" title="Edit section: Provability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One may be interested not only in whether a Turing machine is total, but also in whether this can be proven in a certain logical system, such as <a href="/wiki/First-order_logic" title="First-order logic">first order</a> <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>. </p><p>In a <a href="/wiki/Soundness" title="Soundness">sound</a> proof system, every provably total Turing machine is indeed total, but the converse is not true: informally, for every first-order proof system that is strong enough (including Peano arithmetic), there are Turing machines which are assumed to be total, but cannot be proven as such, unless the system is inconsistent (in which case one can prove anything). The proof of their totality either rests on some assumptions or require another proof system. </p><p>Thus, as one can enumerate all the proofs in the proof system, one can build a Turing machine on input n that goes through the first n proofs and look for a contradiction. If it finds one, it gets into an infinite loop and never halts; otherwise, it halts. If the system is <a href="/wiki/Consistent" class="mw-redirect" title="Consistent">consistent</a>, the Turing machine will halt on every input, but one cannot prove this in a strong enough proof system due to <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a>. </p><p>One can also create a Turing machine that will halt if and only if the proof system is inconsistent, and is thus non-total for a consistent system but cannot be proven such: This is a Turing machine that, regardless of input, enumerates all proofs and halts on a contradiction. </p><p>A Turing machine that goes through <a href="/wiki/Goodstein_sequence" class="mw-redirect" title="Goodstein sequence">Goodstein sequences</a> and halts at zero is total but cannot be proven as such in Peano arithmetic. </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=Decider_(Turing_machine)&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/BlooP_and_FlooP" title="BlooP and FlooP">BlooP and FlooP</a></li> <li><a href="/wiki/Total_functional_programming" title="Total functional programming">Total functional programming</a></li> <li><a href="/wiki/Termination_analysis" title="Termination analysis">Termination analysis</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=Decider_(Turing_machine)&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><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">Sipser, 1996<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (March 2019)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Kozen, 1997<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (March 2019)">page needed</span></a></i>]</sup></span> </li> </ol></div> <ul><li>Brainerd, W.S., <a href="/wiki/Lawrence_Landweber" title="Lawrence Landweber">Landweber, L.H.</a> (1974), <i>Theory of Computation</i>, Wiley.</li> <li><a href="/wiki/Albert_R._Meyer" title="Albert R. Meyer">Meyer, A.R.</a>, <a href="/wiki/Dennis_Ritchie" title="Dennis Ritchie">Ritchie, D.M.</a> (1967), <i><a rel="nofollow" class="external text" href="https://www.researchgate.net/profile/Albert_Meyer/publication/234810406_The_complexity_of_loop_programs/links/00b49517fb0c8b6a2a000000.pdf">The complexity of loop programs</a></i>, Proc. of the ACM National Meetings, 465.</li> <li><a href="/wiki/Michael_Sipser" title="Michael Sipser">Sipser, M.</a> (2006), <i><a href="/wiki/Introduction_to_the_Theory_of_Computation" title="Introduction to the Theory of Computation">Introduction to the Theory of Computation</a></i>, PWS Publishing Co.</li> <li><a href="/wiki/Dexter_Kozen" title="Dexter Kozen">Kozen, D.C.</a> (1997), <i>Automata and Computability</i>, Springer.</li> <li>Ohlebusch, E. (2002), <i>Advanced Topics in Term Rewriting</i>, Springer.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output 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style="width:100%;padding:0;background:transparent;color:inherit;"><div style="padding:0px"><table class="navbox-columns-table" style="border-spacing: 0px; text-align:left;width:100%;"><tbody><tr><td class="navbox-abovebelow" style="font-weight:bold;"><a href="/wiki/Chomsky_hierarchy" title="Chomsky hierarchy">Chomsky hierarchy</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Formal_grammar" title="Formal grammar">Grammars</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Formal_language" title="Formal language">Languages</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Abstract_machine" title="Abstract machine">Abstract machines</a></td></tr><tr style="vertical-align:top"><td class="navbox-list" style="padding:0px;text-align: center;width:10em;"><div> <ul><li>Type-0</li> <li>—</li> <li>Type-1</li> <li>—</li> <li>—</li> <li>—</li> <li>—</li> <li>—</li> <li>Type-2</li> <li>—</li> <li>—</li> <li>Type-3</li> <li>—</li> <li>—</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;width:10em;"><div> <ul><li><a href="/wiki/Unrestricted_grammar" title="Unrestricted grammar">Unrestricted</a></li> <li>(no common name)</li> <li><a href="/wiki/Context-sensitive_grammar" title="Context-sensitive grammar">Context-sensitive</a></li> <li><span style="white-space:nowrap;">Positive <a href="/wiki/Range_concatenation_grammars" class="mw-redirect" title="Range concatenation grammars">range concatenation</a></span></li> <li><a href="/wiki/Indexed_grammar" title="Indexed grammar">Indexed</a></li> <li>—</li> <li><a href="/wiki/Linear_context-free_rewriting_system" class="mw-redirect" title="Linear context-free rewriting system">Linear context-free rewriting systems</a></li> <li><a href="/wiki/Tree-adjoining_grammar" title="Tree-adjoining grammar">Tree-adjoining</a></li> <li><a href="/wiki/Context-free_grammar" title="Context-free grammar">Context-free</a></li> <li><a href="/wiki/Deterministic_context-free_grammar" title="Deterministic context-free grammar">Deterministic context-free</a></li> <li><a href="/wiki/Nested_word" title="Nested word">Visibly pushdown</a></li> <li><a href="/wiki/Regular_grammar" title="Regular grammar">Regular</a></li> <li>—</li> <li><a href="/wiki/Non-recursive_grammar" class="mw-redirect" title="Non-recursive grammar">Non-recursive</a></li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;width:10em;"><div> <ul><li><a href="/wiki/Recursively_enumerable_language" title="Recursively enumerable language">Recursively enumerable</a></li> <li><a href="/wiki/Recursive_language" title="Recursive language">Decidable</a></li> <li><a href="/wiki/Context-sensitive_language" title="Context-sensitive language">Context-sensitive</a></li> <li><span style="white-space:nowrap;">Positive <a href="/wiki/Range_concatenation_language" class="mw-redirect" title="Range concatenation language">range concatenation</a><sup>*</sup></span></li> <li><a href="/wiki/Indexed_language" title="Indexed language">Indexed</a><sup>*</sup></li> <li>—</li> <li><a href="/wiki/Linear_context-free_rewriting_language" class="mw-redirect" title="Linear context-free rewriting language">Linear context-free rewriting language</a></li> <li><a href="/wiki/Tree-adjoining_grammar" title="Tree-adjoining grammar">Tree-adjoining</a></li> <li><a href="/wiki/Context-free_language" title="Context-free language">Context-free</a></li> <li><a href="/wiki/Deterministic_context-free_language" title="Deterministic context-free language">Deterministic context-free</a></li> <li><a href="/wiki/Nested_word" title="Nested word">Visibly pushdown</a></li> <li><a href="/wiki/Regular_language" title="Regular language">Regular</a></li> <li><a href="/wiki/Star-free_language" title="Star-free language">Star-free</a></li> <li><a href="/wiki/Finite_language" class="mw-redirect" title="Finite language">Finite</a></li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;width:10em;"><div> <ul><li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a class="mw-selflink selflink">Decider</a></li> <li><a href="/wiki/Linear_bounded_automaton" title="Linear bounded automaton">Linear-bounded</a></li> <li><a href="/wiki/PTIME" class="mw-redirect" title="PTIME">PTIME</a> Turing Machine</li> <li><a href="/wiki/Nested_stack_automaton" title="Nested stack automaton">Nested stack</a></li> <li><a href="/wiki/Thread_automaton" title="Thread automaton">Thread automaton</a></li> <li>restricted <a href="/wiki/Tree_stack_automaton" title="Tree stack automaton">Tree stack automaton</a></li> <li><a href="/wiki/Embedded_pushdown_automaton" title="Embedded pushdown automaton">Embedded pushdown</a></li> <li><a href="/wiki/Pushdown_automaton" title="Pushdown automaton">Nondeterministic pushdown</a></li> <li><a href="/wiki/Deterministic_pushdown_automaton" title="Deterministic pushdown automaton">Deterministic pushdown</a></li> <li><a href="/wiki/Nested_word" title="Nested word">Visibly pushdown</a></li> <li><a href="/wiki/Finite-state_machine" title="Finite-state machine">Finite</a></li> <li><a href="/wiki/Aperiodic_finite_state_automaton" title="Aperiodic finite state automaton">Counter-free (with aperiodic finite monoid)</a></li> <li><a href="/wiki/Deterministic_acyclic_finite_state_automaton" title="Deterministic acyclic finite state automaton">Acyclic finite</a></li></ul> </div></td></tr></tbody></table></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><span style="white-space:nowrap;">Each category of languages, except those marked by a <sup>*</sup>, is a <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a> of the category directly above it.</span> <span 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