On a New Type of Information Processing for Efficient Management of Complex Systems
Victor Korotkikh
Central Queensland University
Galina Korotkikh
Central Queensland University Full text:
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Last modified: December 18, 2007
Abstract
It is a challenge to manage complex systems efficiently without confronting NPhard problems. To address the situation we consider the description of complex systems in terms of selforganization processes of prime integer relations [1] and suggest to use the processes for information processing. In particular, we propose that for a given problem selforganization processes of prime integer relations could be used to efficiently build a computing system demonstrating the optimal performance for the problem.
The description is realized through the unity of two equivalent forms, i.e., arithmetical and geometrical [1], [2].
In the arithmetical form a complex system is characterized by hierarchical correlation structures built in accordance with selforganization processes of prime integer relations. The arithmetical form reveals nonlocal correlations without reference to signalling as well as the distances and local times of the parts. Thus, the arithmetical form suggests that parts of a complex systems may be far apart in space and time and yet remain interconnected with instantaneous effect on each other, but no signalling.
Through the arithmetical form of the description we become aware of the hierarchical network of prime integer relations, i.e., a set of mutually selfconsistent elements built by the processes. Arithmetic ensures that not even a minor change can be made to any element of the network. The hierarchical network of prime integer relations is a causal structure. As parts of a system change, the prime integer relations cause the other parts to change accordingly.
Specified by two parameters the geometrical form arises as the selforganization processes of prime integer relations find isomorphic realization in terms of transformations of twodimensional geometrical patterns [1]. As a result, hierarchical structures of prime integer relations defining the correlation structures of a complex system become equivalently represented by hierarchical structures of geometrical patterns determining the dynamics of the system and revealing its complexity. The quantitative description of the system turns out to be about the description of these geometrical patterns [1], [2].
The equivalence of the forms unites the dynamics and structure of a complex
system. Based on the fact that arithmetic behind a prime integer relation makes it sensitive even to a minor change of coefficients, the equivalence introduces a new principle. Notably, a breaking of a prime integer relation leads to a collapse of a corresponding system, as some of the relationships disappear. The principle states that the dynamics of the parts is determined to produce precisely the geometrical patterns corresponding to the prime integer relations, which provide the correlation structures of the system. If the dynamics is even slightly different, then some of the relationships are not in place and the system collapses.
To measure the complexity of a system in terms of selforganization processes of prime integer relations a concept of complexity, called the structural complexity, is introduced [1]. Starting with the integers, the selforganization processes of prime integer relations progress to different levels and thus produce a hierarchical complexity order. The higher the level selforganization processes progress to, the greater is the structural complexity of a corresponding system. The description allows to compare complexities of systems in terms of structure  by hierarchical structures of prime integer relations and dynamics  by hierarchical structures of geometrical patterns.
The correlation structures of a complex system contain information about the parts. By changing some parts the information can be processed as the other parts change in accordance with the prime integer relations. This shows the importance of selforganization processes of prime integer relations for information processing. Namely, for a given problem they could be used to efficiently build the correlation structures of a computing system in information processing demonstrating the optimal performance for the problem.
We suggest the hierarchical network of prime integer relations as a new medium for information processing and investigate its navigating properties. It would be important if for a given problem the performance of a system could behave as a concave function of its structural complexity. Guided by this property the performance global maximum could be efficiently found. It would be also beneficial, if at the global maximum the structural complexities of the system and the problem could be related through an optimality condition.
Since the correlation structures of a system are completely determined by prime integer relations, which are equivalent to twodimensional geometrical patterns, the entropy of the system, measuring its information content, can be connected with the areas of the patterns. Thus, in our approach there is a general connection between entropy and area.
Computational experiments have been conducted [3] to test the navigating properties. The results raise the possibility of an optimality condition of complex systems: if the structural complexity of a system is in a certain relationship with the structural complexity of a problem, then the system demonstrates the optimal performance for the problem. The optimality condition presents the structural complexity of a system as a key to its optimization. From its perspective the optimization of a system could be all about the control of the structural complexity of the system to make it consistent with the structural complexity of the problem.
Importantly, the experiments indicate that the performance of a complex system may indeed behave as a concave function of the structural complexity. Therefore, once the structural complexity could be controlled as a single entity, the optimization of a complex system would be potentially reduced to a onedimensional concave optimization irrespective of the number of variables involved its description. This might open a way to a new type of information processing for efficient management of complex systems.
References
[1]. V. Korotkikh, "A Mathematical Structure for Emergent Computation", Kluwer Academic Publishers, Dordrecht/Boston/London, 1999;
http://www.zentralblattmath.org/zmath/search/?an=0947.90100
[2]. V. Korotkikh and G. Korotkikh, “Description of Complex Systems in terms of SelfOrganization Processes of Prime Integer Relation”, in Complexus Mundi: Emergent Patterns in Nature, M. M. Novak (ed.), World Scientific, New Jersey/London, 2006, pp. 6372, arXiv:nlin/0509008. http://arxiv.org/PS_cache/nlin/pdf/0509/0509008v4.pdf;
V. Korotkikh and G. Korotkikh, “On an Irreducible Theory of Complex Systems”, InterJournal of Complex Systems, 2006, 1751, arXiv:nlin/0606023.
http://arxiv.org/PS_cache/nlin/pdf/0606/0606023v2.pdf
[3]. V. Korotkikh, G. Korotkikh and Darryl Bond, “On Optimality Condition of Complex Systems: Computational Evidence”, arXiv.cs/0504092.
http://arxiv.org/PS_cache/cs/pdf/0504/0504092v1.pdf;
V. Korotkikh and G. Korotkikh, "On Principles in Engineering of Distributed Computing Systems", Soft Computing Journal, vol. 2, No. 2, 2008, pp. 201206.
(http://www.springerlink.com/content/161498jx816v023x/fulltext.pdf).


