复杂性

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Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, meaning there is no reasonable higher instruction to define the various possible interactions.引用错误：没有找到与</ref>对应的<ref>标签

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The term is generally used to characterize something with many parts where those parts interact with each other in multiple ways, culminating in a higher order of emergence greater than the sum of its parts. The study of these complex linkages at various scales is the main goal of complex systems theory.

这个术语通常用来描述具有许多部分的事物，这些部分以多种方式相互作用，最终以大于其部分之和的更高级别出现。在不同尺度上研究这些复杂连杆机构是复杂系统理论的主要目标。

Science 模板:As of takes a number of approaches to characterizing complexity; Zayed et al.引用错误：没有找到与</ref>对应的<ref>标签

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reflect many of these. Neil Johnson 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".引用错误：没有找到与</ref>对应的<ref>标签

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Overview

Definitions of complexity often depend on the concept of a "system" – a set of parts or elements that have relationships among them differentiated from relationships with other elements outside the relational regime. Many definitions tend to postulate or assume that complexity expresses a condition of numerous elements in a system and numerous forms of relationships among the elements. However, what one sees as complex and what one sees as simple is relative and changes with time.

复杂性的定义往往取决于”系统”的概念，这是一组部件或要素，它们之间的关系不同于与关系制度之外的其他要素的关系。许多定义倾向于假设或假设复杂性表达了系统中众多元素的条件和元素之间众多形式的关系。然而，人们所认为的复杂和简单是相对的，并且随着时间的推移而变化。

Warren Weaver posited in 1948 two forms of complexity: disorganized complexity, and organized complexity.引用错误：没有找到与</ref>对应的<ref>标签

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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".^[1] Weaver's 1948 paper has influenced subsequent thinking about complexity.引用错误：没有找到与</ref>对应的<ref>标签

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The approaches that embody concepts of systems, multiple elements, multiple relational regimes, and state spaces might be summarized as implying that complexity arises from the number of distinguishable relational regimes (and their associated state spaces) in a defined system.

体现系统、多元素、多关系体系和状态空间概念的方法可以概括为: 复杂性来自于一个已定义系统中可区分的关系体系(及其相关的状态空间)的数量。

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

有些定义涉及表达复杂现象或模型或数学表达式的算法基础，如本文后面所述。

Disorganized vs. 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.

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".

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.

在韦弗看来，无组织的复杂性是由于特定系统拥有非常多的部件，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.^[2]

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".

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.引用错误：没有找到与</ref>对应的<ref>标签

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Sources and factors

There are generally rules which can be invoked to explain the origin of complexity in a given system.

通常有一些规则可以用来解释给定系统中复杂性的起源。

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.

无组织复杂性的根源是感兴趣系统中的大量部件，以及系统中各要素之间缺乏相关性。

In the case of self-organizing living systems, usefully organized complexity comes from beneficially mutated organisms being selected to survive by their environment for their differential reproductive ability or at least success over inanimate matter or less organized complex organisms. See e.g. Robert Ulanowicz's treatment of ecosystems.^[3]

In the case of self-organizing living systems, usefully organized complexity comes from beneficially mutated organisms being selected to survive by their environment for their differential reproductive ability or at least success over inanimate matter or less organized complex organisms. See e.g. Robert Ulanowicz's treatment of ecosystems.

就自我组织的生命系统而言，有效组织的复杂性来自有益的突变生物体，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。

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) ，使用多带图灵机比使用单带图灵机的计算复杂度更小。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。

Varied meanings

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:

In computational complexity theory, the amounts of resources required for the execution of algorithms 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 size of the input (usually measured in bits), using the most efficient algorithm, and the space complexity of a problem equal to the volume of the memory 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 classification of computational problems by complexity class (such as P, NP, etc.). An axiomatic approach to computational complexity was developed by Manuel Blum. 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.

In algorithmic information theory, the Kolmogorov complexity (also called descriptive complexity, algorithmic complexity or algorithmic entropy) of a string is the length of the shortest binary program that outputs that string. Minimum message length is a practical application of this approach. Different kinds of Kolmogorov complexity are studied: the uniform complexity, prefix complexity, monotone complexity, time-bounded Kolmogorov complexity, and space-bounded Kolmogorov complexity. An axiomatic approach to Kolmogorov complexity based on Blum axioms (Blum 1967) was introduced by Mark Burgin in the paper presented for publication by Andrey Kolmogorov.^[4] The axiomatic approach encompasses other approaches to Kolmogorov complexity. It is possible to treat different kinds of Kolmogorov complexity as particular cases of axiomatically defined generalized Kolmogorov complexity. Instead of proving similar theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily deduce all such results from one corresponding theorem proved in the axiomatic setting. This is a general advantage of the axiomatic approach in mathematics. The axiomatic approach to Kolmogorov complexity was further developed in the book (Burgin 2005) and applied to software metrics (Burgin and Debnath, 2003; Debnath and Burgin, 2003).

In information theory, information fluctuation complexity is the fluctuation of information about information entropy. It is derivable from fluctuations in the predominance of order and chaos in a dynamic system and has been used as a measure of complexity in many diverse fields.

In information processing, complexity is a measure of the total number of properties transmitted by an object and detected by an observer. Such a collection of properties is often referred to as a state.

In physical systems, complexity is a measure of the probability of the state vector of the system. This should not be confused with entropy; it is a distinct mathematical measure, one in which two distinct states are never conflated and considered equal, as is done for the notion of entropy in statistical mechanics.

In dynamical systems, statistical complexity measures the size of the minimum program able to statistically reproduce the patterns (configurations) contained in the data set (sequence)^[5]^[6]. While the algorithmic complexity implies a deterministic description of an object (it measures the information content of an individual sequence), the statistical complexity, like forecasting complexity^[7], implies a statistical description, and refers to an ensemble of sequences generated by a certain source. Formally, the statistical complexity reconstructs a minimal model comprising the collection of all histories sharing a similar probabilistic future, and measures the entropy of the probability distribution of the states within this model. It is a computable and observer-independent measure based only on the internal dynamics of the system, and has been used in studies of emergence and self-organization.^[8]. While the algorithmic complexity implies a deterministic description of an object (it measures the information content of an individual sequence), the statistical complexity, like forecasting complexity, implies a statistical description, and refers to an ensemble of sequences generated by a certain source. Formally, the statistical complexity reconstructs a minimal model comprising the collection of all histories sharing a similar probabilistic future, and measures the entropy of the probability distribution of the states within this model. It is a computable and observer-independent measure based only on the internal dynamics of the system, and has been used in studies of emergence and self-organization.^[9]

|last1=Prokopenko |first1=M. |last2=Boschetti |first2=F. |last3=Ryan |first3=A. |year=2009 |title=An information-theoretic primer on complexity, self-organisation and emergence |journal=Complexity |volume=15 |issue=1 |pages=11–28 |doi=10.1002/cplx.20249 |bibcode=2009Cmplx..15a..11P }}</ref>

1 m.2 Boschetti | first2 f.3瑞安。关于复杂性、自我组织和突现的信息理论入门 | 期刊复杂性 | 第15卷 | 第1页 | 第11-28页 | doi 10.1002 / cplx. 20249 | bibcode 2009Cmplx. 15 a. . 11 p } / ref

In mathematics, Krohn–Rhodes complexity is an important topic in the study of finite semigroups and automata.

In Network theory complexity is the product of richness in the connections between components of a system,^[10] and defined by a very unequal distribution of certain measures (some elements being highly connected and some very few, see complex network).

In software engineering, programming complexity 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.

In abstract sense – Abstract Complexity, is based on visual structures perception ^[11] It is complexity of binary string defined as a square of features number divided by number of elements (0's and 1's). Features comprise here all distinctive arrangements of 0's and 1's. Though the features number have to be always approximated the definition is precise and meet intuitive criterion.

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:

A complex adaptive system has some or all of the following attributes:^[12]

- The number of parts (and types of parts) in the system and the number of relations between the parts is non-trivial – however, there is no general rule to separate "trivial" from "non-trivial";

- The system has memory or includes feedback;

- The system can adapt itself according to its history or feedback;

- The relations between the system and its environment are non-trivial or non-linear;

- The system can be influenced by, or can adapt itself to, its environment;

- The system is highly sensitive to initial conditions.

Study

Complexity has always been a part of our environment, and therefore many scientific fields have dealt with complex systems and phenomena. From one perspective, that which is somehow complex – displaying variation without being random – is most worthy of interest given the rewards found in the depths of exploration.

复杂性一直是我们环境的一部分，因此许多科学领域都涉及到复杂的系统和现象。从一个角度来看，这是某种复杂的-显示变化而不是随机的-是最值得关注的，因为在深度探索中发现的回报。

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.引用错误：没有找到与</ref>对应的<ref>标签 This means that complex is the opposite of independent, while complicated is the opposite of simple.

</ref> This means that complex is the opposite of independent, while complicated is the opposite of simple.

/ ref 这意味着复杂是独立的对立面，而复杂是简单的对立面。

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.^[13] 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。

Topics

Behaviour

The behavior of a complex system is often said to be due to emergence and self-organization. Chaos theory has investigated the sensitivity of systems to variations in initial conditions as one cause of complex behaviour.

一个复杂系统的行为通常被认为是由于涌现和自我组织。混沌理论已经研究了系统对初始条件变化的敏感性作为复杂行为的原因之一。

Mechanisms

Recent developments around artificial life, evolutionary computation and genetic algorithms have led to an increasing emphasis on complexity and complex adaptive systems.

最近围绕人工生命、进化计算和遗传算法的发展使人们越来越重视复杂性和复杂适应系统。

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.

在社会科学中，研究宏观属性的出现是从微观属性出发的，在社会学中又称为宏观-微观视角。这个话题通常被认为是社会复杂性，常常与社会科学中计算机模拟的使用有关，例如。计算社会学。

Systems

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Systems theory has long been concerned with the study of complex systems (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-dimensional, 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 systems (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-dimensional, non-linear, and difficult to model. In specific circumstances, they may exhibit low-dimensional behaviour.

系统理论长期以来一直关注复杂系统的研究(近年来，复杂性理论和复杂系统也被用作该领域的名称)。这些系统存在于各种学科的研究中，包括生物学、经济学、社会研究和技术。近年来，复杂性已经成为现实世界社会认知系统和新兴系统学研究的一个自然领域。复杂系统往往是高维的、非线性的、难以建模的。在特定情况下，他们可能表现出低维度的行为。

Data

In information theory, algorithmic information theory is concerned with the complexity of strings of data.

在信息论中，算法信息论关注的是数据串的复杂性。

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-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.

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”可以使计算机输出一个非常复杂的字符串，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。

These algorithmic measures of complexity tend to assign high values to random noise. However, those studying complex systems would not consider randomness as complexity模板:Who.

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。

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.

在信息论中，熵也有时被用来表示复杂性，但熵的随机性也很高。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.^[14]

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。

The complexity measures broadly cover:

复杂性指标大致涵盖:

the overlaps in feature values from differing classes.

the separability of the classes.

measures of geometry, topology, and density of manifolds. 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.^[15]

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 to determine for which data sets filtering (or removing suspected noisy instances from the training set) is the most beneficial^[16] 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 to determine for which data sets filtering (or removing suspected noisy instances from the training set) is the most beneficial and could be expanded to other areas.

实例硬度是一种自下而上的方法，它首先寻求识别可能被错误分类的实例(或者，换句话说，哪些实例是最复杂的)。然后，可能被错误分类的实例的特征根据一组硬度测量值的输出进行测量。硬度测量是基于一些监督式学习技术，如测量不同意的邻居的数量或分配给输入功能的类标签的可能性。复杂性度量提供的信息已经被用于元学习，以确定哪些数据集过滤(或者从训练集中去除可疑的噪音实例)是最有益的，并且可以扩展到其他领域。

In molecular recognition

A recent study based on molecular simulations and compliance constants describes molecular recognition as a phenomenon of organisation.^[17]

A recent study based on molecular simulations and compliance constants describes molecular recognition as a phenomenon of organisation.

最近一项基于分子模拟和顺应常数的研究将分子识别描述为一种组织现象。

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.

即使是像碳水化合物这样的小分子，识别过程也不能被预测或设计，即使假设每个单独的氢键的强度是确切知道的。

Applications

Computational complexity theory is the study of the complexity of problems – that is, the difficulty of 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]\displaystyle{ 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.

计算复杂性理论是研究问题的复杂性，也就是解决问题的难度。问题可以根据算法(通常是计算机程序)解决问题所需的时间(作为问题大小的函数)按复杂程度分类。有些问题很难解决，而有些则很容易。例如，一些困难的问题需要算法花费指数量的时间来解决问题的大小。以旅行推销员问题为例。这个问题可以用时间数学 o (n ^ 2 ^ n) / math (其中 n 是要访问的网络的大小-货郎担必须访问一次的城市数)来解决。随着城市网络规模的扩大，寻找路线所需的时间呈指数增长(超过倍)。

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. 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.

即使一个问题在原则上是可以计算解决的，但在实际操作中可能没有那么简单。这些问题可能需要大量的时间或过多的空间。计算复杂性可以从许多不同的方面来研究。计算复杂性可以根据时间，内存或其他资源来解决问题。在分析复杂性问题时，时间和空间是两个最重要和最普遍的考虑因素。

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 intractable.

有一类问题，虽然原则上是可以解决的，但它们需要很多时间或空间，因此试图解决它们是不切实际的。这些问题被称为棘手的。

There is another form of complexity called hierarchical complexity. It is orthogonal to the forms of complexity discussed so far, which are called horizontal complexity.

还有另一种形式的复杂性称为等级复杂性。它与迄今为止所讨论的复杂性的形式是正交的，即所谓的横向复杂性。

References

↑ 引用错误：无效<ref>标签；未给name属性为Weaver的引用提供文字

↑ "Sir James Lighthill and Modern Fluid Mechanics", by Lokenath Debnath, The University of Texas-Pan American, US, Imperial College Press:

:

, Singapore, page 31. Online at http://cs5594.userapi.com/u11728334/docs/25eb2e1350a5/Lokenath_Debnath_Sir_James_Lighthill_and_mode.pdfhttps://en.wikipedia.org/wiki/Defekte_Weblinks?dwl={{{url}}} Seite nicht mehr abrufbar], Suche in Webarchiven: Kategorie:Wikipedia:Weblink offline (andere Namensräume)[http://timetravel.mementoweb.org/list/2010/Kategorie:Wikipedia:Vorlagenfehler/Vorlage:Toter Link/URL_fehlt

↑ Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997
↑ 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
↑ Crutchfield, J.P.; Young, K. (1989). "Inferring statistical complexity". Physical Review Letters. 63 (2): 105–108. Bibcode:1989PhRvL..63..105C. doi:10.1103/PhysRevLett.63.105. PMID 10040781.
↑ Crutchfield, J.P.; Shalizi, C.R. (1999). "Thermodynamic depth of causal states: Objective complexity via minimal representations". Physical Review E. 59 (1): 275–283 热力学深度的因果状态: 客观的复杂性通过最小的表述. Bibcode:1999PhRvE..59..275C. doi:10.1103/PhysRevE.59.275. {{cite journal}}: Text "卷59" ignored (help); Text "杂志物理评论 e" ignored (help); Text "问题1" ignored (help); Text "页275-283" ignored (help); line feed character in |pages= at position 8 (help)
↑ Grassberger, P. (1986). "Toward a quantitative theory of self-generated complexity". International Journal of Theoretical Physics. 25 (9): 907–938. Bibcode:1986IJTP...25..907G. doi:10.1007/bf00668821.
↑ . Bibcode:1999PhRvE..59..275C. doi:10.1103/PhysRevE.59.275. {{cite journal}}: Cite journal requires |journal= (help); Missing or empty |title= (help)
↑ Prokopenko, M.; Boschetti, F.; Ryan, A. (2009). "An information-theoretic primer on complexity, self-organisation and emergence". Complexity. 15 (1): 11–28. Bibcode:2009Cmplx..15a..11P. doi:10.1002/cplx.20249. {{cite journal}}: Text "bibcode 1999PhRvE. . 59. . 275 c } / ref.虽然算法的复杂性意味着对一个对象的确定性描述(它测量单个序列的信息含量) ，但统计复杂性，如预测复杂性，意味着统计描述，并指的是由某个源生成的一组序列。在形式上，统计复杂性重构了一个最小模型，包括所有共享一个相似的概率未来的历史的集合，并且测量了这个模型中状态概率分布的熵。它是一个可计算的、独立于观测者的度量，仅仅基于系统的内部动力学，并且已经被用于涌现和自我组织的研究。这是一个很好的例子" ignored (help); Text "doi 10.1103 / PhysRevE. 59.275" ignored (help)
↑ A complex network analysis example: "Complex Structures and International Organizations" (Grandjean, Martin (2017). "Analisi e visualizzazioni delle reti in storia. L'esempio della cooperazione intellettuale della Società delle Nazioni". Memoria e Ricerca (2): 371–393. doi:10.14647/87204. See also: French version).
↑ Mariusz Stanowski (2011) Abstract Complexity Definition, Complicity 2, p.78-83 [1]
↑ 引用错误：无效<ref>标签；未给name属性为Neil Johnson的引用提供文字
↑ Bastardas-Boada, Albert. "Complexics as a meta-transdisciplinary field". Congrès Mondial Pour la Pensée Complexe. Les Défis d'Un Monde Globalisé. (Paris, 8-9 Décembre). Unesco.
↑ Ho, T.K.; Basu, M. (2002). "Complexity Measures of Supervised Classification Problems". IEEE Transactions on Pattern Analysis and Machine Intelligence 24 (3), pp 289–300.
↑ Smith, M.R.; Martinez, T.; Giraud-Carrier, C. (2014). "An Instance Level Analysis of Data Complexity". Machine Learning, 95(2): 225–256.
↑ Sáez, José A.; Luengo, Julián; Herrera, Francisco (2013). "Predicting Noise Filtering Efficacy with Data Complexity Measures for Nearest Neighbor Classification". Pattern Recognition. 46: 355–364. doi:10.1016/j.patcog.2012.07.009.
↑ Jorg Grunenberg (2011). "Complexity in molecular recognition". Phys. Chem. Chem. Phys. 13 (21): 10136–10146. Bibcode:2011PCCP...1310136G. doi:10.1039/c1cp20097f. PMID 21503359.

External links

模板:Wikiquote

Complexity Measures – an article about the abundance of not-that-useful complexity measures.

Exploring Complexity in Science and Technology – Introductory complex system course by Melanie Mitchell

Santa Fe Institute focusing on the study of complexity science: Lecture Videos

UC Four Campus Complexity Videoconferences – Human Sciences and Complexity

模板:Chaos theory

Category:Abstraction

类别: 抽象

Category:Chaos theory

范畴: 混沌理论

Category:Complex systems theory

范畴: 复杂系统理论

Category:Holism

分类: 整体论

Category:Systems

类别: 系统

Category:Transdisciplinarity

类别: 跨学科研究

This page was moved from wikipedia:en:Complexity. Its edit history can be viewed at 复杂性/edithistory

[Weaver-1] 引用错误：无效<ref>标签；未给name属性为Weaver的引用提供文字

[2] "Sir James Lighthill and Modern Fluid Mechanics", by Lokenath Debnath, The University of Texas-Pan American, US, Imperial College Press:
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, Singapore, page 31. Online at http://cs5594.userapi.com/u11728334/docs/25eb2e1350a5/Lokenath_Debnath_Sir_James_Lighthill_and_mode.pdfhttps://en.wikipedia.org/wiki/Defekte_Weblinks?dwl={{{url}}} Seite nicht mehr abrufbar], Suche in Webarchiven: Kategorie:Wikipedia:Weblink offline (andere Namensräume)[http://timetravel.mementoweb.org/list/2010/Kategorie:Wikipedia:Vorlagenfehler/Vorlage:Toter Link/URL_fehlt

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[3] Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997

[4] 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

[5] Crutchfield, J.P.; Young, K. (1989). "Inferring statistical complexity". Physical Review Letters. 63 (2): 105–108. Bibcode:1989PhRvL..63..105C. doi:10.1103/PhysRevLett.63.105. PMID 10040781.

[6] Crutchfield, J.P.; Shalizi, C.R. (1999). "Thermodynamic depth of causal states: Objective complexity via minimal representations". Physical Review E. 59 (1): 275–283 热力学深度的因果状态: 客观的复杂性通过最小的表述. Bibcode:1999PhRvE..59..275C. doi:10.1103/PhysRevE.59.275. {{cite journal}}: Text "卷59" ignored (help); Text "杂志物理评论 e" ignored (help); Text "问题1" ignored (help); Text "页275-283" ignored (help); line feed character in |pages= at position 8 (help)

[7] Grassberger, P. (1986). "Toward a quantitative theory of self-generated complexity". International Journal of Theoretical Physics. 25 (9): 907–938. Bibcode:1986IJTP...25..907G. doi:10.1007/bf00668821.

[8] . Bibcode:1999PhRvE..59..275C. doi:10.1103/PhysRevE.59.275. {{cite journal}}: Cite journal requires |journal= (help); Missing or empty |title= (help)

[9] Prokopenko, M.; Boschetti, F.; Ryan, A. (2009). "An information-theoretic primer on complexity, self-organisation and emergence". Complexity. 15 (1): 11–28. Bibcode:2009Cmplx..15a..11P. doi:10.1002/cplx.20249. {{cite journal}}: Text "bibcode 1999PhRvE. . 59. . 275 c } / ref.虽然算法的复杂性意味着对一个对象的确定性描述(它测量单个序列的信息含量) ，但统计复杂性，如预测复杂性，意味着统计描述，并指的是由某个源生成的一组序列。在形式上，统计复杂性重构了一个最小模型，包括所有共享一个相似的概率未来的历史的集合，并且测量了这个模型中状态概率分布的熵。它是一个可计算的、独立于观测者的度量，仅仅基于系统的内部动力学，并且已经被用于涌现和自我组织的研究。这是一个很好的例子" ignored (help); Text "doi 10.1103 / PhysRevE. 59.275" ignored (help)

[10] A complex network analysis example: "Complex Structures and International Organizations" (Grandjean, Martin (2017). "Analisi e visualizzazioni delle reti in storia. L'esempio della cooperazione intellettuale della Società delle Nazioni". Memoria e Ricerca (2): 371–393. doi:10.14647/87204. See also: French version).

[11] Mariusz Stanowski (2011) Abstract Complexity Definition, Complicity 2, p.78-83 [1]

[Neil_Johnson-12] 引用错误：无效<ref>标签；未给name属性为Neil Johnson的引用提供文字

[13] Bastardas-Boada, Albert. "Complexics as a meta-transdisciplinary field". Congrès Mondial Pour la Pensée Complexe. Les Défis d'Un Monde Globalisé. (Paris, 8-9 Décembre). Unesco.

[14] Ho, T.K.; Basu, M. (2002). "Complexity Measures of Supervised Classification Problems". IEEE Transactions on Pattern Analysis and Machine Intelligence 24 (3), pp 289–300.

[15] Smith, M.R.; Martinez, T.; Giraud-Carrier, C. (2014). "An Instance Level Analysis of Data Complexity". Machine Learning, 95(2): 225–256.

[16] Sáez, José A.; Luengo, Julián; Herrera, Francisco (2013). "Predicting Noise Filtering Efficacy with Data Complexity Measures for Nearest Neighbor Classification". Pattern Recognition. 46: 355–364. doi:10.1016/j.patcog.2012.07.009.

[17] Jorg Grunenberg (2011). "Complexity in molecular recognition". Phys. Chem. Chem. Phys. 13 (21): 10136–10146. Bibcode:2011PCCP...1310136G. doi:10.1039/c1cp20097f. PMID 21503359.

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复杂性

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