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<li id="toc-Mutually_recursive_data_types" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mutually_recursive_data_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Mutually recursive data types</span> </div> </a> <ul id="toc-Mutually_recursive_data_types-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Theory</span> </div> </a> <button aria-controls="toc-Theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Theory subsection</span> </button> <ul id="toc-Theory-sublist" class="vector-toc-list"> <li id="toc-Isorecursive_types" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isorecursive_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Isorecursive types</span> </div> </a> <ul id="toc-Isorecursive_types-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equirecursive_types" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equirecursive_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Equirecursive types</span> </div> </a> <ul id="toc-Equirecursive_types-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Recursive_type_synonyms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Recursive_type_synonyms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Recursive type synonyms</span> </div> </a> <ul id="toc-Recursive_type_synonyms-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 class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div 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href="https://ko.wikipedia.org/wiki/%EC%9E%AC%EA%B7%80_%EC%9E%90%EB%A3%8C%ED%98%95" title="재귀 자료형 – Korean" lang="ko" hreflang="ko" data-title="재귀 자료형" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rekurz%C3%ADv_adatt%C3%ADpus" title="Rekurzív adattípus – Hungarian" lang="hu" hreflang="hu" data-title="Rekurzív adattípus" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%86%8D%E5%B8%B0%E3%83%87%E3%83%BC%E3%82%BF%E5%9E%8B" title="再帰データ型 – Japanese" lang="ja" hreflang="ja" data-title="再帰データ型" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tipo_recursivo" title="Tipo recursivo – Portuguese" lang="pt" hreflang="pt" data-title="Tipo recursivo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BA%D1%83%D1%80%D1%81%D0%B8%D0%B2%D0%BD%D0%B8%D0%B9_%D1%82%D0%B8%D0%BF_%D0%B4%D0%B0%D0%BD%D0%B8%D1%85" title="Рекурсивний тип даних – Ukrainian" lang="uk" hreflang="uk" data-title="Рекурсивний тип даних" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%80%92%E5%BD%92%E6%95%B0%E6%8D%AE%E7%B1%BB%E5%9E%8B" title="递归数据类型 – Chinese" lang="zh" hreflang="zh" data-title="递归数据类型" 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vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Data type that refers to itself in its definition</div> <p>In computer <a href="/wiki/Programming_language" title="Programming language">programming languages</a>, a <b>recursive data type</b> (also known as a <b>recursively-defined</b>, <b>inductively-defined</b> or <b>inductive data type</b>) is a <a href="/wiki/Data_type" title="Data type">data type</a> for values that may contain other values of the same type. Data of recursive types are usually viewed as <a href="/wiki/Directed_graph" title="Directed graph">directed graphs</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="Don't you mean a tree? (July 2022)">citation needed</span></a></i>]</sup>. </p><p>An important application of recursion in computer science is in defining dynamic data structures such as Lists and Trees. Recursive data structures can dynamically grow to an arbitrarily large size in response to runtime requirements; in contrast, a static array's size requirements must be set at compile time. </p><p>Sometimes the term "inductive data type" is used for <a href="/wiki/Algebraic_data_type" title="Algebraic data type">algebraic data types</a> which are not necessarily recursive. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recursive_data_type&action=edit&section=1" title="Edit section: Example"><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">See also: <a href="/wiki/Recursion_(computer_science)#Recursive_data_structures_(structural_recursion)" title="Recursion (computer science)">Recursion (computer science) § Recursive data structures (structural recursion)</a></div> <p>An example is the <a href="/wiki/List_(computing)" class="mw-redirect" title="List (computing)">list</a> type, in <a href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a>: </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="kr">data</span><span class="w"> </span><span class="kt">List</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Nil</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kt">Cons</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">(</span><span class="kt">List</span><span class="w"> </span><span class="n">a</span><span class="p">)</span> </pre></div> <p>This indicates that a list of a's is either an empty list or a <b>cons cell</b> containing an 'a' (the "head" of the list) and another list (the "tail"). </p><p>Another example is a similar singly linked type in Java: </p> <div class="mw-highlight mw-highlight-lang-java mw-content-ltr" dir="ltr"><pre><span></span><span class="kd">class</span> <span class="nc">List</span><span class="o"><</span><span class="n">E</span><span class="o">></span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="n">E</span><span class="w"> </span><span class="n">value</span><span class="p">;</span> <span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">E</span><span class="o">></span><span class="w"> </span><span class="n">next</span><span class="p">;</span> <span class="p">}</span> </pre></div> <p>This indicates that non-empty list of type E contains a data member of type E, and a reference to another List object for the rest of the list (or a null reference to indicate that this is the end of the list). </p> <div class="mw-heading mw-heading3"><h3 id="Mutually_recursive_data_types">Mutually recursive data types</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recursive_data_type&action=edit&section=2" title="Edit section: Mutually recursive data types"><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">Further information: <a href="/wiki/Mutual_recursion#Data_types" title="Mutual recursion">Mutual recursion § Data types</a></div> <p>Data types can also be defined by <a href="/wiki/Mutual_recursion" title="Mutual recursion">mutual recursion</a>. The most important basic example of this is a <a href="/wiki/Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">tree</a>, which can be defined mutually recursively in terms of a forest (a list of trees). Symbolically: </p> <pre>f: [t[1], ..., t[k]] t: v f </pre> <p>A forest <i>f</i> consists of a list of trees, while a tree <i>t</i> consists of a pair of a value <i>v</i> and a forest <i>f</i> (its children). This definition is elegant and easy to work with abstractly (such as when proving theorems about properties of trees), as it expresses a tree in simple terms: a list of one type, and a pair of two types. </p><p>This mutually recursive definition can be converted to a singly recursive definition by inlining the definition of a forest: </p> <pre>t: v [t[1], ..., t[k]] </pre> <p>A tree <i>t</i> consists of a pair of a value <i>v</i> and a list of trees (its children). This definition is more compact, but somewhat messier: a tree consists of a pair of one type and a list another, which require disentangling to prove results about. </p><p>In <a href="/wiki/Standard_ML" title="Standard ML">Standard ML</a>, the tree and forest data types can be mutually recursively defined as follows, allowing empty trees:<sup id="cite_ref-FOOTNOTEHarper1998_1-0" class="reference"><a href="#cite_note-FOOTNOTEHarper1998-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-highlight mw-highlight-lang-sml mw-content-ltr" dir="ltr"><pre><span></span><span class="kr">datatype</span> <span class="nd">'a</span> <span class="kt">tree</span> <span class="p">=</span> <span class="nc">Empty</span> <span class="p">|</span> <span class="nc">Node</span> <span class="kr">of</span> <span class="nd">'a</span> <span class="n">*</span> <span class="nd">'a</span> <span class="n">forest</span> <span class="kr">and</span> <span class="nd">'a</span> <span class="kt">forest</span> <span class="p">=</span> <span class="nc">Nil</span> <span class="p">|</span> <span class="nc">Cons</span> <span class="kr">of</span> <span class="nd">'a</span> <span class="n">tree</span> <span class="n">*</span> <span class="nd">'a</span> <span class="n">forest</span> </pre></div><p>In Haskell, the tree and forest data types can be defined similarly:</p><div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="kr">data</span><span class="w"> </span><span class="kt">Tree</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Empty</span> <span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kt">Node</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="kt">Forest</span><span class="w"> </span><span class="n">a</span><span class="p">)</span> <span class="kr">data</span><span class="w"> </span><span class="kt">Forest</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Nil</span> <span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kt">Cons</span><span class="w"> </span><span class="p">(</span><span class="kt">Tree</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="kt">Forest</span><span class="w"> </span><span class="n">a</span><span class="p">)</span> </pre></div> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recursive_data_type&action=edit&section=3" title="Edit section: Theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Type_theory" title="Type theory">type theory</a>, a recursive type has the general form μα.T where the <a href="/wiki/Type_variable" title="Type variable">type variable</a> α may appear in the type T and stands for the entire type itself. </p><p>For example, the natural numbers (see <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>) may be defined by the Haskell datatype: </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="kr">data</span><span class="w"> </span><span class="kt">Nat</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Zero</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kt">Succ</span><span class="w"> </span><span class="kt">Nat</span> </pre></div> <p>In type theory, we would say: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nat=\mu \alpha .1+\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>a</mi> <mi>t</mi> <mo>=</mo> <mi>μ</mi> <mi>α</mi> <mn>.1</mn> <mo>+</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nat=\mu \alpha .1+\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c441aa78f783b3798a06b688e218f9801529388f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.589ex; height:2.676ex;" alt="{\displaystyle nat=\mu \alpha .1+\alpha }"></span> where the two arms of the <a href="/wiki/Sum_type" class="mw-redirect" title="Sum type">sum type</a> represent the Zero and Succ data constructors. Zero takes no arguments (thus represented by the <a href="/wiki/Unit_type" title="Unit type">unit type</a>) and Succ takes another Nat (thus another element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \alpha .1+\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mi>α</mi> <mn>.1</mn> <mo>+</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \alpha .1+\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea41fbf58e9873244ca36c56c9d11b2156339765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.027ex; height:2.676ex;" alt="{\displaystyle \mu \alpha .1+\alpha }"></span>). </p><p>There are two forms of recursive types: the so-called isorecursive types, and equirecursive types. The two forms differ in how terms of a recursive type are introduced and eliminated. </p> <div class="mw-heading mw-heading3"><h3 id="Isorecursive_types">Isorecursive types</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recursive_data_type&action=edit&section=4" title="Edit section: Isorecursive types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With isorecursive types, the recursive type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \alpha .T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mi>α</mi> <mo>.</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \alpha .T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e536f74795348aa746443152f30b594270a0237e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.56ex; height:2.676ex;" alt="{\displaystyle \mu \alpha .T}"></span> and its expansion (or <i>unrolling</i>) <span class="mwe-math-element"><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[\mu \alpha .T/\alpha ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">[</mo> <mi>μ</mi> <mi>α</mi> <mo>.</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>α</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T[\mu \alpha .T/\alpha ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbc5122f9ac522ed8a510aa156993f362d194ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.14ex; height:2.843ex;" alt="{\displaystyle T[\mu \alpha .T/\alpha ]}"></span> (where the notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X[Y/Z]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Z</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X[Y/Z]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00989088ec91a5662c617a728e9d8e5095ba39d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.89ex; height:2.843ex;" alt="{\displaystyle X[Y/Z]}"></span> indicates that all instances of Z are replaced with Y in X) are distinct (and disjoint) types with special term constructs, usually called <i>roll</i> and <i>unroll</i>, that form an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> between them. To be precise: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle roll:T[\mu \alpha .T/\alpha ]\to \mu \alpha .T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>o</mi> <mi>l</mi> <mi>l</mi> <mo>:</mo> <mi>T</mi> <mo stretchy="false">[</mo> <mi>μ</mi> <mi>α</mi> <mo>.</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>α</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>μ</mi> <mi>α</mi> <mo>.</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle roll:T[\mu \alpha .T/\alpha ]\to \mu \alpha .T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b6f8d541d2489945bdb60a01c590421274564d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.813ex; height:2.843ex;" alt="{\displaystyle roll:T[\mu \alpha .T/\alpha ]\to \mu \alpha .T}"></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 unroll:\mu \alpha .T\to T[\mu \alpha .T/\alpha ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mi>n</mi> <mi>r</mi> <mi>o</mi> <mi>l</mi> <mi>l</mi> <mo>:</mo> <mi>μ</mi> <mi>α</mi> <mo>.</mo> <mi>T</mi> <mo stretchy="false">→</mo> <mi>T</mi> <mo stretchy="false">[</mo> <mi>μ</mi> <mi>α</mi> <mo>.</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>α</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle unroll:\mu \alpha .T\to T[\mu \alpha .T/\alpha ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8146ae5a168a59afea5f32932b583a4b67aa17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.538ex; height:2.843ex;" alt="{\displaystyle unroll:\mu \alpha .T\to T[\mu \alpha .T/\alpha ]}"></span>, and these two are <a href="/wiki/Inverse_function" title="Inverse function">inverse functions</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Equirecursive_types">Equirecursive types</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recursive_data_type&action=edit&section=5" title="Edit section: Equirecursive types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under equirecursive rules, a recursive type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \alpha .T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mi>α</mi> <mo>.</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \alpha .T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e536f74795348aa746443152f30b594270a0237e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.56ex; height:2.676ex;" alt="{\displaystyle \mu \alpha .T}"></span> and its unrolling <span class="mwe-math-element"><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[\mu \alpha .T/\alpha ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">[</mo> <mi>μ</mi> <mi>α</mi> <mo>.</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>α</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T[\mu \alpha .T/\alpha ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbc5122f9ac522ed8a510aa156993f362d194ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.14ex; height:2.843ex;" alt="{\displaystyle T[\mu \alpha .T/\alpha ]}"></span> are <i>equal</i> – that is, those two type expressions are understood to denote the same type. In fact, most theories of equirecursive types go further and essentially specify that any two type expressions with the same "infinite expansion" are equivalent. As a result of these rules, equirecursive types contribute significantly more complexity to a type system than isorecursive types do. Algorithmic problems such as type checking and <a href="/wiki/Type_inference" title="Type inference">type inference</a> are more difficult for equirecursive types as well. Since direct comparison does not make sense on an equirecursive type, they can be converted into a canonical form in O(n log n) time, which can easily be compared.<sup id="cite_ref-canonicalize_2-0" class="reference"><a href="#cite_note-canonicalize-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Isorecursive types capture the form of self-referential (or mutually referential) type definitions seen in nominal <a href="/wiki/Object-oriented" class="mw-redirect" title="Object-oriented">object-oriented</a> programming languages, and also arise in type-theoretic semantics of objects and <a href="/wiki/Class_(computer_science)" class="mw-redirect" title="Class (computer science)">classes</a>. In functional programming languages, isorecursive types (in the guise of datatypes) are common too.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Recursive_type_synonyms">Recursive type synonyms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recursive_data_type&action=edit&section=6" title="Edit section: Recursive type synonyms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Original_research plainlinks metadata ambox ambox-content ambox-Original_research" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>possibly contains <a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research">original research</a></b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Recursive_data_type&action=edit">improve it</a> by <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verifying</a> the claims made and adding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a>. Statements consisting only of original research should be removed.</span> <span class="date-container"><i>(<span class="date">January 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/TypeScript" title="TypeScript">TypeScript</a>, recursion is allowed in type aliases.<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> Thus, the following example is allowed. </p> <div class="mw-highlight mw-highlight-lang-typescript mw-content-ltr" dir="ltr"><pre><span></span><span class="kr">type</span><span class="w"> </span><span class="nx">Tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="nx">Tree</span><span class="p">[];</span> <span class="kd">let</span><span class="w"> </span><span class="nx">tree</span><span class="o">:</span><span class="w"> </span><span class="kt">Tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">]];</span> </pre></div> <p>However, recursion is not allowed in type synonyms in <a href="/wiki/Miranda_programming_language" class="mw-redirect" title="Miranda programming language">Miranda</a>, <a href="/wiki/OCaml" title="OCaml">OCaml</a> (unless <code>-rectypes</code> flag is used or it's a record or variant), or Haskell; so, for example the following Haskell types are illegal: </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="kr">type</span><span class="w"> </span><span class="kt">Bad</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="p">(</span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="kt">Bad</span><span class="p">)</span> <span class="kr">type</span><span class="w"> </span><span class="kt">Evil</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Bool</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="kt">Evil</span> </pre></div> <p>Instead, they must be wrapped inside an algebraic data type (even if they only has one constructor): </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="kr">data</span><span class="w"> </span><span class="kt">Good</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Pair</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="kt">Good</span> <span class="kr">data</span><span class="w"> </span><span class="kt">Fine</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Fun</span><span class="w"> </span><span class="p">(</span><span class="kt">Bool</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="kt">Fine</span><span class="p">)</span> </pre></div> <p>This is because type synonyms, like <a href="/wiki/Typedef" title="Typedef">typedefs</a> in C, are replaced with their definition at compile time. (Type synonyms are not "real" types; they are just "aliases" for convenience of the programmer.) But if this is attempted with a recursive type, it will loop infinitely because no matter how many times the alias is substituted, it still refers to itself, e.g. "Bad" will grow indefinitely: <code>Bad</code> → <code>(Int, Bad)</code> → <code>(Int, (Int, Bad))</code> → <code>...</code> . </p><p>Another way to see it is that a level of indirection (the algebraic data type) is required to allow the isorecursive type system to figure out when to <i>roll</i> and <i>unroll</i>. </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=Recursive_data_type&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Recursive_definition" title="Recursive definition">Recursive definition</a></li> <li><a href="/wiki/Algebraic_data_type" title="Algebraic data type">Algebraic data type</a></li> <li><a href="/wiki/Inductive_type" title="Inductive type">Inductive type</a></li> <li><a href="/wiki/Node_(computer_science)" title="Node (computer science)">Node (computer science)</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=Recursive_data_type&action=edit&section=8" 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-FOOTNOTEHarper1998-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHarper1998_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHarper1998">Harper 1998</a>.</span> </li> <li id="cite_note-canonicalize-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-canonicalize_2-0">^</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 class="citation citeseerx cs1">"Numbering Matters: First-Order Canonical Forms for Second-Order Recursive Types". <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=10.1.1.4.2276">10.1.1.4.2276</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=CiteSeerX&rft.atitle=Numbering+Matters%3A+First-Order+Canonical+Forms+for+Second-Order+Recursive+Types&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.4.2276%23id-name%3DCiteSeerX&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARecursive+data+type" 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"><a rel="nofollow" class="external text" href="https://dl.acm.org/doi/10.1145/3428291">Revisiting iso-recursive subtyping | Proceedings of the ACM on Programming Languages</a></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"><a rel="nofollow" class="external text" href="https://devblogs.microsoft.com/typescript/announcing-typescript-3-7/#more-recursive-type-aliases">(More) Recursive Type Aliases - Announcing TypeScript 3.7 - TypeScript</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recursive_data_type&action=edit&section=9" title="Edit section: Sources"><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="CITEREFHarper1998" class="citation cs2">Harper, Robert (1998), <a rel="nofollow" class="external text" href="https://web.archive.org/web/19991001064742/https://www.cs.cmu.edu/~rwh/introsml/core/datatypes.htm"><i>Datatype Declarations</i></a>, archived from <a rel="nofollow" class="external text" href="https://www.cs.cmu.edu/~rwh/introsml/core/datatypes.htm">the original</a> on 1999-10-01</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Datatype+Declarations&rft.date=1998&rft.aulast=Harper&rft.aufirst=Robert&rft_id=https%3A%2F%2Fwww.cs.cmu.edu%2F~rwh%2Fintrosml%2Fcore%2Fdatatypes.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARecursive+data+type" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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<li><a href="/wiki/Byte" title="Byte">Byte</a></li> <li><a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">Trit</a></li> <li><a href="/wiki/Ternary_numeral_system#Tryte" title="Ternary numeral system">Tryte</a></li> <li><a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Word</a></li> <li><a href="/wiki/Bit_array" title="Bit array">Bit array</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Numeric</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">Arbitrary-precision or bignum</a></li> <li><a href="/wiki/Complex_data_type" title="Complex data type">Complex</a></li> <li><a href="/wiki/Decimal_data_type" title="Decimal data type">Decimal</a></li> <li><a href="/wiki/Fixed-point_arithmetic" title="Fixed-point arithmetic">Fixed point</a></li> <li><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating point</a> <ul><li>Reduced precision <ul><li><a href="/wiki/Minifloat" title="Minifloat">Minifloat</a></li> <li><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half precision</a></li> <li><a href="/wiki/Bfloat16_floating-point_format" title="Bfloat16 floating-point format">bfloat16</a></li></ul></li> <li><a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">Single precision</a></li> <li><a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">Double precision</a></li> <li><a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">Quadruple precision</a></li> <li><a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">Octuple precision</a></li> <li><a href="/wiki/Extended_precision" title="Extended precision">Extended precision</a> <ul><li><a href="/wiki/Long_double" title="Long double">Long double</a></li></ul></li></ul></li> <li><a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">Integer</a> <ul><li><a href="/wiki/Signedness" title="Signedness">signedness</a></li></ul></li> <li><a href="/wiki/Interval_arithmetic#Implementations" title="Interval arithmetic">Interval</a></li> <li><a href="/wiki/Rational_data_type" title="Rational data type">Rational</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pointer_(computer_programming)" title="Pointer (computer programming)">Pointer</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Memory_address" title="Memory address">Address</a> <ul><li><a href="/wiki/Physical_address" title="Physical address">physical</a></li> <li><a href="/wiki/Virtual_address_space" title="Virtual address space">virtual</a></li></ul></li> <li><a href="/wiki/Reference_(computer_science)" title="Reference (computer science)">Reference</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plain_text" title="Plain text">Text</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Character_(computing)" title="Character (computing)">Character</a></li> <li><a href="/wiki/String_(computer_science)" title="String (computer science)">String</a> <ul><li><a href="/wiki/Null-terminated_string" title="Null-terminated string">null-terminated</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_data_type" title="Composite data type">Composite</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_data_type" title="Algebraic data type">Algebraic data type</a> <ul><li><a href="/wiki/Generalized_algebraic_data_type" title="Generalized algebraic data type">generalized</a></li></ul></li> <li><a href="/wiki/Array_data_type" class="mw-redirect" title="Array data type">Array</a></li> <li><a href="/wiki/Associative_array" title="Associative array">Associative array</a></li> <li><a href="/wiki/Class_(computer_programming)" title="Class (computer programming)">Class</a></li> <li><a href="/wiki/Dependent_type" title="Dependent type">Dependent</a></li> <li><a href="/wiki/Intuitionistic_type_theory#Equality_type" title="Intuitionistic type theory">Equality</a></li> <li><a href="/wiki/Inductive_type" title="Inductive type">Inductive</a></li> <li><a href="/wiki/Intersection_type" title="Intersection type">Intersection</a></li> <li><a href="/wiki/List_(abstract_data_type)" title="List (abstract data type)">List</a></li> <li><a href="/wiki/Object_(computer_science)" title="Object (computer science)">Object</a> <ul><li><a href="/wiki/Metaobject" title="Metaobject">metaobject</a></li></ul></li> <li><a href="/wiki/Option_type" title="Option type">Option type</a></li> <li><a href="/wiki/Product_type" title="Product type">Product</a></li> <li><a href="/wiki/Record_(computer_science)" title="Record (computer science)">Record or Struct</a></li> <li><a href="/wiki/Refinement_type" title="Refinement type">Refinement</a></li> <li><a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">Set</a></li> <li><a href="/wiki/Union_type" title="Union type">Union</a> <ul><li><a href="/wiki/Tagged_union" title="Tagged union">tagged</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_data_type" title="Boolean data type">Boolean</a></li> <li><a href="/wiki/Bottom_type" title="Bottom type">Bottom type</a></li> <li><a href="/wiki/Container_(abstract_data_type)" title="Container (abstract data type)">Collection</a></li> <li><a href="/wiki/Enumerated_type" title="Enumerated type">Enumerated type</a></li> <li><a href="/wiki/Exception_handling" title="Exception handling">Exception</a></li> <li><a href="/wiki/Function_type" title="Function type">Function type</a></li> <li><a href="/wiki/Opaque_data_type" title="Opaque data type">Opaque data type</a></li> <li><a class="mw-selflink selflink">Recursive data type</a></li> <li><a href="/wiki/Semaphore_(programming)" title="Semaphore (programming)">Semaphore</a></li> <li><a href="/wiki/Stream_(computing)" title="Stream (computing)">Stream</a></li> <li><a href="/wiki/Strongly_typed_identifier" title="Strongly typed identifier">Strongly typed identifier</a></li> <li><a href="/wiki/Top_type" title="Top type">Top type</a></li> <li><a href="/wiki/Type_class" title="Type class">Type class</a></li> <li><a href="/wiki/Empty_type" title="Empty type">Empty type</a></li> <li><a href="/wiki/Unit_type" title="Unit type">Unit type</a></li> <li><a href="/wiki/Void_type" title="Void type">Void</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_data_type" title="Abstract data type">Abstract data type</a></li> <li><a href="/wiki/Boxing_(computer_science)" class="mw-redirect" title="Boxing (computer science)">Boxing</a></li> <li><a href="/wiki/Data_structure" title="Data structure">Data structure</a></li> <li><a href="/wiki/Generic_programming" title="Generic programming">Generic</a></li> <li><a href="/wiki/Kind_(type_theory)" title="Kind (type theory)">Kind</a> <ul><li><a href="/wiki/Metaclass" 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