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A complex system is thereby characterised by its inter-dependencies, whereas a complicated system is characterised by its layers. However, “a characterization of what is complex is possible”. Ultimately Johnson adopts the definition of “complexity science” as “the study of the phenomena which emerge from a collection of interacting objects”. 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. 1948 two forms of complexity: disorganized complexity, and organized complexity. Phenomena of ‘disorganized complexity’ are treated using probability theory and statistical mechanics, while ‘organized complexity’ deals with phenomena that escape such approaches and confront “dealing simultaneously with a sizable number of factors which are interrelated into an organic whole”.

Weaver’s 1948 paper has influenced subsequent thinking about 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. An example of organized complexity is a city neighborhood as a living mechanism, with the neighborhood people among the system’s parts. 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 with one tape are used. This shows that tools of activity can be an important factor of complexity. 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. Different kinds of Kolmogorov complexity are studied: the uniform complexity, prefix complexity, monotone complexity, time-bounded Kolmogorov complexity, and space-bounded Kolmogorov complexity. The axiomatic approach encompasses other approaches to Kolmogorov complexity. It is possible to treat different kinds of Kolmogorov complexity as particular cases of axiomatically defined generalized Kolmogorov complexity.