· 8 years ago · Nov 15, 2017, 08:34 AM
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2objc[5232]: Class JavaLaunchHelper is implemented in both /Library/Java/JavaVirtualMachines/jdk1.8.0_20.jdk/Contents/Home/bin/java (0x1073d74c0) and /Library/Java/JavaVirtualMachines/jdk1.8.0_20.jdk/Contents/Home/jre/lib/libinstrument.dylib (0x10749e4e0). One of the two will be used. Which one is undefined.
3 <html class="client-nojs" lang="en" dir="ltr"> <meta charset="UTF-8"/> <script>document.documentElement.className = document.documentElement.className.replace( /(^|\s)client-nojs(\s|$)/, "$1client-js$2" );</script> mw.user.tokens.set({"editToken":"+\\","patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"});/*@nomin*/ });mw.loader.load(["site","mediawiki.page.startup","mediawiki.user","mediawiki.hidpi","mediawiki.page.ready","mediawiki.searchSuggest","ext.gadget.teahouse","ext.gadget.ReferenceTooltips","ext.gadget.watchlist-notice","ext.gadget.DRN-wizard","ext.gadget.charinsert","ext.gadget.refToolbar","ext.gadget.extra-toolbar-buttons","ext.gadget.switcher","ext.centralauth.centralautologin","mmv.head","mmv.bootstrap.autostart","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.eventLogging.subscriber","ext.wikimediaEvents","ext.navigationTiming","ext.uls.eventlogger","ext.uls.init","ext.uls.interface","ext.centralNotice.geoIP","ext.centralNotice.startUp","skins.vector.js"]);});</script> <script async="" src="/w/load.php?debug=false&lang=en&modules=startup&only=scripts&skin=vector"></script> <link rel="stylesheet" href="/w/load.php?debug=false&lang=en&modules=ext.gadget.charinsert-styles&only=styles&skin=vector"/> <meta name="generator" content="MediaWiki 1.31.0-wmf.6"/> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/en/thumb/8/82/Complexity-map_castellani_w.jpg/1200px-Complexity-map_castellani_w.jpg"/> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Complexity_theory&action=edit"/> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"/> <link rel="search" type="application/opensearchdescription+xml" href="/w/opensearch_desc.php" title="Wikipedia (en)"/> <link rel="license" href="//creativecommons.org/licenses/by-sa/3.0/"/> <link rel="dns-prefetch" href="//login.wikimedia.org"/> <!--[if lt IE 9]><script src="/w/load.php?debug=false&lang=en&modules=html5shiv&only=scripts&skin=vector&sync=1"></script><![endif]--> <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Complexity_theory rootpage-Complexity_theory vector-experimental-print-styles vector-nav-directionality skin-vector action-view"> <div id="mw-page-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <div class="mw-indicators mw-body-content"> <h1 id="firstHeading" class="firstHeading" lang="en">Complexity theory</h1> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> <div id="jump-to-nav" class="mw-jump"> </div> <p><b>Complexity theory</b> (or complexity science) is the study of <a href="/wiki/Complexity" title="Complexity">Complexity</a> in regard to <a href="/wiki/Complex_systems" class="mw-redirect" title="Complex systems">Complex systems</a>. It may refer to:</p> <li><a href="/wiki/Complexity_theory_and_organizations" title="Complexity theory and organizations">Complexity theory and organizations</a>, the application of complexity theory to strategy</li> <li><a href="/wiki/Complex_adaptive_system" title="Complex adaptive system">Complex adaptive system</a>, a special case of complex systems</li> </ul> <div class="center"> </div> <table> <td style="padding-right:2em"> <li><a href="/wiki/Biological_organisation" title="Biological organisation">Biological organisation</a></li> <li><a href="/wiki/Cognitive_model#Dynamical_systems" title="Cognitive model">Cognitive modeling</a></li> <li><a href="/wiki/Complex_adaptive_system" title="Complex adaptive system">Complex adaptive system</a></li> <li><a href="/wiki/Complexity" title="Complexity">Complexity</a></li> <li><a href="/wiki/Dual-phase_evolution" title="Dual-phase evolution">Dual-phase evolution</a></li> </td> <ul> <li><a href="/wiki/Dynamical_systems_theory" title="Dynamical systems theory">Dynamical systems theory</a> <li><a href="/wiki/Attractor" title="Attractor">Attractor</a> and <a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a> <li><a href="/wiki/Lorenz_system" title="Lorenz system">Lorenz system</a>, <a href="/wiki/Limit_cycle" title="Limit cycle">Limit cycle</a>, <a href="/wiki/Edge_of_chaos" title="Edge of chaos">Edge of chaos</a></li> </li> </li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Network_science" title="Network science">Network science</a></li> </ul> <td style="padding-right:2em"> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a> <li><a href="/wiki/Emergence" title="Emergence">Emergence</a></li> </ul> <li><a href="/wiki/Systems_theory" title="Systems theory">Systems theory</a></li> </td> </table> <ul> <ul> <li><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Basin_of_attraction">Basin of attraction (Scholarpedia)</a></li> </ul> </ul> NewPP limit report Cached time: 20171106211607 Dynamic content: false Real time usage: 0.037 seconds Preprocessor generated node count: 0/1500000 Template argument size: 0/2097152 bytes Expensive parser function count: 0/500 Lua memory usage: 585 KB/50 MB <!-- 100.00% 22.412 1 Template:See --> <!-- Saved in parser cache with key enwiki:pcache:idhash:6132-0!canonical and timestamp 20171106211607 and revision id 805616512 <noscript><img src="//en.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;" /></noscript></div> <div class="printfooter"> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Complex_systems_theory" title="Category:Complex systems theory">Complex systems theory</a></li></ul></div></div> <div class="visualClear"></div> </div> <h2>Navigation menu</h2> <div id="mw-head"> <h3 id="p-personal-label">Personal tools</h3> <li id="pt-anonuserpage">Not logged in</li><li id="pt-anontalk"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li><li id="pt-anoncontribs"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-createaccount"><a href="/w/index.php?title=Special:CreateAccount&returnto=Complexity+theory" title="You are encouraged to create an account and log in; however, it is not mandatory">Create account</a></li><li id="pt-login"><a href="/w/index.php?title=Special:UserLogin&returnto=Complexity+theory" title="You're encouraged to log in; however, it's not mandatory. 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4 <html class="client-nojs" lang="en" dir="ltr"> <meta charset="UTF-8"/> <script>document.documentElement.className = document.documentElement.className.replace( /(^|\s)client-nojs(\s|$)/, "$1client-js$2" );</script> mw.user.tokens.set({"editToken":"+\\","patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"});/*@nomin*/ });mw.loader.load(["ext.cite.a11y","ext.math.scripts","site","mediawiki.page.startup","mediawiki.user","mediawiki.hidpi","mediawiki.page.ready","mediawiki.toc","mediawiki.searchSuggest","ext.gadget.teahouse","ext.gadget.ReferenceTooltips","ext.gadget.watchlist-notice","ext.gadget.DRN-wizard","ext.gadget.charinsert","ext.gadget.refToolbar","ext.gadget.extra-toolbar-buttons","ext.gadget.switcher","ext.centralauth.centralautologin","mmv.head","mmv.bootstrap.autostart","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.eventLogging.subscriber","ext.wikimediaEvents","ext.navigationTiming","ext.uls.eventlogger","ext.uls.init","ext.uls.interface","ext.centralNotice.geoIP","ext.centralNotice.startUp","skins.vector.js"]);});</script> <script async="" src="/w/load.php?debug=false&lang=en&modules=startup&only=scripts&skin=vector"></script> <link rel="stylesheet" href="/w/load.php?debug=false&lang=en&modules=ext.gadget.charinsert-styles&only=styles&skin=vector"/> <meta name="generator" content="MediaWiki 1.31.0-wmf.7"/> <link rel="alternate" href="android-app://org.wikipedia/http/en.m.wikipedia.org/wiki/Complexity"/> <link rel="edit" title="Edit this page" href="/w/index.php?title=Complexity&action=edit"/> <link rel="shortcut icon" href="/static/favicon/wikipedia.ico"/> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"/> <link rel="canonical" href="https://en.wikipedia.org/wiki/Complexity"/> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> </head> <div id="mw-head-base" class="noprint"></div> <a id="top"></a> <div id="siteNotice" class="mw-body-content"><!-- CentralNotice --></div> </div> <div id="bodyContent" class="mw-body-content"> <div id="contentSub"></div> Jump to: <a href="#mw-head">navigation</a>, <a href="#p-search">search</a> <div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><div class="mw-parser-output"><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Complexity_(disambiguation)" class="mw-disambig" title="Complexity (disambiguation)">Complexity (disambiguation)</a>.</div> <tr> <div style="width:52px"><a href="/wiki/File:Crystal_Clear_app_kedit.svg" class="image"><img alt="Crystal Clear app kedit.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/40px-Crystal_Clear_app_kedit.svg.png" width="40" height="40" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/60px-Crystal_Clear_app_kedit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/80px-Crystal_Clear_app_kedit.svg.png 2x" data-file-width="128" data-file-height="128" /></a></div> <td class="mbox-text"> </td> </table> <p>The stem of the word "complexity" - <i>complex</i> - combines the Latin roots <i>com</i> (meaning "together") and <i>plex</i> (meaning "woven"). Contrast "complicated" where <i>plic</i> (meaning "folded") refers to many layers. A complex system is thereby characterised by its inter-dependencies, whereas a complicated system is characterised by its layers.</p> <p><a href="/wiki/Science" title="Science">Science</a> as of 2010<sup class="plainlinks noprint asof-tag update" style="display:none;"><a class="external text" href="//en.wikipedia.org/w/index.php?title=Complexity&action=edit">[update]</a></sup> takes a number of approaches to characterizing complexity; Zayed <i>et al.</i><sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup> reflect many of these. <a href="/wiki/Neil_F._Johnson" title="Neil F. Johnson">Neil Johnson</a> states that "even among scientists, there is no unique definition of complexity – and the scientific notion has traditionally been conveyed using particular examples..." Ultimately Johnson adopts the definition of "complexity science" as "the study of the phenomena which emerge from a collection of interacting objects".<sup id="cite_ref-Neil_Johnson_4-0" class="reference"><a href="#cite_note-Neil_Johnson-4">[4]</a></sup></p> <div id="toc" class="toc"> <h2>Contents</h2> <ul> <li class="toclevel-1 tocsection-2"><a href="#Disorganized_vs._organized"><span class="tocnumber">2</span> <span class="toctext">Disorganized vs. organized</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Varied_meanings"><span class="tocnumber">4</span> <span class="toctext">Varied meanings</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Topics"><span class="tocnumber">6</span> <span class="toctext">Topics</span></a> <li class="toclevel-2 tocsection-7"><a href="#Behaviour"><span class="tocnumber">6.1</span> <span class="toctext">Behaviour</span></a></li> <li class="toclevel-2 tocsection-9"><a href="#Simulations"><span class="tocnumber">6.3</span> <span class="toctext">Simulations</span></a></li> <li class="toclevel-2 tocsection-11"><a href="#Data"><span class="tocnumber">6.5</span> <span class="toctext">Data</span></a></li> </ul> <li class="toclevel-1 tocsection-13"><a href="#Applications"><span class="tocnumber">7</span> <span class="toctext">Applications</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#References"><span class="tocnumber">9</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-17"><a href="#External_links"><span class="tocnumber">11</span> <span class="toctext">External links</span></a></li> </div> <h2><span class="mw-headline" id="Overview">Overview</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complexity&action=edit&section=1" title="Edit section: Overview">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <p><a href="/wiki/Warren_Weaver" title="Warren Weaver">Warren Weaver</a> posited in 1948 two forms of complexity: disorganized complexity, and organized complexity.<sup id="cite_ref-Weaver_5-0" class="reference"><a href="#cite_note-Weaver-5">[5]</a></sup> Phenomena of 'disorganized complexity' are treated using probability theory and statistical mechanics, while 'organized complexity' deals with phenomena that escape such approaches and confront "dealing simultaneously with a sizable number of factors which are interrelated into an organic whole".<sup id="cite_ref-Weaver_5-1" class="reference"><a href="#cite_note-Weaver-5">[5]</a></sup> Weaver's 1948 paper has influenced subsequent thinking about complexity.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6">[6]</a></sup></p> <p>Some definitions relate to the algorithmic basis for the expression of a complex phenomenon or model or mathematical expression, as later set out herein.</p> <p>One of the problems in addressing complexity issues has been formalizing the intuitive conceptual distinction between the large number of variances in relationships extant in random collections, and the sometimes large, but smaller, number of relationships between elements in systems where constraints (related to correlation of otherwise independent elements) simultaneously reduce the variations from element independence and create distinguishable regimes of more-uniform, or correlated, relationships, or interactions.</p> <p>In Weaver's view, disorganized complexity results from the particular system having a very large number of parts, say millions of parts, or many more. Though the interactions of the parts in a "disorganized complexity" situation can be seen as largely random, the properties of the system as a whole can be understood by using probability and statistical methods.</p> <p>Organized complexity, in Weaver's view, resides in nothing else than the non-random, or correlated, interaction between the parts. These correlated relationships create a differentiated structure that can, as a system, interact with other systems. The coordinated system manifests properties not carried or dictated by individual parts. The organized aspect of this form of complexity vis-a-vis to other systems than the subject system can be said to "emerge," without any "guiding hand".</p> <h2><span class="mw-headline" id="Sources_and_factors">Sources and factors</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complexity&action=edit&section=3" title="Edit section: Sources and factors">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <p>The source of disorganized complexity is the large number of parts in the system of interest, and the lack of correlation between elements in the system.</p> <p>Complexity of an object or system is a relative property. For instance, for many functions (problems), such a computational complexity as time of computation is smaller when multitape <a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a> are used than when Turing machines with one tape are used. <a href="/wiki/Random_Access_Machine" class="mw-redirect" title="Random Access Machine">Random Access Machines</a> allow one to even more decrease time complexity (Greenlaw and Hoover 1998: 226), while inductive Turing machines can decrease even the complexity class of a function, language or set (Burgin 2005). This shows that tools of activity can be an important factor of complexity.</p> <p>In several scientific fields, "complexity" has a precise meaning:</p> <li>In <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>, the <a href="/wiki/Computational_resource" title="Computational resource">amounts of resources</a> required for the execution of <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> is studied. The most popular types of computational complexity are the time complexity of a problem equal to the number of steps that it takes to solve an instance of the problem as a function of the <a href="/wiki/Problem_size" class="mw-redirect" title="Problem size">size of the input</a> (usually measured in bits), using the most efficient algorithm, and the space complexity of a problem equal to the volume of the <a href="/wiki/Computer_storage" class="mw-redirect" title="Computer storage">memory</a> used by the algorithm (e.g., cells of the tape) that it takes to solve an instance of the problem as a function of the size of the input (usually measured in bits), using the most efficient algorithm. This allows to classify computational problems by <a href="/wiki/Complexity_class" title="Complexity class">complexity class</a> (such as <a href="/wiki/P_(complexity)" title="P (complexity)">P</a>, <a href="/wiki/NP_(complexity)" title="NP (complexity)">NP,</a> etc.). An axiomatic approach to computational complexity was developed by <a href="/wiki/Manuel_Blum" title="Manuel Blum">Manuel Blum</a>. It allows one to deduce many properties of concrete computational complexity measures, such as time complexity or space complexity, from properties of axiomatically defined measures.</li> <li>In <a href="/wiki/Information_processing" title="Information processing">information processing</a>, complexity is a measure of the total number of <a href="/wiki/Property" title="Property">properties</a> transmitted by an object and detected by an <a href="/wiki/Observation" title="Observation">observer</a>. Such a collection of properties is often referred to as a <a href="/wiki/State_(computer_science)" title="State (computer science)">state</a>.</li> <li>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Krohn%E2%80%93Rhodes_complexity" class="mw-redirect" title="Krohn–Rhodes complexity">Krohn–Rhodes complexity</a> is an important topic in the study of finite <a href="/wiki/Semigroup" title="Semigroup">semigroups</a> and <a href="/wiki/Automata_theory" title="Automata theory">automata</a>.</li> <li>In <a href="/wiki/Software_engineering" title="Software engineering">software engineering</a>, <a href="/wiki/Programming_complexity" title="Programming complexity">programming complexity</a> is a measure of the interactions of the various elements of the software. This differs from the computational complexity described above in that it is a measure of the design of the software.</li> </ul> <ul> <ul> <li>The system has memory or includes <a href="/wiki/Feedback" title="Feedback">feedback</a>;</li> <li>The relations between the system and its environment are non-trivial or non-linear;</li> <li>The system is highly sensitive to initial conditions.</li> </li> <h2><span class="mw-headline" id="Study">Study</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complexity&action=edit&section=5" title="Edit section: Study">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <p>The use of the term complex is often confused with the term complicated. In today's systems, this is the difference between myriad connecting "stovepipes" and effective "integrated" solutions.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11">[11]</a></sup> This means that complex is the opposite of independent, while complicated is the opposite of simple.</p> <h2><span class="mw-headline" id="Topics">Topics</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complexity&action=edit&section=6" title="Edit section: Topics">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <p>The behavior of a complex system is often said to be due to emergence and <a href="/wiki/Self-organization" title="Self-organization">self-organization</a>. Chaos theory has investigated the sensitivity of systems to variations in initial conditions as one cause of complex behaviour.</p> <p>Recent developments around <a href="/wiki/Artificial_life" title="Artificial life">artificial life</a>, <a href="/wiki/Evolutionary_computation" title="Evolutionary computation">evolutionary computation</a> and <a href="/wiki/Genetic_algorithm" title="Genetic algorithm">genetic algorithms</a> have led to an increasing emphasis on complexity and <a href="/wiki/Complex_adaptive_systems" class="mw-redirect" title="Complex adaptive systems">complex adaptive systems</a>.</p> <p>In <a href="/wiki/Social_science" title="Social science">social science</a>, the study on the emergence of macro-properties from the micro-properties, also known as macro-micro view in <a href="/wiki/Sociology" title="Sociology">sociology</a>. The topic is commonly recognized as <a href="/wiki/Social_complexity" title="Social complexity">social complexity</a> that is often related to the use of computer simulation in social science, i.e.: <a href="/wiki/Computational_sociology" title="Computational sociology">computational sociology</a>.</p> <div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Complex_system" title="Complex system">Complex system</a></div> <h3><span class="mw-headline" id="Data">Data</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complexity&action=edit&section=11" title="Edit section: Data">edit</a><span class="mw-editsection-bracket">]</span></span></h3> <p>Complex strings are harder to compress. While intuition tells us that this may depend on the <a href="/wiki/Codec" title="Codec">codec</a> used to compress a string (a codec could be theoretically created in any arbitrary language, including one in which the very small command "X" could cause the computer to output a very complicated string like "18995316"), any two <a href="/wiki/Turing_completeness" title="Turing completeness">Turing-complete</a> languages can be implemented in each other, meaning that the length of two encodings in different languages will vary by at most the length of the "translation" language – which will end up being negligible for sufficiently large data strings.</p> <p><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Information entropy</a> is also sometimes used in information theory as indicative of complexity.</p> <p>The complexity measures broadly cover:</p> <li>the overlaps in feature values from differing classes.</li> <li>measures of geometry, topology, and density of <a href="/wiki/Manifold" title="Manifold">manifolds</a>. Instance hardness is another approach seeks to characterize the data complexity with the goal of determining how hard a data set is to classify correctly and is not limited to binary problems.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13">[13]</a></sup></li> <p>Instance hardness is a bottom-up approach that first seeks to identify instances that are likely to be misclassified (or, in other words, which instances are the most complex). The characteristics of the instances that are likely to be misclassified are then measured based on the output from a set of hardness measures. The hardness measures are based on several supervised learning techniques such as measuring the number of disagreeing neighbors or the likelihood of the assigned class label given the input features. The information provided by the complexity measures has been examined for use in <a href="/wiki/Meta_learning_(computer_science)" title="Meta learning (computer science)">meta learning</a> to determine for which data sets filtering (or removing suspected noisy instances from the training set) is the most beneficial<sup id="cite_ref-14" class="reference"><a href="#cite_note-14">[14]</a></sup> and could be expanded to other areas.</p> <p>A recent study based on molecular simulations and compliance constants describes <a href="/wiki/Molecular_recognition" title="Molecular recognition">molecular recognition</a> as a phenomenon of organisation.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15">[15]</a></sup> Even for small molecules like <a href="/wiki/Carbohydrates" class="mw-redirect" title="Carbohydrates">carbohydrates</a>, the recognition process can not be predicted or designed even assuming that each individual <a href="/wiki/Hydrogen_bond" title="Hydrogen bond">hydrogen bond</a>'s strength is exactly known.</p> <p>Computational complexity theory is the study of the complexity of problems – that is, the difficulty of <a href="/wiki/Problem_solving" title="Problem solving">solving</a> them. Problems can be classified by complexity class according to the time it takes for an algorithm – usually a computer program – to solve them as a function of the problem size. Some problems are difficult to solve, while others are easy. For example, some difficult problems need algorithms that take an exponential amount of time in terms of the size of the problem to solve. Take the <a href="/wiki/Travelling_salesman_problem" title="Travelling salesman problem">travelling salesman problem</a>, for example. It can be solved in time <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" > <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo stretchy="false">)</mo> </mrow> </semantics> <p>Even though a problem may be computationally solvable in principle, in actual practice it may not be that simple. These problems might require large amounts of time or an inordinate amount of space. <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Computational complexity</a> may be approached from many different aspects. Computational complexity can be investigated on the basis of time, memory or other resources used to solve the problem. Time and space are two of the most important and popular considerations when problems of complexity are analyzed.</p> <p>There is another form of complexity called <a href="/wiki/Model_of_hierarchical_complexity" title="Model of hierarchical complexity">hierarchical complexity</a>. It is orthogonal to the forms of complexity discussed so far, which are called horizontal complexity.</p> <p>Bejan and Lorente also argued that all the optimality (max,min) statements have limited ad-hoc applicability, and are unified under the Constructal law of design and evolution in nature.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17">[17]</a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18">[18]</a></sup></p> <div class="div-col columns column-width" style="-moz-column-width: 18em; -webkit-column-width: 18em; column-width: 18em;"> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Complex_systems" class="mw-redirect" title="Complex systems">Complex systems</a></li> <li><a href="/wiki/Constructal_law" title="Constructal law">Constructal law</a></li> <li><a href="/wiki/Digital_morphogenesis" title="Digital morphogenesis">Digital morphogenesis</a></li> <li><a href="/wiki/Emergence" title="Emergence">Emergence</a></li> <li><a href="/wiki/Game_complexity" title="Game complexity">Game complexity</a></li> <li><a href="/wiki/Interconnectedness" title="Interconnectedness">Interconnectedness</a></li> <li><a href="/wiki/Model_of_hierarchical_complexity" title="Model of hierarchical complexity">Model of hierarchical complexity</a></li> <li><a href="/wiki/Network_science" title="Network science">Network science</a></li> <li><a href="/wiki/Novelty_theory" class="mw-redirect" title="Novelty theory">Novelty theory</a></li> <li><a href="/wiki/Process_architecture" title="Process architecture">Process architecture</a></li> <li><a href="/wiki/Sociology_and_complexity_science" class="mw-redirect" title="Sociology and complexity science">Sociology and complexity science</a></li> <li><a href="/wiki/Thorngate%27s_postulate_of_commensurate_complexity" title="Thorngate's postulate of commensurate complexity">Thorngate's postulate of commensurate complexity</a></li> <li><a href="/wiki/Volatility,_uncertainty,_complexity_and_ambiguity" title="Volatility, uncertainty, complexity and ambiguity">Volatility, uncertainty, complexity and ambiguity</a></li> <li><a href="/wiki/Zero-Force_Evolutionary_Law" title="Zero-Force Evolutionary Law">Zero-Force Evolutionary Law</a></li> </div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation journal">Antunes, Ricardo; Gonzalez, Vicente (3 March 2015). <a rel="nofollow" class="external text" href="http://www.mdpi.com/2075-5309/5/1/209">"A Production Model for Construction: A Theoretical Framework"</a>. <i>Buildings</i>. <b>5</b> (1): 209–228. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="//doi.org/10.3390%2Fbuildings5010209">10.3390/buildings5010209</a><span class="reference-accessdate">. Retrieved <span class="nowrap">17 March</span> 2015</span>. <q>Vastly present in the literature, the word “complex†seems to stand for a supernatural force supposedly responsible for disturbances, a scary ghost haunting projects. With no absolute definition of what complexity means, the only consensus among researchers is that there is no agreement about the specific definition of complexity [66]. However, a characterization of what is complex is possible. A structure is complex; if composed of several interconnected pieces [67], with dynamic networks of interactions, and their relationships are not aggregations of the individual static entities [68].</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Buildings&rft.atitle=A+Production+Model+for+Construction%3A+A+Theoretical+Framework&rft.volume=5&rft.issue=1&rft.pages=209-228&rft.date=2015-03-03&rft_id=info%3Adoi%2F10.3390%2Fbuildings5010209&rft.aulast=Antunes&rft.aufirst=Ricardo&rft.au=Gonzalez%2C+Vicente&rft_id=http%3A%2F%2Fwww.mdpi.com%2F2075-5309%2F5%2F1%2F209&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplexity" class="Z3988"><span style="display:none;"> </span></span></span></li> <li id="cite_note-Neil_Johnson-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Neil_Johnson_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Neil_Johnson_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation book">Johnson, Neil F. (2009). "Chapter 1: Two's company, three is complexity". <a rel="nofollow" class="external text" href="http://www.uvm.edu/rsenr/nr385se/readings/complexity.pdf"><i>Simply complexity: A clear guide to complexity theory</i></a> <span style="font-size:85%;">(PDF)</span>. Oneworld Publications. p. 3. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:BookSources/978-1780740492" title="Special:BookSources/978-1780740492">978-1780740492</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1%3A+Two%27s+company%2C+three+is+complexity&rft.btitle=Simply+complexity%3A+A+clear+guide+to+complexity+theory&rft.pages=3&rft.pub=Oneworld+Publications&rft.date=2009&rft.isbn=978-1780740492&rft.aulast=Johnson&rft.aufirst=Neil+F.&rft_id=http%3A%2F%2Fwww.uvm.edu%2Frsenr%2Fnr385se%2Freadings%2Fcomplexity.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplexity" class="Z3988"><span style="display:none;"> </span></span></span></li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation book">Johnson, Steven (2001). <i>Emergence: the connected lives of ants, brains, cities, and software</i>. New York: Scribner. p. 46. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:BookSources/0-684-86875-X" title="Special:BookSources/0-684-86875-X">0-684-86875-X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Emergence%3A+the+connected+lives+of+ants%2C+brains%2C+cities%2C+and+software&rft.place=New+York&rft.pages=46&rft.pub=Scribner&rft.date=2001&rft.isbn=0-684-86875-X&rft.aulast=Johnson&rft.aufirst=Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplexity" class="Z3988"><span style="display:none;"> </span></span></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"><cite class="citation book">Jacobs, Jane (1961). <i>The Death and Life of Great American Cities</i>. New York: Random House.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Death+and+Life+of+Great+American+Cities&rft.place=New+York&rft.pub=Random+House&rft.date=1961&rft.aulast=Jacobs&rft.aufirst=Jane&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplexity" class="Z3988"><span style="display:none;"> </span></span></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">Mariusz Stanowski (2011) Abstract Complexity Definition, Complicity 2, p.78-83 <a rel="nofollow" class="external autonumber" href="https://ejournals.library.ualberta.ca/index.php/complicity/article/view/11156">[1]</a></span></li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Ho, T.K.; Basu, M. (2002). "<a rel="nofollow" class="external text" href="http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=990132&tag=1">Complexity Measures of Supervised Classification Problems</a>". IEEE Transactions on Pattern Analysis and Machine Intelligence 24 (3), pp 289-300.</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">Saez, J.; Luengo, J.; Herrera, F. (2013). "<a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science/article/pii/S0031320312003251">Predicting Noise Filtering Efficacy with Data Complexity Measures for Nearest Neighbor Classification</a>". Pattern Recognition 46 (1) pp 355-364.</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 autonumber" href="http://www.constructal.org/en/art/Phil.%20Trans.%20R.%20Soc.%20B%20%282010%29%20365,%201335%961347.pdf">[2]</a> Bejan A., Lorente S., The Constructal Law of Design and Evolution in Nature. <i>Philosophical Transactions of the Royal Society B, Biological Science</i>, Vol. 365, 2010, pp. 1335–1347.</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">Kim S., Lorente S., Bejan A., Milter W., Morse J. (2008) The Emergence of Vascular Design in Three Dimensions, <i>Journal of Applied Physics</i>, Vol. 103, 123511.</span></li> </div> <h2><span class="mw-headline" id="Further_reading">Further reading</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complexity&action=edit&section=16" title="Edit section: Further reading">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <ul> <li><cite class="citation book">Waldrop, M. Mitchell (1992). <i>Complexity: The Emerging Science at the Edge of Order and Chaos</i>. New York: Simon & Schuster. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:BookSources/978-0-671-76789-1" title="Special:BookSources/978-0-671-76789-1">978-0-671-76789-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complexity%3A+The+Emerging+Science+at+the+Edge+of+Order+and+Chaos&rft.place=New+York&rft.pub=Simon+%26+Schuster&rft.date=1992&rft.isbn=978-0-671-76789-1&rft.aulast=Waldrop&rft.aufirst=M.+Mitchell&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplexity" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Solé, R. V.; B. C. Goodwin (2002). <i>Signs of Life: How Complexity Pervades Biology</i>. Basic Books. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-01928-1" title="Special:BookSources/978-0-465-01928-1">978-0-465-01928-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Signs+of+Life%3A+How+Complexity+Pervades+Biology&rft.pub=Basic+Books&rft.date=2002&rft.isbn=978-0-465-01928-1&rft.aulast=Sol%C3%A9&rft.aufirst=R.+V.&rft.au=B.+C.+Goodwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplexity" class="Z3988"><span style="display:none;"> </span></span></li> <li>Burgin, M. (1982) Generalized Kolmogorov complexity and duality in theory of computations, Notices of the Russian Academy of Sciences, v.25, No. 3, pp. 19–23</li> <li>Mitchell, M. (2009). Complexity: A Guided Tour. Oxford University Press, Oxford, UK.</li> </ul> <h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complexity&action=edit&section=17" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span></h2> <tr> <td class="mbox-text plainlist">Wikiquote has quotations related to: <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Complexity" class="extiw" title="q:Special:Search/Complexity">Complexity</a></b></i></td> </table> <tr> <td class="mbox-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/complexity" class="extiw" title="wiktionary:Special:Search/complexity">complexity</a></b></i> in Wiktionary, the free dictionary.</td> </table> <li><a rel="nofollow" class="external text" href="http://bactra.org/notebooks/complexity-measures.html">Complexity Measures</a> – an article about the abundance of not-that-useful complexity measures.</li> <li><a rel="nofollow" class="external text" href="http://www.santafe.edu/">Santa Fe Institute</a> focusing on the study of complexity science: <a rel="nofollow" class="external text" href="http://www.santafe.edu/research/videos/catalog/">Lecture Videos</a></li> </ul> <table class="nowraplinks collapsible collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"> <th scope="col" class="navbox-title" colspan="3"> <ul> <li class="nv-talk"><a href="/wiki/Template_talk:Chaos_theory" title="Template talk:Chaos theory"><abbr title="Discuss this template" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none;">t</abbr></a></li> </ul> <div id="Chaos_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></div> </tr> <th scope="row" class="navbox-group" style="width:1%">Chaos theory</th> <div style="padding:0em 0.25em"> <li><a href="/wiki/Anosov_diffeomorphism" title="Anosov diffeomorphism">Anosov diffeomorphism</a></li> <li><a href="/wiki/Butterfly_effect" title="Butterfly effect">Butterfly effect</a></li> <li><a class="mw-selflink selflink">Complexity</a></li> <li><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical system</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Synchronization_of_chaos" title="Synchronization of chaos">Synchronization of chaos</a></li> </ul> </td> <div><a href="/wiki/File:C%C3%B4ne_textileII.png" class="image" title="Conus textile shell"><img alt="Conus textile shell" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/C%C3%B4ne_textileII.png/60px-C%C3%B4ne_textileII.png" width="60" height="108" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/C%C3%B4ne_textileII.png/90px-C%C3%B4ne_textileII.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/C%C3%B4ne_textileII.png/120px-C%C3%B4ne_textileII.png 2x" data-file-width="2344" data-file-height="4211" /></a> <a href="/wiki/File:Circle_map_bifurcation.jpeg" class="image" title="Circle map with black Arnold tongues"><img alt="Circle map with black Arnold tongues" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Circle_map_bifurcation.jpeg/60px-Circle_map_bifurcation.jpeg" width="60" height="120" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Circle_map_bifurcation.jpeg/90px-Circle_map_bifurcation.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Circle_map_bifurcation.jpeg/120px-Circle_map_bifurcation.jpeg 2x" data-file-width="450" data-file-height="900" /></a></div> </tr> <th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Chaos_theory" title="Chaos theory">Chaotic maps</a><br /> <td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"> <ul> <li><a href="/wiki/Arnold%27s_cat_map" title="Arnold's cat map">Arnold's cat map</a></li> <li><a href="/wiki/Complex_quadratic_polynomial" title="Complex quadratic polynomial">Complex quadratic map</a></li> <li><a href="/wiki/Coupled_map_lattice" title="Coupled map lattice">Coupled map lattice</a></li> <li><a href="/wiki/Double_scroll_attractor" class="mw-redirect" title="Double scroll attractor">Double scroll attractor</a></li> <li><a href="/wiki/Duffing_map" title="Duffing map">Duffing map</a></li> <li><a href="/wiki/Dynamical_billiards" title="Dynamical billiards">Dynamical billiards</a> <li><a href="/wiki/Outer_billiard" title="Outer billiard">outer</a></li> </li> <li><a href="/wiki/Gauss_iterated_map" title="Gauss iterated map">Gauss map</a></li> <li><a href="/wiki/H%C3%A9non_map" title="Hénon map">Hénon map</a></li> <li><a href="/wiki/Ikeda_map" title="Ikeda map">Ikeda map</a></li> <li><a href="/wiki/Kaplan%E2%80%93Yorke_map" title="Kaplan–Yorke map">Kaplan–Yorke map</a></li> <li><a href="/wiki/Lorenz_system" title="Lorenz system">Lorenz system</a></li> <li><a href="/wiki/Rabinovich%E2%80%93Fabrikant_equations" title="Rabinovich–Fabrikant equations">Rabinovich–Fabrikant equations</a></li> <li><a href="/wiki/Standard_map" title="Standard map">Standard map</a></li> <li><a href="/wiki/Tent_map" title="Tent map">Tent map</a></li> <li><a href="/wiki/Van_der_Pol_oscillator" title="Van der Pol oscillator">Van der Pol oscillator</a></li> </ul> </td> <tr> <td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"> <ul> <li><a href="/wiki/Chua%27s_circuit" title="Chua's circuit">Chua's circuit</a></li> <li><a href="/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam%E2%80%93Tsingou_problem" title="Fermi–Pasta–Ulam–Tsingou problem">FPUT problem</a></li> </ul> </td> <tr> <td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"> <ul> <li><a href="/wiki/Mary_Cartwright" title="Mary Cartwright">Mary Cartwright</a></li> <li><a href="/wiki/Mitchell_Feigenbaum" title="Mitchell Feigenbaum">Mitchell Feigenbaum</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Martin Gutzwiller</a></li> <li><a href="/wiki/Michel_H%C3%A9non" title="Michel Hénon">Michel Hénon</a></li> <li><a href="/wiki/Sofia_Kovalevskaya" title="Sofia Kovalevskaya">Sofia Kovalevskaya</a></li> <li><a href="/wiki/Edward_Norton_Lorenz" title="Edward Norton Lorenz">Edward Norton Lorenz</a></li> <li><a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoît Mandelbrot</a></li> <li><a href="/wiki/Edward_Ott" title="Edward Ott">Edward Ott</a></li> <li><a href="/wiki/Mary_Rees" title="Mary Rees">Mary Rees</a></li> <li><a href="/wiki/David_Ruelle" title="David Ruelle">David Ruelle</a></li> <li><a href="/wiki/Oleksandr_Mykolayovych_Sharkovsky" title="Oleksandr Mykolayovych Sharkovsky">Oleksandr Mykolayovych Sharkovsky</a></li> <li><a href="/wiki/Floris_Takens" title="Floris Takens">Floris Takens</a></li> <li><a href="/wiki/Mary_Tsingou" title="Mary Tsingou">Mary Tsingou</a></li> <li><a href="/wiki/James_A._Yorke" title="James A. 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href="https://fi.wikipedia.org/wiki/Kompleksisuus" title="Kompleksisuus – Finnish" lang="fi" hreflang="fi" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv"><a href="https://sv.wikipedia.org/wiki/Komplexitet" title="Komplexitet – Swedish" lang="sv" hreflang="sv" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr"><a href="https://tr.wikipedia.org/wiki/Karma%C5%9F%C4%B1kl%C4%B1k" title="Karmaşıklık – Turkish" lang="tr" hreflang="tr" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-zh"><a href="https://zh.wikipedia.org/wiki/%E5%A4%8D%E6%9D%82" title="å¤æ‚ – Chinese" lang="zh" hreflang="zh" class="interlanguage-link-target">ä¸æ–‡</a></li> </ul> </div> </div> <ul id="footer-info"> <li id="footer-info-copyright">Text is available under the <a rel="license" 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