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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. A complex system is thereby characterised by its inter-dependencies, whereas a complicated system is characterised by its layers. Complexity 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. 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. Warren Weaver posited in 1948 two forms of complexity: disorganized complexity, and organized complexity.
Some definitions relate to the algorithmic basis for the expression of a complex phenomenon or model or mathematical expression, as later set out herein. 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. 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”. The number of parts does not have to be very large for a particular system to have emergent properties. There are generally rules which can be invoked to explain the origin of complexity in a given system. 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. 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. Complexity of an object or system is a relative property.
Turing machines are used than when Turing machines with one tape are used. In computational complexity theory, the amounts of resources required for the execution of algorithms is studied. 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. Rhodes complexity is an important topic in the study of finite semigroups and automata. In software engineering, programming complexity is a measure of the interactions of the various elements of the software.