If we will construct the optimal separation border between patterns i and j then it will be possible to find, that this border at the same time separates another couples of patterns with an error R, not exceeding the permitted value. For example, the solving border (see Fig.1) created for separation of patterns 2 and 5 may as well be used for separation of patterns 1 and 4 from patterns 3 and 6. 
 

 

     Let us imagine that we consider the losses of «aim missing» type equal to R0 as admissible. 
    Such losses may occur, when the separation border lies at the distance d between the control point q and mathematical expectation of i-th pattern, equal to *i. If the control point q is remotes from math expectation of i-th pattern by the distance higher than d, then we can think that it does not belong to pattern i. 
     Every coordinate of properties space makes its own contribution and, if at least one l-th coordinate gives the distance r(ql,*il) for pattern i higher or equal to d, then i-th pattern may be excluded the list of competitors. 
     This consideration creates the basis of by-coordinate exclusion method, which consists in the following. The projections of point q and distribution of all patterns at every coordinate separately are examined. The distances r between the point ql and mathematical expectations *il for all k patterns are defined for each l-th coordinate. The patterns, satisfying the condition r(ql,*il)*d are excluded from the list of patterns, pretending to include the point q. The same procedure is repeated for remaining patterns with application of projection to the second coordinate and so on until the moment, when list of pretenders will contain only the fixed number k1 of patterns. For these most strong pretenders the calculation of distances and estimation of expected losses is performed in n-dimension space with using of optimal decision rules. 
 

    This method reflects aspiration to correct recognition of training sample, but with use of coverings of training sat of each pattern by simple figures, which are becoming complicated as required [3,4].  
    One of variants of this method provides use as covering figures of a set hypersphere. For each of k of patterns the sphere of minimal radius covering all its training realizations is under construction. Value of radiuses of these spheres and distances between their centers allows defining patterns, which spheres are not crossed with spheres of other patterns. It is considered such spheres reference (see pattern 1 in a fig. 2), and their centers and radiuses are remembered in quality   "of the standards of the first generation ". 

 

    If two patterns are crossed, but in the field of crossing there was no realization of training sat, such crossing is considered fictitious, the centers and radiuses of these spheres as are brought in to the list of the standards of the first generation. Thus the area of crossing is considered to belonging sphere with smaller radius (sphere 2 in a fig. 2).  
    If in a zone of crossing there were points only of one pattern, this zone is considered belonging to this pattern (sphere 3 in a fig. 2). The point will be considered concerning to an pattern 4, if it gets in sphere 4 and does not get in sphere 3.  
    If the area of crossing contains points of different patterns, for these points the standards of the "second generation " (sphere 4,5 and 6) are under construction. If also they are crossed, for points from a zone of crossing the standards of the "third generation" are under construction. The procedure of splitting of the standards proceeds before reception of given reliability of recognition of a training sequence.  
    The experience has shown, what even in very complex cases for good recognition of training sat it is enough on the average no more than three generations of the standards.  
    At recognition of new realizations the process of check begins with spheres of the minimal diameter.  
 

    Usually on a grade level the decisive rule is under construction, which parameters are defined and are rigidly fixed under the information contained in training sat. What control realizations would not be showed then for recognition a decisive rule does not vary. But other approach is possible also: a decisive rule to build directly during recognition, leaning on the information contained as in the trainee, and in control sats.  
    In a fig. 3. objects of training sat of two patterns are designated by points and by crosses, and the realizations of control sat are designated by small points. If to approximate clots of points of training sample by unimodal distributions, the decisive border will represent a line a. Such decisive rule constructed on not representative sample, will give many mistakes at recognition of control sample.  
 

 

    If training and control sat to consider in common, it is possible to construct decisive border as a line b, which distinguishes training sat also well, as well as the line a, but reduces quantity of mistakes on control sat. Such decision turns out if to apply criteria taxonomy: the large distances between taxons and small distances inside taxons. It also is necessary in a basis of acceptance of the decisions by a method of the taxonomic decision functions.  
    Mix of training and control objects are exposed taxonomy on k taxons with the help of algorithms such as FOREL or KRAB. If in some taxon there are points of training sample only one i-th of an pattern, all control points which have got in this taxon, concern also to an pattern i. If in taxon there are points from k ' of different patterns, such taxon is broken on k ' finer taxons. This procedure proceeds so long as in everyone taxon there will be no training points only of one pattern. The control points, which have got in taxon, not containing of training points, are considered belonging to new, (k+1)-th pattern. At desire they can be attributed to one of k of patterns, the connection to which will give the greatest value of criterion of taxonomy quality.  
    The described algorithm allows to use not only information from training sat, but also from control. It provides higher stability of algorithm ÒRF to such frequently to meeting unpleasant phenomenon, as small of training sample.  
 

    This algorithm concerns to a class " of Logic Decisive Rules " [6]. In its basis the hypothesis of "projective compactness " lays. Let's consider its work on an example of recognition of two patterns.  
    As the elementary logic statement J in this algorithm the expression of the following kind refers to as:  
  à) X (a) =x        for an attribute X, measured in a nominal scale.  
  á) X (a) < =x    for an attribute in any stronger scale.  
In these expressions x - fixed ("threshold") value. Let's name the statements X (a) = /x and X (a) > x additional to above-stated and to designate through J-. As conjunction of length l on the elementary statements we shall name expression of a kind  
S = J (1) & J (2) &... &J(l). 
    Set conjunctions we shall represent as a dichotomizing tree D, in which to each branch there corresponds one conjunction. The final tops of a tree contain names of objects of training sat, past here from initial top of a tree. If in some final top there are objects of different patterns, it is considered, that this top belongs to that pattern, whose objects in it are more than others.  
    The work on check of the validity or falsity of expressions S in relation to object q is organized with the help of the table T (i, j). To top of a tree the line of the table is compared i-th to number i (i.e. statement J), in which the characteristics j of this top are described:  
Ò (i, 1) - number of a lines occupied by the given top,  
Ò (i, 2) - number of a line, where it is necessary to pass, if J is true,  
Ò (i, 3) - number of a line, where it is necessary to pass, if J is false,  
Ò (i, 4) - number of an attribute, on which is constructed the statement J,  
Ò (i, 5) - threshold x, used at construction of the statement J,  
Ò (i, 6) - quantity of objects of the first pattern in i-that top,  
Ò (i, 7) - quantity of objects of the second pattern in i-that top.  
    The final tops of a tree will be marked with that at them T (i, 4) =0.  
    The movement on a tree with the help of the table T (i, j) is carried out as follows. Let we are in top i. If T (i, 4) = /0, whether that we check satisfies object q to conditions of the statement J, reflected in T (i, 4) and T (i, 5). If yes, we go in top T(i,2), if is not present - in top T(i, 3). If the next top is final, contents of its elements T (i, 6) and T (i, 7) are checked. If will appear, that T (i, 6) > T (i, 7), the object q will be recognized as realization of the first pattern and on the contrary. If T (i, 6) = T (i, 7), the decision is accepted for the benefit of this or that pattern with probability 0,5.  
    At a choice of variants of threshold value x and their combinations arises large testing, which reduction is based on a method of escalating " best to best ". On a first step best in sense of some criterion G the elementary statement J (11) and its addition J (12) = J (11-) is under construction. With the help of these statements the training sat is divided into two groups: group of objects U (11), satisfying J (11), and group U (12), satisfying J (12). If the group U (11) contains objects of different patterns, for it the statement J (21) and its addition J (22), best from the point of view of criterion G is searched. The same procedure for group U (12) gives the best statement J (23) and its addition J (24). In result the training sat is divided into four groups. The process of such division can be presented as a tree D, submitted on a fig. 4.  

                                                          / ----- J (21) - [70i, 3j]  
                                                        / 
                                            / ----- J (11) - [70i, 20j]  
                                          /             \ 
                                        /                 \ ----- J (22) - [0i, 17j] 
                                      / 
                        D - ---- J (1) - [100i, 100j]  
                                     \ 
                                       \                 / ----- J (23) - [20i, 75j] 
                                         \             /  
                                           \ ----- J (12) - [30i, 80j] 
                                                       \  
                                                         \ ----- J (24) - [10i, 5j] 

                                                           Fig. 4. 

     The question on, whether it is necessary further to continue process of escalating of branches, is decided by check of contents of each of the received groups. If in group U will find out h1 of objects of 1-st pattern and h2 of objects of the second pattern and min (h1, h2) > = f*m, the division of group U on subgroups should be continued. Here f - positive constant, smaller 1, and m - common number of objects of training sample. Otherwise group U forms final top and new object getting in it, concerns to a that pattern, whose training realizations it is more in this top than.  
     If to accept f=0,03, at m=200 it is required to make the further division only for top J (23).  
    The logic decisive rules have appeared a very effective means of the decision of tasks of recognition. They can work with different scale of attributes. It is not terrible, if any attribute at new object is not known: can appear, that to available attributes it satisfies to the certain laws and is well distinguished.  
    The process of construction LDF is long, but the process of acceptance of the decisions by the found rules is very simple and can be done even manually.  
    The large importance there is an evident and easily interpretive form of representation LDF as the list of conjunctions " If..., then... ". As has shown experience, some customers, having received  LDF, admit, that it is more interesting to them and it is useful, than an opportunity to use it for automatically recognition of new objects. To them become visible and the simple laws are clear which take place in area, investigated by them.  
    At last, the most important advantage LDF is, that the formulation of laws as conjunction corresponds to the form of representation of knowledge in intellectual systems. Hence, the methods of search LDF can be used for automatically extraction of knowledge from the data.