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Overview[ edit ] Definitions of complexity often depend on the concept of a confidential " system " — a set of parts or elements that 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.

However, what one sees as complex and what one sees as simple is relative and changes with time. Warren Weaver posited in two forms of 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". 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.

Of course, most real-world systems, including planetary orbits, eventually become theoretically unpredictable even using Newtonian dynamics; as discovered by modern chaos theory.

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.

A system of organized complexity may be understood in its properties behavior among the properties through modeling and simulationparticularly modeling and simulation with computers.

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.

For instance, for many functions problemssuch a computational complexity as time of computation is smaller when multitape Turing machines are used than when Turing machines with one tape are used. Random Access Machines allow one to even more decrease time complexity Greenlaw and Hoover This shows that tools of activity can be an important factor of complexity.

Varied meanings[ edit ] In several scientific fields, "complexity" has a precise meaning: In computational complexity theorythe 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 bitsusing the most efficient algorithm, and the space complexity of a problem equal to the volume of the memory used by the algorithm e.

This allows classification of computational problems by complexity class such as PNP, etc. An axiomatic approach to computational complexity was developed by Manuel Blum.

It allows one to deduce many properties of concrete computational complexity measures, such as time complexity or space complexity, from properties of axiomatically defined measures.

In algorithmic information theorythe Kolmogorov complexity also called descriptive complexity, algorithmic complexity or algorithmic entropy of a string is the length of the shortest binary program that outputs that string.Case Studies for Educational Leadership: Solving Administrative Dilemmas by Stephen F.

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