Complexity: Difference between revisions

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{{Short description|~~Properties~~ of systems that ~~cannot be simply described or modeled~~}}

{{About||the use in computer science|Computational complexity|other uses|Complexity (disambiguation)}}

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Some definitions relate to the algorithmic basis for the expression of a complex phenomenon or model or mathematical expression, as later set out herein.

== Disorganized ~~vs.~~ organized ==

== Disorganized versus organized ==

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.

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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.

In the case of [[Self-organization|self-organizing]] [[living systems]], usefully organized complexity comes from beneficially mutated organisms being selected to survive by their environment for their differential [[Reproduction|reproductive ability]] or at least success over inanimate [[matter]] or less organized complex organisms. See e.g. [[Robert Ulanowicz]]'s treatment of [[Ecosystem|ecosystems]].<ref>Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997</ref>

In the case of [[Self-organization|self-organizing]] [[living systems]], usefully organized complexity comes from beneficially mutated organisms being selected to survive by their environment for their differential [[Reproduction|reproductive ability]] or at least success over inanimate [[matter]] or less organized complex organisms. See e.g. [[Robert Ulanowicz]]'s treatment of [[Ecosystem|ecosystems]].<ref>Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997</ref> Interestingly, a strong correlation is observed between this organismal complexity and the mean length of genes. <ref name="Muro et al. 2025">{{cite journal |last1=Muro |first1=Enrique M. |last2=Ballesteros |first2=Fernando J. |last3=Luque |first3=Bartolo |last4=Bascompte |first4=Jordi |title=The emergence of eukaryotes as an evolutionary algorithmic phase transition |journal=PNAS |volume=122 |issue=13 |year=2025 |doi=10.1073/pnas.2422968122 |article-number=e2422968122 |pmid=40146859 |doi-access=free |pmc=12002324 |bibcode=2025PNAS..12222968M }}</ref>

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 [[Turing machine]]s are used than when Turing machines with one tape are used. [[Random Access Machine]]s 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.

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* In [[sociology]], [[social complexity]] is a [[conceptual framework]] used in the [[analysis]] of society.

* In [[combinatorial game theory]], measures of [[game complexity]] involve understanding game positions, possible outcomes, and computation required for various game scenarios.

* In [[binary number]], Abstract Complexity Definition (ACD) formalizes complexity as the complexity of a binary structure, expressed by the formula: C = N² / n, where N is the number of detectable regularities (contrasts), and n is the number of basic elements (0 and 1). These regularities are identified with contrasts—tensions arising from interactions of common and differentiating features—and with structural information (as opposed to Shannon’s telecommunication information).The formula incorporates two factors:• the N/n ratio, representing information compression/density, and the number of regularities N, introduced to account for the difficulty of compressing longer structures. This definition applies both analogically and directly wherever a system can be expressed in binary form, e.g., in music. Differences from classical structural complexity definitions: Traditional definitions focus on the number of elements and relations without specifying the nature of these relations. In ACD, relations are explicitly defined as tensions (contrasts) resulting from interactions of common and differentiating features of system elements, with a defined dynamics. This makes the definition constructive and operational—allowing formal calculation of complexity.Intuitive criterion of complexity:Complexity is the number of distinguishable elements and the number of connections between them. In relation to contrast: differentiating features correspond to distinguishable elements, common features correspond to connections between those elements. The more such features exist and the stronger their interactions, the greater the contrast, and thus the higher the complexity.<ref>{{cite book |title=Theory and Practice of Contrast Integrating Science, Art and Philosophy |last=Stanowski| first = Mariusz|publisher=Taylor&Francis|language=en|date=2021}}</ref><ref>{{Cite journal |last=Mariusz |first=Stanowski| date=2011 |title= Abstract Complexity Definition|journal=Complicity: An International Journal of Complexity and Education| language=en|volume=8(2011)|issue=2}}</ref>

Other fields introduce less precisely defined notions of complexity:

@@ Line 1: / Line 1: @@
-{{Short description|Properties of systems that cannot be simply described or modeled}}
+{{Short description|Feature of systems that defy description}}
 {{About||the use in computer science|Computational complexity|other uses|Complexity (disambiguation)}}
@@ Line 75: / Line 75: @@
 Some definitions relate to the algorithmic basis for the expression of a complex phenomenon or model or mathematical expression, as later set out herein.
-== Disorganized vs. organized ==
+== Disorganized versus organized ==
 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.
@@ Line 102: / Line 102: @@
 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.
-In the case of [[Self-organization|self-organizing]] [[living systems]], usefully organized complexity comes from beneficially mutated organisms being selected to survive by their environment for their differential [[Reproduction|reproductive ability]] or at least success over inanimate [[matter]] or less organized complex organisms. See e.g. [[Robert Ulanowicz]]'s treatment of [[Ecosystem|ecosystems]].<ref>Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997</ref>
+In the case of [[Self-organization|self-organizing]] [[living systems]], usefully organized complexity comes from beneficially mutated organisms being selected to survive by their environment for their differential [[Reproduction|reproductive ability]] or at least success over inanimate [[matter]] or less organized complex organisms. See e.g. [[Robert Ulanowicz]]'s treatment of [[Ecosystem|ecosystems]].<ref>Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997</ref> Interestingly, a strong correlation is observed between this organismal complexity and the mean length of genes. <ref name="Muro et al. 2025">{{cite journal |last1=Muro |first1=Enrique M. |last2=Ballesteros |first2=Fernando J. |last3=Luque |first3=Bartolo |last4=Bascompte |first4=Jordi |title=The emergence of eukaryotes as an evolutionary algorithmic phase transition |journal=PNAS |volume=122 |issue=13 |year=2025 |doi=10.1073/pnas.2422968122 |article-number=e2422968122 |pmid=40146859 |doi-access=free |pmc=12002324 |bibcode=2025PNAS..12222968M }}</ref>
 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 [[Turing machine]]s are used than when Turing machines with one tape are used. [[Random Access Machine]]s 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.
@@ Line 128: / Line 128: @@
 * In [[sociology]], [[social complexity]] is a [[conceptual framework]] used in the [[analysis]] of society.
 * In [[combinatorial game theory]], measures of [[game complexity]] involve understanding game positions, possible outcomes, and computation required for various game scenarios.
+* In [[binary number]], Abstract Complexity Definition (ACD) formalizes complexity as the complexity of a binary structure, expressed by the formula: C = N² / n, where N is the number of detectable regularities (contrasts), and n is the number of basic elements (0 and 1). These regularities are identified with contrasts—tensions arising from interactions of common and differentiating features—and with structural information (as opposed to Shannon’s telecommunication information).The formula incorporates two factors:• the N/n ratio, representing information compression/density, and the number of regularities N, introduced to account for the difficulty of compressing longer structures. This definition applies both analogically and directly wherever a system can be expressed in binary form, e.g., in music. Differences from classical structural complexity definitions: Traditional definitions focus on the number of elements and relations without specifying the nature of these relations. In ACD, relations are explicitly defined as tensions (contrasts) resulting from interactions of common and differentiating features of system elements, with a defined dynamics. This makes the definition constructive and operational—allowing formal calculation of complexity.Intuitive criterion of complexity:Complexity is the number of distinguishable elements and the number of connections between them. In relation to contrast: differentiating features correspond to distinguishable elements, common features correspond to connections between those elements. The more such features exist and the stronger their interactions, the greater the contrast, and thus the higher the complexity.<ref>{{cite book |title=Theory and Practice of Contrast Integrating Science, Art and Philosophy |last=Stanowski| first = Mariusz|publisher=Taylor&Francis|language=en|date=2021}}</ref><ref>{{Cite journal |last=Mariusz |first=Stanowski| date=2011 |title= Abstract Complexity Definition|journal=Complicity: An International Journal of Complexity and Education| language=en|volume=8(2011)|issue=2}}</ref>
 Other fields introduce less precisely defined notions of complexity:

Complexity: Difference between revisions

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