critical networks complex system
1. Radicals as a new class of systems
Among the big variety of natural and artificial systems, the systems called radicals [1,2] are of particular interest. The main feature of radicals is their possibility to exist in two states: passive and active. The passive state of a radical assumes impossibility of performing its functions, i.e., it is ‘switched off’ from life. In an active state, the radical functions are considered without any restrictions. Examples of radicals are: elements of the LEGO-designer, alphabetic letters, words of lexicon of a natural language, military divisions, computer programs, technical devices, skills of a person, neurons of the central nervous system, et al. The term radical means root and underlines basic value of such systems in the design of the construction made of radicals.
Originally the term radical was appeared in algebra of polynomials with rational coefficients where it is used for representation of polynomial roots. In chemistry of solutions the term radical is also used for molecules with open valance, which can form clusters. In hieroglyphic languages like Chinese language, the term radical is also used for designation of standard elements of hieroglyphs, forming a whole hieroglyph.
From the point of view of modern mathematics, a radical means the forming element of some algebra. Any other element of such algebra can be expressed through radicals of this algebra.
2. Neuroradicals as the new type of modeling
Radicals are always organized in sets, aggregates, systems, media, etc. We will use the term environment of radicals because it is more than others corresponds to using the term activation of the environment of radicals. Activation of the environment of radicals means that some radicals of the environment are in the active state. Thus all other radicals of the environment are in the passive state. Each time in the environment of radicals some operating functional system called systemkvant is formed from all active radicals of environment (Fig. 1).
The term systemkvant is widely used in physiology  at P.K. Anokhin's scientific school for designation of the functional system of an organism, which is activated for performing a current problem of an organism.
Mathematically, the radical environment corresponds to the system of generators of some algebras. Thus, such system is a multiple set with each element represented by infinite number of copies.
Physically, the radical environment when all radicals of the environment are passive, corresponds to physical vacuum in its modern understanding. Any activation of such vacuum leads to a birth of some physical particles or more complex structures which are systemkvants for the environment of radicals.
From all radicals, the environment of neural cells of a brain, called neurons, is of particular interest for us. Neurons are organized in a network so that each neuron is connected with thousands other neurons. If to activate a part of neurons, a systemkvant is formed of them. This systemkvant from neurons is usually called a pattern. It is an activated part of a brain which, in turn, activates some functional system of an organism for solving the next problem of live beings. Signals from one neuron to another are transferred only through an active neuron systemkvant. All the other neurons are passive and signals are not transferred trough them. They can be considered as being ‘switched off’.
Neurons of a brain play different roles but the majority of them are symbolical models of the world external to a brain. According to Voronkov G. S. , elementary sensory of a brain, which makes its most part, is a model of the world of live beings.
For the modern stage of modeling, analysis and synthesis of Complex Network Critical Systems (СNСS) it is offered to use the environment of artificial neural elements which we will call environment neuroradicals. Activation in such an environment means a choice of a systemkvant from active neuroradicals, being the symbolical model.
Mathematically, from the point of view of mathematical logic, a neuroradical environment represents a semantic network made of predicate symbols (containers) and subject symbols (unique persons) (see Fig. 2). Communications between symbols correspond to single and multiple predicate formulas . Thus, a neuroradical environment is a system of generators in Boolean algebra of a formulas language of mathematical logic. Essentially, a neuroradical environment is a basic part of a Boolean lattice of all formulas of mathematical logic, reflecting the problem area of СNСS.
A neuroradical environment is not only the modeling environment, but also the environment for solving various problems of СNСS analysis and synthesis. Operating the activation of a neurons environment, it is possible to give rise and destroy, analyze and optimize various systemkvants in a neuroradical environment. Thereby neuroradical environment allows conducting R&D works in symbolical form, i.e., a neuroradical environment is a peculiar symbolical R&D stand.
The first important feature of a neuroradical environment is its information redundancy. This quality of a neuroradical environment provides information stability (information safety) at solving problems of analysis and synthesis of СNCS under conditions of incomplete and inexact information.
The second important feature of neuroradical environment is its absence of conflict. It is provided by the presence of the expert system in neuroradical environment, built in it in the form of a network of ultra containers. Such expert system is responsible for performance of various requirements to СNСS and represents the distributed knowledge base. Thus, a neuroradical environment from the mathematical point of view is a Boolean a lattice of formulas of mathematical logic with operations of Boolean algebra. Besides, secondary binary relation of implications is defined on such a Boolean lattice, i.e., the knowledge base reflecting knowledge of experts, legislative documents, state standards, technology requirements and many other things connected with СNСS.
Take advantage of a formalism of mathematical computer science [1,2]. We will compare to radicals of a problem area of СSСS the subject symbols (terms) of logic of predicates, and to properties, types, classes of these radicals – predicate symbols. For visualization, we will consider some objects of modeling (radicals) of a problem area which we will designate by terms . We will compare these terms with neuroradical (points) and designate they by the same symbols. Let these objects possess properties which are described by single predicates . To each predicate a symbol in the working area of IS is comparable with a neuroradical with the same designations. We will interpret them as the concepts (subsets) describing some properties of objects of the subject domain of СNСS.
The fact of belonging of a point to set we will represent graphically with an arrow from the top with a set symbol to the top with a symbol of point . If one set is a subset of another set, we will represent it graphically with an arrow from a symbol of a larger set to a symbol of a smaller set, .
As a result of such representation of the problem area of СNСS there will be a symbolical model of the problem area in which arrows correspond to references. Neuroradical environment is a some kind of a semantic network of various concepts, for example, in the form of the hypertext. Thus neuroradicals-terms form classical mathematical models of objects of a problem area, and the neuroradicals-concepts connected in a network are some kind of coordinate hierarchical system of concepts.
The constructed structure of a neuroradical environment realizes ideas of topological coordinatization and filtrations of sets [1,2]. Each concept of a network is an element of an atomic scale of a corresponding lattice of concepts (subsets). Cross references of such concepts realize an idea of product of lattices of concepts and, thereby, idea of parallel (duplicating) information channels similar to visual, acoustical, olfactory, flavoring, touch channels in natural intelligence (brain).
In the offered symbolical model, instead of functional record of predicate we will use information record in the form of chain .
Along with topological coordinatization, in modeling the neuroradical environment functional communications between neuroradicals are realized. For example, algebraic operations over points or over concepts. As a result, a neuroradical environment realizes an idea of algebraic system in A.I. Maltsev's sense. Thus, neuroradicals represent only generators of corresponding algebras, while other necessary elements of algebras are obtained from generators of arising problems.
Along with single predicates, in neuroradicals environment the various multiple predicates describing a problem area of СNСS are used. As a whole environment a neuroradical is the distributed database and knowledge base of a problem area of СNСS.
3. Neurocomputer as a new computer architecture
The third important feature of a neuroradical environment is obligatory presence of an activating system for a neuroradical environment. Together with an activating system, neuroradical environment forms a new architecture of the computer which is called neurocomputer. Such expanded understanding of the term neurocomputer does not contradict its habitual understanding. In particular, a neurocomputer has two subsystems: an activating subsystem and a modeling subsystem in the form of a neuroradical environment.
Neurocomputer architecture is aimed at intellectual systemkvantization of СNСS. In turn, intellectualization of СNСS is caused by necessity of supporting its information-system safety (ISS). ISS concept  is the main characteristic of СNСS. We will remind, that ISS includes two parts: the information part of safety and the system part of safety of a complex system. Information safety provides the unconditional decision of problems of life cycle of СNСS without dependence on completeness and the form of representation of the entrance information. System safety is an unconditional preservation of the kernel of СNСS, i.e., integrity of a complex system at solving any private problem of life cycle.
The basic approach to support СNСS ISS is intellectualization of such a system. System intellectualization means creation of an intellectual computer superstructure, some kind of brain СNСS, including:
1) creation of computer symbolical model of the whole problem area (a world picture) of СNСS;
2) creation of information-program equipment of this model by means of supporting СNСS ISS.
Intellectualization means the property of such superstructure to develop and expand a circle of regular problems of ISS solved by it at the expense of self-training to solving some supernumerary problems for it.
Offered computer symbolical modeling is based on the concept of a neuroradical environment. Normalized neuroradical environment provides solution of both regular and supernumerary problems of life cycle of system. Мathematically, normalized neuroradical environment represents an ultra plural semantic model of a database and combined with it by ultra operational production model of the knowledge base of the problem area of СNСS .
Neuroradical environment demands creation of means of its activation and regulation, i.e., means of allocation of schemes, navigation in the environment, means of the analysis and synthesis of schemes, schemes-inquiries and schemes-answers are defined. Special schemes of radicals are intended for search of answers to inquiries – the activators. The schemes representing problems, methods of their solving, and also the scheme of an estimation and classification of other schemes  are entered.
During life cycle of СNСS there are problems of the resolution of conflicts to provide ISS. It becomes by means of operating influences from specialized schemes of radicals – the controllers.
The concept of a radical allows us to look at the whole problem area of СNСS, i.e., on the complex system and its environment taking into account the system life cycle, as on the media of radicals. Radicals of such environment, owing to possibility of their activation and removing, are components of complex system and its environment, their communication, a problem of life cycle, means and methods of solving such problems, experts, standard documents and many other things.
Thereby the modeling subsystem of a neurocomputer is an information-superfluous symbolical model of a problem area of a complex system in the form of neuroradical environment. The activating subsystem is responsible for ISS-decision of problems of life cycle of a system and constantly resolves conflicts in a neuroradical environment. Using the model of the problem area, the activating subsystem should create in each moment in a modeling subsystem a systemkvant (model) which defines a cluster in the environment of radicals of СNСS, providing ISS-behavior of СNСS.
In conclusion, we should note once again that in an activating subsystem of a neurocomputer, means for solving various problems connected with ISS of СNСS on the basis of a neuroradical environment are realized. The main role in such a subsystem is played by activators and regulators. Activators are specialized radicals of a neurocomputer, each aimed at solving the class of problems. The work of activators is based on the use of the distributed database and knowledge base of the problem area in the form of a neuroradical environment. Controllers are the radicals which are responsible for support of integrity and absence of conflicts of a neuroradical environment. Besides in an activating subsystem are widely used command (painting) neuroradicals which isolate separate schemes in a neuroradical environment and serve as the navigation means in a neuroradical environment, means of preservation of experience of the previous decision of problems, etc. .
The idea of neurocomputers will completely be coordinated with modern lines of supercomputers, GRID calculations, аgents systems, computer clusters, etc.
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