3Vladimir Krasilenko, Alexander Nikolsky, Sergei Pavlov: THE ASSOCIATIVE D-MEMORIES BASED ON MATRIX-TENSOR EQUIVALENTAL MODELS II.58486688 UDK 4.93:7.5 THE ASSOCIATIVE D-MEMORIES BASED ON MATRIX-TENSOR EQUIVALENTAL MODELS Vladimir Krasilenko, Alexander Nikolsky, Sergei Pavlov 4ｨ 99ｨｬｨ ｨｰ8ｨｨ9ｨ 6ｨｰ97 78ｨｨｨｪｨｨ74 769ｨｰ86ｨｪｨｨ7 ｨ c96ｨｨｨ ｨｰｨｨ9ｨｪ6ｨｦ 7ｨ ｨｬ7ｨｰｨｨ ｨｨ ｨｪｨｦ86ｨｪｨｪ4 9ｨｰｨｦ 57 9ｨｮｨｬ8ｨｪ4 ｨｨ46ｨ 8ｨ 3ｨｪｨｨｨｦ ｨｪｨ 69ｨｪ69 ｨｬｨ ｨｰ8ｨｨ ﾂｨｪ6-ｨｰｨｪ468ｨｪ4 5ｨｺ9ｨｨ9ｨ 5ｨｪｨｰｨｪ69ｨｰｨｪ4 ｨｬ65ｨｦ. 38563ｨｪ4 ｨｬ65ｨｨ 9 ｨ ｨ 7ｨｰｨｨ9ｨｪ6-5ｨｺ9ｨｨ9ｨ 5ｨｪｨｰｨｪ69ｨｰｨｪ4ｨｬ 9493ｨｨ9ｨ ｨｪｨｨｨｬ, 796576ｨｴｨｨ ｨｮ95ｨｨ ﾂｨｨｨｰｨｹ 6ｨ ｨｹｨｬ 7ｨ ｨｬ7ｨｰｨｨ 78ｨｨ 8ｨ ｨｪｨｪｨｨｨｨ ｨｨ 8ｨ 97ｨｪｨ 9ｨ ｨｪｨｨｨｨ 9496ｨｺ6ｨｺ6885ｨｨ869ｨ ｨｪ4 9ｨｮｨｬ8ｨｪ4 6ｨ 8ｨ 469, 9ｨｺ56 ﾂｨ 7 9ｨｮ8ｨ ｨ ｨｨ6ｨｪｨｪ6 ｨｨ ｨｬｨｪ668ｨ ｨ ｨｨ6ｨｪｨｪ6 ｨｨ46ｨ 8ｨ 3ｨｪｨｨ7. ｨ 9ｨｮ3ｨ 6ｨｰ97 ｨ 8ｨｨｨｰｨｺｨｰｨｮ84 ｨｨ 9ｨｬ63ｨｪ4 8ｨ 5ｨｨ4ｨ ｨｨｨｨ ｨｪｨｦ86ｨｪｨｪ4 9ｨｰｨｦ ｨｨ ｨ 96ｨｨｨ ｨｰｨｨ9ｨｪ6ｨｦ 7ｨ ｨｬ7ｨｰｨｨ, ｨ 6ｨｰ97 6ｨｪｨｺｨｨ ｨｨ 76ｨｺｨ 4ｨ ｨｰ5ｨｦ. The new principles of development of associative memory (AM) and neural networks (NN) for D-mage based on matrixtensor equivalental models (MTEMs) are considered. The given models use -D image adaptive-equivalentive weighting in order to increase memory capacity while storing highly correlated Dimages. The architectures of NN and AM are also suggested. Basic devices for implementations on the base of matrix-tensor equivalental model are matrix-tensor equivalentors (MTE). The MTEMs for two level and multilevel D-images are shown. The results of recognition processes modeling in such AM are given on the example of D-image recognition with the number of neuron from - thousand. A successful recognition of correlated image is achieved. Key words: optical neural networks, associative memory, matrix-tensor equivalentor, equivalental model, matrix multilevel logic.. INTRODUCTION The main perspective tendency of development of information-calculating systems and computer technologies is making them intellectual and similar to human thinking and perception []. Demonstration of the intelligence - an associative call from the memory by fragments (features), classification of patterns, their recognition, training, adaptation to the situation, (processed data, necessary precision, etc.), auto-regulation and auto-control - appears in highly developed systems (biological and technical) spontaneously. As the models of intellectual systems in the field of neurophysiology, cognitive psychology, artificial intelligence (AI), the next models of auto-processing nets are mainly used: connection models, neural nets models, models with parallel distributed processing, models with activity distribution []. Though there are differences and approaches peculiar to different fields of knowledge, where such models are used, basic researches reveal a number of important general principles besides spontaneous intellectual qualities. The principle of parallel information process on all the levels of working up in intellectual systems and the concept of active (not passive) memory (the functions of storage and associative processing are distributed among elements, which form a system) are related to such features. The model of neural net or the neural model is a connectional model, which imitates biophysical processes of working up of information in nervous system like in auto-processing system. The last one shows global system behavior (considered being "intelligent") caused by simultaneous local interactions of its numerous elements []. The great interest to neural model forces to revalue the fundamental theses in many fields of knowledge, including computer techniques. The neural models, concepts and paradigms with their new principles and wide parallel processing are almost the only way of development in creation of new nontraditional computer technique and active dynamic systems [3]. In science neural models, as the models of brain activity and cognitive processes, most probably will cause prospective results in neurophysiology, pathologophysiology, neurology, psychiatry, the appearance of the systems with increased computing resources and intelligent qualities for solving problems of medical informatics and diagnostics. Many failures on the way of improving of artificial intelligence appeared in recent years because firstly, the chosen computing techniques were not adequate to solve the important and complicated problems, and secondly, simple and not perfect neural models and nets were applied. Today, an active development of mathematical logic, especially matrix (multiciphered, fuzzy, neural) [4-8], accumulation of data about continual (analog) and obviously nonlinear functions of neurons [9-], elaboration of the neural net theory, neurobiology and neurocybernetic, and adequate algebrologic instruments for mathematical description and modeling [- 5], development of optical technologies create conditions for building technical systems, adequate in resources and architecture almost to any problem of artificial intelligence. The integration of data of neurophysiology, neuroanatomy, neurogenetic, neurochemistry, neurolinguistics i.e. neuroscience will become a basis of further development of neural intelligence and neural computers. Generalization of data of neural science makes possible to mark out a number of basic neurointellectual structures, among which associative memory of D-images takes an important place. It makes possible 45
358486688 to form a hypothesis about the object by a set of features, fragments of image and to extract an integrated pattern from the memory. Nervous system has a faculty to self-organization. Talking about active character of recognition processes in associative memory, processes of training and synthesis of internal images, the possibility to choose, to throw off (decrease) or on the contrary to add (increase) separated features, fragments, to change their weights while forming adequate internal images are meant here. In many neural models (with the exception [5]) used for optical realization of different associative devices and neural nets, only carrier sets Cb {ｨC ; } N and Cu { ; } N are used in bipolar (b) and unipolar (u) coding respectively [,3]. Questions connected with increase of neural nets and associative memory (AM) capacity, especially storing large and greatly correlated images, were solving for a long time by many authors [6-8]. The methods and realization of effective recognition of greatly correlated vectors of memory, based on weighing input images [7-8] and weight of interconnections [5, 6], were proposed in [6-8]. But these propositions concerned only coding of binary images and one-dimensional images. Models, proposed in [3-5] and called equivalental, are more general and good for representation of bipolar and unipolar signals, including multilevel signals. The connections, especially braking, are described more natural there. In such models the basic operation is a standard equivalency of vectors. The models are suitable for different methods of weighing. Considering their prospects, in this work we will show how to build associative (auto-associative and hetero-associative) memory of correlated -D images, including multilevel (gray scale) images, on the basis of matrix-tensor models.. CONCEPTUAL BACKGROUND AND THEORY.. Basic neurological operations of normilixed equivalence For mathematical description of neural net associative memory (NNAM) algebro-logical operations will be considered (equivalental algebra [3-5]), combining linear algebra and neural bio-logic (NBL) [4,6]. Neural bio-logic is an integration (gnoseologically developed and specified) of known logic: multivalued, hybrid, continuous, controlled continuous [], fuzzy etc. At the same time, the integrated operations in fuzzy logic are: operation of fuzzy negation, t- norms and s-norms, and they have the relation of dualism according to general form of De Morgan principle. The examples of t-norms are: logical multiplication (min), algebraic multiplication ( a 66 b), limited multiplication etc. The examples of s-norms are logical sum (max), algebraic sum ( a + b ｨC a 66 b), limited sum ( ﾄ ( a + b) ), contrast sum etc. [] The basic operations of NBL, used in equivalental models NNAM [3-5], are binary operations of equivalence and nonequivalence, which have a few variants. The variants of these equivalence operations on a carrier set are shown on fig. (a, b, c) respectively for: ﾄ C ﾅ eq a 6ﾕ5 b max{ min( ab, ), min( a, b) }; eq a 6ﾕ5 b a 66 b+ a66 b ; eq 3 a 6ﾕ5 + b + a ｨC b. () a E E3,5,8,6,4,,8,6,4, eq ﾅ a6ﾕ5 b eq a 6ﾕ5 b eq 3 a 6ﾕ5 + b,6,4, Figure - Operations of equivalence,8,5,5 u [ ; ] N E a a 46 ISSN 67-374 ｰ4ｨ 665ｨｺｨｰ86ｨｪ6ｨｺｨ. 5ｨｪ68ｨｬｨ ｨｰｨｨｨｺｨ. 778ｨ 956ｨｪｨｪ7 ｱ,
3Vladimir Krasilenko, Alexander Nikolsky, Sergei Pavlov: THE ASSOCIATIVE D-MEMORIES BASED ON MATRIX-TENSOR EQUIVALENTAL MODELS Their negations are first, second and third nonequivalence respectively. In general case for scalar variables a, b ﾊ ﾄ C [ A, B] - continuous line segment, signals ﾄ themselves and their functions and segment C can be brought to segments [-D,D] (or [-,]) in bipolar coding and [,D] (or [,]) in unipolar coding. Further a carrier set ﾄ C [, ] and its variables will be considered. Besides, for easier transformations we can limit ourselves at first to operation of equivalence (nonequivalence) of the second type, namely ( 6ﾕ5 ) and ( 6ﾕ5 ) respectively. Extending this basic operation ( ) for vector, matrix case of variables and making a corresponding normalization for quality registration of the components in vector or matrix, we will get normalized equivalence ( 9) of two sets of variables A a ij m ﾁ n and B b ij m ﾁ n A 6ﾕ5 n B m n m ﾁ n ( a 6ﾕ5 b ) ij ij and normalized nonequivalence: i j A 6ﾕ5 B n m n m ﾁ n ( a 6ﾕ5 b ) ij ij as integrated i j negation of normalized equivalence. If excitation matrix (input D image) is S inp, weight matrix of connections of k, l-th neuron-equivalent with input image is T k, l, then namely this neuron-equivalent will complete operation ( 6ﾕ5 ), because the signal on its output n ｦﾂ kl will be: ｦﾂ kl S inp 6ﾕ5 T, and () n kl ｦﾂ kl ﾊ ﾄ C [, ] The operation ( 6ﾕ5 ) will be completed by dual neuron-nonequivalent with the signal ｦﾂ kl ﾊ ﾄ C on its n output: ｦﾂ kl ｨC ｦﾂ kl S inp 6ﾕ5 T n kl (3) Let us mark out, that with the help of properties of equivalence operations [5] the signals (direct and dual) can be also represented on the output of integrated (dual) neuronequivalent (nonequivalent) in such a way: ｦﾂ kl S inp 6ﾕ5 T,,, n k, l ｦﾂ kl S inp 6ﾕ5 T n k, l ｦﾂ kl S inp 6ﾕ5 T n kl ｦﾂ kl S inp 6ﾕ5 T,, (4) n kl ｦﾂ kl S inp 6ﾕ5 T n kl ｦﾂ kl S inp 6ﾕ5 T n kl Therefore, if such integrated neurons have dual inputs and dual outputs, then the last ones can be defined (each one) in four ways considering () ﾂ (4). And that, maybe, provides correspondent vitality of neural nets at the expense of such a repeated backup. Normalized equivalence ( 6ﾕ5 ) and nonequivalence ( ) are n 6ﾕ5 n more general new complementary metrics in matrix space R. + In particular, ( 6ﾕ5 n ) is a normalized metric distance d ( A, B) 6ｨ8 m ﾁ n, and for A and B ﾊ {, } N it turns into normalized distance of Hamming d n ( A, B) 6ｨ8 N. The variants of operations of equivalence and nonequivalence depend on different types of operations of t-norms and s-norms used in them and integrated operations of crossing and joining up in fuzzy logic. Depending on type, variants of equivalent algebra (EA) [5], as a new algebro-logical instrument for creation of equivalental theory of NNAM on the basis of matrix NBL... Nonelinear transformation The basic operation of NBL with variable a i, j from A ﾄ a i, j range of continuous normalized set can be an operation of integrated nonlinear m ﾁ n C [, ] m ﾁ n transformation ｦﾐ( a, ｦﾁ) with coefficient ｦﾁ: ｦﾑ' ( a, ｦﾁ) a ｦﾁ, ｦﾑ'' ( a, ｦﾁ) ( a) ｦﾁ ( ｨC a) ｦﾁ, ｦﾑ''' ( a, ｦﾁ) ｦﾑ' ( a, ｦﾁ) ( ｨC a) ｦﾁ, ｦﾑ'''' ( a, ｦﾁ) ｦﾑ'' ( a, ｦﾁ) ( ｨC a) ｦﾁ ｨC ( ｨC a) ｦﾁ, a ﾊ [, ], ｦﾁ,, ｭ ﾞ. One should mark out that for ｦﾁ operation p''' is in kernel a negation operation in continuous and fuzzy logic, p'' is also a negation operation, and ｦﾑ'''' p'. Let us introduce two more iteration equivalental operations of nonlinear transformation, defined in the next way: ｦﾃ( a, ｦﾁ) 6ﾕ5 max( a, ｦﾁ) 6ﾕ5 max( a, ｦﾁ) 6ﾕ5 ｭ 6ﾕ5 max( a, ｦﾁ), ｦﾁ ｨC time ｦﾃ( a, ｦﾁ) 6ﾕ5 min( a, ｦﾁ) 6ﾕ5 min( a, ｦﾁ) 6ﾕ5 ｭ 6ﾕ5 min( a, ｦﾁ).(5) ｦﾁ ｨC time From (5) can be seen that the second operation is a negation of the fist one and vise versa. Besides, for a >, 5 fist ｦﾃ > and second ｦﾃ > functions will be: ｦﾃ > ( a, ｦﾁ) 6ﾕ5 a 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5 a ｦﾁ ｨC time 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5 a, (6) ｦﾃ > ( a, ｦﾁ) 6ﾕ5 a 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5 a 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5 a, and for ｦﾁ ｨC time ｦﾁ ｨC time a<,5 these functions ｦﾃ < and ｦﾃ < will be: ｦﾃ > ( a, ｦﾁ) 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5 a, ｦﾁ ｨC time ｦﾃ > ( a, ｦﾁ) 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5a. (7) ｦﾁ ｨC time ｦﾁ ｨC time 47
358486688 Hence it is obvious that for all a ﾊ [, ], ｦﾃ( a, ｦﾁ) ﾝ, 5 and ｦﾃ( a, ｦﾁ) ﾜ, 5. By analogy let us determine these functions of variable a : ｦﾃ( a, ｦﾁ) ｦﾃ( a, ｦﾁ), ｦﾃ( a, ｦﾁ) ｦﾃ( a, ｦﾁ). (8) From (8) it follows that these functions are symmetric concerning variable aa ( ). Weighing (equivalently) these functions of variable, we will get nonlinear iterative transformation, reducing ratio between signals a and a. We call it competitive nonlinear transformation and let us define it taking into account properties of the operations: aｦﾁ kn a 6ﾕ5 ｦﾃ( a, ｦﾁ) a 6ﾕ5 ｦﾃ( a, ｦﾁ) a 6ﾕ5 ｦﾃ( a, ｦﾁ) a 6ﾕ5 ｦﾃ( a, ｦﾁ), aｦﾁ kn a 6ﾕ5 ｦﾃ( a, ｦﾁ) a 6ﾕ5 ｦﾃ( a, ｦﾁ) a 6ﾕ5 ｦﾃ( a, ｦﾁ) a 6ﾕ5 ｦﾃ( a, ｦﾁ). (9) Expression (9) can be written in more convenient way taking into consideration (6) and (7): aｦﾁ kn Or in another way: 6ｦﾃ( a, ｦﾁ + ), for a > 5,, 6 63ｦﾃ( a, ｦﾁ + ), for a < 5,, aｦﾁ 6ｦﾃ( a, ｦﾁ + ), for a > 5,, kn 6 63ｦﾃ( a, ｦﾁ + ), for a <, 5. aｦﾁ kn 6 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5 a, for a >, 5, ( ｦﾁ + ) ｨC time 6 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5 a, for a <, 5, 63 ( ｦﾁ + ) ｨC time aｦﾁ 6 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5 a, for a >, 5, kn 6 63 6ﾕ5 a 6ﾕ5 ｭ 6ﾕ5 a, for a <, 5. () It can be seen from () that any variable a or a of the range >,5 sums equivalently with itself ｦﾁ times at competitive nonlinear transformation, and a variable of the range <,5 - sums nonequivalently ｦﾁ times with itself. At the same time the indicators for ( ｦﾁ + ) equivalence (nonequivalence) in corresponding ranges for variable aｦﾁ kn and its negation aｦﾁ kn respectively are "" and "" for ranges (>,5) and (<,5) in a case and "" and "" for the same ranges respectively in a case. Hence, we can see that different nonlinear transformations can be also expressed by operations of equivalence (nonequivalence) or by t-norms in case of ｦﾑ i ( a, ｦﾁ) transformation..3 Matrix-tenzor equivalental models (MTEM) of NNAM For simplification at first let us consider NNAM for associative recognition and reading of two-level (binary) -D images. Input, output, and q th trained images are respectively S inp Sinp ij ﾊ {, } N m ﾁ n, S out Sout ij ﾊ {, } N, S q Sq ij ﾊ {, } N, where q ﾊ { ﾂ Q}. Then tensor of weights of interconnections q from input neurons to output will be T ﾊ [, ] N ﾁ N, Q q q where T i, j, i', j' --- for simple NNAM, Q ( S ij, 6ﾕ5 S i', j' ) q where ﾜ i ﾜ m, ﾜ j ﾜ n, ﾜ i' ﾜ m, ﾜ j' ﾜ n. Considering determination of normalized equivalency ( 6ﾕ5 ) the component of interconnection tensor can be assign n as: q Ti, j, i', j' Si, j6ﾕ5 n Si', j', where Si, j Si', j' ( S ij, ｭS i, j ｭSQ i, j, S Q ( i', j' ｭS i', j' ) t, where T i, j, i', j' ﾊ {, 6ｨ8 Q ｭ ( Q ｨC ) 6ｨ8 Q, }. () To determine a new state of k, l-th neuron S out in (t+) moment it is necessary to consider contributions of all weights, and it means to calculate normalized equivalence between S inp () t and matrix (k, l-th tensor plane) T k, l [ T ij ] N k, l ﾍ[ x] 6, if x ﾝ x 6 63, if x < x and to take its threshold function The expression for net recalculation is: or considering duality: (function of activation) [5]. Sk, l out ( t + ) ﾍ[ S inp () t 6ﾕ5 T n k, l] () Sk, l out ( t + ) ﾍ[ S inp () t 6ﾕ5 T and n k, l] Sk, l out ( t + ) ﾍ[ S inp () t 6ﾕ5 T.(3) n k, l] ﾍ[ S inp () t 6ﾕ5 T n kl, ] Considering () expression () (let it be basic) can be transformed: Sk, l out ( t + ) ﾍ Q Q Q, Q --- S q ( kl, 6ﾕ5 ｦﾂ q ) ﾍ[ Sk, l6ﾕ5n ｦﾂ ] q Q where ｦﾂ q S inp () t S q n 6ﾕ5 ; ｦﾂ ( ｦﾂ, ｭｦﾂ q, ｭｦﾂ Q ) t. Combining all k, l-th outputs into S out ( t + ) matrix, one can write an expression for recalculation of equivalental model of simple NNAM, increasing threshold scalar operator 48 ISSN 67-374 ｰ4ｨ 665ｨｺｨｰ86ｨｪ6ｨｺｨ. 5ｨｪ68ｨｬｨ ｨｰｨｨｨｺｨ. 778ｨ 956ｨｪｨｪ7 ｱ,
3Vladimir Krasilenko, Alexander Nikolsky, Sergei Pavlov: THE ASSOCIATIVE D-MEMORIES BASED ON MATRIX-TENSOR EQUIVALENTAL MODELS ﾍ( x) to matrix pel-by-pel ﾍ( x) : Q Q S out ( t + ) ﾍ[ net k, l ] ﾍ[ [ Sk, l6ﾕ5n ｦﾂ ]] Q Q [ ﾍ kl [ Sk, l6ﾕ5n ｦﾂ ]] m ﾁ n N..4. MTEM of adaptive-equivalental NNAM (4) To improve the properties of the models, namely MTEM of NNAM during recognition of greatly correlated patterns, we will introduce two weighing coefficients ｦﾁp k, l and ｦﾃｦﾂ ( q, p) (in essence ｦﾁ matrix and ｦﾂkn p vector) and modify training so as weights of interconnections could be calculated in the next way: q ｦﾁｦﾃ, Tijkl,,, Q q p q --- T, (5) Q ( ijkl,,, 6ﾕ5 F k, l ( ｦﾁ p k, l ｦﾃｦﾂ ( q, p) )) q p q where F k, l - equivalental operation with variables ｦﾁ p and ｦﾃ( p) of p-th degree from coefficients 69 Q 6 + q ｦﾁ k, l 6--- S 63 ;. The Q kl, 6ﾕ5, 5 ｦﾂ 6 63 q S inp () t S q n 6ﾕ5 6 q first coefficient takes into consideration the "equivalence" (similarity) of the input images with one of the stored pattern images. The second one accounts the "equivalence" of the k-th and l-th pixel of the pattern images. The process of recalculation of MTEM of NNAM with such dual adaptive-equivalental weighing comes to matrixtensor procedures with operation of "equivalence": S out( t + ) ﾍ Q --- 6 p F q 69 N --- ( S q inp() t ﾁ T), N --- ( S ｦﾁ 6ﾕ5 inp ﾁ q T) 6 ﾁ 6ﾕ5 6 6 63 6ﾕ5 q tr65 ﾁ T 66, 67 here Sｦﾁ inp S inp 6ﾕ5 ｦﾁ p, p F q q p q p ｦﾂ ｦﾁkn 6ﾕ5 ｦﾂ kn (the other types of equivalence operations are possible). The expression for recalculation of the state of one k, l-th neuron (not the whole matrix) in MTEM of NNAM with adaptive-equivalental weighing has more evident form: then we can decompose this image into k-images, each having only two levels. Each two-level or -bit image is referred to as a bit plane. Using different codes for transformation of multilevel image in digit-bit plane, we can complete thus analog-digital transformation of images in any needed code: unit position code, unit normal code at morphological threshold decomposition, binary code at irredandunt coding, alternating code, codes of Fibonacci and others. The devices, which implement such transformation parallel for all pixels, called analog-digit image converters are considered in [, ]. For our MTEM of NNAM we used mainly two types of coding in AD-transformations of multilevel images. In the first case we used morphological threshold decomposition with programmed number of levels or bit planes and low (high) threshold levels. At that the whole set of digit levels ﾂ 55 for every gray level image (or for every of three main colors R, G, B) was transformed in programmed (as a rule 8 in our experiments) number of bit ordered planes. That led to actual compression of information, and multilevel image, having 56 levels, was transformed in multilevel image with less number of levels. But levels of the last one remained the same and that is why total dynamic range did not change. Applying the operation of pel-by-pel logic sum of corresponding ordered bit planes we form (of 8 planes) a result prepared (converted) bit plane (two-level image) (RCBP). Thus any input image and all trained multilevel images are represented by their RPBPs. The last ones are used as input image and trained images correspondingly in NNAM for twolevel images. In the second case we used 8-digit ADC of picture type (virtual) for transformation of multilevel images. For that purpose, a corresponding subprogram was written. In hardware realization and in this program we use a new algorithm of ADC and a new mathematical model on the basis on neural logic, but we will not describe then here in detail (we will do that in another work). One should note that all 8 bit planes (input, standard, output, gray level image) are processed parallel (or consecutively, virtually) and used during recognition with the help of 8 independent (possibly dependent) NNAM for two-level, two-gradation images. Hence, MTEM of NNAM with AE weighing for multilevel images in the first case of ADC does not differ from the one described in section.4., as since RPBP, similar by shape, but prepared one was used instead of ordinary D two-level images. For the second case the model consists of several two-level images, i.e. it is a superposition, combination of digit-by-digit bit planed models. Sk, l ﾍ out( t + ) Sk q, l 6ﾕ5 ( ｦﾁ p 6ﾕ5 S n q n inp 6ﾕ5 S q ) p N n N kn. (6) 3.SYSTEM DESIGN AND PROPOSED IMPLE- MENTATIONS.5. MTEM of NNAM with weighing for multilevel images There are basically two types of still images: two-level and multilevel. A two-level image is often called a "blackand-white" image, where as a multilevel image is usually called a gray level image. If we represent each pixel of a multilevel image by means of as k-bit binary code word, 3.. Systems of NNAM Fig. shows the structure scheme of neural net associative memory for multilevel D images with the first variant of transformation of input and trained multi-gradation (colour) D images into a result bit plane. Analog-digit converter (ADC) of picture type (PT) and digit-analog converters (DAC) of picture type perform necessary trans- 49
358486688 formations of multilevel images in a set of bit planes and visa versa. Schemes of coding and decoding of picture type (C of PT and DC of PT) complete transformation of the set of bit planes S out ﾂ S 7 out into one result two-level image RCBPinp and inversion of output RCBP out into a set of bit planes S out ﾂ S 7 out respectively. Trained multilevel images S q, being input into memory with the help of ADC of PT and C of PT, are transformed into RCBP q. That is why in ordinary addressed memory of page type only two-level (binary) images are stored. For the second variant of coding, according to section.5., the structure scheme will differ from the one on fig. only because it does not have coder and decoder, and the number of blocs of NNAM and blocs of memory of trained images MTI will be equal to the number of bit planes. Every i-th bit plane Si inp of input and S i ｭS i q ｭSi Q of trained images is processed on i-th NNAM i, which forms i-th bit plane Si out of output image on its output. All NNAM ﾂ NNAM 7 can work independently and jointly. In the last case the internal result measure of equivalence can be a certain complicated function (of vector normalized equivalence type) of particular i-th measures of equivalence of i-th bit planes. Taking into consideration limitation connected with the size of the article, let's concentrate on realization of basic associative memory with two-level -D images. 3.. Implementations As can be seen from section for simultaneous parallel calculation of all components of ｦﾂ Q vector or ｦﾂｦﾁ vector, net vector vector-matrix and matrix-tensor procedures with operation of equivalence are necessary to form result sets of values, data proportional to equivalence. Besides, for simultaneous calculation of all coefficients ｦﾁ ij, all functions p F q q p, ｦﾂ kn, ｦﾁ p, matrixes of (D-array) elements are required for component-by-component calculation of equivalence operations, nonlinear transformations, threshold processing, sum and so on. That is why proposed MTEMs of NNAM are easier represented on modern and progressive matrix architecture, multifunctional elements of matrix logic [4,7,3,] and processors of picture type [3, 4, 5]. One should mention that normalization may not be done, it is necessary to change threshold, taking function of activation. The device, which performs operation over S matrix and T tensor, namely S ﾁ ｹ T, will be called matrix-tensor equivalentor (MTE). According to elaborated in paragraph models (MTEMs of NNAM) the basic structure element for them will be just these MTEs. If ( 6ﾕ5) equivalency is used, then MTE can be built on ordinary digit matrix-tensor multipliers, including optical, as: (7) Fig. 3 shows architecture of NNAM on the basis of two matrix-tensor eqivalentors MTE and MTE. The first nonlinear converter NC of matrix type is used for competitive nonlinear transformation (expression (9) section..), and the second one NC of PT is used for threshold procession according to activation function. Input commutator-multiplexer M and output commutator-demultiplexer D are intended to input, output and form back propagation of iteration recalculations. ｹ ﾁ S ﾁ T S ﾁ T + S ﾁ T S inp multilevel D image ADC of PT. Sinp 7 Sinp input Neuro netvorks Coder image associative of memory. Decoder PT two-level D-images of RCBPinp. PT RCBPout teaching images memory teaching images (MTI) q ﾊ{,...Q} ｦﾁ (MTEM NNAM) S out 7 S out DA C of PT S out multilevel D image RCBP Figure - Schematic block diagram of associative memory for multilevel D-images 5 ISSN 67-374 ｰ4ｨ 665ｨｺｨｰ86ｨｪ6ｨｺｨ. 5ｨｪ68ｨｬｨ ｨｰｨｨｨｺｨ. 778ｨ 956ｨｪｨｪ7 ｱ,
3Vladimir Krasilenko, Alexander Nikolsky, Sergei Pavlov: THE ASSOCIATIVE D-MEMORIES BASED ON MATRIX-TENSOR EQUIVALENTAL MODELS Feedback Loop Sinp M q T [ ｦﾂ ] [ ] S ( t) () t int NC S out MTE MTE x x ｦﾂ p kn q T tr [ ﾍ ] DM S out NN AM D-images Figure 3 - Block diagram of NNAM Light SLM S inp Q times SLM 3 SLM 4 Phｧ h s SLM S ` S q All Q teaching images S q BS SLM 3 Sinp Q multiplaxing S inp SLM SLM LA SLM 4 S All Q teaching images S q Computer a) b) S Q Figure 4 - Optical system for MTE a) and inputting in SLM images b) In a fig.4. one of possible variants of optical realization MTE is shown. Input images Sinp are recorded from the managing computer in first spatial light modulators (SLM ) Q times, i.e. multiplicited. This multiplication is carried out at recording in SLM in the computer. Contrastly inverted input image S inp is also recorded in SLM 3 Q times. On the second SLM all Q trained images (S, ｭSq) are recorded and on the fourth SLM 4 - all Q trained contrastly inverted images ( S, ｭS q ). The lens array LA serves for spatial integration within each q-zone thus, on an input of every q-photodetector of array PDA a signal will be proportional to normalized equivalence of an input image with the trained S q. Non-linear transformation insfead of blocks NC and NC are shown in the circuit on fig3. They are carried out over the PDA output signals in the computer. Instead of PDA it is possible to use commercial liquid crystal television (LCTV) with 5x5 pixels and 3 ms refresh rates [6]. In this case it is possible to work with D-imagesdimantion 3x3 pixels and x teaching D-images, since on every LCTV it is necessary to record all Q images by dimension mxn. And it puts forward restriction on m ﾁ n< 5 ﾁ 5 6ｨ8 Q at given Q or restriction on Q < ( 5 ﾁ 5) 6ｨ8 ( m ﾁ n) with the given format of the images mxn D. Let's estimate productivity of such NNAM on the basis of optical realization of MTE. The recording in all SLM (LCTV) (fig4.) and updating given in each iteration is made simultaneously. Nonlinear processing simulating the work of NC and NC (fig.3) and recordings of the processed data in SLM can be combined. Let's consider therefore, that one iterative recalculation of a network the system has executed in 5 ms. At dimension of matrix of the input image 3x3 pixels or " 3 elements of a vector (image) the matrix of weight coefficients will have " 6 components or connections. It means that the system in 5 ms calculated as a matter of fact 6 connections. Thus, the rating show that productivity of NNAM realized on the simplest traditional circuits (fig.4) and on a basis slowly working LCTV archives 66 7 of connections/s. There is a quite real opportunity to increase capacity of SLM up to 5x5 (9 line pairs/ mm) pixel resolution and 5ms response time [7]. In this 5
358486688 case at dimension of D - image equal x pixels(and Q) in 5ms already 8 connections are calculated. Therefore ratings will show productivity of NNAM at a level 66 of connections/s. The common time of reading from associative memory does not exceed ( ｨC 3)ｦﾓ ter ﾖ 8c.The usage of other more high-speed The suggested matrix-tensor equivalental models MTEM s for construction on their basis of neuro-net associative memory of two-level and multilevel D-images, comprising models with doubled weighting, are of great importance, since they provide the opportunity to implement the whole number of competitive NNAP systems with increased capacity, productivity and fast acting. The result of modeling and experiment prove the validy of the theoretical work. Since suggested MTEM s are realized on the base of matrix tenzor (vectormatrix) procedures and multipliers and equivalentors it is possible to make a conclusion about necessity and, possibility of construction of complete digital optical NN associative memory and optical pattern of recognition systems. optoelectronic matrixes [3-5], calculating necessary operation of equivalence and matrix-tenzor procedures will allow to reduce significantly the time of reference to NNAM. In this work we wont concentrate on realizations in details, as our purpose is to show new most general principles of their realization. 4. RESULTS OF MODELING For modeling we developed a program realization of MTEMs models. Algorithm is realized in a program product on a low class PC. Technical characteristics this product: - any quantity of segments (elements, neurons) of a processed image with the dimensions in X x Y up to 3 neurons; - time of recognition of a image - part of a second to second; - a number of iterations (steps) necessary for recognition of a image - to 5; - training period depends only on time of data base insert and is significantly shorter than that of other well-known neuro - net paradigms; - correlation between the quantity of sample vectors and number of neurons in the network (network capacity) is -,5 times more (3%) than that in Hopfield networks (4-5%); - the level of distortion of image which permits its authentic recognition is up to 3%; - the number of gradation of images under process is 56 in each color, as well as black - and -white image; - quantity of digit layers at the process of image coding- 8 beat layers; - a possibility of space-invariant recognition of images; Mathematical models used are equivalental models. Metric system for comparison of images is non - Euclidean. A possibility can be fore seen of an adaptive regulation of the network for renewing and extending data base and for the optimization criterion and metric parameters. We shall gave it's characteristics below. The results of modeling are shown on fig. 5-8. a) 5. CONCLUSION b) for differ- Figure 5 - Output of neurons open layer ent p: a) p; b) p3. q p ｦﾂｦﾁ 5 ISSN 67-374 ｰ4ｨ 665ｨｺｨｰ86ｨｪ6ｨｺｨ. 5ｨｪ68ｨｬｨ ｨｰｨｨｨｺｨ. 778ｨ 956ｨｪｨｪ7 ｱ,
3Vladimir Krasilenko, Alexander Nikolsky, Sergei Pavlov: THE ASSOCIATIVE D-MEMORIES BASED ON MATRIX-TENSOR EQUIVALENTAL MODELS Figure 6 - Two-level D-images recognition results Figure 8 - Multilevel D-images invariant recognition results 6. REFERENCES Figure 7 - Multilevel -D images recognition on base of RCBP result. N.M. et al..amosov Neurostructures and Intelligent Robots. Naukova Dumka, Kiev, 99.. D. A. Redgia, G. G. Satton Autoprocessing Nets and Their Significance for Biomedical Researches. TIIER, vol. 76, 6, pp. 46-59, 988. 3. Freeman James a., D.M. Skapura Neural Networks: Algorithms, Applications and Programming Techniques, Addison- Wesley Publishing Company, 99. 4. V.G. Krasilenko, O.K. Kolesnitsky, A.K. Boguhvalsky "Creation Opportunities of Optoelectronic Continuous Logic Neural Elements, Which are Universal Circuitry Macrobasis of Optical Neural Networks ". Proc. SPIE, Vol. 7, pp. 8-7, 995. 5. V.I. Levin. "Continuous Logic, Its Generalization and Application". Automatica and Telemechanica, 8, pp. 3-, 99. 6. V.G. Krasilenko et. al. "Lines of Optoelectronic Neural Elements with Optical Inputs/Outputs Based on BISPIN-devices for Optical Neural Networks". Proc. SPIE, Vol. 7, pp. - 7, 995. 7. V.G. Krasilenko, A.T.Magas "Fundamentals of Design of Multifunctional Devices of Matrix Multiciphered Logic with Fast Programmed Adjusting". Measuring and Computer Technique in Technological Processes. 4, pp. 3-, 999. 8. A.S. Abdul Awwal, "Khan M. Iftekharuddin. Computer Arithmetic for Optical Computing. Special Section". Optical Eng. Vol. 38, 3, 999. 9. E.N. Sokolov, G. G.Vaytkyavichus. Neurointelligence: from Neuron to Neurocomputer. M.: Nauka, -38 p, 989.. N.V. Pozin Modeling of Neural Structures. M.: Nauka, - p. 97.. Yu.G. Antomonov Principles of Neurodynamics. Naukova 53
358486688 Dumka, Kiev, 974.. L.N. Volgin. "Complementary Algebra and Relative Models of Neural Structures with Coding of Channel Numbers". Electrical Modeling, Vol. 6, 3, pp. 5-5. 994. 3. V.G. Krasilenko, A.K.Bogakhvalskiy, A.T.Magas. "Equivalental Models of Neural Networks and Their Effective Optoelectronic Implementations Based on Matrix Multivalued Elements". Proc. SPIE, Vol. 355, pp. 7-36, 996. 4. V.G. Krasilenko et. al. "Applications of Nonlinear Correlation Functions and Equivalence Models in Advanced Neuronets". Proc. SPIE, Vol. 337, pp. -, 997. 5. V.G. Krasilenko et. al. "Continuous Logic Equivalental Models of Hamming Network Architectures with Adaptive-Correlated Weighting". Proc. SPIE, Vol. 34, pp. 398-48, 997. 6. Guo Doughui, Chen Zhenxiang, Lui Ruitaugh, Wu Boxi. "A Weighted Bipolar Neural Networks with High Capacity of Stable Storage". Proc. SPIE, Vol. 3, pp. 68-68. 7. B.Kiselyov, N.Kulakov, A.Mikaelian, V.Shkitin. 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A.Huang, "About Architecture of Optical Digit Computing Machine"//TIIER.-Vol. 7, 7.-pp. 34-4, 984. 78 6.74 36665868 96 ﾁ3587 5934 83566 6843448 757568 73489368 8.8.9ｨ 8ｨｺｨ 569, 85ｨｹ-4ｨ ｨ ｨｨ 8ｨｪｨ ｨｪ, 8.8.88ｨ 9ｨｨ ﾂｨｺ69 385ｨ ｨ ｨｰ97 ｨｬｨｰ6ｨｨｨｺｨ 67ｨｰｨｨｨｬｨｨ4ｨ ｨｨｨｨ 9ｨｬ ｨ 89ｨ ｨｨｨｨ ｨｺ6ｨｬ7ｨｨｨｨ6ｨｪｨｪ66 ｨｬｨｨｨｺ867868ｨ ｨｬｨｬｨｪ66 ｨｮ9ｨｰ86ｨｦ9ｨｰ9ｨ ｨｮ78ｨ 95ｨｪｨｨ7. ｨｪｨ 69ｨｪ69ｨ ｨｪｨ ｨｪｨ ｨｺ6ｨｨ869ｨ ｨｪｨｨｨｨ ｨｪ6ｨｬ869 78669 ｨｺ6ｨｪ ﾂｨｪ66 ｨ 9ｨｰ6ｨｬｨ ｨｰｨ. ｨ ｨｪｨｪｨ 7 ｨｬｨｰ6ｨｨｨｺｨ 79657ｨｰ ｨｮｨｬｨｪｨｹ3ｨ ｨｰｨｹ ｨ 77ｨ 8ｨ ｨｰｨｪ4 4ｨ ｨｰ8ｨ ｨｰ4, ｨｪ6ｨ 6ｨｨｨｬ4 ｨｪｨ 8ｨ 5ｨｨ4ｨ ｨｨ6 9ｨｬ ｨ 89ｨ ｨｨｨｨ. 38ｨｨ9ｨｪ 78ｨｨｨｬ8 ｨｨ9765ｨｹ469ｨ ｨｪｨｨ7. 38676ｨｪｨｮｨｰｨｹ97 ｨｬｨｰ6ｨｨｨｺｨ 67ｨｰｨｨｨｬｨ 67 9ｨｬ ｨ 89ｨ 67 ｨｺ6ｨｬ7ｨｨ6ｨｦｨｪ66 ｨｬ6ｨｺ867868ｨ ｨｬｨｪ66 78ｨｨ9ｨｰ866 ｨｺ8ｨｮ9ｨ ｨｪｨｪ7. 6ｨｪｨ 4ｨ 9ｨｪ69ｨ ｨｪｨ ｨｪｨ ｨｺ6ｨｮ9ｨ ｨｪｨｪ6 ｨｪ6ｨｬ869 78669 ｨｺ6ｨｪ966 ｨ 9ｨｰ6ｨｬｨ ｨｰｨ. ｨ ｨｪｨ ｨｬｨｰ6ｨｨｨｺｨ 9657 4ｨｬｨｪ3ｨｮ9ｨ ｨｰｨｨ ｨ 7ｨ 8ｨ ｨｰｨｪ6 9ｨｨｨｰ8ｨ ｨｰｨｨ, ｨｪ6ｨ 6ｨｪ6 ｨｪｨ 8ｨ 5ｨ 66 9ｨｬ ｨ 89ｨ 67. 38ｨｨ9ｨｪ6 78ｨｨｨｺ5ｨ 77 9ｨｨｨｺ68ｨｨ9ｨｰｨ ｨｪｨｪ7. The optimization method of microcommand's addressing circuits for compositional control unit is proposed. Method based on the encoding of transition's numbers of hard-logic automata. Given method allows to reduce the number of LSI-outputs in the addressing circuits. The method is illustrated by example. 778ｨ 9576ｨｴ ｨｮ9ｨｰ86ｨｦ9ｨｰ96 (77) ｨｨ8696ｨｦ 9ｨｨ9ｨｰｨｬ4 ｨｬ63ｨｰ ｨ 4ｨｰｨｹ 8ｨ 5ｨｨ469ｨ ｨｪ6 9 9ｨｨ ｨｺ6ｨｬ7ｨｨｨｨ6ｨｪｨｪ66 ｨｬｨｨｨｺ867868ｨ ｨｬｨｬｨｪ66 ｨｮ9ｨｰ86ｨｦ9ｨｰ9ｨ ｨｮ78ｨ 95ｨｪｨｨ7 (877) []. ｨｪｨ 9ｨｰ67ｨｴ 98ｨｬ7 57 8ｨ 5ｨｨ4ｨ ｨｨｨｨ 56ｨｨ ﾂ9ｨｺｨｨ 9ｨｬ 77 3ｨｨ86ｨｺ6 78ｨｨｨｬｨｪ76ｨｰ97 7868ｨ ｨｬｨｬｨｨ8ｨｮｨｬ4 56ｨｨ ﾂ9ｨｺｨｨ ｨｮ9ｨｰ86ｨｦ9ｨｰ9ｨ (397) []. 5ｨｪｨ ﾂｨｨｨｰ5ｨｹｨｪｨ 7 9ｨｰ6ｨｨｨｬ69ｨｰｨｹ 397 94449ｨ ｨｰ ｨｪ6ｨ 6ｨｨｨｬ69ｨｰｨｹ ｨｮｨｬｨｪｨｹ3ｨｪｨｨ7 ﾂｨｨ95ｨ ｨｬｨｨｨｺ869ｨｬ 9 9ｨｬ 77, 5ｨｨｨ 6 4ｨ ｨｬｨｪ4 ｨｬｨｨｨｺ869ｨｬ 7868ｨ ｨｬｨｬｨｨ8ｨｮｨｬ6ｨｦ 56ｨｨ ﾂ9ｨｺ6ｨｦ ｨｬｨ ｨｰ8ｨｨ4 (39) ｨｨ 7868ｨ ｨｬｨｬｨｨ8ｨｮｨｬ6ｨｦ ｨｬｨ ｨｰ8ｨｨ ﾂｨｪ6ｨｦ 56ｨｨｨｺｨｨ (39) ｨ 65 394ｨｬｨｨ 357. ｨｪｨ 9ｨｰ67ｨｴｨｦ 8ｨ ｨ 6ｨｰ 785ｨ ｨ ｨｰ97 ｨｬｨｰ6 67ｨｰｨｨｨｬｨｨ4ｨ ｨｨｨｨ 9ｨｰ6ｨｨｨｬ69ｨｰｨｨ 877, ｨｺ6ｨｰ684ｨｦ 78ｨｨｨｬｨｪｨｨｨｬ 78ｨｨ 9ｨｨｨｪｨｰ4 56ｨ 66 77 ｨｪｨ 9 ﾂｨｰ ﾂｨｨｨｺ [3]. 3ｨｮ9ｨｰｨｹ 57 8ｨ 9ｨｬ4 ｨ 568ｨｨｨｰｨｬｨ (58) 765ｨｮ ﾂｨｪ6 ｨｬｨｪ639ｨｰ96 678ｨ ｨｰ68ｨｪ4 5ｨｨｨｪｨｦｨｪ4 7ｨｦ (9) C { ｦﾋ, ｭ, ｦﾋ g }, ｦﾋ g ﾊ C - 769569ｨ ｨｰ5ｨｹｨｪ69ｨｰｨｹ 678ｨ ｨｰ68ｨｪ4 983ｨｨｨｪ 58 ｨｬ3ｨｮ ｨｺ6ｨｰ684ｨｬｨｨ ｨｪｨｰ ｨｮ9569ｨｪ4 983ｨｨｨｪ []. 8ｨ 3ｨ 7 9 ｦﾋ g ﾊ C ｨｨｨｬｨｰ 9 Ik g (K, ｭ,K g ) ｨｨ 6ｨｨｨｪ 946 O g. 678ｨ ｨｰ68ｨｪ4 983ｨｨｨｪｨ 58 4ｨ 7ｨｨ949ｨ 6ｨｰ97 ｨｬｨｨｨｺ86ｨｺ6ｨｬｨ ｨｪ4 (8) Y t 6ﾛ7 Y, Y { y, ｭ, y N } - ｨｬｨｪ639ｨｰ96 ｨｬｨｨｨｺ86678ｨ ｨｨｨｦ. ｨｮ9569ｨｪ4 983ｨｨｨｪｨ 58 4ｨ 7ｨｨ949ｨ 6ｨｰ97 55ｨｬｨｪｨｰ4 ｨｬｨｪ639ｨｰ9ｨ 56ｨｨ ﾂ9ｨｺｨｨ ｨｮ9569ｨｨｨｦ X { x, ｭ, x L }. 3ｨｮ9ｨｰｨｹ 9 785ｨ ｨｺｨ 36ｨｦ 9 ｨｬｨｨｨｺ86ｨｺ6ｨｬｨ ｨｪ4 ｨｨｨｬ6ｨｰ 969ｨｪｨｨ ｨ 89ｨ, ｨｰ6 9ｨｰｨｹ 95ｨｨ 8 Y t 4ｨ 7ｨｨ9ｨ ｨｪｨ 9 983ｨｨｨｪ b i, ｨ 8 - Y g 9 95ｨｮ6ｨｴｨｦ 4ｨ ｨｪｨｦ 983ｨｨｨｪ b j, ｨｰ6 AY ( g ) AY ( t ) +, () A(Y k ) - ｨ 89 ｨｬｨｨｨｺ86ｨｺ6ｨｬｨ ｨｪ4 Y k ( k ﾊ { t, g} ). 5ｨｰ6ｨｬ 95ｨｮ ﾂｨ 57 ｨｨｨｪｨｰ878ｨｰｨ ｨｨｨｨ 58 78ｨｨｨｬｨｪｨｨｨｬ6 877 (4ｨｨ9.), 789ｨｰｨ 9576ｨｴ 96ｨ 6ｨｦ ｨｺ6ｨｬ7ｨｨｨｨ6 ｨ 9ｨｰ6ｨｬｨ ｨｰｨ 9 "39ｨｰｨｺ6ｨｦ" 56ｨｨｨｺ6ｨｦ S (85, RG) ｨｨ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ 9 "7868ｨ ｨｬｨｬｨｨ8ｨｮｨｬ6ｨｦ 56ｨｨｨｺ6ｨｦ" S (56, 3). 877 ｨｮｨｪｨｺｨｨ6ｨｪｨｨ8ｨｮｨｰ 95ｨｮ6ｨｴｨｨｨｬ 6ｨ 8ｨ 46ｨｬ. 36 9ｨｨｨｪｨ 5ｨｮ "3ｨｮ9ｨｺ" 9 8ｨｨ9ｨｰ8 7ｨ ｨｬ7ｨｰｨｨ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S RG ｨｨ 9 9 ﾂｨｰ ﾂｨｨｨｺ 56 4ｨ ｨｪ697ｨｰ97 ｨｪｨｮ594 ｨｺ. 3ｨｮ9ｨｰｨｹ 9 ｨｬ6ｨｬｨｪｨｰ 98ｨｬｨｪｨｨ t 9 RG ｨｪｨ 67ｨｰ97 ｨｺ6 K(a m ) ｨｰｨｺｨｮｨｴ6 969ｨｰ67ｨｪｨｨ7 a m ﾊ A ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S, 8 {a, ｭ,a M }. 3ｨｮ9ｨｰｨｹ 9 ｨｬ6ｨｬｨｪｨｰ 98ｨｬｨｪｨｨ t 9 56 ｨｪｨ 6ｨｨｨｰ97 ｨ 89 A(Y i ) ｨｬｨｨｨｺ86ｨｺ6ｨｬｨ ｨｪ4, 967ｨｴｨｦ 9 9 ｦﾋ g ﾊ C. 395ｨｨ 8 Y i ｨｪ 7957ｨｰ97 9466ｨｬ 9 ｦﾋ g, ｨｰ6 ｨｨ4 ｨｮ78ｨ 9576ｨｴｨｦ 7ｨ ｨｬ7ｨｰｨｨ 73 9ｨｬ9ｨｰ 54 ISSN 67-374 ｰ4ｨ 665ｨｺｨｰ86ｨｪ6ｨｺｨ. 5ｨｪ68ｨｬｨ ｨｰｨｨｨｺｨ. 778ｨ 956ｨｪｨｪ7 ｱ,
38.8.9ｨ 8ｨｺｨ 569, 85ｨｹ-4ｨ ｨ ｨｨ 8ｨｪｨ ｨｪ, 8.8.88ｨ 9ｨｨ ﾂｨｺ69: 36665868 96 ﾁ3587 5934 83566 6843448 757568 73489368 9 ｨｬｨｨｨｺ86ｨｺ6ｨｬｨ ｨｪｨ ｨｬｨｨ y n ﾊ y i 94ｨ ｨｨ8ｨ ｨｰ97 9ｨｨｨｪｨ 5 y. 36 y ｨｺ 9683ｨｨｨｬ6ｨｬｨｮ 56 78ｨｨｨ ｨ 957ｨｰ97 ｨｨｨｪｨｨｨ, ｨ 89ｨｮ7 6 ﾂ8ｨｪｨｮ6 ｨｬｨｨｨｺ86ｨｺ6ｨｬｨ ｨｪｨｮ 9 ｦﾋ g. 38ｨｨ 5ｨｰ6ｨｬ 969ｨｰ67ｨｪｨｨ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S ｨｪ ｨｬｨｪ7ｨｰ97. 395ｨｨ 946 9 ｦﾋg 69ｨｰｨｨｨｪｨｮｨｰ, ｨｰ6 9ｨｨｨｪｨ 5 y ｨｪ 68ｨｬｨｨ8ｨｮｨｰ97. 5ｨｰ6ｨｬ 95ｨｮ ﾂｨ ｨｺ6ｨｬｨ ｨｨｨｪｨ ｨｨ6ｨｪｨｪｨ 7 9ｨｬｨ 85 68ｨｬｨｨ8ｨｮｨｰ ｨｮｨｪｨｺｨｨｨｨ 8 8(X, T), 76 ｨｺ6ｨｰ684ｨｬ 9 56 4ｨ ｨｪ69ｨｨｨｰ97 ｨ 89 96ｨ 6 ﾂ8ｨｪ6ｨｦ 9 ｦﾋ g ﾊC, ｨｨ 88(X, T), ｨｨ4ｨｬｨｪ76ｨｴｨｨ 969ｨｰ67ｨｪｨｨ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S. 59ｨｹ T{T, ｭ,T R } - ｨｬｨｪ639ｨｰ96 9ｨｪｨｮｨｰ8ｨｪｨｪｨｨ 78ｨｬｨｪｨｪ4, ｨｺ6ｨｨ8ｨｮ6ｨｴｨｨ 969ｨｰ67ｨｪｨｨ7 a m ﾊ A, R ]log M[. 8ｨｮｨｪｨｺｨｨ6ｨｪｨｨ869ｨ ｨｪｨｨ 4ｨ 983ｨ ｨｰ97 78ｨｨ 69ｨｰｨｨ3ｨｪｨｨｨｨ ｨｺ6ｨｪ ﾂｨｪ66 969ｨｰ67ｨｪｨｨ7 ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S 7695 68ｨｬｨｨ869ｨ ｨｪｨｨ7 99 ｨｬｨｨｨｺ86ｨｺ6ｨｬｨ ｨｪ 966ｨｰ9ｨｰ9ｨｰ9ｨｮ6ｨｴｨｦ 5ｨｰ6ｨｬｨｮ 969ｨｰ67ｨｪｨｨ6 9. ｨｪ6ｨｬ8ｨ 786ｨ 9 ｨｮｨｪｨｺｨｨｨｨ 8 ｨｨ ｦﾗ. 6ｨ ｨｺ6ｨｦ 766 76863ｨ ｨｰ 877 9 786ｨ 8ｨ 469ｨ ｨｪｨｨｨｬ ｨｪ6ｨｬ8ｨ 786ｨ (8ｨｨ9. ), ｨｮｨｪｨｺｨｨ6ｨｪｨｨ8ｨｮ6ｨｴｨｨ 95ｨｮ6ｨｴｨｨｨｬ 6ｨ 8ｨ 46ｨｬ. 395ｨｨ 9ｨｨｨｪｨ 5 y 68ｨｬｨｨ8ｨｮｨｰ97, ｨｰ6 786ｨｨ96ｨｨｨｰ ｨｪｨ 8ｨ ｨｴｨｨ9ｨ ｨｪｨｨ 9 ﾂｨｰ ﾂｨｨｨｺｨ. 786ｨｰｨｨ9ｨｪ6ｨｬ 95ｨｮ ﾂｨ 9ｨｬｨ 85 ｨｪ68ｨｬｨｨ8ｨｮｨｰ 78ｨｬｨｪｨｪ4 Z, ｨｺ6ｨｨ8ｨｮ6ｨｴｨｨ ｨｪ6ｨｬ8ｨ 78669 ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S. 386ｨ 8ｨ 469ｨ ｨｰ5ｨｹ ｨｺ669 38 68ｨｬｨｨ8ｨｮｨｰ 78ｨｬｨｪｨｪ4 88(Z) ｨｨ ｦﾗ ｦﾗ(Z), ｨｮ78ｨ 9576ｨｴｨｨ 56 ｨｨ RG 966ｨｰ9ｨｰ9ｨｰ9ｨｪｨｪ6, ﾂｨｰ6 78ｨｨ96ｨｨｨｰ ｨｺ 6 ﾂ8ｨｪ6ｨｬｨｮ 786ｨｮ 877. ｧｷ ｧｬｧｳ Z ｧｱｧｬ ｧｶ ｧｳｧｴ ｧｵｧｱ Y y ｧｷ ｧｬｧｳ ｧｶ ｧｱｧ蟋罘ﾜ ｧｳｧｴ ｧｵｧｱ Y RG ｧｴ RG ｧｴ 4ｨｨ9ｨｮｨｪ6ｨｺ - 5ｨｰ8ｨｮｨｺｨｰｨｮ8ｨｪｨ 7 9ｨｬｨ 877 9 786ｨ 8ｨ 469ｨ ｨｪｨｨｨｬ ｨｪ6ｨｬ8ｨ 786ｨ 4ｨｨ9ｨｮｨｪ6ｨｺ - 5ｨｰ8ｨｮｨｺｨｰｨｮ8ｨｪｨ 7 9ｨｬｨ ｨｺ6ｨｬ7ｨｨｨｨ6ｨｪｨｪ66 ｨｬｨｨｨｺ867868ｨ ｨｬｨｬｨｪ66 ｨｮ9ｨｰ86ｨｦ9ｨｰ9ｨ ｨｮ78ｨ 95ｨｪｨｨ7 4ｨ 487ｨｪ69ｨｰｨｹ ｨ 89ｨ ｨｬｨｨｨｺ86ｨｺ6ｨｬｨ ｨｪ4 R 67857ｨｰ97 ｨｺｨ ｨｺ ｨｮｨｪｨｺｨｨ7 6ｨｰ ﾂｨｨ95ｨ M 678ｨ ｨｰ68ｨｪ4 983ｨｨｨｪ 58: R ]log M [. 6ｨ ｨｺｨｨｨｬ 6ｨ 8ｨ 46ｨｬ, 9ｨｬｨ 85 ｨｨｨｬｨｰ R R + R () 94669. 3ｨｮ9ｨｰｨｹ t 397 - ﾂｨｨ956 94669 397 9ｨｬ4 85, ｨｰ6ｨ 78ｨｨ 94765ｨｪｨｪｨｨｨｨ ｨｮ9569ｨｨ7 R > t 397 (3) ｨｪ6ｨ 6ｨｨｨｬ6 94765ｨｪｨｨｨｰｨｹ 8ｨ 93ｨｨ8ｨｪｨｨ 397 76 946ｨ ｨｬ [3]. 6ｨｰ6 78ｨｨ96ｨｨｨｰ ｨｺ ｨｮ95ｨｨ ﾂｨｪｨｨ6 ﾂｨｨ95ｨ 397 9 9ｨｬ. 3ｨｮ9ｨｰｨｹ ｨ 9ｨｰ6ｨｬｨ ｨｰ S 786ｨｨ496ｨｨｨｰ H 78669, ﾂｨｰ6 67857ｨｰ 5ｨｨｨｪｨｮ 6 787ｨｬ6ｨｦ 9ｨｰ8ｨｮｨｺｨｰｨｮ8ｨｪ6ｨｦ ｨｰｨ ｨ 5ｨｨ4 (356), ｨｨ R 3 ]log H[. 38ｨｨ 94765ｨｪｨｪｨｨｨｨ ｨｮ9569ｨｨ7 R 3 ﾜ t 397 (4) 9ｨｰ6ｨｨｨｬ69ｨｰｨｹ 9ｨｬ4 877 ｨｬ63ｨｪ6 ｨｮｨｬｨｪｨｹ3ｨｨｨｰｨｹ. 57 5ｨｰ66 9 ｨ ｨｪｨｪ6ｨｦ 8ｨ ｨ 6ｨｰ 785ｨ ｨ ｨｰ97 999ｨｰｨｨ 786ｨ 8ｨ 469ｨ ｨｰ5ｨｹ ｨｪｨ 9ｨｰ67ｨｴｨｦ 8ｨ ｨ 6ｨｰ 785ｨ ｨ ｨｰ97 ｨｬｨｰ6ｨｨｨｺｨ 9ｨｨｨｪｨｰ4ｨ 877 9 786ｨ 8ｨ 469ｨ ｨｪｨｨｨｬ ｨｪ6ｨｬ8ｨ 786ｨ, 9ｨｺ56 ﾂｨ 7 95ｨｮ6ｨｴｨｨ 5ｨｰｨ 74:.868ｨｬｨｨ869ｨ ｨｪｨｨ ｨｬｨｪ639ｨｰ9ｨ 9 C, ｨ 89ｨ ｨｨ7 ｨｬｨｨｨｺ86ｨｺ6ｨｬｨ ｨｪ 965ｨ 9ｨｪ6 (), 68ｨｬｨｨ869ｨ ｨｪｨｨ 9683ｨｨｨｬ66 73 ｨｨ 356 ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S. 6ｨｰ6ｨｰ 5ｨｰｨ 7 94765ｨｪ7ｨｰ97 965ｨ 9ｨｪ6 ｨｬｨｰ6ｨｨｨｺ ｨｨ4 8ｨ ｨ 6ｨｰ4 [].. 86ｨｨ869ｨ ｨｪｨｨ ｨｪ6ｨｬ869 h 78669 F h 96ｨｨ ﾂｨｪ4ｨｬｨｨ ｨｺ6ｨ ｨｬｨｨ K(F h ) 8ｨ 487ｨｪ69ｨｰｨｨ R 3. 868ｨｬｨｨ869ｨ ｨｪｨｨ ｨｬｨｪ639ｨｰ9ｨ ｨｺ6ｨｨ8ｨｮ6ｨｴｨｨ 78ｨｬｨｪｨｪ4 Z. 386ｨ ｨｬ F h, 966ｨｰ9ｨｰ9ｨｰ9ｨｮ6ｨｴｨｨｨｬ 6ｨｨｨｪｨ ｨｺ694ｨｬ 7ｨ 8ｨ ｨｬ < 8, ｦﾗ > 9ｨｰｨ 9ｨｨｨｰ97 9 966ｨｰ9ｨｰ9ｨｰ9ｨｨ 6ｨｨｨｪ ｨｪ6ｨｬ8. 3. 868ｨｬｨｨ869ｨ ｨｪｨｨ 786ｨ 8ｨ 469ｨ ｨｪｨｪ6ｨｦ 356 ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S ｨｮｨ 5ｨｪｨｨｨｬ 9ｨｰ65ｨ 69 8 h ｨｨ ｦﾗ h, 9683ｨ ｨｴｨｨ ｨｮｨｪｨｺｨｨｨｨ ｨｮ78ｨ 95ｨｪｨｨ7 RG ｨｨ 56, ｨｨ 9966ｨｬ 9ｨｰ65ｨ ｨ Z h, 9683ｨ ｨｴｨｨ ｨｮｨｪｨｺｨｨｨｨ ｨｮ78ｨ 95ｨｪｨｨ7 38. 868ｨｬｨｨ869ｨ ｨｪｨｨ 9ｨｨ9ｨｰｨｬ4 ｨｮｨｪｨｺｨｨｨｦ Z Z (X, T). 4. 868ｨｬｨｨ869ｨ ｨｪｨｨ ｨｰｨ ｨ 5ｨｨ4 38 9 96ｨ ｨｬｨｨ Z ｨｨ 946ｨ ｨｬｨｨ 8 ｨｨ ｦﾗ. 868ｨｬｨｨ869ｨ ｨｪｨｨ 9ｨｨ9ｨｰｨｬ ｨｮｨｪｨｺｨｨｨｦ 88(Z) ｨｨ ｦﾗ ｦﾗ(Z). 5. 5ｨｨｨｪｨｰ4 56ｨｨ ﾂ9ｨｺ6ｨｦ 9ｨｬ4 877 9 4ｨ ｨ ｨｪｨｪ6ｨｬ 55ｨｬｨｪｨｰｨｪ6ｨｬ ｨ ｨ 4ｨｨ9. 5ｨｬｨ 85 8ｨ 5ｨｨ4ｨｮｨｰ97 ｨｪｨ 39 ｨｨ5ｨｨ 39, 9ｨｬ4 38 ｨｨ 73 - ｨｪｨ 357. 4ｨ 99ｨｬ6ｨｰ8ｨｨｨｬ 78ｨｨｨｬｨｪｨｪｨｨ 5ｨｰ6ｨｦ ｨｬｨｰ6ｨｨｨｺｨｨ ｨｪｨ 78ｨｨｨｬ8 9ｨｨｨｪｨｰ4ｨ 877, ｨ 9ｨｰ6ｨｬｨ ｨｰ S ｨｺ6ｨｰ6866 4ｨ ｨ ｨｪ 356 (6ｨ ｨ 5.). 55
358486688 6ｨ ｨ 5ｨｨｨ - 387ｨｬｨ 7 9ｨｰ8ｨｮｨｺｨｰｨｮ8ｨｪｨ 7 ｨｰｨ ｨ 5ｨｨｨ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S 877 S a m K(a m ) a s K(a s ) Xh ｦﾗ h 8h h 6ｨ ｨ 5ｨｨｨ - 386ｨ 8ｨ 469ｨ ｨｪｨｪｨ 7 787ｨｬｨ 7 9ｨｰ8ｨｮｨｺｨｰｨｮ8ｨｪｨ 7 ｨｰｨ ｨ 5ｨｨｨ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S 877 S a m K(a m ) Xh K(Y h ) Zh h a x D D 3 D 4 x - a a 3 x x D D 3 D 5 D 6 a x x Z 3 a x x D D 4 D 6 3 x x Z 3 a a x 3 D D 3 D 4 4 a 3 x 3 D D 3 D 6 5 a x 3-4 x 3 Z Z 3 5 a x D D 3 D 5 6 x Z 6 a 3 a 3 x x 4 D D 3 D 5 D 6 7 a 3 x x 4 Z 3 7 a x x 4 - D 4 D 5 8 x x 4 Z Z 3 8 59ｨｹ 8ｨｨ9ｨｰ8 RG ｨｨ 9 ﾂｨｰ ﾂｨｨｨｺ 56 ｨｨｨｬｨｰ 9 D-ｨｰｨｨ7ｨ, ｦﾗ {D,D }, 8{D 3, ｭ,D 6 }, R, R 4, R 6. 6ｨｰｨ ｨｺ, 965ｨ 9ｨｪ6 () 397 9ｨｬ4 85 653ｨｪ4 ｨｨｨｬｨｰｨｹ 6 94669. 696ｨｪｨ 7 356 ｨｨｨｬｨｰ H 8 9ｨｰ86ｨｺ, 6ｨｪｨ ｨｺ6 78 F ｨｨ F 4, F ｨｨ F 7 9697ｨ ｨ 6ｨｰ. 38ｨｨ996ｨｨｨｬ 786ｨ ｨｬ 95ｨｮ6ｨｴｨｨ ｨｪ6ｨｬ8ｨ : F ｨｨ F 4, F ｨｨ F 7, F 3 3, F 5 4, F 6 5, F 8 6. 6ｨ ｨｺｨｨｨｬ 6ｨ 8ｨ 46ｨｬ, ｨｨｨｬｨｰ97 H 6 8ｨ 45ｨｨ ﾂｨｪ4 ｨｪ6ｨｬ869, 57 ｨｺ6ｨｨ869ｨ ｨｪｨｨ7 ｨｺ6ｨｰ684 69ｨｰｨ ｨｰ6 ﾂｨｪ6 R 3 ]log H [ (5) 78ｨｬｨｪｨｪ4 Z q ﾊ Z, ｨｰ6 9ｨｰｨｹ Z{Z,Z,Z 3 }. 3ｨｮ9ｨｰｨｹ K(F )K(F 4 ), K(F )K(F 7 ), K(F 3 ), K(F 5 ), K(F 6 ), K(F 8 ). 4ｨ 5ｨｨ4ｨ ｨｨ7 769ｨｰ86ｨｪｨｪ6ｨｦ 356 (6ｨ ｨ 5.) 786ｨｨ496ｨｨｨｰ97 55ｨｬｨｪｨｰｨ 8ｨｪ6. 5ｨｨ9ｨｰｨｬｨ ｨｮｨｪｨｺｨｨｨｦ Z 9 6ｨ ｨｴｨｬ 95ｨｮ ﾂｨ ｨｨｨｬｨｰ 95ｨｮ6ｨｴｨｨｨｦ 9ｨｨ: H Z r V Crh Am Xh ( r h, ｭ, R 3 ), (6) 6ｨ ｨ 5ｨｨｨ 3-6ｨ ｨ 5ｨｨｨ 786ｨ 8ｨ 469ｨ ｨｰ57 ｨｺ669 ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S 877 S Zh 8 h ｦﾗ h h D 3 D 4 D, 4 D 3 D 5 D 6 D, 7 D 4 D 6 D 3 D 3 D 6 D 5 D 3 D 5 D 6 D 4 D 5-8 678694 8ｨ 5ｨｨ4ｨ ｨｨｨｨ ｨｺ6ｨｬｨ ｨｨｨｪｨ ｨｨ6ｨｪｨｪ4 9ｨｬ 76 ｨｰｨ ｨ 5ｨｨｨ ｨｬ 9 ｨ ｨ 4ｨｨ9 397, 69ｨｰｨ ｨｰ6 ﾂｨｪ6 6ｨｰ8ｨ 3ｨｪ4 9 7 ﾂｨ ｨｰｨｨ []. 59ｨｹ 5ｨｰｨｨ 9678694 ｨｪ 8ｨ 99ｨｬｨ ｨｰ8ｨｨ9ｨ 6ｨｰ97. 96ｨｨ ﾂ9ｨｺｨ 7 9ｨｬｨ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S 78ｨｨ9ｨｪｨ ｨｪｨ 8ｨｨ9.3, 49ｨｹ 85 9ｨｨｨｪｨｰ4ｨｨ869ｨ ｨｪｨ 76 9ｨｨ9ｨｰｨｬ (6). Crh - ｨ ｨｮ59ｨ 78ｨｬｨｪｨｪｨ 7, 8ｨ 9ｨｪｨ 7 ｨｨｨｪｨｨ, 95ｨｨ ｨｨ ｨｰ65ｨｹｨｺ6 95ｨｨ 9 h-ｨｦ 9ｨｰ86ｨｺ 356 Z r. Am - ｨｺ6ｨｪｨｲ6ｨｪｨｺｨｨ7 78ｨｬｨｪｨｪ4 T r ﾊ T, 966ｨｰ9ｨｰ9ｨｰ9ｨｮ6ｨｴｨ 7 ｨｺ6ｨｮ 969ｨｰ67ｨｪｨｨ7 a m ﾊ A ｨｨ4 h-ｨｦ 9ｨｰ86ｨｺｨｨ 356; Xh - ｨｺ6ｨｪｨｲ6ｨｪｨｺｨｨ7 56ｨｨ ﾂ9ｨｺｨｨ ｨｮ9569ｨｨｨｦ, 678576ｨｴｨ 7 h-ｨｦ 786 ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S. ｨ 78ｨｨｨｬ8, ｨｨ4 ｨｰｨ ｨ 5. ｨｨｨｬｨｬ Z A 3 x ﾅ A 3 x x 4 T T T 3 x ﾅ T T x x 4. x x x3 x4 T T 3 4 5 6 PLA ｧｬｧｳ 3 Z Z Z 3 3 y clock ROM ｧｱｧｬ & 3 4 5 6 ｧｱｧ蟋罘ﾜ D3 D4 D5 D6 D D D D D RD C RG T T 6ｨ ｨ 5ｨｨｨ 38 877 S (6ｨ ｨ 5.3) 9683ｨｨｨｰ 9ｨｰ65ｨ 4 Z h (96ｨｨ ﾂｨｪ4ｨｦ ｨｺ6 9ｨｰ86ｨｺｨｨ), 8 h ｨｨ ｦﾗ h (ｨｮｨｪｨｺｨｨｨｨ ｨｮ78ｨ 95ｨｪｨｨ7 RG ｨｨ 56), h - ｨｪ6ｨｬ8 9ｨｰ86ｨｺｨｨ 356. 6ｨ ｨ 5ｨｨｨ 38 7957ｨｰ97 ｨｰｨ ｨ 5ｨｨｨｦ ｨｨ9ｨｰｨｨｨｪｨｪ69ｨｰｨｨ ｨｮｨｪｨｺｨｨｨｦ 8 ｨｨ ｦﾗ. 395ｨｨ 38 8ｨ 5ｨｨ4ｨｮｨｰ97 ｨｪｨ 357, ｨｰ6 9ｨｨ9ｨｰｨｬ4 ｨｮｨｪｨｺｨｨｨｦ 8 ｨｨ ｦﾗ 6ｨｪｨｪｨ ﾂｨｪ6 4ｨ ｨ 6ｨｰ97 ｨｰｨ ｨ 5ｨｨｨｦ 38. 4ｨｨ9ｨｮｨｪ6ｨｺ 3-96ｨｨ ﾂ9ｨｺｨ 7 9ｨｬｨ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S 877 S ｨｰｨｬｨｰｨｨｨｬ, ﾂｨｰ6 9ｨｨｨｪ86ｨｪｨｨ4ｨ ｨｨ7 RG C y *Clock, Clock - 9ｨｨｨｪｨ 5 9ｨｨｨｪ86ｨｪｨｨ4ｨ ｨｨｨｨ 877. 56 ISSN 67-374 ｰ4ｨ 665ｨｺｨｰ86ｨｪ6ｨｺｨ. 5ｨｪ68ｨｬｨ ｨｰｨｨｨｺｨ. 778ｨ 956ｨｪｨｪ7 ｱ,
33..967ｨｪ9ｨｺｨｨｨｦ, 5..36769: 9 88366 9336366 6587556349 37449 53637 98 9489686 3449 5689 58897 ﾁ363 699569ｨ ｨｪｨｨ7 ｨ 9ｨｰ6869 76ｨｺｨ 4ｨ 5ｨｨ, ﾂｨｰ6 78ｨｨ 94765ｨｪｨｪｨｨｨｨ ｨｮ9569ｨｨｨｦ (3) ｨｨ (4) ｨｬｨｰ6 ｨｺ6ｨｨ869ｨ ｨｪｨｨ7 ｨｪ6ｨｬ869 78669 79657ｨｰ ｨｮｨｬｨｪｨｹ3ｨｨｨｰｨｹ 9ｨｰ6ｨｨｨｬ69ｨｰｨｹ 8ｨ 5ｨｨ4ｨ ｨｨｨｨ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S. 38ｨｨ 5ｨｰ6ｨｬ 5ｨｺ6ｨｪ6ｨｬｨｨ7 76943ｨ ｨｰ97 76 ｨｬ8 869ｨｰｨ 8ｨ 4ｨｪ69ｨｰｨｨ R - t 397 ｨｨ ｨｬ63ｨｰ 69ｨｰｨｨｨ ｨｰｨｹ 35-4% 76 98ｨ 9ｨｪｨｪｨｨ6 9 ｨｰ8ｨ ｨｨｨｨ6ｨｪｨｪ6ｨｦ 8ｨ 5ｨｨ4ｨ ｨｨｨｦ ｨ 9ｨｰ6ｨｬｨ ｨｰｨ S. 38563ｨｪｨｪ4ｨｦ ｨｬｨｰ6 78ｨｨ96ｨｨｨｰ ｨｺ ｨ 65 ｨｬ5ｨｪｨｪ4ｨｬ 9ｨｬｨ ｨｬ 877 ｨｨ4-4ｨ 99ｨｪｨｨ7 38, ｨｮ95ｨｨ ﾂｨｨ9ｨ 6ｨｴ6 98ｨｬ7 68ｨｬｨｨ869ｨ ｨｪｨｨ7 ｨｬｨｨｨｺ86678ｨ ｨｨｨｦ. 6ｨ ｨｺｨｨｨｬ 6ｨ 8ｨ 46ｨｬ, ｨｬｨｰ6 ｨｺ6ｨｨ869ｨ ｨｪｨｨ7 ｨｪ6ｨｬ869 78669 78ｨｨｨｬｨｪｨｨｨｬ, 95ｨｨ ｨｺ8ｨｨｨｰ8ｨｨｨｬ 5ｨｺｨｰｨｨ9ｨｪ69ｨｰｨｨ 9ｨｬ4 77 7957ｨｰ97 ｨｬｨｨｨｪｨｨｨｬｨｮｨｬ 9ｨｰ6ｨｨｨｬ69ｨｰｨｨ. 3343 ﾁ35 55498. 9ｨ 8ｨｺｨ 569 8.8. 3ｨ 5ｨ ｨｰｨｨｨｪ8.. 5ｨｨｨｪｨｰ4 ｨｬｨｨｨｺ867868ｨ ｨｬｨｬｨｪ4 ｨｮ9ｨｰ86ｨｦ9ｨｰ9 ｨｮ78ｨ 95ｨｪｨｨ7. - 8ｨｨ9: 68 8 7ｨｺ8ｨ ｨｨｨｪ4, 997-56 9.. 56569ｨｹ9.. 386ｨｺｨｰｨｨ869ｨ ｨｪｨｨ ｨｨ8694 9ｨｨ9ｨｰｨｬ ｨｪｨ 69ｨｪ69 7868ｨ ｨｬｨｬｨｪ4 56ｨｨ ﾂ9ｨｺｨｨ ｨｨｨｪｨｰ8ｨ 5ｨｹｨｪ4 9ｨｬ. -.: 687 ﾂｨ 7 5ｨｨｨｪｨｨ7-65ｨｺ6ｨｬ, - 636 9. 3. 9ｨ 8ｨｺｨ 569 8.8. 5ｨｨｨｪｨｰ4 ｨｮ9ｨｰ86ｨｦ9ｨｰ9 ｨｮ78ｨ 95ｨｪｨｨ7 ｨｪｨ 7868ｨ ｨｬｨｬｨｨ8ｨｮｨｬ4 56ｨｨ ﾂ9ｨｺｨｨ ｨｮ9ｨｰ86ｨｦ9ｨｰ9ｨ. - 6ｨｪｨｺ: 6ｨｪ67, - 6 9. 78 68.53.6 9 88366 9336366 6587556349 37449 53637 98 9489686 3449 5689 3..967ｨｪ9ｨｺｨｨｨｦ, 5..36769 4ｨ 99ｨｬ6ｨｰ8ｨｪｨ 4ｨ ｨ ﾂｨ 6ｨ ｨｲｨｨｨｪｨｪｨｨ7 ｨ ｨｪ9ｨ ｨｬｨ 57 ｨｨ9ｨｺｨｮ99ｨｰ9ｨｪｨｪ4 ｨｪｨｦ86ｨｪｨｪ4 9ｨｰｨｦ, 69ｨｮｨｴ9ｨｰ9576ｨｴｨｨ 6ｨ 8ｨ ｨ 6ｨｰｨｺｨｮ ｨｬｨｪ66ｨｬ8ｨｪ4 9ｨｨｨｪｨ 569, 4ｨ ｨ ｨｪｨｪ4 9 9ｨｨ ｨｨ9ｨｺ8ｨｰｨｪ4 98ｨｬｨｪｨｪ4 8769. 38563ｨｪ ｨ 568ｨｨｨｰｨｬ 6ｨ ｨｲｨｨｨｪｨｪｨｨ7 9ｨｰｨｦ ｨｪｨ ｨｺ6ｨｪ ﾂｨｪ6ｨｦ 94ｨ 68ｨｺ, ｨｨ 6ｨｺｨ 4ｨ ｨｪｨ 6 67ｨｰｨｨｨｬｨ 5ｨｹｨｪ69ｨｰｨｹ. 4ｨ 48ｨ ｨ 6ｨｰｨ ｨｪ4 8ｨｺｨｮ88ｨｪｨｰｨｪ4 786ｨｮ84, 78ｨｪｨ 4ｨｪｨ ﾂｨｪｨｪ4 57 8ｨ ｨ 6ｨｰ4 9 8ｨ 5ｨｹｨｪ6ｨｬ 98ｨｬｨｪｨｨ. 38563ｨｪ ｨ 568ｨｨｨｰｨｬ 57 6ｨｪｨｺｨｨ "9ｨｺ5ｨ ｨ " ｨｺｨ 36ｨｦ ｨｨ4 9ｨｰｨｦ, 967ｨｴｨｨ 9 ｨ ｨｪ9ｨ ｨｬｨ 5ｨｹ, 9 6ｨ ｨｲｨｨｨｪｨｪｨｪｨｮ6 6ｨｪｨｺｨｮ. 5ｨｨｨｪｨｰ4ｨｨ869ｨ ｨｪｨ ｨ 8ｨｨｨｰｨｺｨｰｨｮ8ｨ ｨｪｨｦ86ｨｪｨｪ6ｨｦ ｨｬｨｰｨ 9ｨｰｨｨ, 83ｨ 6ｨｴｨｦ 8ｨ 99ｨｬｨ ｨｰ8ｨｨ9ｨ ｨｬｨｮ6 786ｨ 5ｨｬｨｮ. 4ｨ 49ｨｨ9ｨ ｨｬ4ｨｦ 766 79657ｨｰ 76949ｨｨｨｰｨｹ ｨｰ6 ﾂｨｪ69ｨｰｨｹ 83ｨｪｨｨ7 4ｨ ｨ ﾂ 786ｨｪｨｨ869ｨ ｨｪｨｨ7, ｨｨ5ｨｹｨｰ8ｨ ｨｨｨｨ, 95ｨ 3ｨｨ9ｨ ｨｪｨｨ7, 5ｨｬｨｮ57ｨｨｨｨ, 6ｨ 8ｨ ｨｰｨｪ66 ｨｬ65ｨｨ869ｨ ｨｪｨｨ7 ｨｨ ｨｨｨｬ 766ｨ ｨｪ4, 57 83ｨｪｨｨ7 ｨｺ6ｨｰ684 ｨｨ9765ｨｹ4ｨｮ6ｨｰ ｨｨ9ｨｺｨｮ99ｨｰ9ｨｪｨｪ4 ｨｪｨｦ86ｨｪｨｪ4 9ｨｰｨｨ, 69ｨｪ69ｨ ｨｪｨｪ4 ｨｪｨ 7ｨ 8ｨ ｨｨｨｬ 6ｨ ｨｮ ﾂｨｪｨｨ7 9 ｨｮ ﾂｨｨｨｰ5ｨｬ. 457ｨｪｨｮｨｰ6 4ｨ ｨ ﾂｨｮ 6ｨ 'ｨｪｨ ｨｪｨｪ7 ｨ ｨｪ9ｨ ｨｬｨ 56 3ｨｰｨｮ ﾂｨｪｨｨ ｨｪｨｦ86ｨｪｨｪｨｨ ｨｬ83, ｨｴ6 46ｨｦ9ｨｪ66ｨｰｨｹ 6ｨ 86ｨ ｨｺｨｮ ｨ ｨ ｨ ｨｰ69ｨｨｨｬ68ｨｪｨｨ 9ｨｨｨｪｨ 569, 7ｨｺ6 4ｨ ｨ ｨｪ6 ｨｮ 9ｨｨ576 ｨｨ9ｨｺ8ｨｰｨｪｨｨ ﾂｨ 969ｨｨ 8769. 5ｨ 78676ｨｪ69ｨ ｨｪ6 ｨ 568ｨｨｨｰｨｬ 6ｨ 'ｨｪｨ ｨｪｨｪ7 ｨｬ83 ｨｪｨ ｨｺ6ｨｪ96ｨｦ 9ｨｨｨ 686, 6 69ｨｪ6 ｨｦ66 67ｨｰｨｨｨｬｨ 5ｨｹｨｪ69ｨｰｨｹ. 486ｨ 5ｨｪ6 8ｨｺｨｮ8ｨｪｨｰｨｪ6 786ｨｮ8ｨｨ 57 86ｨ 6ｨｰｨｨ 9 8ｨ 5ｨｹｨｪ6ｨｬｨｮ ﾂｨ 96. 5ｨ 78676ｨｪ69ｨ ｨｪ6 ｨ 568ｨｨｨｰｨｬ 57 66ｨｪｨｺｨｨ "9ｨｪ9ｨｺｨｮ" ｨｺ63ｨｪ67 4 ｨｬ83, ｨｴ6 967ｨｰｨｹ 6 ｨ ｨｪ9ｨ ｨｬｨ 56, 9 6ｨ 'ｨｪｨ ｨｪｨｮ 66ｨｪｨｺｨｮ. 5ｨｨｨｪｨｰ469ｨ ｨｪ6 ｨ 86ｨｰｨｺｨｰｨｮ8ｨｮ ｨｪｨｦ86ｨｪｨｪ67 ｨｬｨｰｨ ｨｬ836, ｨｴ6 89'74ｨｮ 857ｨｪｨｮｨｰｨｮ 786ｨ 5ｨｬｨｮ. 366, ｨｴ6 89ｨｨ9ｨ ｨｰｨｹ97, 9657 769ｨｨｨｴｨｨｨｰｨｨ ｨｰ6 ﾂｨｪ69ｨｰｨｹ 89'74ｨ ｨｪｨｪ7 4ｨ ｨ ﾂ 786ｨｪｨｮ9ｨ ｨｪｨｪ7, 65ｨｹｨｰ8ｨ 67, 45ｨ 3ｨｮ9ｨ ｨｪｨｪ7, ｨｬｨｮ5767, 49686ｨｰｨｪ66 ｨｬ6569ｨ ｨｪｨｪ7 ｨｰｨ 766ｨ ｨｪｨｨ 7ｨｬ, 57 89'74ｨ ｨｪｨｪ7 7ｨｺｨｨ 9ｨｨｨｺ68ｨｨ9ｨｰ69ｨｮ6ｨｰｨｹ 3ｨｰｨｮ ﾂｨｪ6 ｨｪｨｦ86ｨｪｨｪ6 ｨｬ836, ｨｴ6 4ｨ 9ｨｪ69ｨ ｨｪ6 ｨｪｨ 7ｨ 8ｨ ｨｨｨｬ6 ｨｪｨ 9 ﾂｨ ｨｪｨｪ7 4 ｨｮ ﾂｨｨｨｰ5ｨｬ. The problem of combining of an ensemble of artificial neural networks that process multivariate signals presented as discrete time series is considered. An algorithm for combining of the networks on a finite sample is proposed and its optimality is proven. Recurrent procedures for real-time processing are developed. An algorithm for assessing the "contribution" of each member network into the combined estimate is proposed. Architecture of a neural meta-network for the considered problem's solution is synthesized. The presented approach provides improvement of the solution's precision in extrapolation, filtering, smoothing, emulation, reverse modeling and other similar problems that can be solved using artificial neural networks based on supervised learning paradigm.. 356888 349934 ｨｪｨ 9ｨｰ67ｨｴ 98ｨｬ7 ｨｨ9ｨｺｨｮ99ｨｰ9ｨｪｨｪ4 ｨｪｨｦ86ｨｪｨｪ4 9ｨｰｨｨ (65) ｨｪｨ 35ｨｨ 3ｨｨ86ｨｺ6 78ｨｨｨｬｨｪｨｪｨｨ 9 4ｨ ｨ ﾂｨ 6ｨ 8ｨ ｨ 6ｨｰｨｺｨｨ ｨｨｨｪ68ｨｬｨ ｨｨｨｨ ｨｰｨ ｨｺｨｨ, ｨｺｨ ｨｺ 786ｨｪｨｨ869ｨ ｨｪｨｨ, ｨｨ5ｨｹｨｰ8ｨ ｨｨ7, 95ｨ 3ｨｨ9ｨ ｨｪｨｨ, 5ｨｬｨｮ57ｨｨ7, 6ｨ 8ｨ ｨｰｨｪ6 ｨｬ65ｨｨ869ｨ ｨｪｨｨ, ｨ ｨ 7ｨｰｨｨ9ｨｪ6 ｨｮ78ｨ 95ｨｪｨｨ, ｨｰ.. 9 4ｨ ｨ ﾂｨ, ｨｺ6ｨｰ684 ｨｬ6ｨｮｨｰ ｨ 4ｨｰｨｹ 83ｨｪ4 9 8ｨ ｨｬｨｺｨ ｨｪｨｦ869ｨｰ96ｨｦ 7ｨ 8ｨ ｨｨｨｬ4 6ｨ ｨｮ ﾂｨｪｨｨ7 9 ｨｮ ﾂｨｨｨｰ5ｨｬ. 797 ｨｨ9765ｨｹ469ｨ ｨｪｨｨ7 65 9974ｨ ｨｪ, 783 996, 9 ｨｮｨｪｨｨ989ｨ 5ｨｹｨｪ4ｨｬｨｨ ｨ 7786ｨｺ9ｨｨｨｬｨｨ8ｨｮ6ｨｴｨｨｨｬｨｨ 9ｨｬ63ｨｪ69ｨｰ7ｨｬｨｨ, 796576ｨｴｨｨｨｬｨｨ 83ｨ ｨｰｨｹ 69ｨｰｨ ｨｰ6 ﾂｨｪ6 9563ｨｪ4 ｨｪ5ｨｨｨｪｨｦｨｪ4 9ｨｰ6ｨ 9ｨｰｨｨ ﾂ9ｨｺｨｨ 4ｨ ｨ ﾂｨｨ 9 ｨｮ9569ｨｨ7 ｨ 78ｨｨ68ｨｪ6ｨｦ ｨｨ ｨｰｨｺｨｮｨｴｨｦ ｨｪ6785ｨｪｨｪ69ｨｰｨｨ 6 ｨ 8ｨ ｨｺｨｰ8ｨｨ9ｨｰｨｨｨｺｨ 6ｨ 8ｨ ｨ ｨ ｨｰ49ｨ ｨｬ4 9ｨｨｨｪｨ 569. ｨｬ9ｨｰ 9 ｨｰｨｬ 95ｨｮｨｰ 6ｨｰｨｬｨｰｨｨｨｰｨｹ, ﾂｨｰ6 6ｨｪｨ ｨｨ ｨｰｨ 3 4ｨ ｨ ﾂｨ ｨｬ63ｨｰ ｨ 4ｨｰｨｹ 83ｨｪｨ 9 76ｨｬ6ｨｴｨｹ6 8ｨ 45ｨｨ ﾂｨｪ4 9ｨｰｨｦ, 6ｨｰ5ｨｨ ﾂｨ 6ｨｴｨｨ97 ｨ 8ｨｨｨｰｨｺｨｰｨｮ86ｨｦ, ｨｺ65ｨｨ ﾂ9ｨｰ96ｨｬ ｨｪｨｦ86ｨｪ69, ｨ 568ｨｨｨｰｨｬｨ ｨｬｨｨ 6ｨ ｨｮ ﾂｨｪｨｨ7, ｨｪｨ ﾂｨ 5ｨｹｨｪ4ｨｬｨｨ ｨｮ9569ｨｨ7ｨｬｨｨ, 97696ｨ ｨ ｨｬｨｨ 68ｨ ｨｪｨｨ4ｨ ｨｨｨｨ 6ｨ ｨｮ ﾂｨ 6ｨｴｨｦ 94ｨ 68ｨｺｨｨ ｨｰｨ ｨｺ, ﾂｨｰ6 94ｨ 68 ｨｺ6ｨｪｨｺ8ｨｰｨｪ6ｨｦ 9ｨｰｨｨ 57 ｨｺ6ｨｪｨｺ8ｨｰｨｪ6ｨｦ 4ｨ ｨ ﾂｨｨ 7957ｨｰ97 69ｨｰｨ ｨｰ6 ﾂｨｪ6 9563ｨｪ6ｨｦ 786ｨ 5ｨｬ6ｨｦ. 8ｨ ﾂ9ｨｰ96 83ｨｪｨｨ7 4ｨ ｨ ﾂｨｨ ｨｬ63ｨｰ ｨ 4ｨｰｨｹ 9ｨｮｨｴ9ｨｰ9ｨｪｨｪ6 76943ｨｪ6 9 76ｨｬ6ｨｴｨｹ6 ｨｰｨ ｨｺ ｨｪｨ 449ｨ ｨｬ66 ｨ ｨｪ9ｨ ｨｬｨ 57 (ｨｺ6ｨｬｨｨｨｰｨｰｨ ) ｨｪｨｦ869ｨｰｨｦ [-9], ｨｺ6ｨ 6ｨｪｨｨ ｨｨ ｨｰ 3 ｨ ｨｪｨｪ4 6ｨ 8ｨ ｨ ｨ ｨｰ49ｨ 6ｨｰ97 7ｨ 8ｨ 555ｨｹｨｪ6 ｨｪ9ｨｺ65ｨｹｨｺｨｨｨｬｨｨ 65, 946ｨｪ4 9ｨｨｨｪｨ 54 ｨｺ6ｨｰ684 ｨ 5 ｨｪｨｺ6ｨｰ684ｨｬ 6ｨ 8ｨ 46ｨｬ ｨｺ6ｨｬｨ ｨｨｨｪｨｨ8ｨｮ6ｨｰ97 9 6ｨ ｨｲｨｨｨｪｨｪｨｪｨｮ6 6ｨｪｨｺｨｮ, 7896967ｨｴｨｮ6 76 ｨｺｨ ﾂ9ｨｰ9ｨｮ 6ｨｪｨｺｨｨ, 765ｨｮ ﾂｨｪｨｪ4 9 76ｨｬ6ｨｴｨｹ6 56ｨｺｨ 5ｨｹｨｪ4 9ｨｰｨｦ, 967ｨｴｨｨ 9 ｨ ｨｪ9ｨ ｨｬｨ 5ｨｹ ｨｰｨ ｨｺ, ｨｺｨ ｨｺ 5ｨｰ6 76ｨｺｨ 4ｨ ｨｪ6 ｨｪｨ 8ｨｨ9.. ｨ 78ｨ ｨｺｨｰｨｨｨｺ ｨｪｨ ｨｨｨ 65ｨｹ3 8ｨ 97869ｨｰ8ｨ ｨｪｨｪｨｨ 765ｨｮ- ﾂｨｨ5ｨｨ 9ｨ 766ｨ ｨｺ 6ｨ ｨｲｨｨｨｪｨｪｨｨ6 9ｨｰｨｦ ｨ ｨｪ9ｨ ｨｬｨ 57: ｨｬ6ｨｮ5ｨｹｨｪ4ｨｦ ｨｨ 69ｨｪ69ｨ ｨｪｨｪ4ｨｦ ｨｪｨ 9493ｨｪｨｪ6ｨｬ ｨｮ98ｨｪｨｪｨｨｨｨ [3]. 6 6ｨｰ7 9683ｨ ｨｰ5ｨｹｨｪ6 6ｨｪｨｨ 69ｨｰｨ ｨｰ6 ﾂｨｪ6 6ｨｰ5ｨｨ ﾂｨ 6ｨｰ97 8ｨｮ 6ｨｰ 8ｨｮｨ, ｨｨ 6ｨ ｨｲｨｨｨｪ7ｨｰ ｨｰ6, ﾂｨｰ6 6ｨ ｨ 6ｨｪｨｨ ｨｨ9765ｨｹ4ｨｮ6ｨｰ 5ｨｨｨｪｨｦｨｪｨｮ6 ｨｺ6ｨｬｨ ｨｨｨｪｨ ｨｨ6 996ｨｨ ｨｺ6ｨｬ76ｨｪｨｪｨｰ 9 ｨｰ6ｨｦ ｨｨ5ｨｨ ｨｨｨｪ6ｨｦ 68ｨｬ [9]. 6ｨｮ5ｨｹｨｪ4ｨｦ 766 ｨｨｨｬｨｰ 69ｨｰｨ ｨｰ6 ﾂｨｪ6 598ｨｨ9ｨｰｨｨ ﾂ9ｨｺｨｨｨｦ ｨ 8ｨ ｨｺｨｰ8 9 6ｨｰ5ｨｨ ﾂｨｨ 6ｨｰ ｨｬｨ ｨｰｨｬｨ ｨｰｨｨ ﾂ9ｨｺｨｨ ｨ 65 57