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Note that the arbitrary pattern recognition class problems can be reduced to the others, with the satisfaction of compactness type hypothesis. However this doesn't mean that the compactness hypothesis is universal because the pattern recognition problem's solution for the given space or creation of appropriate transformations to the other problems are the equivalent problems.

Now let us formulate the condition F0, which will formalize the additional to the compactness hypothesis properties of classes. We'll consider the case of two classes ( l = 2 ) intending the formalism simplifications.

Particularly, in case of completing of partially defined predicate P, we will base on condition F0. We'll apply a correspondence of considered object properties and the set of binary variables x1,x2,...,xn, and the same time between the partial predicate P - and it's characteristic function f ( x1,x2,...,xn ), and will solve the modeled problem of determining (completing) of the target Boolean function F.

~ Let = (1,2,...,n ).

~ ~ ~ Consider the determination (completion) of function f in. Take the arbitrary, 1. If there exists such ~ ~ ~ ~ ~ ~ ~ ~ a point, 0, that (so the is different of on a subset of the set of properties, ~ ~ ~ ~ ~ where are different and ), then we conclude that logically separates from, and the information, ~ ~ that f ( ) = 1 doesn't affect on the determination of the value of the function f in the point by 1. In the ~ ~ opposite case we'll call allowable in respect to the point and to the set 1 and decide, that information ~ ~ f ( ) = 1 influence on the determination of by one, and the real measure of that is given by the value of the object similarity measures.

Consider the following classes of points of the n --dimensional unit cube:

f ~ 0 -- the set of all, which are allowable for the set 0 and not allowable for 1, ~ 1f -- the set of all, which are allowable for the set 1 and not allowable for 0, f ~ 2 -- the set of all, which are not allowable for the sets 0 and 1, f ~ 3 -- the set of all, which are allowable for both the 0 and 1.

Knowledge Engineering [3] pointed out the general relation of condition F0 with the notion of the reduced disjunctive normal form of Boolean functions. To see this relation let us consider the functions f and its negation f, and let, f f correspondingly are the reduced forms for these functions. Denote by:

f ~ ~ ~ 0 --the collection of all points for which ( ) = 0, ( ) = 1, f f ~ ~ ~ 1f --the collection of all points for which ( ) = 1, ( ) = 0, f f f ~ ~ ~ --the collection of all points for which ( ) = 0, ( ) = 0, 2 f f f ~ ~ ~ 3 --the collection of all points for which ( ) = 1, ( ) = 1, f f Proposition 1. if if,i =

f f Proposition 2. If 0 1 0, then 2 is empty, in opposite case 2.

It is simply to prove this and some of the consecutive propositions and by this reason we omit the complete ~ proofs and give the main idea of that. So, to prove proposition 2 consider an arbitrary point. If ~ 0 1 0, then let us take the distance of to the set 0 1 (which equals the minimal possible ~ ~ distance of from any of the points of 0 1 0 ), which is in some point 0 1. Suppose, ~ ~ ~, without loss of generality, that 0. Then the interval (binary subcube) ( ), constructed on base of ~ ~ points and does not contain points from the set 1. From here, on base of definition of reduced ~ disjunctive normal form implies, that the point is allowable in respect to the set 0.

Proposition 3. If f0 is an appropriate completion of function f, constructed on base of condition F0, then ~ ~ f ~ ~ 0 ( f0( ) = 0 )and 1f ( f0( ) = 1).

f Proposition 4. 0 0 and 1 1f.

As a consequence from these two propositions we get, that the arbitrary completion of function f, which is made on base of condition F0, constructs the function, allowable in respect of f. In terms of pattern recognition problems this means that arbitrary methods of recognition, which are based on the condition F0, couldn't be "false" on the elements of the learning set 0 1. Write out the minimal completions of the function f, constructed on base of the condition F0 :

f 0, if ( x1,x2,...,xn ) f1( x1,x2,...,xn ) = 1 if ( x1,x2,...,xn )1f f f is not det er mined if ( x1,x2,...,xn ) \ ( 0 1f ) = So we get some "enlargement" for the basic function f. There arose a question -- might f1 be the new starting point (learning set, function) for the completion on base of condition F0, and how close we can approach by this steps the final goal The answer gives the Fourth International Conference I.TECH 2006 Proposition 5. If fi+1 is completion of partial function fi, constructed on base of condition F0, i =1,2,..., then f1 fk, k =1,2,....

Let us now analyze the conditions, related to the successful continuation on base of F0 of a partial Boolean function (that is the case of the solvable problems). Let f -- is a partially defined Boolean function and 1,2,..., -- all that functions of class P2( n ) which might appear as a continuation of function f, constructed by the given assumptions. Then we are interested in conditions, when extension f1 is allowable in respect to each of functions 1,2,...,.

Consider the function f0, defined in the following way:

0, if i( x1,x2,...,xn ) = 0,i =1,2,..., f0( x1,x2,...,xn ) = 1, if i( x1,x2,...,xn ) =1,i =1,2,..., is not defined in other cases Denote by 0( f0 ) and 1( f0 ) sets of all n --cube vertices, where function f0 achieves values 0 and f respectively. Then our requirement can be formulated as the following: 0 0( f0 ) and 1f 1( f0 ).

f f f Here 3 = \ ( 0 1f ) and 3 \( 0( f0 ) 1( f0 )) so that this partial continuation doesn't violate the continuality of starting function to the each of functions 1,2,...,. It is to mention that the f conditions 0 0( f0 ) and 1f 1( f0 ) are not convenient, which is related to the applied information on the final goal (the functions 1,2,..., ). Supposing the case of continuation for needs of pattern recognition problems let us show that practically useful conditions of the given type might be formulated.

Consider the structural behavior, when n and suppose a parameter 0 given as. Suppose f0 P2( n ) (note, that the results below are true in more general cases and in more general forms). Here are some preliminary results.

- 1. Consider the concept Hk ( f0 ) introduced by Glagolev [7]. Hk ( f0 ) equals the number of vertices ~ ~ En, where f0( ) =1, and which are covered by (involved in) maximal intervals of function f0 of sizes not exceeding k. It was proven [7] that for almost all functions f0 P2( n ) Hk ( f0 ) = o( 2n ) when n and k k1 = log log n -1.

2. We'll say, [5] that the cube vertices prick the intervals including these vertexes. The set A of vertices of n -dimensional unit cube is a pricking set for the set Bk -all of the k -size intervals, if each k -subcube is pricked at least by one of the vertices of A. Denote by K( n,k ) the minimal number of vertices, forming a pricking set for k -subcubes. By [5] 2n-k K( n,k ) ( n +1)2n-k. We will use the upper bound by this formulae but in our case k k1 = log log n -1 and a better estimation is possible as follows [4] (an ~ extended survey on pricking is included in [6]). Let us denote by Ai( ) the set of all of n -cube vertices, ~ which lay in respect to the given vertex on layers with numbers, i(mod k + 1), k k i = 0,1,...,k, k n. Let E -is an arbitrary k -subcube of an n -cube. Points of subcube E are placed n ~ exactly in the k +1 consecutive layers of E in respect to it's arbitrary vertex. It is correct to post the Knowledge Engineering ~ ~ Proposition 6. Each of the sets Ai( ), En,i = 0,1,...,k are pricking for the set Bk -all of the k -subcubes of n -cube, and 2n-k K( n,k ) 2n / k +1.

Proposition 7. F0 implemented in continuation of almost all functions f0 P2( n ) yields the accuracy, tending ~ to 1 as n, if for the initial function f holds the condition 0 1 Ai( ) at least for any i,i =1,2,...

~ ~ and vertices En, where Ai( ) is constructed for a k [log log n ] -1.

Note, that the proposiition 7 postulation is constructive, and it implies to the "sufficient" learning set, consisted no more that from 2n / log log n points (which is o( 2n )) as n. However, basically, in the pattern recognition problems it is impossible to obtain the learning set arbitrarily. Often it is formed as a random collection of any fixed size of the main collection of the studying objects.

Conclusion Logic Separation is an element of pattern recognition hypotheses and formalisms. Structures appear in this relation based and introduced in terms of Reduced Disjunctive Normal Forms of Boolean Functions. An initial set of properties of these structures were introduced in propositions 1-7.

Bibliography 1. Zhuravlev Yu. I. On an algorithmic approach to the problems of recognition and classification. Problemi Kibernetiki, 33, (1978) 5--68.

2. Zhuravlev Yu. I. Selected Scientific Works, Publishing House Magister, Moscow, (1998) 417p.

3. Aslanyan L. H. On a pattern recognition method based on the separation by the disjunctive normal forms. Kibernetika, 5, (1975), 103--110.

4. Vapnik V. and Chervonenkis A. Theory of Pattern Recognition. Nauka, 1974.

5. Aslanyan L. H. The Discrete Isoperimetric Problem and Related Extremal Problems for Discrete Spaces. Problemy Kibernetiki, 36, (1979), 85--128.

6. Nechiporuk E. I. On topological principles of self-correcting. Problemy Kibernetiki, 21, (1969), 5--102.

7. Graham N., Harary F., Livingston M. and Stout Q. Subcube Fault-Tolerance in Hypercubes. Information and Computation 102 (1993), pp. 280{314.

8. Glagolev V. V. Some Estimations of D.N.F. for functions of Algebra of Logic. Problemy Kibernetiki, 19, (1967), 75--94.

Authors' Information Levon Aslanyan Institute for Informatics and Automation Problems, NAS Armenia, P.Sevak St. 1, Yerevan-14, Armenia; e-mail: lasl@sci.am Juan Castellanos Dpto. Intelligencia Artificial Facultad de Informatica Universidad Politecnica de Madrid 28660 Boadilla del Monte Madrid, Spain; e-mail: J.Castellanos@fi.upm.es Fourth International Conference I.TECH 2006 DNA SIMULATION OF GENETIC ALGORITHMS: FITNETSS COMPUTATIONMaria Calvino, Nuria Gomez, Luis F. Mingo Abstract: In this paper a computational mode is presented base on DNA molecules. This model incorporates the theoretical simulation of the principal operations in genetic algorithms. It defines the way of coding of individuals, crossing and the introduction of the individuals so created into the population. It resolves satisfactorily the problems of fitness coding. It shows also the model projection for the resolution of TSP. This is the basic step that will allow the resolution of larger examples of search problems beyond the scope of exact exponentially sized DNA algorithms like the proposed by Adleman [Adleman, 1994] and Lipton [Lipton, 1995].

Keywords: Genetic Algorithms, Fitness Function, DNA Computing, Evolutionary Computing.

Introduction Deoxyribonucleic acid (DNA) is the chemical inside the nucleus of all cells that carries the genetic instructions for making living organisms. A DNA molecule consists of two strands that wrap around each other to resemble a twisted ladder. The sides are made of sugar and phosphate molecules. The rungs are made of nitrogencontaining chemicals called bases. Each strand is composed of one sugar molecule, one phosphate molecule, and a base. Four different bases are present in DNA - adenine (A), thymine (T), cytosine (C), and guanine (G).

The particular order of the bases arranged along the sugar - phosphate backbone is called the DNA sequence;

the sequence specifies the exact genetic instructions required to create a particular organism with its own unique traits. Each strand of the DNA molecule is held together at its base by a weak bond. The four bases pair in a set manner: Adenine (A) pairs with thymine (T), while cytosine (C) pairs with guanine (G). These pairs of bases are known as Base Pairs (bp).

DNA and RNA computing is a new computational paradigm that harnesses biological molecules to solve computational problems. Research in this area began with an experiment by Leonard Adleman [Adleman, 1994] in 1994 using the tools of molecular biology to solve a hard computational problem. Adlemans experiment solved a simple instance of the Travelling Salesman Problem (TSP) by manipulating DNA. This marked the first solution of a mathematical problem with the tools of biology.

Computing with DNA generated a tremendous amount of excitement by offering a brand new paradigm for performing and viewing computations. The main idea is the encoding of data in DNA strands and the use of tools from molecular biology to execute computational operations. Besides the novelty of this approach, molecular computing has the potential to outperform electronic computers. For example, DNA computers may use a billion times less energy than electronic computers, while storing data in a trillion times less space. Moreover, computing with DNA is highly parallel: in principle there could be billions upon trillions of DNA or RNA molecules undergoing chemical reactions, that is, performing computations, simultaneously. Some advantages of DNA are that it is both self-complementary, allowing single strands to seek and find their own opposite sequences, and can easily be copied. Also, molecular biologists have already established a toolbox of DNA manipulations, including restriction enzyme cutting, ligation, sequencing, amplification, and fluorescent labelling, giving DNA a head start in the arena of non-silicon computing.

This work has been partially supported by Spanish Project TIC2003-9319-c03-03 Neural Networks and Networks of Evolutionary Processors.

Knowledge Engineering Despite the complexity of this technology, the idea behind DNA computing springs from a simple analogy between the following two processes, one biological and one mathematical: the complex structure of a living organism ultimately derives from applying sets of simple instructed operations (such as copying, marking, joining, inserting, deleting, etc.) to information in a DNA sequence, and computation is the result of combining very simple basic arithmetic and logical operations.

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