In general usage, complexity often tends to be used to characterize something with many parts in intricate arrangement. In science there are at this time a number of approaches to characterizing complexity, many of which are reflected in this article. Seth Lloyd of M.I.T. writes that he once gave a presentation which set out 32 definitions of complexity. Seth Lloyd is a Professor of Mechanical engineering at Massachusetts Institute of Technology. [1]

Definitions are often tied to the concept of a ‘system’ – a set of parts or elements which have relationships among them differentiated from relationships with other elements outside the relational regime. System (from Latin systēma, in turn from Greek systēma is a set of interacting or interdependent Entities, real or abstract 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.

Some definitions key on the question of the probability of encountering a given condition of a system once characteristics of the system are specified. Warren Weaver has posited that the complexity of a particular system is the degree of difficulty in predicting the properties of the system if the properties of the system’s parts are given. System (from Latin systēma, in turn from Greek systēma is a set of interacting or interdependent Entities, real or abstract In Weaver's view, complexity comes in two forms: disorganized complexity, and organized complexity. [2] Weaver’s paper has influenced contemporary thinking about complexity. Warren Weaver (b July 17 1894 in Reedsburg Wisconsin d November 24 1978 in New Milford Connecticut) was an American [3]

The approaches which 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.

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

## Disorganized complexity vs. organized complexity

One of the problems in addressing complexity issues has been distinguishing conceptually 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.

Weaver perceived and addressed this problem, in at least a preliminary way, in drawing a distinction between 'disorganized complexity' and 'organized complexity'.

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.

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, for example, with the (relative) simplicity of the planetary orbits – the latter can be known by applying Newton’s laws of motion, though this example involved highly correlated events. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the

Organized complexity, in Weaver's view, resides in nothing else than the non-random, or correlated, interaction between the parts. These non-random, or correlated, relationships create a differentiated structure which can, as a system, interact with other systems. The coordinated system manifests properties not carried by, or dictated by, individual parts. The organized aspect of this form of complexity vis a vis other systems than the subject system can be said to "emerge," without any “guiding hand. For other uses see Emergence (disambiguation, Emergent, and Emergency.

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. Scientific modelling is the process of generating abstract, conceptual, Graphical and or mathematical models. Simulation is the imitation of some real thing state of affairs or process A computer simulation, a computer model or a computational model is a Computer program, or network of computers that attempts to simulate an An example of organized complexity is a city neighborhood as a living mechanism, with the neighborhood people among the system’s parts. [4]

## Sources of complexity

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.

There is no consensus at present on general rules regarding the sources of organized complexity, though the lack of randomness implies correlations between elements. See e. g. Robert Ulanowixz's treatment of ecosystems. [5] Consistent with prior statements here, the number of parts (and types of parts) in the system and the number of relations between the parts would have to be non-trivial – however, there is no general rule to separate “trivial” from “non-trivial.

## Specific meanings of complexity

In several scientific fields, "complexity" has a specific meaning :

• In computational complexity theory, the amounts of resources required for the execution of algorithms is studied. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources In Computational complexity theory, a computational resource is a resource used by some Computational models in the solution of Computational problems In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation 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. In the fields of algorithm analysis and Computational complexity theory, the running time or space requirements of an Algorithm are expressed as a function of In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Computer data storage, often called storage or memory, refers to Computer components devices and recording media that retain digital 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. In the fields of algorithm analysis and Computational complexity theory, the running time or space requirements of an Algorithm are expressed as a function of In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation This allows to classify computational problems by complexity class (such as P, NP . In Computational complexity theory, a complexity class is a set of problems of related complexity In Computational complexity theory, P, also known as PTIME or DTIME ( n O(1 is one of the most fundamental Complexity In Computational complexity theory, NP is one of the most fundamental Complexity classes The abbreviation NP refers to " N on-deterministic . . ). Axiomatic approach to computational complexity was developed by M. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources Blum.
• 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 which outputs that string. Algorithmic information theory is a subfield of Information theory and Computer science that concerns itself with the relationship between computation In Algorithmic information theory (a subfield of Computer science) the Kolmogorov complexity (also known as descriptive complexity, Kolmogorov-Chaitin In Computer programming and some branches of Mathematics, a string is an ordered Sequence of Symbols. Computer programs (also software programs, or just programs) are instructions for a Computer. Different kinds of Kolmogorov complexity are studied: the uniform complexity, prefix complexity, monotone complexity, time-bounded Kolmogorov complexity, and space-bounded Kolmogorov complexity.
• In information processing, complexity is a measure of the total number of properties transmitted by an object and detected by an observer. Information processing is the change (processing of Information in any manner detectable by an observer. Property is any physical or virtual entity that is owned by an individual Observation is either an activity of a living being (such as a Human) which senses and assimilates the Knowledge of a Phenomenon, or the recording of data Such a collection of properties is often referred to as a state. In Computer science and Automata theory, a state is a unique configuration of information in a program or machine
• In physical systems, complexity is a measure of the probability of the state vector of the system. In Physics the word system has a technical meaning namely it is the portion of the physical Universe chosen for analysis Probability is the likelihood or chance that something is the case or will happen System (from Latin systēma, in turn from Greek systēma is a set of interacting or interdependent Entities, real or abstract 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 statistical mechanics. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics
• In mathematics, Krohn-Rhodes complexity is an important topic in the study of finite semigroups and automata. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation

There are different specific forms of complexity:

• In the sense of how complicated a problem is from the perspective of the person trying to solve it, limits of complexity are measured using a term from cognitive psychology, namely the hrair limit. Cognitive psychology is a branch of Psychology that investigates internal mental processes such as problem solving memory and language "The Magical Number Seven Plus or Minus Two Some Limits on Our Capacity for Processing Information" is a 1956 paper by the cognitive psychologist George A
• Unruly complexity denotes situations that do not have clearly defined boundaries, coherent internal dynamics, or simply mediated relations with their external context, as coined by Peter Taylor.
• Complex adaptive system denotes systems which have some or all of the following attributes [6]
• 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; and
• The system can be influenced by, or can adapt itself to, its environment. Complex adaptive systems are special cases of Complex systems They are complex in that they are diverse and made up of multiple interconnected elements and adaptive Feedback is a circular causal Process whereby some proportion of a system's output is returned (fed back to the Input.

## Study of complexity

Complexity has always been a part of our environment, and therefore many scientific fields have dealt with complex systems and phenomena. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Indeed, some would say that only what is somehow complex – what displays variation without being random – is worthy of interest. Randomness is a lack of order Purpose, cause, or predictability

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. [7] This means that complex is the opposite of independent, while complicated is the opposite of simple.

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. In Academia, Pedagogy, Physical sciences, Earth sciences, Human sciences and Social sciences An ant colony is an underground Lair where Ants live Colonies consist of a series of underground chambers connected to each other and the surface of the earth by The human brain controls the Central nervous system (CNS by way of the Cranial nerves and Spinal cord, the Peripheral nervous system (PNS A stock market, or (equity market is a private or public market for the trading of company Stock and derivatives of company One such interndisciplinary group of fields is relational order theories. A number of independent lines of research depict the universe including the social organization of living creatures which is of particular interest to humans as Systems or networks of

## Complexity topics

### Complex behaviour

The behaviour of a complex system is often said to be due to emergence and self-organization. For other uses see Emergence (disambiguation, Emergent, and Emergency. Self-organization is a process of Attraction and repulsion in which the internal organization of a System, normally an open system, increases Chaos theory has investigated the sensitivity of systems to variations in initial conditions as one cause of complex behaviour. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that

One of the main claims in Stephen Wolfram's book A New Kind of Science is that such behaviour can be generated by simple systems, such as the rule 110 cellular automaton. Stephen Wolfram (born August 29, 1959 in London) is a British Physicist, Mathematician and Businessman known for his A New Kind of Science is a Controversial book by Stephen Wolfram, published in 2002 The Rule 110 cellular automaton (often simply Rule 110) is a one-dimensional two-state Cellular automaton with the following rule table Interesting

### Complex mechanisms

Recent developments around artificial life, evolutionary computation and genetic algorithms have led to an increasing emphasis on complexity and complex adaptive systems. Artificial life (commonly Alife or alife) is a field of study and an associated art form which examine Systems related to Life, its processes In Computer science evolutionary computation is a subfield of Artificial intelligence (more particularly Computational intelligence) that involves A genetic algorithm (GA is a Search technique used in Computing to find exact or Approximate solutions to optimization and Search Complex adaptive systems are special cases of Complex systems They are complex in that they are diverse and made up of multiple interconnected elements and adaptive

### Complex 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 social sciences comprise academic disciplines concerned with the study of the social life of human groups and individuals including Anthropology, Communication studies Sociology (from Latin: socius "companion" and the suffix -ology "the study of" from Greek λόγος lógos "knowledge" The topic is commonly recognized as social complexity that is often related to the use of computer simulation in social science, i. Social complexity is the approach to social phenomena that tries to analyze a social system as a Complex system. A computer simulation, a computer model or a computational model is a Computer program, or network of computers that attempts to simulate an e. : computational sociology. Computational sociology is a recently developed branch of Sociology that uses Computation to analyze social phenomena

### Complex systems

Main article: Complex system

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). This article describes complex system as a type of system For other meanings see Complex systems. Systems theory is an Interdisciplinary field of Science and the study of the nature of Complex systems in Nature, Society, and This article describes complex system as a type of system For other meanings see Complex systems. These systems can be biological, economic, technological, etc. System (from Latin systēma, in turn from Greek systēma is a set of interacting or interdependent Entities, real or abstract Foundations of modern biology There are five unifying principles An economy is the realized social system of production exchange distribution and consumption of goods and services of a country or other area Technology is a broad concept that deals with a Species ' usage and knowledge of Tools and Crafts and how it affects a species' ability to control and adapt Recently, complexity is a natural domain of interest of the real world socio-cognitive systems and emerging systemics research. Systemics is the emerging branch of Science that studies Holistic Systems It tries to develop logical mathematical engineering and philosophical paradigms Complex systems tend to be high-dimensional, non-linear and hard to model. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it This article describes the use of the term nonlinearity in mathematics In specific circumstances they may exhibit low dimensional behaviour.

### Complexity in data

In information theory, algorithmic information theory is concerned with the complexity of strings of data. Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information. Algorithmic information theory is a subfield of Information theory and Computer science that concerns itself with the relationship between computation In Computer programming and some branches of Mathematics, a string is an ordered Sequence of Symbols.

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. A codec is a device or program capable of encoding and/or decoding a Digital Data stream or signal. In computability theory, several closely-related terms are used to describe the "computational power" of a computational system (such as an Abstract machine or

These algorithmic measures of complexity tend to assign high values to random noise. In Science, and especially in Physics and Telecommunication, noise is fluctuations in and the addition of external factors to the stream of target However, those studying complex systems would not consider randomness as complexity. Randomness is a lack of order Purpose, cause, or predictability

Information entropy is also sometimes used in information theory as indicative of complexity.

### Complex Project Management

Complex project management is a cross-discipline meta-methodology, that applies complexity theory to the social science of project management. The College of Complex Project Managers defines Complex project management in their Complex Project Manager Competency Standards as "the lifecycle delivery of emergent strategic outcomes through projects which: are usually adaptive system of systems; have high uncertainty in scope definition; are distributed; have ongoing environmental and internal turbulence; are implemented through wave planning; and are unable to be decomposed into elements with clearly defined boundaries" [8]

## Applications of complexity

Computational complexity theory is the study of the complexity of problems - that is, the difficulty of solving them. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources Problem solving forms part of thinking. Considered the most complex of all intellectual functions problem solving has been defined as higher-order Cognitive 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. In Computational complexity theory, a complexity class is a set of problems of related complexity In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation In the fields of algorithm analysis and Computational complexity theory, the running time or space requirements of an Algorithm are expressed as a function of 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. The Travelling salesman problem ( TSP) in Operations research is a problem in discrete or Combinatorial optimization. It can be solved in time O(n22n) (where n is the size of the network to visit - let's say 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. Computational complexity may be approached from many different aspects. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources Computational complexity can be investigated on the basis of time, memory or other resources used to solve the problem. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources 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 intractable. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources

There is another form of complexity called hierarchical complexity. The model of hierarchical complexity is a framework for scoring how complex a Behavior is It is orthogonal to the forms of complexity discussed so far, which are called horizontal complexity

## References

1. ^ Lloyd, Seth, Programming the Universe, Knopf, 2006
2. ^ Weaver, Warren (1948), “Science and Complexity”, American Scientist 36: 536 (Retrieved on 2007-11-21. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that The Command and Control Research Program (CCRP within the Office of the Assistant Secretary of Defense (NII focuses upon (1 improving both the state of the art and the state of the practice The evolution of complexity is an important outcome of the process of Evolution. Combinatorial game theory has several ways of measuring game complexity. Holism in science, or Holistic science, is an approach to Research that emphasizes the study of Complex systems. Interconnectedness is part of the terminology of a worldview which sees a Oneness in all things The model of hierarchical complexity is a framework for scoring how complex a Behavior is Occam's razor (sometimes spelled Ockham's razor) is a principle attributed to the 14th-century English Logician and Franciscan Friar, Process architecture is the structural design of general process systems and applies to fields such as computers (software hardware networks etc Programming Complexity, which is often also referred to as Software Complexity is a term that encompasses numerous properties of a piece of software all of which affect Systems theory is an Interdisciplinary field of Science and the study of the nature of Complex systems in Nature, Society, and In Cybernetics the term variety denotes the total number of distinct states of a System. Cyclomatic complexity is a Software metric (measurement It was developed by Thomas J )
3. ^ Johnson, Steven (2001). Emergence: the connected lives of ants, brains, cities, and software. New York: Scribner, p. 46. ISBN 0-684-86875-X.  .
4. ^ Jacobs, Jane (1961). The Death and Life of Great American Cities. New York: Random House.
5. ^ Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997
6. ^ Johnson, Neil F. (2007). Two’s Company, Three is Complexity: A simple guide to the science of all sciences. Oxford: Oneworld. ISBN 978-1-85168-488-5.
7. ^ (Lissack and Roos, 2000)
8. ^ Dombkins, David. H. (2007). Complex Project Management. North Charleston: BookSurge. ISBN 978-1-4196-7690-1.