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解决复杂性问题的一个问题是将随机集合中现存的大量关系变量与系统中有时较大但较小的元素之间的关系(与其他独立元素的相关性有关)同时减少了元素独立性的变量，并创造了更加统一或相关的关系或相互作用的可区分的制度之间的直观概念区分形式化。
解决复杂性问题的一个问题是将随机集合中现存的大量关系变量与系统中有时较大但较小的元素之间的关系(与其他独立元素的相关性有关)同时减少了元素独立性的变量，并创造了更加统一或相关的关系或相互作用的可区分的制度之间的直观概念区分形式化。
Weaver perceived and addressed this problem, in at least a preliminary way, in drawing a distinction between "disorganized complexity" and "organized complexity".
Weaver perceived and addressed this problem, in at least a preliminary way, in drawing a distinction between "disorganized complexity" and "organized complexity".
在区分“无组织的复杂性”和“有组织的复杂性”时，编织者至少以一种初步的方式察觉并解决了这个问题。
在区分“无组织的复杂性”和“有组织的复杂性”时，编织者至少以一种初步的方式察觉并解决了这个问题。
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.
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.
在韦弗看来，无组织的复杂性是由于特定系统具有非常多的部件，比如数百万个部件，或者更多。虽然在“无组织复杂性”的情况下，各部分之间的相互作用可以看作是很大程度上的随机性，但是系统作为一个整体的性质可以通过使用概率和统计方法来理解。
在韦弗看来，无组织的复杂性是由于特定系统具有非常多的部件，比如数百万个部件，或者更多。虽然在“无组织复杂性”的情况下，各部分之间的相互作用可以看作是很大程度上的随机性，但是系统作为一个整体的性质可以通过使用概率和统计方法来理解。
A prime example of disorganized complexity is a gas in a container, with the gas molecules as the parts. Some would suggest that a system of disorganized complexity may be compared with the (relative) simplicity of planetary orbits – the latter can be predicted by applying Newton's laws of motion. Of course, most real-world systems, including planetary orbits, eventually become theoretically unpredictable even using Newtonian dynamics; as discovered by modern chaos theory.
A prime example of disorganized complexity is a gas in a container, with the gas molecules as the parts. Some would suggest that a system of disorganized complexity may be compared with the (relative) simplicity of planetary orbits – the latter can be predicted by applying Newton's laws of motion. Of course, most real-world systems, including planetary orbits, eventually become theoretically unpredictable even using Newtonian dynamics; as discovered by modern chaos theory.
无组织复杂性的一个典型例子是一个容器中的气体，以气体分子为部件。有些人认为，一个无组织的复杂系统可以与行星轨道的(相对)简单性相比较——后者可以通过应用牛顿运动定律来预测。当然，大多数真实世界的系统，包括行星轨道，最终在理论上变得不可预测，即使使用牛顿动力学; 正如现代混沌理论所发现的那样。
无组织复杂性的一个典型例子是一个容器中的气体，以气体分子为部件。有些人认为，一个无组织的复杂系统可以与行星轨道的(相对)简单性相比较——后者可以通过应用牛顿运动定律来预测。当然，大多数真实世界的系统，包括行星轨道，最终在理论上变得不可预测，即使使用牛顿动力学; 正如现代混沌理论所发现的那样。
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".
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".
在 Weaver 看来，有组织的复杂性仅仅存在于各部分之间的非随机或相关的交互中。这些相互关联的关系创建了一个可以作为一个系统与其他系统交互的差异化结构。协调系统显示的属性不是由单个部分承载或支配的。这种形式的复杂性相对于主体系统以外的其他系统的有组织的方面可以说是“浮现” ，没有任何“指导手”。
在 Weaver 看来，有组织的复杂性仅仅存在于各部分之间的非随机或相关的交互中。这些相互关联的关系创建了一个可以作为一个系统与其他系统交互的差异化结构。协调系统显示的属性不是由单个部分承载或支配的。这种形式的复杂性相对于主体系统以外的其他系统的有组织的方面可以说是“浮现” ，没有任何“指导手”。
The number of parts does not have to be very large for a particular system to have emergent properties. A system of organized complexity may be understood in its properties (behavior among the properties) through modeling and simulation, particularly modeling and simulation with computers. An example of organized complexity is a city neighborhood as a living mechanism, with the neighborhood people among the system's parts.
The number of parts does not have to be very large for a particular system to have emergent properties. A system of organized complexity may be understood in its properties (behavior among the properties) through modeling and simulation, particularly modeling and simulation with computers. An example of organized complexity is a city neighborhood as a living mechanism, with the neighborhood people among the system's parts.
对于一个具有涌现特性的特定系统来说，部件的数量不一定非常大。一个有组织的复杂系统可以从它的属性(属性之间的行为)来理解，通过建模与模拟，特别是计算机的建模与模拟。有组织的复杂性的一个例子是一个城市邻里作为一个生活机制，与邻里的人在系统的部分。
对于一个具有涌现特性的特定系统来说，部件的数量不一定非常大。一个有组织的复杂系统可以从它的属性(属性之间的行为)来理解，通过建模与模拟，特别是计算机的建模与模拟。有组织的复杂性的一个例子是一个城市邻里作为一个生活机制，与邻里的人在系统的部分。
There are generally rules which can be invoked to explain the origin of complexity in a given system.
There are generally rules which can be invoked to explain the origin of complexity in a given system.
就自我组织的生命系统而言，有效组织的复杂性来自于有益的突变生物体，它们被选择在其环境中生存，因为它们具有不同的生殖能力，或者至少在无生命物质或组织较少的复杂生物体上取得成功。参见。罗伯特·尤兰维奇对生态系统的处理。
就自我组织的生命系统而言，有效组织的复杂性来自于有益的突变生物体，它们被选择在其环境中生存，因为它们具有不同的生殖能力，或者至少在无生命物质或组织较少的复杂生物体上取得成功。参见。罗伯特·尤兰维奇对生态系统的处理。
[[Warren Weaver]] posited in 1948 two forms of complexity: disorganized complexity, and organized complexity.<ref name=Weaver>{{Cite journal
[[Warren Weaver]] posited in 1948 two forms of complexity: disorganized complexity, and organized complexity.
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 machines are used than when Turing machines with one tape are used. Random Access Machines 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.
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 machines are used than when Turing machines with one tape are used. Random Access Machines 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.
对象或系统的复杂性是一个相对的属性。例如，对于许多函数(问题)来说，使用多带图灵机比使用单带图灵机的计算复杂度要小。随机存取机器允许一个人甚至更多地降低时间复杂度(Greenlaw 和 Hoover 1998:226) ，而归纳图灵机甚至可以降低函数、语言或集合的复杂度等级(Burgin 2005)。这表明活动工具可以是复杂性的一个重要因素。
对象或系统的复杂性是一个相对的属性。例如，对于许多函数(问题)来说，使用多带图灵机比使用单带图灵机的计算复杂度要小。随机存取机器允许一个人甚至更多地降低时间复杂度(Greenlaw 和 Hoover 1998:226) ，而归纳图灵机甚至可以降低函数、语言或集合的复杂度等级(Burgin 2005)。这表明活动工具可以是复杂性的一个重要因素。
In several scientific fields, "complexity" has a precise meaning:
In several scientific fields, "complexity" has a precise meaning:
在一些科学领域，“复杂性”有着精确的含义:
在一些科学领域，“复杂性”有着精确的含义:
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".<ref name=Weaver/> Weaver's 1948 paper has influenced subsequent thinking about complexity.<ref>{{cite book
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".<ref name=Weaver/> Weaver's 1948 paper has influenced subsequent thinking about complexity.<ref>{{cite book
Other fields introduce less precisely defined notions of complexity:
Other fields introduce less precisely defined notions of complexity:
其他领域则引入了定义不那么精确的复杂性概念:
其他领域则引入了定义不那么精确的复杂性概念:
While this has led some fields to come up with specific definitions of complexity, there is a more recent movement to regroup observations from different fields to study complexity in itself, whether it appears in anthills, human brains, or stock markets, social systems. One such interdisciplinary group of fields is relational order theories.
While this has led some fields to come up with specific definitions of complexity, there is a more recent movement to regroup observations from different fields to study complexity in itself, whether it appears in anthills, human brains, or stock markets, social systems. One such interdisciplinary group of fields is relational order theories.
虽然这已经导致一些领域提出了复杂性的具体定义，但是最近有一种运动重新组合来自不同领域的观察结果来研究复杂性本身，无论它是出现在蚁丘、人类大脑，还是股票市场、社会系统。其中一个跨学科的领域就是关系秩序理论。
虽然这已经导致一些领域提出了复杂性的具体定义，但是最近有一种运动重新组合来自不同领域的观察结果来研究复杂性本身，无论它是出现在蚁丘、人类大脑，还是股票市场、社会系统。其中一个跨学科的领域就是关系秩序理论。
复杂的字符串更难压缩。直觉告诉我们，这可能取决于用于压缩字符串的编解码器(编解码器理论上可以在任何语言中创建，包括一个非常小的命令“ x”可以导致计算机输出非常复杂的字符串，比如“18995316”) ，但是任何两种图灵完整语言都可以在彼此中实现，这意味着不同语言中两种编码器的长度最多只会随着“翻译”语言的长度而变化——这对于足够大数据字符串来说可以忽略不计。
复杂的字符串更难压缩。直觉告诉我们，这可能取决于用于压缩字符串的编解码器(编解码器理论上可以在任何语言中创建，包括一个非常小的命令“ x”可以导致计算机输出非常复杂的字符串，比如“18995316”) ，但是任何两种图灵完整语言都可以在彼此中实现，这意味着不同语言中两种编码器的长度最多只会随着“翻译”语言的长度而变化——这对于足够大数据字符串来说可以忽略不计。
The number of parts does not have to be very large for a particular system to have emergent properties. A system of organized complexity may be understood in its properties (behavior among the properties) through [[model (abstract)|modeling]] and [[simulation]], particularly [[computer simulation|modeling and simulation with computers]]. An example of organized complexity is a city neighborhood as a living mechanism, with the neighborhood people among the system's parts.<ref>{{cite book
The number of parts does not have to be very large for a particular system to have emergent properties. A system of organized complexity may be understood in its properties (behavior among the properties) through [[model (abstract)|modeling]] and [[simulation]], particularly [[computer simulation|modeling and simulation with computers]]. An example of organized complexity is a city neighborhood as a living mechanism, with the neighborhood people among the system's parts.
These algorithmic measures of complexity tend to assign high values to random noise. However, those studying complex systems would not consider randomness as complexity.
These algorithmic measures of complexity tend to assign high values to random noise. However, those studying complex systems would not consider randomness as complexity.
这些复杂度的算法测量倾向于给随机噪声赋予较高的值。然而，那些研究复杂系统的人并不认为随机性就是复杂性。
这些复杂度的算法测量倾向于给随机噪声赋予较高的值。然而，那些研究复杂系统的人并不认为随机性就是复杂性。
Information entropy is also sometimes used in information theory as indicative of complexity, but entropy is also high for randomness. Information fluctuation complexity, fluctuations of information about entropy, does not consider randomness to be complex and has been useful in many applications.
Information entropy is also sometimes used in information theory as indicative of complexity, but entropy is also high for randomness. Information fluctuation complexity, fluctuations of information about entropy, does not consider randomness to be complex and has been useful in many applications.
信息论中有时也会用熵表示复杂性，但是熵的随机性也很高。信息波动的复杂性，熵信息的波动性，不考虑随机性的复杂性，已经在许多应用中得到应用。
信息论中有时也会用熵表示复杂性，但是熵的随机性也很高。信息波动的复杂性，熵信息的波动性，不考虑随机性的复杂性，已经在许多应用中得到应用。
Recent work in machine learning has examined the complexity of the data as it affects the performance of supervised classification algorithms. Ho and Basu present a set of complexity measures for binary classification problems.
Recent work in machine learning has examined the complexity of the data as it affects the performance of supervised classification algorithms. Ho and Basu present a set of complexity measures for binary classification problems.
最近机器学习的工作已经检查了数据的复杂性，因为它影响了监督分类算法的性能。Ho 和 Basu 为二分类问题提出了一套复杂度量方法。
最近机器学习的工作已经检查了数据的复杂性，因为它影响了监督分类算法的性能。Ho 和 Basu 为二分类问题提出了一套复杂度量方法。
The complexity measures broadly cover:
The complexity measures broadly cover:
复杂性指标大致涵盖:
复杂性指标大致涵盖:
== Topics ==
== Topics ==
=== Behaviour ===
=== Behaviour ===
Recent developments around [[artificial life]], [[evolutionary computation]] and [[genetic algorithm]]s have led to an increasing emphasis on complexity and [[complex adaptive systems]].
Recent developments around [[artificial life]], [[evolutionary computation]] and [[genetic algorithm]]s have led to an increasing emphasis on complexity and [[complex adaptive systems]].
=== Simulations ===
=== Simulations ===
In [[social science]], the study on the emergence of macro-properties from the micro-properties, also known as macro-micro view in [[sociology]]. The topic is commonly recognized as [[social complexity]] that is often related to the use of computer simulation in social science, i.e.: [[computational sociology]].
In [[social science]], the study on the emergence of macro-properties from the micro-properties, also known as macro-micro view in [[sociology]]. The topic is commonly recognized as [[social complexity]] that is often related to the use of computer simulation in social science, i.e.: [[computational sociology]].
生物科学理论
生物科学理论
=== Systems ===
=== Systems ===
[[Systems theory]] has long been concerned with the study of [[complex system]]s (in recent times, ''complexity theory'' and ''complex systems'' have also been used as names of the field). These systems are present in the research of a variety disciplines, including [[biology]], [[economics]], social studies and [[technology]]. Recently, complexity has become a natural domain of interest of real world socio-cognitive systems and emerging [[systemics]] research. Complex systems tend to be high-[[dimension]]al, [[non-linearity|non-linear]], and difficult to model. In specific circumstances, they may exhibit low-dimensional behaviour.
[[Systems theory]] has long been concerned with the study of [[complex system]]s (in recent times, ''complexity theory'' and ''complex systems'' have also been used as names of the field). These systems are present in the research of a variety disciplines, including [[biology]], [[economics]], social studies and [[technology]]. Recently, complexity has become a natural domain of interest of real world socio-cognitive systems and emerging [[systemics]] research. Complex systems tend to be high-[[dimension]]al, [[non-linearity|non-linear]], and difficult to model. In specific circumstances, they may exhibit low-dimensional behaviour.
=== Data ===
=== Data ===
In [[information theory]], algorithmic information theory is concerned with the complexity of strings of data.
In [[information theory]], algorithmic information theory is concerned with the complexity of strings of data.
Complex strings are harder to compress. While intuition tells us that this may depend on the [[codec]] 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 [[Turing completeness|Turing-complete]] 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.
Complex strings are harder to compress. While intuition tells us that this may depend on the [[codec]] 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 [[Turing completeness|Turing-complete]] 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.
These algorithmic measures of complexity tend to assign high values to [[signal noise|random noise]]. However, those studying complex systems would not consider randomness as complexity{{who|date=October 2013}}.
These algorithmic measures of complexity tend to assign high values to [[signal noise|random noise]]. However, those studying complex systems would not consider randomness as complexity{{who|date=October 2013}}.
[[Information entropy]] is also sometimes used in information theory as indicative of complexity, but entropy is also high for randomness. [[Information fluctuation complexity]], fluctuations of information about entropy, does not consider randomness to be complex and has been useful in many applications.
[[Information entropy]] is also sometimes used in information theory as indicative of complexity, but entropy is also high for randomness. [[Information fluctuation complexity]], fluctuations of information about entropy, does not consider randomness to be complex and has been useful in many applications.
Recent work in [[machine learning]] has examined the complexity of the data as it affects the performance of [[Supervised learning|supervised]] classification algorithms. Ho and Basu present a set of [[Computational complexity theory|complexity measures]] for [[binary classification]] problems.<ref>Ho, T.K.; Basu, M. (2002). "[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=990132&tag=1 Complexity Measures of Supervised Classification Problems]". IEEE Transactions on Pattern Analysis and Machine Intelligence 24 (3), pp 289–300.</ref>
Recent work in [[machine learning]] has examined the complexity of the data as it affects the performance of [[Supervised learning|supervised]] classification algorithms. Ho and Basu present a set of [[Computational complexity theory|complexity measures]] for [[binary classification]] problems.<ref>Ho, T.K.; Basu, M. (2002). "[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=990132&tag=1 Complexity Measures of Supervised Classification Problems]". IEEE Transactions on Pattern Analysis and Machine Intelligence 24 (3), pp 289–300.</ref>
The complexity measures broadly cover:
The complexity measures broadly cover:
* the overlaps in feature values from differing classes.
* the overlaps in feature values from differing classes.
* the separability of the classes.
* the separability of the classes.
* measures of geometry, topology, and density of [[manifold]]s. 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.<ref>Smith, M.R.; Martinez, T.; Giraud-Carrier, C. (2014). "[https://link.springer.com/article/10.1007%2Fs10994-013-5422-z An Instance Level Analysis of Data Complexity]". Machine Learning, 95(2): 225–256.</ref>
* measures of geometry, topology, and density of [[manifold]]s. 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.<ref>Smith, M.R.; Martinez, T.; Giraud-Carrier, C. (2014). "[https://link.springer.com/article/10.1007%2Fs10994-013-5422-z An Instance Level Analysis of Data Complexity]". Machine Learning, 95(2): 225–256.</ref>
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 [[Meta learning (computer science)|meta learning]] to determine for which data sets filtering (or removing suspected noisy instances from the training set) is the most beneficial<ref>{{cite journal|title= Predicting Noise Filtering Efficacy with Data Complexity Measures for Nearest Neighbor Classification|journal= Pattern Recognition|volume= 46|pages= 355–364|doi= 10.1016/j.patcog.2012.07.009|year= 2013|last1= Sáez|first1= José A.|last2= Luengo|first2= Julián|last3= Herrera|first3= Francisco}}</ref> and could be expanded to other areas.
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 [[Meta learning (computer science)|meta learning]] to determine for which data sets filtering (or removing suspected noisy instances from the training set) is the most beneficial<ref>{{cite journal|title= Predicting Noise Filtering Efficacy with Data Complexity Measures for Nearest Neighbor Classification|journal= Pattern Recognition|volume= 46|pages= 355–364|doi= 10.1016/j.patcog.2012.07.009|year= 2013|last1= Sáez|first1= José A.|last2= Luengo|first2= Julián|last3= Herrera|first3= Francisco}}</ref> and could be expanded to other areas.
=== In molecular recognition ===
=== In molecular recognition ===
A recent study based on molecular simulations and compliance constants describes [[molecular recognition]] as a phenomenon of organisation.<ref>{{cite journal | title=Complexity in molecular recognition | author=Jorg Grunenberg | journal=Phys. Chem. Chem. Phys. | year=2011 | volume=13 | issue=21 | pages= 10136–10146 | doi=10.1039/c1cp20097f| pmid=21503359 | bibcode=2011PCCP...1310136G }}</ref>
A recent study based on molecular simulations and compliance constants describes [[molecular recognition]] as a phenomenon of organisation.<ref>{{cite journal | title=Complexity in molecular recognition | author=Jorg Grunenberg | journal=Phys. Chem. Chem. Phys. | year=2011 | volume=13 | issue=21 | pages= 10136–10146 | doi=10.1039/c1cp20097f| pmid=21503359 | bibcode=2011PCCP...1310136G }}</ref>
Even for small molecules like [[carbohydrates]], the recognition process can not be predicted or designed even assuming that each individual [[hydrogen bond]]'s strength is exactly known.
Even for small molecules like [[carbohydrates]], the recognition process can not be predicted or designed even assuming that each individual [[hydrogen bond]]'s strength is exactly known.
== Applications ==
== Applications ==
Computational complexity theory is the study of the complexity of problems – that is, the difficulty of [[problem solving|solving]] 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 [[travelling salesman problem]], for example. It can be solved in time <math>O(n^2 2^n)</math> (where ''n'' is the size of the network to visit – the number of cities the travelling salesman must visit exactly once). As the size of the network of cities grows, the time needed to find the route grows (more than) exponentially.
Computational complexity theory is the study of the complexity of problems – that is, the difficulty of [[problem solving|solving]] 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 [[travelling salesman problem]], for example. It can be solved in time <math>O(n^2 2^n)</math> (where ''n'' is the size of the network to visit – the number of cities the travelling salesman must visit exactly once). As the size of the network of cities grows, the time needed to find the route grows (more than) exponentially.
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. [[Analysis of algorithms|Computational complexity]] 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.
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. [[Analysis of algorithms|Computational complexity]] 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.
There exist a certain class of problems that although they are solvable in principle they require so much time or space that it is not practical to attempt to solve them. These problems are called [[Computational complexity theory#Intractability|intractable]].
There exist a certain class of problems that although they are solvable in principle they require so much time or space that it is not practical to attempt to solve them. These problems are called [[Computational complexity theory#Intractability|intractable]].
There is another form of complexity called [[Model of hierarchical complexity|hierarchical complexity]]. It is orthogonal to the forms of complexity discussed so far, which are called horizontal complexity.
There is another form of complexity called [[Model of hierarchical complexity|hierarchical complexity]]. It is orthogonal to the forms of complexity discussed so far, which are called horizontal complexity.
== See also ==
== See also ==
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* [[Chaos theory]]
* [[Chaos theory]]
第一名: 弗朗西斯
第一名: 弗朗西斯
* [[Complexity theory (disambiguation)|Complexity theory]] (disambiguation page)
* [[Complexity theory (disambiguation)|Complexity theory]] (disambiguation page)
* [[Cyclomatic complexity]]
* [[Cyclomatic complexity]]
* [[Digital morphogenesis]]
* [[Digital morphogenesis]]
* [[Dual-phase evolution]]
* [[Dual-phase evolution]]
* [[Emergence]]
* [[Emergence]]
* [[Evolution of complexity]]
* [[Evolution of complexity]]
* [[Game complexity]]
* [[Game complexity]]
* [[Holism in science]]
* [[Holism in science]]
* [[Law of Complexity/Consciousness]]
* [[Law of Complexity/Consciousness]]
2008年
2008年
* [[Model of hierarchical complexity]]
* [[Model of hierarchical complexity]]
* [[Names of large numbers]]
* [[Names of large numbers]]
* [[Network science]]
* [[Network science]]
* [[Network theory]]
* [[Network theory]]
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