International Encyclopedia of Systems and Cybernetics

(From Latin: complexus, i.e. containing)

Some definitions

According to common dictionaries

G. KLIR quotes Webster's Third International Dictionary, according to which complexity: "is the quality of being complex" i.e.:

"- having many varied interrelated parts, patterns, or elements and consequently hard to understand fully ", or

"- marked by an involvement of many parts, aspects, details, notions, and necessitating earnest study or examination to understand or cope with" (1993, p.40).

For the American Heritage Dictionary, complexity implies: "… interconnected or interwoven parts… involved and intricate, as in structure". And: "Complex often implies many varying parts. Intricate refers to a pattern of intertwining parts that is difficult to follow. Involved stresses confusion arising from the commingling of parts and the consequent difficulty of separating them". Moreover" Tangled strongly emphasizes the random twisting of many parts".

G. KLAUS and H. LIEBSCHER's german Wörterbuch der Kybernetik gives a more formal definition: "The complexity K of a system is directly proportional to the number n of its elements the number z of the possible states of these elements and the number k of couplings between the elements" (1976, p.314).

French and Spanish common dictionaries are much less precise and do not offer specific insights.

H. SIMON's views

K. BAUSCH has given a most general definition: "A systemic characteristic that stands for a large number of densely connected parts and multiple levels of embeddedness and entanglement"

This definition can easily accomodate for example stellar-planetary systems or ecosystems

1. Systems with numerous components may be considered complex. Thus the cardinality of a set can be used as a measure of complexity.

2. Systems with much interdependence between components are generally considered more complex than others with less interdependence.

3. Undecidable systems (GÖDEL, TURING) can be considered as complex in comparison to deterministic ones.

4. Complexity of systems can be measured by their information content in the sense of SHANNON and WEAVER.

5. In this context, "informational complexity" can be defined by the number of parameters or symbols needed for their definition.

6. "Computational complexity" refers to the number of calculus steps needed to resolve problems corresponding to a specific class (1977, p.507).

Some French views

1) "Complexity of a system depends on two factors:

the number of elements that constitutes it and, the number of interrelations among them" (J.L. LEMOIGNE – 1990a, p.91).

This definition is merely quantitative. It does not say anything about the nature of the interrelations.

2) According to E. MORIN, complexity implies recursivity and can be characterized by "a tangle of myriads of inter-retro-actions" (1984).

Here, the structural and functional nature of complexity is emphasized.

3) J.L. VULLIERME insists on the hierarchical character of complexity, that implies :"… a plurality of logical levels, interdependent but irreductible to each other" (1990, p.147).

Complexity is also in the eye of the observer.

4) B.de HENNIN defines complexity as: "the expression and manifestation of the potential and actual behavior of a system that an observer represents in a model".

Such a model: "… expresses, includes and connects what is probable and what is certain, what is organized and what is dubious" (1993, p.1492).

This author enumerates a series of conditions:

- The need for the system to be able to communicate

- The interactive behavior between the system and its environment (translatable into digital as well as analogic information)

- The presence of an observer interacting with the system

- The existence of a specific project of the observer

- The scientific rationality implicit in the project

- An epistemology in correspondance with the model

5) JL LEMOIGNE writes that it may be "… a property of the presently available representation of the system" (1990 b, p.109).

6) R. VALLEE states that it "… depends on the grain of observation and on the selected resolution level" (1990, p.240).

Many definitions of complexity have thus been proposed. They differ due to the very variable viewpoints of their proponents. It does not seem appropiate to exclude any: each contributes some specific angle, useful for the global understanding of what we mean by complexity.