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id="toc-Problems_in_combinatorial_group_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_in_topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Problems_in_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Problems in topology</span> </div> </a> <ul id="toc-Problems_in_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_in_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Problems_in_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Problems in analysis</span> </div> </a> <ul id="toc-Problems_in_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_about_formal_languages_and_grammars" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Problems_about_formal_languages_and_grammars"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Problems about formal languages and grammars</span> </div> </a> <ul id="toc-Problems_about_formal_languages_and_grammars-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_problems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Other problems</span> </div> </a> <ul id="toc-Other_problems-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">9</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 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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Computational problems no algorithm can solve</div> <p>In <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>, an <b><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable problem</a></b> is a <a href="/wiki/Decision_problem" title="Decision problem">decision problem</a> for which an <a href="/wiki/Effective_method" title="Effective method">effective method</a> (algorithm) to derive the correct answer does not exist. More formally, an undecidable problem is a problem whose language is not a <a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">recursive set</a>; see the article <a href="/wiki/Decidable_language" class="mw-redirect" title="Decidable language">Decidable language</a>. There are <a href="/wiki/Uncountable_set" title="Uncountable set">uncountably</a> many undecidable problems, so the list below is necessarily incomplete. Though undecidable languages are not recursive languages, they may be <a href="/wiki/Subset" title="Subset">subsets</a> of <a href="/wiki/Alan_Turing" title="Alan Turing">Turing</a> recognizable languages: i.e., such undecidable languages may be recursively enumerable. </p><p>Many, if not most, undecidable problems in mathematics can be posed as <a href="/wiki/Word_problem_(mathematics)" title="Word problem (mathematics)">word problems</a>: determining when two distinct strings of symbols (encoding some mathematical concept or object) represent the same object or not. </p><p>For undecidability in axiomatic mathematics, see <a href="/wiki/List_of_statements_undecidable_in_ZFC" class="mw-redirect" title="List of statements undecidable in ZFC">List of statements undecidable in ZFC</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Problems_in_logic">Problems in logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_undecidable_problems&action=edit&section=1" title="Edit section: Problems in logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Hilbert's <a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a>.</li> <li><a href="/wiki/Type_inference" title="Type inference">Type inference</a> and <a href="/wiki/Type_checking" class="mw-redirect" title="Type checking">type checking</a> for the <a href="/wiki/Second-order_lambda_calculus" class="mw-redirect" title="Second-order lambda calculus">second-order lambda calculus</a> (or equivalent).<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></li> <li>Determining whether a first-order sentence in the <a href="/wiki/Logic_of_graphs" title="Logic of graphs">logic of graphs</a> can be realized by a finite undirected graph.<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></li> <li><a href="/wiki/Trakhtenbrot%27s_theorem" title="Trakhtenbrot's theorem">Trakhtenbrot's theorem</a> - Finite satisfiability is undecidable.</li> <li>Satisfiability of first order <a href="/wiki/Horn_clause" title="Horn clause">Horn clauses</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Problems_about_abstract_machines">Problems about abstract machines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_undecidable_problems&action=edit&section=2" title="Edit section: Problems about abstract machines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a> (determining whether a <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> halts on a given input) and the <a href="/wiki/Mortality_(computability_theory)" title="Mortality (computability theory)">mortality problem</a> (determining whether it halts for every starting configuration).</li> <li>Determining whether a Turing machine is a <a href="/wiki/Busy_beaver#Non-computability" title="Busy beaver">busy beaver champion</a> (i.e., is the longest-running among halting Turing machines with the same number of states and symbols).</li> <li><a href="/wiki/Rice%27s_theorem" title="Rice's theorem">Rice's theorem</a> states that for all nontrivial properties of partial functions, it is undecidable whether a given machine computes a partial function with that property.</li> <li>The halting problem for a <a href="/wiki/Register_machine" title="Register machine">register machine</a>: a finite-state automaton with no inputs and two counters that can be incremented, decremented, and tested for zero.</li> <li>Universality of a nondeterministic <a href="/wiki/Pushdown_automaton" title="Pushdown automaton">pushdown automaton</a>: determining whether all words are accepted.</li> <li>The problem whether a <a href="/wiki/Tag_system" title="Tag system">tag system</a> halts.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Problems_about_matrices">Problems about matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_undecidable_problems&action=edit&section=3" title="Edit section: Problems about matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Mortal_matrix_problem" class="mw-redirect" title="Mortal matrix problem">mortal matrix problem</a>.</li> <li>Determining whether a finite set of upper triangular 3 × 3 matrices with nonnegative integer entries generates a free <a href="/wiki/Semigroup" title="Semigroup">semigroup</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2024)">citation needed</span></a></i>]</sup></li> <li>Determining whether two finitely generated subsemigroups of <a href="/wiki/Integer_matrices" class="mw-redirect" title="Integer matrices">integer matrices</a> have a common element.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2024)">citation needed</span></a></i>]</sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Problems_in_combinatorial_group_theory">Problems in combinatorial group theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_undecidable_problems&action=edit&section=4" title="Edit section: Problems in combinatorial group theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Word_problem_for_groups" title="Word problem for groups">word problem for groups</a>.</li> <li>The <a href="/wiki/Conjugacy_problem" title="Conjugacy problem">conjugacy problem</a>.</li> <li>The <a href="/wiki/Group_isomorphism_problem" title="Group isomorphism problem">group isomorphism problem</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Problems_in_topology">Problems in topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_undecidable_problems&action=edit&section=5" title="Edit section: Problems in topology"><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/Simplicial_complex_recognition_problem" title="Simplicial complex recognition problem">Simplicial complex recognition problem</a></div> <ul><li>Determining whether two finite <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complexes</a> are <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a>.</li> <li>Determining whether a finite <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a> is (homeomorphic to) a <a href="/wiki/Manifold" title="Manifold">manifold</a>.</li> <li>Determining whether the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of a finite simplicial complex is trivial.</li> <li>Determining whether two non-<a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> <a href="/wiki/5-manifold" title="5-manifold">5-manifolds</a> are homeomorphic, or if a 5-manifold is homeomorphic to <b>S</b><sup>5</sup>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Problems_in_analysis">Problems in analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_undecidable_problems&action=edit&section=6" title="Edit section: Problems in analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>For functions in certain classes, the problem of determining: whether two functions are equal, known as the zero-equivalence problem (see <a href="/wiki/Richardson%27s_theorem" title="Richardson's theorem">Richardson's theorem</a>);<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> the zeroes of a function; whether the indefinite integral of a function is also in the class.<sup id="cite_ref-stall_5-0" class="reference"><a href="#cite_note-stall-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Of course, some subclasses of these problems are decidable. For example, there is an effective decision procedure for the elementary integration of any function which belongs to a field of transcendental elementary functions, the <a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a>.</li> <li>"The problem of deciding whether the definite contour multiple integral of an elementary meromorphic function is zero over an everywhere real analytic manifold on which it is analytic", a consequence of the <a href="/wiki/Matiyasevich%27s_theorem" class="mw-redirect" title="Matiyasevich's theorem">MRDP theorem</a> resolving <a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a>.<sup id="cite_ref-stall_5-1" class="reference"><a href="#cite_note-stall-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>Determining the domain of a solution to an <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> of the form</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dx}{dt}}=p(t,x),~x(t_{0})=x_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dt}}=p(t,x),~x(t_{0})=x_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5e11a6c6f0c205d6ab3f8f254486cc2b5f995a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.439ex; height:5.509ex;" alt="{\displaystyle {\frac {dx}{dt}}=p(t,x),~x(t_{0})=x_{0},}"></span></dd></dl></dd> <dd>where <i>x</i> is a <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> in <b>R</b><sup>n</sup>, <i>p</i>(<i>t</i>, <i>x</i>) is a vector of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> in <i>t</i> and <i>x</i>, and <i>(t<sub>0</sub>, x<sub>0</sub>)</i> belongs to <b>R</b><sup>n+1</sup>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Problems_about_formal_languages_and_grammars">Problems about formal languages and grammars</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_undecidable_problems&action=edit&section=7" title="Edit section: Problems about formal languages and grammars"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Post_correspondence_problem" title="Post correspondence problem">Post correspondence problem</a>.</li> <li>Determining if a <a href="/wiki/Context-free_grammar" title="Context-free grammar">context-free grammar</a> generates all possible strings, or if it is <a href="/wiki/Ambiguous_grammar" title="Ambiguous grammar">ambiguous</a>.</li> <li>Given two context-free grammars, determining whether they generate the same set of strings, or whether one generates a subset of the strings generated by the other, or whether there is any string at all that both generate.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Other_problems">Other problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_undecidable_problems&action=edit&section=8" title="Edit section: Other problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The problem of determining if a given set of <a href="/wiki/Wang_tile" title="Wang tile">Wang tiles</a> can tile the plane.</li> <li>The problem of determining the <a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a> of a string.</li> <li><a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a>: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers.</li> <li>Determining whether a given initial point with rational coordinates is periodic, or whether it lies in the basin of attraction of a given open set, in a piecewise-linear iterated map in two dimensions, or in a piecewise-linear flow in three dimensions.<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></li> <li>Determining whether a <a href="/wiki/Lambda_calculus#Undecidability_of_equivalence" title="Lambda calculus">λ-calculus</a> formula has a normal form.</li> <li><a href="/wiki/Conway%27s_Game_of_Life#Undecidability" title="Conway's Game of Life">Conway's Game of Life</a> on whether, given an initial pattern and another pattern, the latter pattern can ever appear from the initial one.</li> <li><a href="/wiki/Rule_110" title="Rule 110">Rule 110</a> - most questions involving "can property X appear later" are undecidable.</li> <li>The problem of determining whether a quantum mechanical system has a <a href="/wiki/Spectral_gap_(physics)" title="Spectral gap (physics)">spectral gap</a>.<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></li> <li>Finding the capacity of an information-stable finite state machine channel.<sup id="cite_ref-elkouss_fsmc_10-0" class="reference"><a href="#cite_note-elkouss_fsmc-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>In <a href="/wiki/Network_coding" class="mw-redirect" title="Network coding">network coding</a>, determining whether a network is solvable.<sup id="cite_ref-li_nc_11-0" class="reference"><a href="#cite_note-li_nc-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kuhne_matroid_12-0" class="reference"><a href="#cite_note-kuhne_matroid-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li>Determining whether a player has a winning strategy in a game of <i><a href="/wiki/Magic:_The_Gathering" title="Magic: The Gathering">Magic: The Gathering</a></i>.<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></li> <li>Planning in a <a href="/wiki/Partially_observable_Markov_decision_process" title="Partially observable Markov decision process">partially observable Markov decision process</a>.</li> <li>The problem of planning <a href="/wiki/Air_travel" title="Air travel">air travel</a> from one destination to another, when <a href="/wiki/Fare" title="Fare">fares</a> are taken into account.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li> <li>In the <a href="/wiki/Ray_tracing_(graphics)" title="Ray tracing (graphics)">ray tracing</a> problem for a 3-dimensional system of reflective or refractive objects, determining if a ray beginning at a given position and direction eventually reaches a certain point.<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></li> <li>Determining if a particle path of an ideal fluid on a three dimensional domain eventually reaches a certain region in space.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li></ul> <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=List_of_undecidable_problems&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Lists_of_problems" title="Lists of problems">Lists of problems</a></li> <li><a href="/wiki/List_of_unsolved_problems" class="mw-redirect" title="List of unsolved problems">List of unsolved problems</a></li> <li><a href="/wiki/Reduction_(complexity)" title="Reduction (complexity)">Reduction (complexity)</a></li> <li><a href="/wiki/Unknowability" title="Unknowability">Unknowability</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=List_of_undecidable_problems&action=edit&section=10" 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 mw-references-columns"><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="CITEREFWells1993" class="citation journal cs1">Wells, J. B. (1993). "Typability and type checking in the second-order lambda-calculus are equivalent and undecidable". <i>Tech. Rep. 93-011</i>. Comput. Sci. Dept., Boston Univ.: 176–185. <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=10.1.1.31.3590">10.1.1.31.3590</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Tech.+Rep.+93-011&rft.atitle=Typability+and+type+checking+in+the+second-order+lambda-calculus+are+equivalent+and+undecidable&rft.pages=176-185&rft.date=1993&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.31.3590%23id-name%3DCiteSeerX&rft.aulast=Wells&rft.aufirst=J.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+undecidable+problems" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrahtenbrot1950" class="citation journal cs1"><a href="/wiki/Boris_Trakhtenbrot" title="Boris Trakhtenbrot">Trahtenbrot, B. 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New Series. <b>70</b>: 569–572. <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=0033784">0033784</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Doklady+Akademii+Nauk+SSSR&rft.atitle=The+impossibility+of+an+algorithm+for+the+decision+problem+for+finite+domains&rft.volume=70&rft.pages=569-572&rft.date=1950&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0033784%23id-name%3DMR&rft.aulast=Trahtenbrot&rft.aufirst=B.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+undecidable+problems" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell1993" class="citation cs2"><a href="/wiki/John_Stillwell" title="John Stillwell">Stillwell, John</a> (1993), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=265lbM42REMC&pg=PA247"><i>Classical Topology and Combinatorial Group Theory</i></a>, Graduate Texts in Mathematics, vol. 72, Springer, p. 247, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387979700" title="Special:BookSources/9780387979700"><bdi>9780387979700</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Topology+and+Combinatorial+Group+Theory&rft.series=Graduate+Texts+in+Mathematics&rft.pages=247&rft.pub=Springer&rft.date=1993&rft.isbn=9780387979700&rft.aulast=Stillwell&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D265lbM42REMC%26pg%3DPA247&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+undecidable+problems" 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">Keith O. 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"Computability and Beltrami fields in Euclidean space". <i>Journal de Mathématiques Pures et Appliquées</i>. <b>169</b>: 50-81. <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/2111.03559">2111.03559</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.matpur.2022.11.007">10.1016/j.matpur.2022.11.007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+Math%C3%A9matiques+Pures+et+Appliqu%C3%A9es&rft.atitle=Computability+and+Beltrami+fields+in+Euclidean+space&rft.volume=169&rft.pages=50-81&rft.date=2023&rft_id=info%3Aarxiv%2F2111.03559&rft_id=info%3Adoi%2F10.1016%2Fj.matpur.2022.11.007&rft.aulast=Cardona&rft.aufirst=R.&rft.au=Miranda%2C+E.&rft.au=Peralta-Salas%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+undecidable+problems" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_undecidable_problems&action=edit&section=11" title="Edit section: Bibliography"><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="CITEREFBrookshear1989" class="citation book cs1">Brookshear, J. Glenn (1989). <i>Theory of Computation: Formal Languages, Automata, and Complexity</i>. Redwood City, California: Benjamin/Cummings Publishing Company, Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Computation%3A+Formal+Languages%2C+Automata%2C+and+Complexity&rft.place=Redwood+City%2C+California&rft.pub=Benjamin%2FCummings+Publishing+Company%2C+Inc.&rft.date=1989&rft.aulast=Brookshear&rft.aufirst=J.+Glenn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+undecidable+problems" class="Z3988"></span> Appendix C includes impossibility of algorithms deciding if a grammar contains ambiguities, and impossibility of verifying program correctness by an algorithm as example of Halting Problem.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalava1997" class="citation report cs1">Halava, Vesa (1997). Decidable and undecidable problems in matrix theory (TUCS technical report). Vol. 127. Turku Centre for Computer Science. <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=10.1.1.31.5792">10.1.1.31.5792</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Decidable+and+undecidable+problems+in+matrix+theory&rft.pub=Turku+Centre+for+Computer+Science&rft.date=1997&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.31.5792%23id-name%3DCiteSeerX&rft.aulast=Halava&rft.aufirst=Vesa&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+undecidable+problems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoretH._D._Shapiro1991" class="citation book cs1">Moret, B. M. E.; H. D. Shapiro (1991). <i>Algorithms from P to NP, volume 1 - Design and Efficiency</i>. Redwood City, California: Benjamin/Cummings Publishing Company, Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithms+from+P+to+NP%2C+volume+1+-+Design+and+Efficiency&rft.place=Redwood+City%2C+California&rft.pub=Benjamin%2FCummings+Publishing+Company%2C+Inc.&rft.date=1991&rft.aulast=Moret&rft.aufirst=B.+M.+E.&rft.au=H.+D.+Shapiro&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+undecidable+problems" class="Z3988"></span> Discusses intractability of problems with algorithms having exponential performance in Chapter 2, "Mathematical techniques for the analysis of algorithms."</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberger2005" class="citation book cs1">Weinberger, Shmuel (2005). <i>Computers, rigidity, and moduli</i>. Princeton, NJ: Princeton University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computers%2C+rigidity%2C+and+moduli&rft.place=Princeton%2C+NJ&rft.pub=Princeton+University+Press&rft.date=2005&rft.aulast=Weinberger&rft.aufirst=Shmuel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+undecidable+problems" class="Z3988"></span> Discusses undecidability of the word problem for groups, and of various problems in topology.</li></ul> <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=List_of_undecidable_problems&action=edit&section=12" 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="CITEREFPoonen2012" class="citation arxiv cs1"><a href="/wiki/Bjorn_Poonen" title="Bjorn Poonen">Poonen, Bjorn</a> (2 April 2012). "Undecidable problems: a sampler". <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/1204.0299">1204.0299</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Undecidable+problems%3A+a+sampler&rft.date=2012-04-02&rft_id=info%3Aarxiv%2F1204.0299&rft.aulast=Poonen&rft.aufirst=Bjorn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+undecidable+problems" class="Z3988"></span></li></ul>    </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=List_of_undecidable_problems&oldid=1259072052">https://en.wikipedia.org/w/index.php?title=List_of_undecidable_problems&oldid=1259072052</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:Mathematics-related_lists" title="Category:Mathematics-related lists">Mathematics-related lists</a></li><li><a href="/wiki/Category:Theory_of_computation" title="Category:Theory of computation">Theory of computation</a></li><li><a href="/wiki/Category:Computability_theory" title="Category:Computability theory">Computability theory</a></li><li><a href="/wiki/Category:Undecidable_problems" title="Category:Undecidable problems">Undecidable problems</a></li><li><a href="/wiki/Category:Lists_of_problems" title="Category:Lists of problems">Lists of problems</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_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_April_2024" title="Category:Articles with unsourced statements from April 2024">Articles with unsourced statements from April 2024</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 23 November 2024, at 05:44<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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