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Complexity in general usage is the opposite of simplicity. Complexity in specific usage is the opposite of independence, while complication is the opposite of simplicity.

Contents 1 Study of complexity 1.1 Complex systems 1.1.1 Complex mechanisms 1.1.2 Complex simulations 1.1.3 Complex behaviour 1.2 Complexity in data 1.3 Complexity of problems 2 Specific meanings 3 Quotes about complexity 4 See also 5 References 6 External links

Complexity has always been a part of our environment, and therefore many scientific fields have dealt with complex systems and phenomena. Indeed, some would say that only what is somehow complex – what displays variation without being random – is worthy of interest.

The use of the term complex is often confused with the term complicated. In today’s systems, this is the difference between a myriad of connecting “stovepipes” and effective “integrated” solutions. (Lissack and Roos, 2000) 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.

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). These systems can be biological, economic, technological, etc. Recently, complexity is a natural domain of interest of the real world socio-cognitive systems and emerging systemics research.

Complex systems tend to be high-dimensional, non-linear and hard to model. In specific circumstances they may exhibit low dimensional behaviour.

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

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.

The behaviour 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.

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.

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-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 random noise. However, those studying complex systems would not consider randomness as complexity.

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

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 to solve them as function of the problem size. For example, the travelling salesman problem can be solved in time O(n²2ⁿ) (where n is the size of the network to visit).

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.

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.

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

• In computational complexity theory, the time complexity of a problem is 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. This allows to classify problems by complexity class (such as P, NP ... ) such analysis also exists for space, that is, the memory used by the algorithm.
• In algorithmic information theory, the Kolmogorov complexity (also called descriptive complexity or algorithmic entropy) of a string is the length of the shortest binary program which outputs that string.
• 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 is often confused with entropy, but is a distinct Mathematical analysis of the probability of the state of the system, where two distinct states are never conflated and considered equal as in statistical mechanics.
• In mathematics, Krohn-Rhodes complexity is an important topic in the study of finite semigroups and automata.
• 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.
• Specified complexity is a term used in intelligent design theory, first coined by William Dembski.
• Irreducible complexity is a term used in arguments against the generally accepted theory of biological evolution, being a concept popularized by the biochemist Michael Behe.
• 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.

• "The complexity of a document is proportional to the number of fingers that you need to read it." DeMarco's Law is a paraphrase from Tom DeMarco. For example, 'The complexity of a computer program is proportional to the number of fingers you need to read it.'
• "The essence of tyranny is the denial of complexity" Jacob Burkhardt, Swiss historian.
• "Some days I will say yes, and then odd days it seems things say yes to me. And stranger still, there are those times when I become a yes." (And they are moments of the Calm) -Kevin Hart, quoted by Mark Taylor in 'The Moment of Complexity'

• Chaos theory
• Complexity theory (disambiguation page)
• Command and Control Research Program
• Interconnectedness
• Important publications on computational complexity in computer science
• Occam's razor
• Programming Complexity
• Holism in science
• Game complexity

• Roger Lewin. Complexity: Life at the Edge of Chaos. Macmillan, 1992.
• M. Mitchell Waldrop, Complexity: The Emerging Science at the Edge of Order and Chaos. Simon and Schuster, 1992.
• R. V. Solé and B. C. Goodwin, Signs of Life: How Complexity Pervades Biology, Basic Books, 2001.
• Micahel R. Lissack and Johan Roos. The Next Common Sense, The e-Manager’s Guide to Mastering Complexity. Nicholas Brealey Publishing.
• Complexity Theory and Network Centric Warfare by James Moffat (CCRP, 2003)
• Complexity, Networking, and Effects Based Approaches to Operations by Edward Smith (CCRP, 2006)
• Complexity, Global Politics, and National Security by Alberts and Czerwinski (CCRP, 1997)
• Coping with the Bounds by Czerwinski (CCRP, 1998)

• Center for Models of Life - Niels Bohr Institute
• The Command and Control Research Program
• Quantifying Complexity Theory - classification of complex systems
• Living Roadmap of Complexity Sciences - at ONCE-CS
• Complexity Measures - an article about the abundance of not-that-useful complexity measures.
• VisualComplexity.com - A visual exploration on mapping complex networks
• Santa Fe Institute - an institute dedicated to complexity research
• Complex Systems Society - sponsoring the annual International/European Complexity Conferences such as ECCS'07 abd ECCS'07
• New England Complex Systems Institute
• Modelling Complex Socio-Technical Systems using Morphological Analysis From the Swedish Morphological Society
• UCLA Human Complex Systems Program
• The Unicist Research Institute- basic and applied research on an ontological approach to complexity.
• The University of Michigan's Center for the Study of Complex Systems, a research group on complex systems, esp. in the social sciences.
• UC Four Campus Complexity Videoconferences - Human Sciences and Complexity
• The Simplicity Cycle by Dan Ward is a graphical exploration of the relationship between complexity, goodness and time.