There are different senses of 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 is often confused with entropy, but is a distinct 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 computer science, the study of how much time and memory a computer algorithm may take is the field of computational complexity theory. (For finite state machines in automata theory, KrohnRhodes complexity is used.)
 In social science, the study on the emergence of macroproperties from the microproperties, also known as macromicro view in sociology. The topic is commonly recognized as social complexity that oftenly related to the use of computer simulation in social science, i.e.: computational sociology

Complexity is often used as a shorthand for the field that developed in the late 1980s around the use of mathematical and computational modeling of biological, economic and technological systems known as "complex systems" (sometimes complex adaptive systems). These systems tend to exhibit highdimensionality, nonlinearity, and often, sensitive dependence of initial conditions.
 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.
 In mathematics, KrohnRhodes complexity is an important topic in the study of finite semigroups and automata.
Quote about complexity
 "When I hear the word 'complexity', I don't exactly reach for my hammer, but I suspect my eyes narrow. It has the dangerous allure of an incantation, threatening to acquire the same blithe explanatory role that 'adaptation' once did in biology". Philip Ball , Nature Materials 3, 78 (2004), doi:10.1038/nmat1069
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