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Kind (type theory) - Wikipedia
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<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">Type of types in a type system</div> <p>In the area of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a> known as <a href="/wiki/Type_theory" title="Type theory">type theory</a>, a <b>kind</b> is the type of a <a href="/wiki/Type_constructor" title="Type constructor">type constructor</a> or, less commonly, the type of a <a href="/wiki/Higher-order_type_operator" class="mw-redirect" title="Higher-order type operator">higher-order type operator</a>. A kind system is essentially a <a href="/wiki/Simply_typed_lambda_calculus" title="Simply typed lambda calculus">simply typed lambda calculus</a> "one level up", endowed with a primitive type, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span> and called "type", which is the kind of any <a href="/wiki/Data_type" title="Data type">data type</a> which does not need any <a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">type parameters</a>. </p><p>A kind is sometimes confusingly described as the "type of a <a href="/wiki/Data_type" title="Data type">(data) type</a>", but it is actually more of an <a href="/wiki/Arity" title="Arity">arity</a> specifier. Syntactically, it is natural to consider polymorphic types to be type constructors, thus non-polymorphic types to be <a href="/wiki/Nullary" class="mw-redirect" title="Nullary">nullary</a> type constructors. But all nullary constructors, thus all monomorphic types, have the same, simplest kind; namely <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>. </p><p>Since higher-order type operators are uncommon in <a href="/wiki/Programming_language" title="Programming language">programming languages</a>, in most programming practice, kinds are used to distinguish between data types and the types of constructors which are used to implement <a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">parametric polymorphism</a>. Kinds appear, either explicitly or implicitly, in languages whose type systems account for parametric polymorphism in a programmatically accessible way, such as <a href="/wiki/C%2B%2B" title="C++">C++</a>,<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 href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a> and <a href="/wiki/Scala_(programming_language)" title="Scala (programming language)">Scala</a>.<sup id="cite_ref-higherkinds_2-0" class="reference"><a href="#cite_note-higherkinds-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kind_(type_theory)&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>, pronounced "type", is the kind of all <a href="/wiki/Data_type" title="Data type">data types</a> seen as <a href="/wiki/Nullary" class="mw-redirect" title="Nullary">nullary</a> type constructors, and also called proper types in this context. This normally includes function types in <a href="/wiki/Functional_programming_language" class="mw-redirect" title="Functional programming language">functional programming languages</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *\rightarrow *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *\rightarrow *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3138429c944b039139b6200d57a9727a2c1136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.939ex; height:1.843ex;" alt="{\displaystyle *\rightarrow *}" /></span> is the kind of a <a href="/wiki/Unary_function" title="Unary function">unary</a> <a href="/wiki/Type_constructor" title="Type constructor">type constructor</a>, e.g. of a <a href="/wiki/List_type" class="mw-redirect" title="List type">list type</a> constructor.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *\rightarrow *\rightarrow *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *\rightarrow *\rightarrow *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f834008af5a6f4d907b29af594544b438b46e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.716ex; height:1.843ex;" alt="{\displaystyle *\rightarrow *\rightarrow *}" /></span> is the kind of a <a href="https://en.wiktionary.org/wiki/binary" class="extiw" title="wiktionary:binary">binary</a> type constructor (via <a href="/wiki/Currying" title="Currying">currying</a>), e.g. of a <a href="/wiki/Pair_type" class="mw-redirect" title="Pair type">pair type</a> constructor, and also that of a <a href="/wiki/Function_type" title="Function type">function type</a> constructor (not to be confused with the result of its application, which itself is a function type, thus of kind <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (*\rightarrow *)\rightarrow *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (*\rightarrow *)\rightarrow *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96958b84d6ce5c17fe6594cc8355bcb9664aed84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.525ex; height:2.843ex;" alt="{\displaystyle (*\rightarrow *)\rightarrow *}" /></span> is the kind of a higher-order type operator from unary type constructors to proper types.<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="Kinds_in_Haskell">Kinds in Haskell</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kind_(type_theory)&action=edit&section=2" title="Edit section: Kinds in Haskell"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>(<i>Note</i>: Haskell documentation uses the same arrow for both function types and kinds.) </p><p>The kind system of <a href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell 98</a><sup id="cite_ref-haskell98_4-0" class="reference"><a href="#cite_note-haskell98-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> includes exactly two kinds: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>, pronounced "type" is the kind of all <a href="/wiki/Data_type" title="Data type">data types</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}\rightarrow k_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→<!-- → --></mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{1}\rightarrow k_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9070b7720610d2dd3bbfc0005d1e5962b7a40e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.145ex; height:2.509ex;" alt="{\displaystyle k_{1}\rightarrow k_{2}}" /></span> is the kind of a <a href="/wiki/Unary_function" title="Unary function">unary</a> <a href="/wiki/Type_constructor" title="Type constructor">type constructor</a>, which takes a type of kind <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376315fd4983f01dada5ec2f7bebc48455b14a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{1}}" /></span> and produces a type of kind <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}" /></span>.</li></ul> <p>An inhabited type (as proper types are called in Haskell) is a type which has values. For instance, ignoring <a href="/wiki/Type_class" title="Type class">type classes</a> which complicate the picture, <code>4</code> is a value of type <code>Int</code>, while <code>[1, 2, 3]</code> is a value of type <code>[Int]</code> (list of Ints). Therefore, <code>Int</code> and <code>[Int]</code> have kind <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>, but so does any function type, for instance <code>Int -> Bool</code> or even <code>Int -> Int -> Bool</code>. </p><p>A type constructor takes one or more type arguments, and produces a data type when enough arguments are supplied, i.e. it supports <a href="/wiki/Partial_application" title="Partial application">partial application</a> thanks to currying.<sup id="cite_ref-haskell-2010_5-0" class="reference"><a href="#cite_note-haskell-2010-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></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> This is how Haskell achieves parametric types. For instance, the type <code>[]</code> (list) is a type constructor - it takes a single argument to specify the type of the elements of the list. Hence, <code>[Int]</code> (list of Ints), <code>[Float]</code> (list of Floats) and even <code>[[Int]]</code> (list of lists of Ints) are valid applications of the <code>[]</code> type constructor. Therefore, <code>[]</code> is a type of kind <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 *\rightarrow *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *\rightarrow *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3138429c944b039139b6200d57a9727a2c1136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.939ex; height:1.843ex;" alt="{\displaystyle *\rightarrow *}" /></span>. Because <code>Int</code> has kind <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>, applying <code>[]</code> to it results in <code>[Int]</code>, of kind <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>. The 2-<a href="/wiki/Tuple" title="Tuple">tuple</a> constructor <code>(,)</code> has kind <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 *\rightarrow *\rightarrow *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *\rightarrow *\rightarrow *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f834008af5a6f4d907b29af594544b438b46e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.716ex; height:1.843ex;" alt="{\displaystyle *\rightarrow *\rightarrow *}" /></span>, the 3-tuple constructor <code>(,,)</code> has kind <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 *\rightarrow *\rightarrow *\rightarrow *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *\rightarrow *\rightarrow *\rightarrow *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0dbe4bc477068f60bca49ed7b8f64e6507073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.492ex; height:1.843ex;" alt="{\displaystyle *\rightarrow *\rightarrow *\rightarrow *}" /></span> and so on. </p> <div class="mw-heading mw-heading3"><h3 id="Kind_inference">Kind inference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kind_(type_theory)&action=edit&section=3" title="Edit section: Kind inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Standard Haskell does not allow <a href="/w/index.php?title=Polymorphic_kind&action=edit&redlink=1" class="new" title="Polymorphic kind (page does not exist)">polymorphic kinds</a>. This is in contrast to <a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">parametric polymorphism</a> on types, which is supported in Haskell. For instance, in the following example: </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">z</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Leaf</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kt">Fork</span><span class="w"> </span><span class="p">(</span><span class="kt">Tree</span><span class="w"> </span><span class="n">z</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="kt">Tree</span><span class="w"> </span><span class="n">z</span><span class="p">)</span> </pre></div> <p>the kind of <code>z</code> could be anything, including <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>, but also <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 *\rightarrow *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *\rightarrow *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3138429c944b039139b6200d57a9727a2c1136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.939ex; height:1.843ex;" alt="{\displaystyle *\rightarrow *}" /></span> etc. Haskell by default will always infer kinds to be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>, unless the type explicitly indicates otherwise (see below). Therefore the type checker will reject the following use of <code>Tree</code>: </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">FunnyTree</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Tree</span><span class="w"> </span><span class="kt">[]</span><span class="w"> </span><span class="c1">-- invalid</span> </pre></div> <p>because the kind of <code>[]</code>, <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 *\rightarrow *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *\rightarrow *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3138429c944b039139b6200d57a9727a2c1136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.939ex; height:1.843ex;" alt="{\displaystyle *\rightarrow *}" /></span> does not match the expected kind for <code>z</code>, which is always <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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" /></span>. </p><p>Higher-order type operators are allowed however. For instance: </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">App</span><span class="w"> </span><span class="n">unt</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Z</span><span class="w"> </span><span class="p">(</span><span class="n">unt</span><span class="w"> </span><span class="n">z</span><span class="p">)</span> </pre></div> <p>has kind <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 (*\rightarrow *)\rightarrow *\rightarrow *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">→<!-- → --></mo> <mo>∗<!-- ∗ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (*\rightarrow *)\rightarrow *\rightarrow *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366eb512d4a250727231b14d5feca3be0e75c502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.301ex; height:2.843ex;" alt="{\displaystyle (*\rightarrow *)\rightarrow *\rightarrow *}" /></span>, i.e. <code>unt</code> is expected to be a unary data constructor, which gets applied to its argument, which must be a type, and returns another type. </p><p><a href="/wiki/Glasgow_Haskell_Compiler" title="Glasgow Haskell Compiler">GHC</a> has the extension <code>PolyKinds</code>, which, together with <code>KindSignatures</code>, allows polymorphic kinds. For example: </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="p">(</span><span class="n">z</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Leaf</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kt">Fork</span><span class="w"> </span><span class="p">(</span><span class="kt">Tree</span><span class="w"> </span><span class="n">z</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="kt">Tree</span><span class="w"> </span><span class="n">z</span><span class="p">)</span> <span class="kr">type</span><span class="w"> </span><span class="kt">FunnyTree</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">Tree</span><span class="w"> </span><span class="kt">[]</span><span class="w"> </span><span class="c1">-- OK</span> </pre></div> <p>Since GHC 8.0.1, types and kinds are merged.<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> </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=Kind_(type_theory)&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/System_F-omega" class="mw-redirect" title="System F-omega">System F-omega</a></li> <li><a href="/wiki/Pure_type_system" title="Pure type system">Pure type system</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=Kind_(type_theory)&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output 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.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="CITEREFPierce2002" class="citation book cs1">Pierce, Benjamin (2002). <i><a href="/wiki/Types_and_Programming_Languages" title="Types and Programming Languages">Types and Programming Languages</a></i>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-16209-1" title="Special:BookSources/0-262-16209-1"><bdi>0-262-16209-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Types+and+Programming+Languages&rft.pub=MIT+Press&rft.date=2002&rft.isbn=0-262-16209-1&rft.aulast=Pierce&rft.aufirst=Benjamin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKind+%28type+theory%29" class="Z3988"></span>, chapter 29, "Type Operators and Kinding"</li></ul> </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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www2.cs.uregina.ca/~saxton/CS115/Notes/Templates/func.html">"CS 115: Parametric Polymorphism: Template Functions"</a>. <i>www2.cs.uregina.ca</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www2.cs.uregina.ca&rft.atitle=CS+115%3A+Parametric+Polymorphism%3A+Template+Functions&rft_id=http%3A%2F%2Fwww2.cs.uregina.ca%2F~saxton%2FCS115%2FNotes%2FTemplates%2Ffunc.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKind+%28type+theory%29" class="Z3988"></span></span> </li> <li id="cite_note-higherkinds-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-higherkinds_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120610022753/http://adriaanm.github.com/files/higher.pdf">"Generics of a Higher Kind"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="https://adriaanm.github.com/files/higher.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2012-06-10<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-06-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Generics+of+a+Higher+Kind&rft_id=http%3A%2F%2Fadriaanm.github.com%2Ffiles%2Fhigher.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKind+%28type+theory%29" 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"> Pierce (2002), chapter 32</span> </li> <li id="cite_note-haskell98-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-haskell98_4-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.haskell.org/onlinereport/decls.html#sect4.1.1">Kinds - The Haskell 98 Report</a></span> </li> <li id="cite_note-haskell-2010-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-haskell-2010_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.haskell.org/onlinereport/haskell2010/haskellch4.html#x10-65017x3">"Chapter 4 Declarations and Binding"</a>. <i>Haskell 2010 Language Report</i><span class="reference-accessdate">. Retrieved <span class="nowrap">23 July</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Haskell+2010+Language+Report&rft.atitle=Chapter+4+Declarations+and+Binding&rft_id=http%3A%2F%2Fwww.haskell.org%2Fonlinereport%2Fhaskell2010%2Fhaskellch4.html%23x10-65017x3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKind+%28type+theory%29" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMiran" class="citation web cs1">Miran, Lipovača. <a rel="nofollow" class="external text" href="http://learnyouahaskell.com/making-our-own-types-and-typeclasses">"Learn You a Haskell for Great Good!"</a>. <i>Making Our Own Types and Typeclasses</i><span class="reference-accessdate">. Retrieved <span class="nowrap">23 July</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Making+Our+Own+Types+and+Typeclasses&rft.atitle=Learn+You+a+Haskell+for+Great+Good%21&rft.aulast=Miran&rft.aufirst=Lipova%C4%8Da&rft_id=http%3A%2F%2Flearnyouahaskell.com%2Fmaking-our-own-types-and-typeclasses&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKind+%28type+theory%29" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://downloads.haskell.org/~ghc/8.0.1/docs/html/users_guide/glasgow_exts.html#ghc-flag--XTypeInType">"9.1. Language options — Glasgow Haskell Compiler Users Guide"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=9.1.+Language+options+%E2%80%94+Glasgow+Haskell+Compiler+Users+Guide&rft_id=https%3A%2F%2Fdownloads.haskell.org%2F~ghc%2F8.0.1%2Fdocs%2Fhtml%2Fusers_guide%2Fglasgow_exts.html%23ghc-flag--XTypeInType&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKind+%28type+theory%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl 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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 href="/wiki/Recursive_data_type" title="Recursive data type">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 class="mw-selflink selflink">Kind</a> <ul><li><a href="/wiki/Metaclass" title="Metaclass">metaclass</a></li></ul></li> <li><a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">Parametric polymorphism</a></li> <li><a href="/wiki/Primitive_data_type" title="Primitive data type">Primitive data type</a></li> <li><a href="/wiki/Interface_(object-oriented_programming)" title="Interface (object-oriented programming)">Interface</a></li> <li><a href="/wiki/Subtyping" title="Subtyping">Subtyping</a></li> <li><a href="/wiki/Type_constructor" title="Type constructor">Type constructor</a></li> <li><a href="/wiki/Type_conversion" title="Type conversion">Type conversion</a></li> <li><a href="/wiki/Type_system" title="Type system">Type system</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li> <li><a href="/wiki/Variable_(computer_science)" title="Variable (computer science)">Variable</a></li></ul> </div></td></tr></tbody></table></div> <!-- NewPP limit report Parsed by mw‐web.eqiad.next‐6857976fc‐54g89 Cached time: 20250306000936 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.274 seconds Real time usage: 0.580 seconds Preprocessor visited node count: 733/1000000 Post‐expand include size: 24916/2097152 bytes Template 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