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class="vector-toc-numb">2</span> <span>Higher-kinded polymorphism</span> </div> </a> <ul id="toc-Higher-kinded_polymorphism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multi-parameter_type_classes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Multi-parameter_type_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Multi-parameter type classes</span> </div> </a> <ul id="toc-Multi-parameter_type_classes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functional_dependencies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Functional_dependencies"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Functional dependencies</span> </div> </a> <ul id="toc-Functional_dependencies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Type_classes_and_implicit_parameters" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Type_classes_and_implicit_parameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Type classes and implicit parameters</span> </div> </a> <ul id="toc-Type_classes_and_implicit_parameters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_approaches_to_operator_overloading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_approaches_to_operator_overloading"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other approaches to operator overloading</span> </div> </a> <ul id="toc-Other_approaches_to_operator_overloading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_notions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_notions"> <div 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<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">Type system in computer science</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">This article is about polymorphic type systems in computer science. For the mathematical class of order-types of a given cardinality, see <a href="/wiki/Glossary_of_set_theory#T" title="Glossary of set theory">Glossary of set theory § T</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Class_(computer_programming)" title="Class (computer programming)">class (computer programming)</a>.</div> <p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, a <b>type class</b> is a <a href="/wiki/Type_system" title="Type system">type system</a> construct that supports <a href="/wiki/Ad_hoc_polymorphism" title="Ad hoc polymorphism">ad hoc polymorphism</a>. This is achieved by adding constraints to type variables in <a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">parametrically polymorphic</a> types. Such a constraint typically involves a type class <code>T</code> and a <a href="/wiki/Type_variable" title="Type variable">type variable</a> <code>a</code>, and means that <code>a</code> can only be instantiated to a type whose members support the overloaded operations associated with <code>T</code>. </p><p>Type classes were first implemented in the <a href="/wiki/Haskell_programming_language" class="mw-redirect" title="Haskell programming language">Haskell programming language</a> after first being proposed by <a href="/wiki/Philip_Wadler" title="Philip Wadler">Philip Wadler</a> and Stephen Blott as an extension to "eqtypes" in <a href="/wiki/Standard_ML" title="Standard ML">Standard ML</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><sup id="cite_ref-wadler88how_2-0" class="reference"><a href="#cite_note-wadler88how-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and were originally conceived as a way of implementing <a href="/wiki/Operator_overloading" title="Operator overloading">overloaded arithmetic and equality operators</a> in a principled fashion.<sup id="cite_ref-kaes88parametric_3-0" class="reference"><a href="#cite_note-kaes88parametric-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-wadler88how_2-1" class="reference"><a href="#cite_note-wadler88how-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In contrast with the "eqtypes" of Standard ML, overloading the equality operator through the use of type classes in Haskell does not need extensive modification of the <a href="/wiki/Compiler_frontend" class="mw-redirect" title="Compiler frontend">compiler frontend</a> or the underlying type system.<sup id="cite_ref-appel91standard_4-0" class="reference"><a href="#cite_note-appel91standard-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_class&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Type classes are defined by specifying a set of function or constant names, together with their respective types, that must exist for every type that belongs to the class. In Haskell, types can be parameterized; a type class <code>Eq</code> intended to contain types that admit equality would be declared in the following way: </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="kr">class</span><span class="w"> </span><span class="kt">Eq</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="kr">where</span> <span class="w"> </span><span class="p">(</span><span class="o">==</span><span class="p">)</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">-></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">Bool</span> <span class="w"> </span><span class="p">(</span><span class="o">/=</span><span class="p">)</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">-></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">Bool</span> </pre></div> <p>where <code>a</code> is one instance of the type class <code>Eq</code>, and <code>a</code> defines the function signatures for 2 functions (the equality and inequality functions), which each take 2 arguments of type <code>a</code> and return a Boolean. </p><p>The type variable <code>a</code> has <a href="/wiki/Kind_(type_theory)" title="Kind (type theory)">kind</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <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> (<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> is also known as <code>Type</code> in the latest <a href="/wiki/Glasgow_Haskell_Compiler" title="Glasgow Haskell Compiler">Glasgow Haskell Compiler</a> (GHC) release),<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> meaning that the kind of <code>Eq</code> is </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="kt">Eq</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="kt">Type</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="kt">Constraint</span> </pre></div> <p>The declaration may be read as stating a "type <code>a</code> belongs to type class <code>Eq</code> if there are functions named <code>(==)</code>, and <code>(/=)</code>, of the appropriate types, defined on it". A programmer could then define a function <code>elem</code> (which determines if an element is in a list) in the following way: </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="nf">elem</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="kt">Eq</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">=></span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="kt">Bool</span> <span class="nf">elem</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="kt">[]</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="kt">False</span> <span class="nf">elem</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="kt">:</span><span class="n">xs</span><span class="p">)</span><span class="w"> </span><span class="ow">=</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">elem</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="n">xs</span> </pre></div> <p>The function <code>elem</code> has the type <code>a -> [a] -> Bool</code> with the context <code>Eq a</code>, which constrains the types which <code>a</code> can range over to those <code>a</code> which belong to the <code>Eq</code> type class. (Haskell <code> => </code> can be called a 'class constraint'.) </p><p>Any type <code>t</code> can be made a member of a given type class <code>C</code> by using an <i>instance declaration</i> that defines implementations of all of <code>C</code>'s methods for the particular type <code>t</code>. For example, if a new data type <code>t</code> is defined, this new type can be made an instance of <code>Eq</code> by providing an equality function over values of type <code>t</code> in any way that is useful. Once this is done, the function <code>elem</code> can be used on <code>[t]</code>, that is, lists of elements of type <code>t</code>. </p><p>Type classes are different from <a href="/wiki/Class_(computer_programming)" title="Class (computer programming)">classes</a> in <a href="/wiki/Object-oriented_programming" title="Object-oriented programming">object-oriented programming</a> languages. In particular, <code>Eq</code> is not a type: there is no such thing as a <i>value</i> of type <code>Eq</code>. </p><p>Type classes are closely related to parametric <a href="/wiki/Polymorphism_(computer_science)" title="Polymorphism (computer science)">polymorphism</a>. For example, the type of <code>elem</code> as specified above would be the parametrically polymorphic type <code>a -> [a] -> Bool</code> were it not for the type class constraint "<code>Eq a =></code>". </p> <div class="mw-heading mw-heading2"><h2 id="Higher-kinded_polymorphism">Higher-kinded polymorphism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_class&action=edit&section=2" title="Edit section: Higher-kinded polymorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A type class need not take a type variable of <a href="/wiki/Kind_(type_theory)" title="Kind (type theory)">kind</a> <code>Type</code> but can take one of any kind. These type classes with higher kinds are sometimes called constructor classes (the constructors referred to are type constructors such as <code>Maybe</code>, rather than data constructors such as <code>Just</code>). An example is the <a href="/wiki/Monad_(functional_programming)" title="Monad (functional programming)"><code>Monad</code></a> class: </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="kr">class</span><span class="w"> </span><span class="kt">Monad</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="kr">where</span> <span class="w"> </span><span class="n">return</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="n">a</span> <span class="w"> </span><span class="p">(</span><span class="o">>>=</span><span class="p">)</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="n">b</span><span class="p">)</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="n">b</span> </pre></div> <p>That m is applied to a type variable indicates that it has kind <code>Type -> Type</code>, i.e., it takes a type and returns a type, the kind of <code>Monad</code> is thus: </p> <div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre><span></span><span class="kt">Monad</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="p">(</span><span class="kt">Type</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="kt">Type</span><span class="p">)</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="kt">Constraint</span> </pre></div> <div class="mw-heading mw-heading2"><h2 id="Multi-parameter_type_classes">Multi-parameter type classes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_class&action=edit&section=3" title="Edit section: Multi-parameter type classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Type classes permit multiple type parameters, and so type classes can be seen as relations on types.<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> For example, in the <a href="/wiki/Glasgow_Haskell_Compiler" title="Glasgow Haskell Compiler">GHC</a> <a href="/wiki/Standard_library" title="Standard library">standard library</a>, the class <code>IArray</code> expresses a general immutable array interface. In this class, the type class constraint <code>IArray a e</code> means that <code>a</code> is an array type that contains elements of type <code>e</code>. (This restriction on polymorphism is used to implement <a href="/wiki/Boxing_(computer_programming)" title="Boxing (computer programming)">unboxed</a> array types, for example.) </p><p>Like <a href="/wiki/Multimethod" class="mw-redirect" title="Multimethod">multimethods</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="Multimethod like lisp's one are evaluated at runtime, and haskell checks types at compile time. Also, there is a thread on haskell mailing list (http://www.haskell.org/pipermail/haskell-cafe/2007-August/030097.html) which seams to show that implementing static multimethod needs universal types. (June 2013)">citation needed</span></a></i>]</sup> multi-parameter type classes support calling different implementations of a method depending on the types of multiple arguments, and indeed return types. Multi-parameter type classes do not require searching for the method to call on every call at runtime;<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><sup class="reference nowrap"><span title="Page / location: minute 25:12">: minute 25:12 </span></sup> rather the method to call is first compiled and stored in the dictionary of the type class instance, just as with single-parameter type classes. </p><p>Haskell code that uses multi-parameter type classes is not portable, as this feature is not part of the Haskell 98 standard. The popular Haskell implementations, GHC and <a href="/wiki/Hugs_(interpreter)" title="Hugs (interpreter)">Hugs</a>, support multi-parameter type classes. </p> <div class="mw-heading mw-heading2"><h2 id="Functional_dependencies">Functional dependencies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_class&action=edit&section=4" title="Edit section: Functional dependencies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In Haskell, type classes have been refined to allow the programmer to declare functional dependencies between type parameters—a concept <a href="/wiki/Functional_dependency" title="Functional dependency">inspired from relational database theory</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> That is, the programmer can assert that a given assignment of some subset of the type parameters uniquely determines the remaining type parameters. For example, a general <a href="/wiki/Monad_(functional_programming)" title="Monad (functional programming)">monad</a> <code>m</code> which carries a state parameter of type <code>s</code> satisfies the type class constraint <code>Monad.State s m</code>. In this constraint, there is a functional dependency <code>m -> s</code>. This means that for a given monad <code>m</code> of type class <code>Monad.State</code>, the state type accessible from <code>m</code> is uniquely determined. This aids the compiler in <a href="/wiki/Type_inference" title="Type inference">type inference</a>, as well as aiding the programmer in <a href="/w/index.php?title=Type-directed_programming&action=edit&redlink=1" class="new" title="Type-directed programming (page does not exist)">type-directed programming</a>. </p><p><a href="/wiki/Simon_Peyton_Jones" title="Simon Peyton Jones">Simon Peyton Jones</a> has objected to the introduction of functional dependencies in Haskell on grounds of complexity.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Type_classes_and_implicit_parameters">Type classes and implicit parameters</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_class&action=edit&section=5" title="Edit section: Type classes and implicit parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Type classes and implicit parameters are very similar in nature, although not quite the same. A polymorphic function with a type class constraint such as: </p> <div class="mw-highlight mw-highlight-lang-hs mw-content-ltr" dir="ltr"><pre><span></span><span class="nf">sum</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="kt">Num</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">=></span><span class="w"> </span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="n">a</span> </pre></div> <p>can be intuitively treated as a function that implicitly accepts an instance of <code>Num</code>: </p> <div class="mw-highlight mw-highlight-lang-hs mw-content-ltr" dir="ltr"><pre><span></span><span class="nf">sum_</span><span class="w"> </span><span class="ow">::</span><span class="w"> </span><span class="kt">Num_</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="ow">-></span><span class="w"> </span><span class="n">a</span> </pre></div> <p>The instance <code>Num_ a</code> is essentially a record that contains the instance definition of <code>Num a</code>. (This is in fact how type classes are implemented under the hood by the Glasgow Haskell Compiler.) </p><p>However, there is a crucial difference: implicit parameters are more <i>flexible</i>; different instances of <code>Num Int</code> can be passed. In contrast, type classes enforce the so-called <i>coherence</i> property, which requires that there should only be one unique choice of instance for any given type. The coherence property makes type classes somewhat antimodular, which is why orphan instances (instances that are defined in a module that neither contains the class nor the type of interest) are strongly discouraged. However, coherence adds another level of safety to a language, providing a guarantee that two disjoint parts of the same code will share the same instance.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>As an example, an ordered <a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">set</a> (of type <code>Set a</code>) requires a <a href="/wiki/Total_order" title="Total order">total ordering</a> on the elements (of type <code>a</code>) to function. This can be evidenced by a constraint <code>Ord a</code>, which defines a comparison operator on the elements. However, there can be numerous ways to impose a total order. Since set algorithms are generally intolerant of changes in the ordering once a set has been constructed, passing an incompatible instance of <code>Ord a</code> to functions that operate on the set may lead to incorrect results (or crashes). Thus, enforcing coherence of <code>Ord a</code> in this particular scenario is crucial. </p><p>Instances (or "dictionaries") in <a href="/wiki/Scala_(programming_language)" title="Scala (programming language)">Scala</a> type classes are just ordinary values in the language, rather than a completely separate kind of entity.<sup id="cite_ref-scala-typeclasses_12-0" class="reference"><a href="#cite_note-scala-typeclasses-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> While these instances are by default supplied by finding appropriate instances in scope to be used as the implicit parameters for explicitly-declared implicit formal parameters, that they are ordinary values means that they can be supplied explicitly, to resolve ambiguity. As a result, Scala type classes do not satisfy the coherence property and are effectively a <a href="/wiki/Syntactic_sugar" title="Syntactic sugar">syntactic sugar</a> for implicit parameters. </p><p>This is an example taken from the Cats documentation:<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-highlight mw-highlight-lang-scala mw-content-ltr" dir="ltr"><pre><span></span><span class="c1">// A type class to provide textual representation</span> <span class="k">trait</span><span class="w"> </span><span class="nc">Show</span><span class="p">[</span><span class="nc">A</span><span class="p">]</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">def</span><span class="w"> </span><span class="nf">show</span><span class="p">(</span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="nc">A</span><span class="p">):</span><span class="w"> </span><span class="nc">String</span> <span class="p">}</span> <span class="c1">// A polymorphic function that works only when there is an implicit </span> <span class="c1">// instance of Show[A] available</span> <span class="k">def</span><span class="w"> </span><span class="nf">log</span><span class="p">[</span><span class="nc">A</span><span class="p">](</span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="nc">A</span><span class="p">)(</span><span class="k">implicit</span><span class="w"> </span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="nc">Show</span><span class="p">[</span><span class="nc">A</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="n">show</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1">// An instance for String</span> <span class="k">implicit</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="n">stringShow</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nc">Show</span><span class="p">[</span><span class="nc">String</span><span class="p">]</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">def</span><span class="w"> </span><span class="nf">show</span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="nc">String</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span> <span class="p">}</span> <span class="c1">// The parameter stringShow was inserted by the compiler.</span> <span class="n">scala</span><span class="o">></span><span class="w"> </span><span class="n">log</span><span class="p">(</span><span class="s">"a string"</span><span class="p">)</span> <span class="n">a</span><span class="w"> </span><span class="n">string</span> </pre></div> <p><a href="/wiki/Coq_(software)" class="mw-redirect" title="Coq (software)">Coq</a> (version 8.2 onward) also supports type classes by inferring the appropriate instances.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Recent versions of <a href="/wiki/Agda_(programming_language)" title="Agda (programming language)">Agda</a> 2 also provide a similar feature, called "instance arguments".<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Other_approaches_to_operator_overloading">Other approaches to operator overloading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_class&action=edit&section=6" title="Edit section: Other approaches to operator overloading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Standard_ML" title="Standard ML">Standard ML</a>, the mechanism of "equality types" corresponds roughly to Haskell's built-in type class <code>Eq</code>, but all equality operators are derived automatically by the compiler. The programmer's control of the process is limited to designating which type components in a structure are equality types and which type variables in a polymorphic type range over equality types. </p><p>SML's and <a href="/wiki/OCaml" title="OCaml">OCaml</a>'s modules and functors can play a role similar to that of Haskell's type classes, the principal difference being the role of type inference, which makes type classes suitable for <i>ad hoc</i> polymorphism.<sup id="cite_ref-dreyer06modular_17-0" class="reference"><a href="#cite_note-dreyer06modular-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The object oriented subset of <a href="/wiki/OCaml" title="OCaml">OCaml</a> is yet another approach which is somewhat comparable to the one of type classes. </p> <div class="mw-heading mw-heading2"><h2 id="Related_notions">Related notions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_class&action=edit&section=7" title="Edit section: Related notions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An analogous notion for overloaded data (implemented in <a href="/wiki/Glasgow_Haskell_Compiler" title="Glasgow Haskell Compiler">GHC</a>) is that of <a href="/wiki/Type_family" title="Type family">type family</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/C%2B%2B" title="C++">C++</a>, notably <a href="/wiki/C%2B%2B20" title="C++20">C++20</a>, has support for type classes using <a href="/wiki/Concepts_(C%2B%2B)" title="Concepts (C++)">Concepts (C++)</a>. As an illustration, the above mentioned Haskell example of typeclass Eq would be written as </p> <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre><span></span><span class="k">template</span><span class="w"> </span><span class="o"><</span><span class="k">typename</span><span class="w"> </span><span class="nc">T</span><span class="o">></span> <span class="k">concept</span><span class="w"> </span><span class="nc">Equal</span><span class="w"> </span><span class="o">=</span> <span class="w"> </span><span class="k">requires</span><span class="w"> </span><span class="p">(</span><span class="n">T</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">T</span><span class="w"> </span><span class="n">b</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="n">std</span><span class="o">::</span><span class="n">convertible_to</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="p">;</span> <span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="o">-></span><span class="w"> </span><span class="n">std</span><span class="o">::</span><span class="n">convertible_to</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="p">;</span> <span class="p">};</span> </pre></div> <p>In <a href="/wiki/Clean_(programming_language)" title="Clean (programming language)">Clean</a> typeclasses are similar to Haskell, but have a slightly different <a href="/wiki/Syntax_(programming_languages)" title="Syntax (programming languages)">syntax</a>. </p><p><a href="/wiki/Rust_(programming_language)" title="Rust (programming language)">Rust</a> supports <a href="/wiki/Trait_(computer_programming)" title="Trait (computer programming)">traits</a>, which are a limited form of type classes with coherence.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Mercury_(programming_language)" title="Mercury (programming language)">Mercury</a> has typeclasses, although they are not exactly the same as in Haskell.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag needs further explanation. (May 2018)">further explanation needed</span></a></i>]</sup> </p><p>In <a href="/wiki/Scala_(programming_language)" title="Scala (programming language)">Scala</a>, type classes are a <a href="/wiki/Programming_idiom" title="Programming idiom">programming idiom</a> that can be implemented with existing language features such as implicit parameters, not a separate language feature per se. Because of the way they are implemented in Scala, it is possible to explicitly specify which type class instance to use for a type at a particular place in the code, in case of ambiguity. However, this is not necessarily a benefit as ambiguous type class instances can be error-prone. </p><p>The proof assistant <a href="/wiki/Coq_(software)" class="mw-redirect" title="Coq (software)">Coq</a> has also supported type classes in recent versions. Unlike in ordinary programming languages, in Coq, any laws of a type class (such as the monad laws) that are stated within the type class definition, must be mathematically proved of each type class instance before using them. </p> <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=Type_class&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 mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMorris2013" class="citation thesis cs1">Morris, John G. (2013). <a rel="nofollow" class="external text" href="https://jgbm.github.io/pubs/morris-dissertation.pdf"><i>Type Classes and Instance Chains: A Relational Approach</i></a> <span class="cs1-format">(PDF)</span> (PhD). Department of Computer Science, Portland State University. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.15760%2Fetd.1010">10.15760/etd.1010</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Type+Classes+and+Instance+Chains%3A+A+Relational+Approach&rft.inst=Department+of+Computer+Science%2C+Portland+State+University&rft.date=2013&rft_id=info%3Adoi%2F10.15760%2Fetd.1010&rft.aulast=Morris&rft.aufirst=John+G.&rft_id=https%3A%2F%2Fjgbm.github.io%2Fpubs%2Fmorris-dissertation.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-wadler88how-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-wadler88how_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-wadler88how_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWadlerBlott1989" class="citation book cs1">Wadler, P.; Blott, S. (1989). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/2710954">"How to make ad-hoc polymorphism less ad hoc"</a>. <i>Proceedings of the 16th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL '89)</i>. Association for Computing Machinery. pp. <span class="nowrap">60–</span>76. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F75277.75283">10.1145/75277.75283</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0897912942" title="Special:BookSources/0897912942"><bdi>0897912942</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15327197">15327197</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=How+to+make+ad-hoc+polymorphism+less+ad+hoc&rft.btitle=Proceedings+of+the+16th+ACM+SIGPLAN-SIGACT+Symposium+on+Principles+of+Programming+Languages+%28POPL+%2789%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E60-%3C%2Fspan%3E76&rft.pub=Association+for+Computing+Machinery&rft.date=1989&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15327197%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F75277.75283&rft.isbn=0897912942&rft.aulast=Wadler&rft.aufirst=P.&rft.au=Blott%2C+S.&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F2710954&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-kaes88parametric-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-kaes88parametric_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKaes1988" class="citation conference cs1">Kaes, Stefan (March 1988). "Parametric overloading in polymorphic programming languages". <i>Proceedings 2nd European Symposium on Programming Languages</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-19027-9_9">10.1007/3-540-19027-9_9</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Parametric+overloading+in+polymorphic+programming+languages&rft.btitle=Proceedings+2nd+European+Symposium+on+Programming+Languages&rft.date=1988-03&rft_id=info%3Adoi%2F10.1007%2F3-540-19027-9_9&rft.aulast=Kaes&rft.aufirst=Stefan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-appel91standard-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-appel91standard_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAppelMacQueen1991" class="citation book cs1">Appel, A.W.; MacQueen, D.B. (1991). "Standard ML of New Jersey". In Maluszyński, J.; Wirsing, M. (eds.). <i>Programming Language Implementation and Logic Programming. PLILP 1991</i>. Lecture Notes in Computer Science. Vol. 528. Springer. pp. <span class="nowrap">1–</span>13. <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=10.1.1.55.9444">10.1.1.55.9444</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-54444-5_83">10.1007/3-540-54444-5_83</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-54444-5" title="Special:BookSources/3-540-54444-5"><bdi>3-540-54444-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Standard+ML+of+New+Jersey&rft.btitle=Programming+Language+Implementation+and+Logic+Programming.+PLILP+1991&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E13&rft.pub=Springer&rft.date=1991&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.55.9444%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1007%2F3-540-54444-5_83&rft.isbn=3-540-54444-5&rft.aulast=Appel&rft.aufirst=A.W.&rft.au=MacQueen%2C+D.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://hackage.haskell.org/package/base-4.9.0.0/docs/Data-Kind.html#t:Type"><code>Type</code></a> from <code>Data.Kind</code> appeared in version 8 of the <a href="/wiki/Glasgow_Haskell_Compiler" title="Glasgow Haskell Compiler">Glasgow Haskell Compiler</a></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"><a href="/wiki/Haskell%27" class="mw-redirect" title="Haskell'">Haskell'</a> page <i><a rel="nofollow" class="external text" href="http://prime.haskell.org/wiki/MultiParamTypeClasses">MultiParamTypeClasses</a></i>.</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">In GHC, the C Core uses Girard & Reynold's System F type signatures to identify a typed case for processing in the optimization phases. -- Simon Peyton-Jones "<a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=uR_VzYxvbxg">Into the Core - Squeezing Haskell into Nine Constructors"</a> Erlang User Conference, Sep 14, 2016</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJones2000" class="citation book cs1">Jones, Mark P. (2000). <a rel="nofollow" class="external text" href="http://web.cecs.pdx.edu/~mpj/pubs/fundeps.html">"Type Classes with Functional Dependencies"</a>. In Smolka, G. (ed.). <i>Programming Languages and Systems. ESOP 2000</i>. Lecture Notes in Computer Science. Vol. 1782. Springer. pp. <span class="nowrap">230–</span>244. <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=10.1.1.26.7153">10.1.1.26.7153</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-46425-5_15">10.1007/3-540-46425-5_15</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-46425-5" title="Special:BookSources/3-540-46425-5"><bdi>3-540-46425-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Type+Classes+with+Functional+Dependencies&rft.btitle=Programming+Languages+and+Systems.+ESOP+2000&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E230-%3C%2Fspan%3E244&rft.pub=Springer&rft.date=2000&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.26.7153%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1007%2F3-540-46425-5_15&rft.isbn=3-540-46425-5&rft.aulast=Jones&rft.aufirst=Mark+P.&rft_id=http%3A%2F%2Fweb.cecs.pdx.edu%2F~mpj%2Fpubs%2Ffundeps.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="/wiki/Haskell%27" class="mw-redirect" title="Haskell'">Haskell'</a> page <i><a rel="nofollow" class="external text" href="http://prime.haskell.org/wiki/FunctionalDependencies">FunctionalDependencies</a></i>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeyton_Jones2006" class="citation web cs1"><a href="/wiki/Simon_Peyton_Jones" title="Simon Peyton Jones">Peyton Jones, Simon</a> (2006). <a rel="nofollow" class="external text" href="https://mail.haskell.org/pipermail/haskell-prime/2006-February/000289.html">"MPTCs and functional dependencies"</a>. <i>Haskell-prime mailing list</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Haskell-prime+mailing+list&rft.atitle=MPTCs+and+functional+dependencies&rft.date=2006&rft.aulast=Peyton+Jones&rft.aufirst=Simon&rft_id=https%3A%2F%2Fmail.haskell.org%2Fpipermail%2Fhaskell-prime%2F2006-February%2F000289.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKmett2015" class="citation audio-visual cs1">Kmett, Edward (January 21, 2015). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=hIZxTQP1ifo"><i>Type Classes vs. the World</i></a> (video). Boston Haskell Meetup. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211221/hIZxTQP1ifo">Archived</a> from the original on 2021-12-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Type+Classes+vs.+the+World&rft.pub=Boston+Haskell+Meetup&rft.date=2015-01-21&rft.aulast=Kmett&rft.aufirst=Edward&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DhIZxTQP1ifo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-scala-typeclasses-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-scala-typeclasses_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOliveiraMoorsOdersky2010" class="citation book cs1">Oliveira, Bruno C.d.S.; Moors, Adriaan; Odersky, Martin (2010). <a rel="nofollow" class="external text" href="http://ropas.snu.ac.kr/~bruno/papers/TypeClasses.pdf">"Type Classes as Objects and Implicits"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the ACM International Conference on Object Oriented Programming Systems Languages and Applications (OOPSLA '10)</i>. Association for Computing Machinery. pp. <span class="nowrap">341–</span>360. <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=10.1.1.205.2737">10.1.1.205.2737</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1869459.1869489">10.1145/1869459.1869489</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781450302036" title="Special:BookSources/9781450302036"><bdi>9781450302036</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207183083">207183083</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Type+Classes+as+Objects+and+Implicits&rft.btitle=Proceedings+of+the+ACM+International+Conference+on+Object+Oriented+Programming+Systems+Languages+and+Applications+%28OOPSLA+%2710%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E341-%3C%2Fspan%3E360&rft.pub=Association+for+Computing+Machinery&rft.date=2010&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.205.2737%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A207183083%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F1869459.1869489&rft.isbn=9781450302036&rft.aulast=Oliveira&rft.aufirst=Bruno+C.d.S.&rft.au=Moors%2C+Adriaan&rft.au=Odersky%2C+Martin&rft_id=http%3A%2F%2Fropas.snu.ac.kr%2F~bruno%2Fpapers%2FTypeClasses.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</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://danielwestheide.com/blog/2013/02/06/the-neophytes-guide-to-scala-part-12-type-classes.html">"The Neophyte's Guide to Scala Part 12: Type classes - Daniel Westheide"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Neophyte%27s+Guide+to+Scala+Part+12%3A+Type+classes+-+Daniel+Westheide&rft_id=http%3A%2F%2Fdanielwestheide.com%2Fblog%2F2013%2F02%2F06%2Fthe-neophytes-guide-to-scala-part-12-type-classes.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://typelevel.org/cats/typeclasses.html">typelevel.org, Scala Cats</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCastéranSozeau2014" class="citation web cs1">Castéran, P.; Sozeau, M. (2014). <a rel="nofollow" class="external text" href="http://www.labri.fr/perso/casteran/CoqArt/TypeClassesTut/typeclassestut.pdf">"A Gentle Introduction to Type Classes and Relations in Coq"</a> <span class="cs1-format">(PDF)</span>. <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=10.1.1.422.8091">10.1.1.422.8091</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Gentle+Introduction+to+Type+Classes+and+Relations+in+Coq&rft.date=2014&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.422.8091%23id-name%3DCiteSeerX&rft.aulast=Cast%C3%A9ran&rft.aufirst=P.&rft.au=Sozeau%2C+M.&rft_id=http%3A%2F%2Fwww.labri.fr%2Fperso%2Fcasteran%2FCoqArt%2FTypeClassesTut%2Ftypeclassestut.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="http://wiki.portal.chalmers.se/agda/pmwiki.php?n=ReferenceManual.ModellingTypeClassesWithInstanceArguments">Modelling Type Classes With Instance Arguments</a>".</span> </li> <li id="cite_note-dreyer06modular-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-dreyer06modular_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDreyerHarperChakravarty2007" class="citation book cs1">Dreyer, Derek; Harper, Robert; Chakravarty, Manuel M.T. (2007). "Modular Type Classes". <a rel="nofollow" class="external text" href="http://citeseer.ist.psu.edu/751836.html"><i>Proceedings of the 34th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL '07)</i></a>. pp. 63–70. See p. 63. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1190216.1190229">10.1145/1190216.1190229</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1595935755" title="Special:BookSources/978-1595935755"><bdi>978-1595935755</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1828213">1828213</a>. TR-2006-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Modular+Type+Classes&rft.btitle=Proceedings+of+the+34th+Annual+ACM+SIGPLAN-SIGACT+Symposium+on+Principles+of+Programming+Languages+%28POPL+%2707%29&rft.pages=63-70.+See+p.+63&rft.date=2007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1828213%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F1190216.1190229&rft.isbn=978-1595935755&rft.aulast=Dreyer&rft.aufirst=Derek&rft.au=Harper%2C+Robert&rft.au=Chakravarty%2C+Manuel+M.T.&rft_id=http%3A%2F%2Fciteseer.ist.psu.edu%2F751836.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</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/haskellwiki/GHC/Type_families">"GHC/Type families - HaskellWiki"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=GHC%2FType+families+-+HaskellWiki&rft_id=http%3A%2F%2Fwww.haskell.org%2Fhaskellwiki%2FGHC%2FType_families&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTuron2017" class="citation report cs1">Turon, Aaron (2017). <a rel="nofollow" class="external text" href="https://aturon.github.io/tech/2017/02/06/specialization-and-coherence/">Specialization, coherence, and API evolution</a> (Report).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Specialization%2C+coherence%2C+and+API+evolution&rft.date=2017&rft.aulast=Turon&rft.aufirst=Aaron&rft_id=https%3A%2F%2Faturon.github.io%2Ftech%2F2017%2F02%2F06%2Fspecialization-and-coherence%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></span> </li> </ol></div></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 .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeyton_JonesJonesMeijer1997" class="citation journal cs1"><a href="/wiki/Simon_Peyton_Jones" title="Simon Peyton Jones">Peyton Jones, Simon</a>; Jones, Mark; Meijer, Erik (May 1997). <a rel="nofollow" class="external text" href="http://research.microsoft.com/~simonpj/Papers/type-class-design-space">"Type classes: an exploration of the design space"</a>. <i>Proc. ACM SIGPLAN Haskell Workshop</i>. <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=10.1.1.1085.8703">10.1.1.1085.8703</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+ACM+SIGPLAN+Haskell+Workshop&rft.atitle=Type+classes%3A+an+exploration+of+the+design+space&rft.date=1997-05&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.1085.8703%23id-name%3DCiteSeerX&rft.aulast=Peyton+Jones&rft.aufirst=Simon&rft.au=Jones%2C+Mark&rft.au=Meijer%2C+Erik&rft_id=http%3A%2F%2Fresearch.microsoft.com%2F~simonpj%2FPapers%2Ftype-class-design-space&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_class&action=edit&section=9" title="Edit section: External links"><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 class="citation book cs1"><a rel="nofollow" class="external text" href="http://www.haskell.org/tutorial/classes.html">"5. Type Classes and Overloading"</a>. <i>A Gentle Introduction to Haskell</i>. June 2000. Version 98.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=5.+Type+Classes+and+Overloading&rft.btitle=A+Gentle+Introduction+to+Haskell&rft.date=2000-06&rft_id=http%3A%2F%2Fwww.haskell.org%2Ftutorial%2Fclasses.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AType+class" class="Z3988"></span></li> <li>Advanced Functional Programming course at Utrecht University, 74 lecture slides on <a rel="nofollow" class="external text" href="http://www.cs.uu.nl/wiki/pub/Afp/CourseSchedule/AFP-2008.3-12.pdf">Advanced Type Classes</a>. 2005-06-07.</li> <li><a rel="nofollow" class="external text" href="http://okmij.org/ftp/Computation/typeclass.html">Implementing, and Understanding Type Classes</a>. 2014-11-13.</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 ul{margin:0;padding:0}.mw-parser-output .hlist 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