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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="Data_transformation_(computing)" title="Data transformation (computing)">Data transformation</a></th></tr><tr><th class="sidebar-heading" style="background:#E7C6A5; padding:2px">
Concepts</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Metadata" title="Metadata">Metadata</a></li>
<li><a href="Data_element" title="Data element">Data element</a></li>
<li><a href="Data_mapping" title="Data mapping">Data mapping</a></li>
<li><a href="Data_migration" title="Data migration">Data migration</a></li>
<li><a href="Data_transformation_(computing)" title="Data transformation (computing)">Data transformation</a></li>
<li><a href="Model_transformation" title="Model transformation">Model transformation</a></li>
<li><a href="Macro_(computer_science)" title="Macro (computer science)">Macro</a></li>
<li><a href="Preprocessor" title="Preprocessor">Preprocessor</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#E7C6A5; padding:2px">
<a href="Transformation_language" title="Transformation language">Transformation languages</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="ATLAS_Transformation_Language" title="ATLAS Transformation Language">ATL</a></li>
<li><a href="AWK" title="AWK">AWK</a></li>
<li><a href="MOFM2T" class="mw-redirect" title="MOFM2T">MOFM2T</a></li>
<li><a href="QVT" title="QVT">QVT</a></li>
<li><a href="XML_transformation_language" title="XML transformation language">XML languages</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#E7C6A5; padding:2px">
Techniques and transforms</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Identity_transform" title="Identity transform">Identity transform</a></li>
<li><a href="Data_refinement" class="mw-redirect" title="Data refinement">Data refinement</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#E7C6A5; padding:2px">
Applications</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Data_conversion" title="Data conversion">Data conversion</a></li>
<li><a href="Data_migration" title="Data migration">Data migration</a></li>
<li><a href="Data_integration" title="Data integration">Data integration</a></li>
<li><a href="Extract%2C_transform%2C_load" title="Extract, transform, load">Extract, transform, load</a> (ETL)</li>
<li><a href="Web_template_system" title="Web template system">Web template system</a></li></ul></td>
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Related</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Data_wrangling" title="Data wrangling">Data wrangling</a></li>
<li><a href="Transformation_language" title="Transformation language">Transformation languages</a></li></ul></td>
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<p><b>Refinement</b> is a generic term of computer science that encompasses various approaches for producing <a href="Correctness_(computer_science)" title="Correctness (computer science)">correct</a> computer programs and simplifying existing programs to enable their formal verification.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Program_refinement">Program refinement</h2></div>
<p>In <a href="Formal_methods" title="Formal methods">formal methods</a>, <b>program refinement</b> is the <a href="Formal_verification" title="Formal verification">verifiable</a> transformation of an <i>abstract</i> (high-level) <a href="Formal_specification" title="Formal specification">formal specification</a> into a <i>concrete</i> (low-level) <a href="Executable_program" class="mw-redirect" title="Executable program">executable program</a>. <i><a href="Stepwise_refinement" class="mw-redirect" title="Stepwise refinement">Stepwise refinement</a></i> allows this process to be done in stages. Logically, refinement normally involves <a href="Logical_consequence" title="Logical consequence">implication</a>, but there can be additional complications.
</p><p>The progressive just-in-time preparation of the product backlog (requirements list) in <a href="Agile_software_development" title="Agile software development">agile software development</a> approaches, such as <a href="Scrum_(software_development)" title="Scrum (software development)">Scrum</a>, is also commonly described as refinement.<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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Data_refinement">Data refinement</h2></div>
<p><b>Data refinement</b> is used to convert an abstract data model (in terms of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a> for example) into implementable <a href="Data_structures" class="mw-redirect" title="Data structures">data structures</a> (such as <a href="Array_data_structure" class="mw-redirect" title="Array data structure">arrays</a>). Operation refinement converts a <a href="Specification" class="mw-redirect" title="Specification">specification</a> of an operation on a system into an implementable <a href="Computer_program" title="Computer program">program</a> (e.g., a <a href="Procedure_(computer_science)" class="mw-redirect" title="Procedure (computer science)">procedure</a>). The <a href="Postcondition" title="Postcondition">postcondition</a> can be strengthened and/or the <a href="Precondition" title="Precondition">precondition</a> weakened in this process. This reduces any <a href="Nondeterministic_algorithm" title="Nondeterministic algorithm">nondeterminism</a> in the specification, typically to a completely <a href="Deterministic" class="mw-redirect" title="Deterministic">deterministic</a> implementation.
</p><p>For example, <i>x</i> ∈ {1,2,3} (where <i>x</i> is the value of the <a href="Variable_(programming)" class="mw-redirect" title="Variable (programming)">variable</a> <i>x</i> after an operation) could be refined to <i>x</i> ∈ {1,2}, then <i>x</i> ∈ {1}, and implemented as <i>x</i>&nbsp;:= 1. Implementations of <i>x</i>&nbsp;:= 2 and <i>x</i>&nbsp;:= 3 would be equally acceptable in this case, using a different route for the refinement. However, we must be careful not to refine to <i>x</i> ∈ {} (equivalent to <i>false</i>) since this is unimplementable; it is impossible to select a <a href="Element_(mathematics)" title="Element (mathematics)">member</a> from the <a href="Empty_set" title="Empty set">empty set</a>.
</p><p>The term <a href="Reification_(computer_science)" title="Reification (computer science)">reification</a> is also sometimes used (coined by <a href="Cliff_Jones_(computer_scientist)" title="Cliff Jones (computer scientist)">Cliff Jones</a>). <a href="Retrenchment_(computing)" title="Retrenchment (computing)">Retrenchment</a> is an alternative technique when formal refinement is not possible. The opposite of refinement is <a href="Abstraction_(computer_science)" title="Abstraction (computer science)">abstraction</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Refinement_calculus">Refinement calculus</h2></div>
<p><a href="Refinement_calculus" title="Refinement calculus">Refinement calculus</a> is a <a href="Formal_system" title="Formal system">formal system</a> (inspired from <a href="Hoare_logic" title="Hoare logic">Hoare logic</a>) that promotes program refinement. The FermaT Transformation System is an industrial-strength implementation of refinement. The <a href="B-Method" title="B-Method">B-Method</a> is also a <a href="Formal_method" class="mw-redirect" title="Formal method">formal method</a> that extends refinement calculus with a component language: it has been used in industrial developments.
</p>
<div class="mw-heading mw-heading2"><h2 id="Refinement_types">Refinement types</h2></div>
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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Refinement_type" title="Refinement type">Refinement type</a></div>
<p>In <a href="Type_theory" title="Type theory">type theory</a>, a <b>refinement type</b><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><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><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> is a type endowed with a predicate which is assumed to hold for any element of the refined type. Refinement types can express <a href="Precondition" title="Precondition">preconditions</a> when used as <a href="Function_argument" class="mw-redirect" title="Function argument">function arguments</a> or <a href="Postcondition" title="Postcondition">postconditions</a> when used as <a href="Return_type" title="Return type">return types</a>: for instance, the type of a function which accepts natural numbers and returns natural numbers greater than 5 may be written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {N} \rightarrow \{n:\mathbb {N} \,|\,n>5\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
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<mi mathvariant="double-struck">N</mi>
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<mo stretchy="false">→<!-- → --></mo>
<mo fence="false" stretchy="false">{</mo>
<mi>n</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {N} \rightarrow \{n:\mathbb {N} \,|\,n&gt;5\}}</annotation>
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</math></span><img src="./cb75138a80e82b05d953d03eb48736bf009db47c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.919ex; height:2.843ex;" alt="{\displaystyle f:\mathbb {N} \rightarrow \{n:\mathbb {N} \,|\,n>5\}}" loading="lazy"></span>. Refinement types are thus related to <a href="Behavioral_subtyping" title="Behavioral subtyping">behavioral subtyping</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Reification_(computer_science)" title="Reification (computer science)">Reification (computer science)</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFCho2009" class="citation book cs1">Cho, L (2009). "Adopting an Agile Culture a User Experience Team's Journey". <i>2009 Agile Conference</i>. pp.&nbsp;<span class="nowrap">416–</span>421. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FAGILE.2009.76">10.1109/AGILE.2009.76</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-7695-3768-9</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&nbsp;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:38580329">38580329</a>.</cite></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFFreemanPfenning1991" class="citation conference cs1">Freeman, T.; Pfenning, F. (1991). <a rel="nofollow" class="external text" href="https://www.cs.cmu.edu/~fp/papers/pldi91.pdf">"Refinement types for ML"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the ACM Conference on Programming Language Design and Implementation</i>. pp.&nbsp;<span class="nowrap">268–</span>277. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F113445.113468">10.1145/113445.113468</a>.</cite></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"><cite id="CITEREFHayashi1993" class="citation conference cs1">Hayashi, S. (1993). "Logic of refinement types". <i>Proceedings of the Workshop on Types for Proofs and Programs</i>. pp.&nbsp;<span class="nowrap">157–</span>172. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a>&nbsp;<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.38.6346">10.1.1.38.6346</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-58085-9_74">10.1007/3-540-58085-9_74</a>.</cite></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFDenney1998" class="citation conference cs1">Denney, E. (1998). "Refinement types for specification". <i>Proceedings of the IFIP International Conference on Programming Concepts and Methods</i>. Vol.&nbsp;125. Chapman &amp; Hall. pp.&nbsp;<span class="nowrap">148–</span>166. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a>&nbsp;<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.4988">10.1.1.22.4988</a></span>.</cite></span>
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