Regularization, Recognition and Complexity Estimation Methods of Automata Models of Discrete Dynamical Systems in Control Problem

: In paper are considered laws of functioning of discrete determined dynamical systems and specific processes of functioning of such systems. As basic mathematical model of laws of functioning of systems are used automata models with a fundamentally new extension of these models to models with a countable infinite sets of states. This expansion is possible thanks to the proposed and developed by Tverdohlebov V. A. the mathematical apparatus of geometrical images of automaton mappings. Are presented results of development of regularization methods for partially set automata models of systems based on use of geometrical images of automatons mappings and numerical interpolation methods. Also in paper are considered a problem of complexity estimation of laws in a whole and specific processes of functioning of dynamic systems. For these purpose are used recurrent models and methods and also a specific mathematical apparatus of discrete riv-functions. Is spent classification by complexity estimations of automata models


Introduction
In the theory of experiments with automatons initial base is decoding of a contained of "black box". Initial data is the information on variants of a contained of black box. Under the general scheme of carrying out of experiment to a black box (to contents of a black box) are put influences, reaction to these influences is observed and on these supervision is construction logic conclusions. In control problems the automaton and family of automatons are set and it is required to define, contents of a black box is this allocated automaton or the automaton from the set family. In case of diagnosing it is supposed, that contents of a black box is the element of the set family of automatons and it is required to define what it is the automaton. By E. Mure [1], A. Gill [2], T. Hibbard [3] and other authors solve following problems: criteria of existence of the decision of a problem of recognition of a contained of black box are found; the basic method of construction of experiment on automaton recognition in the set family of automatons, including construction minimum on length of simple unconditional experiment (E. Mure, A. Gill) is developed. Further essentially important expansion of approaches and methods of technical diagnosing was representation of laws of functioning of automatons by geometrical images, i.e. numerical mathematical structures in the form of discrete numerical graphics [4]. If the automatons presented for the decision of control and diagnosing problems in their geometrical images to combine with analytically set curves search and construction of control and diagnostic experiments can be carried out on the basis of the decision of systems of the equations for the geometrical curves set analytically.
In the basic works, containing development of the automata theory, the problem of regularization of automata on the basis of the uniform approach is not considered. There are problems, at which decision used methods assume completely set laws of functioning of automatons, but in initial data these laws are presented partially. Fundamental mathematical results on regularization of partially set graphics are presented by classical methods of interpolation of Newton, Lagrange, Gauss, Bessel, spline-interpolation methods etc. Inapplicability of these methods for partially set automatons is connected with the symbolical form of the presentation of automatons by tables, matrixes, graphs, systems of the logic equations, etc. Presentation of laws of functioning of automata by numerical structures, offered and developed by V. A. Tverdokhlebov (see, for example, [4,5]), allows to use classical methods of interpolation in the automata theory. In paper are developed methods of interpolation for partially set laws of functioning of the automatons, presented by geometrical images.
One of fundamentals, making mathematical models of large-scale systems are the algorithms, realized by system according to its target mission. For an algorithmically solvable class of problems there is an infinite set of algorithms (solving a class of problems), which can be ordered on complexity. Realization of algorithms is generally connected with complexity of algorithm in the system, defining the major indicators: performance, a memory size, reliability, expenditure of energy etc. Number of variants of concept of complexity is sufficiently great and continues to increase in works of many researchers. For example, estimations of algorithms on their belong to NP and P classes (detail review see, for example, in work [6] and one of contemporary papers [7], in which Radoslaw Hofman show, that P not equal NP), Kolmogorov сomplexity [8], complexity from below, from above, complexity on the average, bit complexity (one of basic work in this area is [9]), multiplicate complexity, algebraic complexity, there are very large amount of works on asymptotical estimations of complexity (see, for example works [10,11]), approximation complexity of analysis of graph structures [12,13] and for identifying codes [14] etc.
In the given work with use of the apparatus of geometrical images of automatons [4], it is offered and is investigated the estimation of complexity of laws of functioning of the discrete determined dynamic systems (automatons) on the basis of discrete riv-functions of kind In this work is offered the estimation of complexity of laws of functioning of the discrete determined dynamic systems (automatons) on the basis of geometrical representation of laws and use of discrete riv-functions. Is carried out the analysis of more than 10 million discrete rivfunctions. In clause are considered the riv-functions, containing more of 20 billion of discrete graphs, on which are synthesised laws of functioning of automatons. Specificity of all considered riv-functions is defined. As complexity indicators are considered the minimum and maximum number of states at the minimal automaton from the set of automatons, defined by riv-function.

Geometrical Images of Lows of Functioning of Automatons
It is known, that the apparatus of continuous numerical mathematics effectively uses infinite sets. In this connection Tverdohlebov V. A. was developed the new approach to construction of models of complex systems and methods of the analysis of such models, which are stated in works [4,5]. A developed principle is placing of discrete structures on continuous geometrical curves, set analytically. For this purpose instead of next-state function and output function of automaton is considered a automaton mapping, i.e. symbolical mathematical structures of a kind (input sequence, output sequence). Geometrical image γ s of laws of functioning (see works [4,5]) of initial finite determined automaton A s = (S, X, Y, δ, λ, s) with sets of states S, input signals X, output signals Y and next-state function δ: S×X→S and functions of outputs λ: S×X→Y it is defined on the basis of introduction of a linear order ω in automata mapping Automaton mapping ρ s (set of pairs) is ordered by linear order ω, defined on the basis of an order ω 1 on X * and set by following rules: Rule 1. On set Х some linear order ω 1 (which we will designate 1 ≺ ) is entered Rule 2. An order ω 1 on Х we will extend to a linear order on set Х * , believing, that a. For any words  1 ω′ -an order on s ρ′ , induced rather ω 1 on X * . Having a linear order ω 2 , defined on set Y and having placed set of points ρ s in system of coordinates D 1 with an axis of abscisses (X * , ω 1 ) and an axis of ordinates (Y, ω 2 ), we receive a geometrical image γ s of laws of functioning of initial finite determined automaton A s = (S, X, Y, δ, λ, s). Linear orders ω 1 and ω 2 allow to replace elements of sets X * and Y by their numbers r 1 (p) and r 2 (p) on these orders. As a result are defined two forms of geometrical images, first, as symbolical structure in system of coordinates D 1 , and secondly, as numerical structure in system of coordinates with integer or real positive semiaxes.

The Method of Recognition of Automatons by Their Geometrical Images
Let automaton A 0 is mathematical model of efficient technical system and the family of automatons represents set I of failures of technical system. We will assume, that these automatons are set by geometrical image γ 0 and family of geometrical images In the developed method of recognition geometrical images γ 0 and rely located on analytically set geometrical curve L 0 and family of analytically set geometrical curves Then equality -sets of points of curves, is defined the decision of a problem of the control with use of simple unconditional experiment. Definition 3.1. Let L -a geometrical curve and ∆ -a piece on an axis of abscisses, on which the part of curve L (or all curve L) is defined. This part of a curve we will designate L (∆). Theorem 3.1. Let initial automaton A 0 = (S, X, Y, δ, λ, s 0 ) have a geometrical image γ 0 , located on curve L 0 and -a family of their geometrical images, located accordingly on curves from family is carried out and in a piece ∆ of abscissa axis are defined some points of geometrical image γ 0 and geometrical images from family β, then piece ∆ contains the decision of a problem of recognition of the automaton concerning of family α by simple unconditional experiment.
The proof. Let I = {1, 2, …, k}. We will present system of , are recognized by output sequences on the general input sequence p, i.e. by simple unconditional experiment.
On the basis of the theorem 1 is offered the method (with 4 stages) of recognition of the automaton, which laws of functioning are set by the geometrical images, located on analytically set curves.

The Method of Recognition of Automaton in Pair of Automatons by Geometrical Images
The method consists of following stages: Stage 1. For automaton A 1 = (S 1 , X, Y, δ 1 , λ 1 , s 01 ) and A 2 = (S 2 , X, Y, δ 2 , λ 2 , s 02 ), making an exclusive class and set in the geometrical images d The method is proved by the following theorem. Theorem 4.1. The problem of recognition of the automaton by simple unconditional experiment in pair of automatons A 1 and A 2 where A 1 =(S 1 , X, Y, δ 1 , λ 1 , s 01 ) and A 2 = (S 2 , X, Y, δ 2 , λ 2 , s 02 ), S 1 ∩ S 2 = ∅ , has the decision in only case when when for some Y y ∈ characteristic function φ y satisfies to condition

Interpolation for Regularization of Laws of Functioning of Automatons
The choice and application of a method of interpolation by implication correspond to acceptance and realization of a hypothesis, that the method of interpolation, applied to the numerical graphic, representing partially set geometrical image of the automaton, enough precisely regularize points of a geometrical image, i.e. is enough exact regularize partially set laws of functioning of the automaton. Therefore, validity of the results, received with use of the chosen method of interpolation, is shown to a substantiation of correctness of a hypothesis. In the given paragraph methods of a choice of a hypothesis (a choice of a concrete method of interpolation) are investigated and developed for concrete classes of automatons on an example of a choice of more exact method of interpolation from two methods of interpolation: Newton and Lagrange (under the similar scheme also is carried out the analysis of Gauss, Bessel etc. methods). These methods include following stages: 1 (method of interpolation of Newton with an assessment 0.14 more precisely, than a method of Lagrange).

Problem Statement
The finite determined automaton is set by finite geometrical image of laws of functioning (on a finite section). It is required to define the top and bottom borders for number of states without obvious construction of next-

Descrete Riv-Functions
The by the first 40 digits of number π, makes more than 10 27 .

Estimation of Complexity of Laws and Processes of Functioning of Automatons with Use of Descrete Riv-Functions
The finite The In clause are considered the initial finite determined automatons of type of Mile of kind , according to dynamics equations: s(t+1) = δ(s(t), x(t)), y(t) = λ(s(t), x(t)).
In the given work is used the apparatus of geometrical images of laws of functioning of the automatons, for the first time offered by V. A. Tverdohlebov in 1995г. and later developed in work [4]. Transformation of phase pictures to geometrical images of laws of functioning of the automaton, offered and developed by V. A. Tverdohlebov, has allowed to represent phase pictures by uniform mathematical structuresbroken lines with numerical coordinates of points. To V. A. Tverdohlebov is shown, that sequence of elements from the finite set, combined with linear order on set of input words, defines laws of functioning of the discrete determined dynamic system (automaton). V. A. Tverdohlebov is offered and developed a method of synthesis of laws of functioning of the automaton on the set sequence (see, for example, [4,5] The detailed description of a method of synthesis of the automaton on sequence, and also a method of check of equivalence of states of the automaton by it geometrical image contains in the monography [4]. We will note only the basic moments of a method of synthesis of the automaton on sequence. If as the task of laws of functioning of the automaton is considered the sequence ξ( , and next-state function δ define by rules s 0 =s ε and δ(s p , x) = s px , then function δ appears standard for all automatons with set of input signals X. Specificity of automatons is shown, that on infinite set of states for each automaton classes of equivalent states are allocated. At synthesis of laws of functioning of the automaton is essential the way of regularization of next-state functions δ of the automaton. Various ways of regularization of next-state functions of the automaton are possible: cyclic regularization, regularization in an initial state, state can generate by random generator (from set of possible states), etc. In a case, when , let's name cyclic regularization (or regularization of type 2). In paper are considered the specific families of automatons , С 1 , С 2 ∈ N + , С 2 -С 1 +1 = l, l ≥ 2 and m ∈ N + , m ≥ 2, and for family of automatons  [4]). In discrete riv-function k d H , we will allocate so much initial parts, how many in pairs various broken lines of length m, take places in a rectangle l×m, i.e. l m parts. We will assume, that the geometrical image of the automaton from family , is required to regularization.
In a case, when l m <τ, next-state function δ of the automaton for states with numbers 1, 2, …, l m also is defined by a standard rule, and for states with numbers l m +1, l m +2, …, . In definition 6.12 is offered the specific way of regularization to next-state function. Definition 6. 12. A way of regularization to next-state function δ of the finite determined automaton of type of Mile  will be necessarily is 1-equivalent to any condition from set (S\S′) (since in the table of outputs of the automaton for each column with number l m +1, l m +2, …, there will be an identical column with number from 1 to l m ). We will show, that at use of a way of regularization to nextstate function δ of the automaton all states from set S′ will be 2-recognizable and, besides, any conditions s and s′ where S s ∈ , S s ′ ∈ ′ , which are 1-equivalent, are recognizable conditions. Because any two conditions with numbers from 1 to l m are recognizable, at use of a way 3 of regularization to next-state function δ of the automaton for any states s′ and s″ from set S′, provided that pairs not equivalent, that proves the theorem statement. □

Example of Estimation of Complexity
Example 1. Whereas information on the real law of functioning of concrete complex discrete dynamic system has huge dimension, and the law is known only partially and is required the decision of additional problems on regularization it to completely set law, we will spend an illustration of the offered method of an estimation of complexity on an example, in which the top and bottom borders of riv-function are presented by known mathematical sequences. On fig. 1

Conclusion
In given article on the basis of use of the apparatus of geometrical images of automatons are offered methods and the algorithms, developed for recognition of laws of functioning of discrete determined dynamic systems (automatons), set by the automaton mappings, placed on analytically set geometrical curves. Methods are proved by corresponding theorems. In paper are stated models and the methods developed for interpolation of partially set laws of functioning of automatons, set by the automata mappings placed on geometrical curves, using base points of the interpolation, selected on the basis of selection of autonomous subautomatons.
In paper is offered the estimation of complexity of laws of functioning of the discrete determined dynamic systems (automatons) on the basis of geometrical representation of laws and use of discrete riv-functions. Is carried out the analysis of more than 10 million the discrete riv-functions, formed by fundamental mathematical sequences of length to 80 signs, taken from bank [15]. Also are considered the rivfunctions, containing more of 20 billion of discrete graphs, on which laws of functioning of automatons are synthesised. Specificity of all considered riv-functions is defined. As complexity indicators are considered: k min -the minimum number of conditions at the reduced automaton in family The number of states of modeling system is one of the fundamental characteristics, used at designing and system manufacturing. The offered method of an estimation of complexity of laws of functioning of the discrete determined automatons can be applied to get exact bottom and exact top estimations of number of states at the minimal automaton only on the basis of the analysis of a geometrical image of the automaton, without obvious construction of next-state and output functions of automaton and carrying out the subsequent minimization, which practical realization for automatons with large number of states even with use of modern computing systems is essentially complicated. The basic parameters, used in an offered method, are length of a considered initial piece of a geometrical image of the automaton -d, number of input signals of the automaton -m and number of output signals of the automaton -l. In view of that the basic criteria for reception of estimations in an offered method are only ratios of sizes d, m, l, and in a method recursive procedures of construction aren't used, the method can be used for the big finite sizes d, m, l. Use of the apparatus of geometrical images of the automatons, offered and developed by V. A. Tverdokhlebov (see, for example, [4]), allows to consider geometrical curves and numerical sequences with automata interpretation, i.e. as ways of the task of laws of functioning of automatons. It allows to build automaton models of the discrete determined systems without restrictions on number of states. The offered method is based on use of geometrical representation of laws of functioning of automatons and allows to give concrete estimations on number of states for any, as is wished great, values of sizes d, m, l, that can be used in practice at designing of systems for carrying out analysis on number of states of possible variants of realization of system for purpose of a choice of system with the least number of states.
The number of states of modeling system is one of the fundamental characteristics, used at designing and system manufacturing. The offered method of an estimation of complexity of laws of functioning of the discrete determined automatons can be applied to get exact bottom and exact top estimations of number of states at the minimal automaton only on the basis of the analysis of a geometrical image of the automaton, without obvious construction of next-state and output functions of automaton and carrying out the subsequent minimization, which practical realization for automatons with large number of states even with use of modern computing systems is essentially complicated. The basic parameters, used in an offered method, are length of a considered initial piece of a geometrical image of the automaton -d, number of input signals of the automaton -m and number of output signals of the automaton -l. In view of that the basic criteria for reception of estimations in an offered method are only ratios of sizes d, m, l, and in a method recursive procedures of construction aren't used, the method can be used for the big finite sizes d, m, l. Use of the apparatus of geometrical images of the automatons, offered and developed by V. A. Tverdokhlebov (see, for example, [4]), allows to consider geometrical curves and numerical sequences with automata interpretation, i.e. as ways of the task of laws of functioning of automatons. It allows to build automaton models of the discrete determined systems without restrictions on number of states. The offered method is based on use of geometrical representation of laws of functioning of automatons and allows to give concrete estimations on number of states for any, as is wished great, values of sizes d, m, l, that can be used in practice at designing of systems for carrying out analysis on number of states of possible variants of realization of system for purpose of a choice of system with the least number of states.