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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div class="cdx-message cdx-message--block cdx-message--warning mw-revision"><span class="cdx-message__icon"></span><div class="cdx-message__content"><div id="mw-revision-info"><p><b>This is an <a href="/wiki/Help:Page_history" title="Help:Page history">old revision</a> of this page, as edited by <span id="mw-revision-name"><a href="/wiki/User:Smimram" class="mw-userlink" title="User:Smimram" data-mw-revid="477888734"><bdi>Smimram</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Smimram" class="mw-usertoollinks-talk" title="User talk:Smimram">talk</a> | <a href="/wiki/Special:Contributions/Smimram" class="mw-usertoollinks-contribs" title="Special:Contributions/Smimram">contribs</a>)</span></span> at <span id="mw-revision-date">14:41, 20 February 2012</span><span id="mw-revision-summary"> <span class="comment">(<span class="autocomment"><a href="#Formal_definition">→<bdi dir="ltr">Formal definition</bdi></a></span>)</span></span>. The present address (URL) is a <a href="/wiki/Help:Permanent_link" title="Help:Permanent link">permanent link</a> to this revision, which may differ significantly from the <span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/wiki/Event_structure">current revision</a></span>.</b></p><div id="revision-info-plain" style="display: none;">Revision as of 14:41, 20 February 2012 by <a href="/wiki/User:Smimram" class="mw-userlink" title="User:Smimram" data-mw-revid="477888734"><bdi>Smimram</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Smimram" class="mw-usertoollinks-talk" title="User talk:Smimram">talk</a> | <a href="/wiki/Special:Contributions/Smimram" class="mw-usertoollinks-contribs" title="Special:Contributions/Smimram">contribs</a>)</span> <span class="comment">(<span class="autocomment"><a href="#Formal_definition">→<bdi dir="ltr">Formal definition</bdi></a></span>)</span></div></div><div id="mw-revision-nav">(<a href="/w/index.php?title=Event_structure&diff=prev&oldid=477888734" title="Event structure">diff</a>) <a href="/w/index.php?title=Event_structure&direction=prev&oldid=477888734" title="Event structure">← Previous revision</a> | <a href="/wiki/Event_structure" title="Event structure">Latest revision</a> (<a href="/w/index.php?title=Event_structure&diff=cur&oldid=477888734" title="Event structure">diff</a>) | <a href="/w/index.php?title=Event_structure&direction=next&oldid=477888734" title="Event structure">Newer revision →</a> (<a href="/w/index.php?title=Event_structure&diff=next&oldid=477888734" title="Event structure">diff</a>)</div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In mathematics and computer science, an <b>event structure</b> represents a set of events, some of which can only be performed after another (there is a <i>dependency</i> between the events) and some of which might not be performed together (there is a <i>conflict</i> between the events). </p> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div> <p>An <b>event structure</b> <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 (E,\leq ,\#)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mo>≤</mo> <mo>,</mo> <mi mathvariant="normal">#</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\leq ,\#)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7680f7301368d96d9fdc62ce8a64559a61f79f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.397ex; height:2.843ex;" alt="{\displaystyle (E,\leq ,\#)}"></span> consists of </p> <ul><li>a set <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> of <b>events</b></li> <li>a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> relation on <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> called <b>causal dependency</b>,</li> <li>a <a href="/wiki/Irreflexive" class="mw-redirect" title="Irreflexive">irreflexive</a> <a href="/wiki/Symmetric" class="mw-redirect" title="Symmetric">symmetric</a> relation <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"> <mi mathvariant="normal">#</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f72e8254fd59fa4060c66c9310acbaf6df2ce894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.936ex; height:2.509ex;" alt="{\displaystyle \#}"></span> called <b>incompatibility</b> (or <b>conflict</b>)</li></ul> <p>such that </p> <ul><li><i>finite causes</i>: for every event <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 e\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34778736d9c6d607a4da3d25594b38dd3e8c82ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.7ex; height:2.176ex;" alt="{\displaystyle e\in E}"></span>, the set <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 [e]=\{f\in E|f\leq e\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>e</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>∈</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo>≤</mo> <mi>e</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [e]=\{f\in E|f\leq e\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ebcd813af31fdea8496207125630c2e8321c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.803ex; height:2.843ex;" alt="{\displaystyle [e]=\{f\in E|f\leq e\}}"></span> of predecessors of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> is finite</li> <li><i>hereditary conflict</i>: for every events <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 d,e,f\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d,e,f\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/547baf2527131bf043a74a39cf3386305f3a6631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.262ex; height:2.509ex;" alt="{\displaystyle d,e,f\in E}"></span>, if <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 d\leq e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≤</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\leq e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5db2bbf97298438149cf148c01a0e4daa2f565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle d\leq e}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\#f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi mathvariant="normal">#</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\#f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a700cfe41f39dc4866448f0bc3b4947edcd13f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.43ex; height:2.509ex;" alt="{\displaystyle d\#f}"></span> then <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 e\#f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mi mathvariant="normal">#</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\#f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df61017ff8c6f052635e67512e6277beadde1b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.298ex; height:2.509ex;" alt="{\displaystyle e\#f}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <ul><li><a href="/w/index.php?title=Template:Cite_article&action=edit&redlink=1" class="new" title="Template:Cite article (page does not exist)">Template:Cite article</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r1012311289">.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;right:1em;display:none}</style><div role="note" class="metadata plainlinks asbox stub"><table role="presentation"><tbody><tr class="noresize"><td><span typeof="mw:File"><a href="/wiki/File:E-to-the-i-pi.svg" class="mw-file-description"><img alt="Stub icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/34px-E-to-the-i-pi.svg.png" decoding="async" width="34" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/51px-E-to-the-i-pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/68px-E-to-the-i-pi.svg.png 2x" data-file-width="512" data-file-height="453" /></a></span></td><td><p class="asbox-body">This <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>-related article is a <a href="/wiki/Wikipedia:Stub" title="Wikipedia:Stub">stub</a>. 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