## Fundamental Research in Complex Systems Theorems and Principles

Fundamental research in complex systems is designed to obtain characterizations of complex systems and relationships between quantities that characterize them. When there are well-defined relationships, these are formalized as theorems or principles. More general characterizations and classifications of complex systems are described below in major directions of inquiry. These are only a sample of the ongoing research areas.

A theorem or principle of complex systems should apply to physical, biological, social, and engineered systems. Similar to laws in physics, a law in complex systems should relate various quantities that characterize the system and its context. An example is Newton's 2nd law that relates force, mass, and acceleration. Laws in complex systems relate qualities of system, action, environment, function, and information. Three examples follow.

### Functional Complexity

Given a system whose function we want to specify, for which the environmental (input) variables have a complexity of C(e), and the actions of the system have a complexity of C(a), then the complexity of specification of the function of the system is

where complexity is defined as the logarithm (base 2) of the number of possibilities or, equivalently, the length of a description in bits.

The proof follows from recognizing that complete specification of the function is given by a table whose rows are the actions (C(a) bits) for each possible input, of which there are 2 C(e). Since no restriction has been assumed on the actions, all actions are possible, and this is the minimal length description of the function. Note that this theorem applies to the complexity of description as defined by the observer, so that each of the quantities can be defined by the desires of the observer for descriptive accuracy. This theorem is known in the study of Boolean functions (binary functions of binary variables) but is not widely understood as a basic theorem in complex systems (Bar-Yam 1997).

The implications of this theorem are widespread and significant to science and engineering. The exponential relationship between the complexity of function and the complexity of environmental variables implies that systems that have environmental variables (inputs) with more than a few bits (i.e., 100 bits or more of relevant input) have functional complexities that are greater than the number of atoms in a human being and thus cannot be reasonably specified. Since this is true about most systems that we characterize as "complex," the limitation is quite general. The implications are that fully phenomenological approaches to describing complex systems, such as the behaviorist approach to human psychology, cannot be successful. Similarly, the testing of response or behavioral descriptions of complex systems cannot be performed. This is relevant to various contexts, including testing computer chips, and the effects of medical drugs in double-blind population studies. In each case, the number of environmental variables (inputs) is large enough that all cases cannot be tested.

### Requisite Variety

The Law of Requisite Variety states that the larger the variety of actions available to a control system, the larger the variety of perturbations it is able to compensate (Ashby 1957). Quantitatively, it specifies that a well-adapted system's probability of success in the context of its environment can be bounded:

Qualitatively, this theorem specifies the conditions in which success is possible: a matching between the environmental complexity and the system complexity, where success implies regulation of the impact of the environment on the system.

The implications of this theorem are widespread in relating the complexity of desired function to the complexity of the system that can succeed in the desired function. This is relevant to discussions of the limitations of specific engineered control system structures, of the limitations of human beings, and of human organizational structures.

Note that this theorem, as formulated, does not take into account the possibility of avoidance (actions that compensate for multiple perturbations because they anticipate and thus avoid the direct impact of the perturbations), or the relative measure of the space of success to that of the space of possibilities. These limitations can be compensated for.

### Non-averaging

The Central Limit Theorem specifies that collective or aggregate properties of independent components with bounded probability distributions are Gaussian, distributed with a standard deviation that diminishes as the square root of the number of components. This simple solution to the collective behavior of non-interacting systems does not extend to the study of interacting or interdependent systems. The lack of averaging of properties of complex systems is a statement that can be used to guide the study of complex systems more generally. It also is related to a variety of other formal results, including Simpson's paradox (Simpson 1951), which describes the inability of averaged quantities to characterize the behavior of systems, and Arrow's Dictator Theorem, which describes the generic dynamics of voting systems (Arrow 1963; Meyer and Brown 1998).

The lack of validity of the Central Limit Theorem has many implications that affect experimental and theoretical treatments of complex systems. Many studies rely upon unjustified assumptions in averaging observations that lead to misleading, if not false, conclusions. Development of approaches that can identify the domain of validity of averaging and use more sophisticated approaches (like clustering) when they do not apply are essential to progress in the study of complex systems.

Another class of implications of the lack of validity of the Central Limit Theorem is the recognition of the importance of individual variations between different complex systems, even when they appear to be within a single class. An example mentioned above is the importance of individual differences and the lack of validity of averaging in cognitive science studies. While snowflakes are often acknowledged as individual, research on human beings often is based on assuming their homogeneity.

More generally, we see that the study of complex systems is concerned with their universal properties, and one of their universal properties is individual differences. This apparent paradox, one of many in complex systems (see below), reflects the importance of identifying when universality and common properties apply and when they do not, a key part of the study of complex systems.

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