# Dictionary Definition

complexity n : the quality of being intricate and
compounded; "he enjoyed the complexity of modern computers" [syn:
complexness] [ant:
simplicity]

# User Contributed Dictionary

## English

### Etymology

### Pronunciation

/kəm'plɛk.sɪ.ti/### Noun

- The state of being complex; intricacy; entanglement.
- That which is complex; intricacy; complication.

#### Translations

The state of being complex; intricacy;
entanglement

- Arabic: تعقيد
- Catalan: complexitat
- Chinese: (fùzá xìng)
- Czech: složitost
- Dutch: ingewikkeldheid
- Estonian: keerulisus
- Finnish: monimutkaisuus, mutkikkuus
- French: complexité
- Galician complexidade
- German: Vielfalt
- Greek: περιπλοκότητα
- Hebrew: מורכבות
- Hungarian: bonyolultság
- Italian: complessità
- Japanese: 複雑さ (ふくざつさ, fukuzatsusa)
- Latvian: sarežģītība g Latvian
- Lithuanian: sudėtingumas
- Polish: złożoność
- Portuguese: complexidade
- Romanian: complexitate
- Russian: сложность
- Slovak: zložitosť
- Slovene: zapletenost
- Spanish: complejidad
- Turkish: karmaşıklık

That which is complex; intricacy; complication

- Catalan: complexitat
- Danish: indviklethed, kompleksitet
- Dutch: gecompliceerdheid
- Estonian: keerukus
- Finnish: monimutkaisuus, vaikeus, ongelma, hankaluus
- French: complexité
- Galician complexidade
- German: Komplexität, Schwierigkeit
- Greek: περιπλοκή
- Hungarian: bonyodalom, komplikáció
- Italian: complessità
- Latvian: komplicētība g Latvian
- Lithuanian: painumas
- Norwegian: floke g Norwegian, forvikling g Norwegian
- Polish: zawiłość
- Portuguese: complexidade
- Romanian: complexitate
- Slovak: zložitosť
- Slovene: zapletenost
- Spanish: complejidad
- Swedish: komplikation, krånglighet
- Turkish: karmaşık

- ttbc Bulgarian: усложненост, сложност, обърканост, забърканост, заплетеност

# Extensive Definition

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.

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.
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. In Weaver's view,
complexity comes in two forms: disorganized complexity, and
organized complexity. Weaver’s
paper has influenced contemporary thinking about complexity.

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.

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.”

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.

## 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. 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. 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 to classify computational problems by complexity class (such as P, NP ... ). Axiomatic approach to computational complexity was developed by M. 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. 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. 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 statistical mechanics.

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

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.

- 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
- 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.

## Study of complexity

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 myriad connecting "stovepipes" and effective
"integrated" solutions. 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. One such interndisciplinary group of fields is
relational order theories.

## Complexity topics

### Complex behaviour

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.

### Complex mechanisms

Recent developments around artificial life, evolutionary computation and genetic algorithms have led to an increasing emphasis on complexity and complex adaptive systems.### 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 topic is commonly recognized as social complexity that is often related to the use of computer simulation in social science, i.e.: computational sociology.### Complex systems

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.### Complexity in 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-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.

### 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"## Applications 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 - 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 O(n^2 2^n) (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 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.

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

## See also

- Chaos theory
- Complex project management
- Complexity theory (disambiguation page)
- Command and Control Research Program
- Evolution of complexity
- Game complexity
- Holism in science
- Interconnectedness
- Model of hierarchical complexity
- Occam's razor
- Process architecture
- Programming Complexity
- Systems theory
- Variety (cybernetics)
- Cyclomatic complexity

## References

## Further reading

## External links

- Quantifying Complexity Theory - classification of complex systems
- Complexity Measures - an article about the abundance of not-that-useful complexity measures.
- UC Four Campus Complexity Videoconferences - Human Sciences and Complexity

complexity in Arabic: تعقد

complexity in German: Komplexität

complexity in Spanish: Complejidad

complexity in Persian: پیچیدگی

complexity in French: Complexité

complexity in Polish: Złożoność

complexity in Portuguese: Complexidade

complexity in Finnish: Kompleksisuus

complexity in Chinese: 复杂

# Synonyms, Antonyms and Related Words

abstruseness, arduousness, bothersomeness, burdensomeness, complication, convolution, crabbedness, crampedness, deepness, difficultness, difficulty, esoterica, hairiness, hardness, inscrutability, intricacy, involvement, knottiness, laboriousness, onerousness, oppressiveness, profoundness, profundity, reconditeness, rigor, rigorousness, ruggedness, strenuousness, toilsomeness, toughness, troublesomeness