Complexity

The complexity from the point of view of the information theory

A concept of complexity is defined in Information theory.

Complexity of Kolmogorov

The theory of the Complexité of Kolmogorov defines the complexity of an object finished by the size of the more data-processing little program (with the theoretical direction) which makes it possible to produce this object. Thus, a compressible text has a low complexity and contains little information. It is besides why the utilities of compression general practitioners cannot compress completely random files (operation by impossible nature ), but only of the files which one knows in advance that they comprise some Redondance which results in Corrélation S.

The complexity of Kolmogorov is a discussed subject. One can indeed always give to a computer a construction such as a very particular operation (for example the calculation of pi or the impression of the integral of works of Victor Hugo) there will be coded by a bit. The known concept here is found that a Information is never contained in a message alone, but always in the couple message + decoder taken in an indissociable way. Also the concept of “more theoretical little program” can it be operationally defined in a rigorous and univocal way only with the reference to a machine. One could object that it is enough to take as reference the simple machine more . It is to forget that what we will name simple precisely depends on our lived and our language, both arbitrary. See the articles Razor of Occam and Paradox of the compressor: The library ''' is ''' the language (“the library is the language”).

An additional difficulty lies in the fact that the complexity of Kolmogorov is not décidable: one can give an algorithm producing the wanted object, which proves that the complexity of this object is with more the size of this algorithm. But one cannot write of program which gives the complexity of Kolmogorov of any object that one would like to give him in entry.

The concept, handled with precaution, appears nevertheless at the origin of many theoretical results.

For example, one calls incompressible number within the meaning of Kolmogorov a reality whose development p-adic (binary for more convenience) is comparable with the size of the algorithm which makes it possible to calculate it. In this direction, all rational and the certain irrational ones are compressible, in particular the transcendent numbers like π, E or 0,12345678910111213… whose binary development is infinite but the perfectly calculable continuation of the decimals. On the other hand, the number known as Ω de Chaitin is not it: this number measures the probability that a machine, supplied with a program made up of random bits, stops. The topology of the incompressible numbers is interesting: it is conceived intuitively that this unit is dense in R, but it is impossible of exhiber an incompressible reality, since that would amount giving an effective method of calculating of it. One can thus conjecture that all the incompressible numbers are normal, the continuation of their decimals must be random with the strong direction, at least starting from a certain row.

Other complexities

See algorithmic Complexity, descriptive Complexity, Système complexes.

Complexity P: The class P is the whole of the problems of decisions which are resolvable by a algorithm at polynomial time.

The complexity from the point of view of physics
Intuitively, a system is complex when many ramifications compose it (thus it is not inevitably complicated, since by breaking up it can be simple to include/understand). Two criteria make it possible to characterize this concept more finely: the number and the independence of the parts.

The number and the independence of the parts
A Système complexes is composed of a great number of parts. With this only criterion all the material systems would be complex except the particle S, the Atome S, small the Ion S and small the Molécule S. But a system can have a great number of parts without having a very complicated movement, if all the parts move in the same way for example. The criterion of the independence of the parts is intended to exclude these cases. But it is difficult to define precisely.
As long as one regards a solid as a perfectly rigid body, its parts are not independent from/to each other. A few numbers, some variable of state are enough to completely characterize the state of movement of the solid: position of the Center of inertia, Speed of Translation, speed of Rotation. The movement of each part is completely determined by these numbers. On the other hand, if one studies the Vibration S of the solid, the movements can be much more complicated, because each part can have a movement different from the others. The same applies to a Fluide. To describe these movements one needs much more variables of state, an infinite number in theory. To say here that the parts are independent, it is not to say that they do not interact with the others but only that the knowledge of the Contracting State of does not provide or little information on the Contracting State of other.
There is a share of subjectivity and ambiguity in the appreciation of the independence of the parts: a badly known system can seem just as easily complex, because unexplainable, that very simple, while being satisfied with surface explanations.

The complexity of reality
The simple systems are privileged objects of studies. For a long time they were the only systems for which one could make calculations, but it is not true any more now, thanks to the computers. These are also the only systems that one can characterize well during an experiment and it is an important point for reproducibility (the fact that one can reproduce the same experiment several times and always obtain the same result). This interest of the simplicity explains partly why one finds in all the books and the physics laboratories the same simple geometries (Cercle, Sphère, Cylindre,…).
The examples studied in the books are often simple but reality is much less. One can say that at first approximation the complex systems are all the systems . Complexity is the rule, simplicity the exception. Complexity is a challenge for mathematics applied: to use mathematics to include/understand all that is under our eyes, not to limit itself so that one can trace with the rule and the compass.
All the real systems are complex, or almost all. But the more complex one system is, the more it is difficult to know it with precision. The number of the possible combinations for example poses problem. As the parts are interdependent, the possible states a priori are all the combinations of states of the part. The combinative explosion led to gigantic numbers of possible cases, often more than the number of particles in the known universe, even for systems relatively not very complex. The precise knowledge of the state present of a Système complexes also poses problem. There are far too many variables of state to measure. The complex systems are often badly known and they hold many surprised (emergence of collective ownerships, Car-organization, Nombres of Feigenbaum in the chaotic systems). The Institute of Santa Fe, created by several physicists of which Murray Freezing-Mann and whose official name is Institute for complexity , made study of this type of questions its activity full-time.
The general theory of the systems is sometimes called Systémique.
The Redondance is not the repetition with identical, but the deployment of a multitude of versions different from the same diagram or reason (in English pattern ).
Then, it is possible to model complexity in terms of functional redundancy, as the Chinese restaurant where several functions are carried out in the same place of a structure.
For the complication, the model would be the redundancy structural of a factory where the same function is carried out in several places different from a structure.
1 - The structural redundancy indicates different structures to carry out the same function, like the double brake system of a motor vehicle or several different workshops or different factories to manufacture the same part or the same machine. The structural redundancy characterizes the “complication”. The structural redundancy is illustrated with the double brake system for more safety in modern motor vehicles and with the multiple control circuits electric, hydraulic and tire of the engines of wars to bring back them to the fold with their crew after damage of the combat.
2 - The functional redundancy is that of the multiplicity of functions different carried out in a point from a structure, as a workshop of craftsman who carries out various operations on various materials. The functional redundancy characterizes “complexity” and condition of the car-organization at Henri Atlan. It is the “variety” in the neuropsychiatrist Ross W. Ashby passed to cybernetics.
The complication is about the redundancy structural of a configuration with ( cum ) much of folds (Latin: plico , are , atum : to fold). The complication, multiplication, duplication and replication are same series of the folds and crumplings. It is the multiplicity of the control circuits to carry out the same function.
Complexity is a configuration with ( cum ) a node ( plexus ) of interlacings of tangles. Then, complexity is about the functional redundancy, like a restaurant which presents a menu of 40 different dishes. A machinery woodworking combined of craftsman who saws, planes borer and tutti quanti is representative of this complexity, like an electric drilling machine of amateur with a multiplicity of accessories for various functions.

See too

algorithmic Complexity;

Theory of complexity;

Synergy;

Autopoièse ;

dynamic System;

Holisme .

External bonds

European Programme MCX “Modeling of Complexity”

Association for the Thought Complexes

Complexité and evolution

Blog Walter Baets - Complexity and Innovation

Complexite and information

Site on complexity

the site of Claude Rochet

Interview of Marc Halevy around his book " Sciences and Sens" realized by Denis Failly and who treats complexity

Vidéos

Video
of Edgar Morin realized by Denis Failly around " Intelligence of the complexité" resulting from the Conference of Cerisy
Video
of Jean Louis Moigne realized by Denis Failly around " Intelligence of the complexité" resulting from the Conference of Cerisy 2005

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