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ОБЧИСЛЮВАЛЬШ СИСТЕМИ
UDC 681.4
V.A. YASHCHENKO*
THEORETICAL FOUNDATIONS OF HARDWARE IMPLEMENTATION OF MULTIPLY NEURAL-LIKE GROWING NETWORKS
*The Institute of Mathematical Machines and Systems Problems of the NAS of Ukraine, Kyiv, Ukraine
Анотаця. У роботг розглядаються теоретичт основи апаратног реалгзацИ нейроподгбних зрос-таючих мереж. На основi наведених теорем i тверджень визначет базов1 в1дносини повних век-тор1в, заданих об'еднанням вiдносин гх пiдвекторiв. Урезультатi визначаються операцИ' побудови i функщонування нейроподiбних мереж. Показано збтьшення вiдносног швидкостi обробки тфор-мацИ' при збтьшент гг об'ему.
Ключовi слова: нейроподiбнi зростаючi мережi, повний вектор, тдвектори, теорiя вiдносин век-торiв.
Аннотация. В работе рассматриваются теоретические основы аппаратной реализации нейро-подобных растущих сетей. На основе приведенных теорем и утверждений определены базовые отношения полных векторов, заданных объединением отношений их подвекторов. В результате определяются операции построения и функционирования нейроподобной сети. Показано увеличение относительной скорости обработки информации при увеличении ее объема. Ключевые слова: нейроподобные растущие сети, полный вектор, подвекторы, теория отношений векторов.
Abstract. The theoretical foundations of the hardware implementation of neural-like growing networks are regarded in the paper. On the base of the given theorems and statements the most crucial relations of the full vectors given union relations of their sub-vectors are determined. As a consequence, the operations of construction and functioning of the neural-like network are determined.
The increase of the relative speed of the information processing under the increase of its volume is shown. Keywords: neural-like growing networks, the full vector, subvector, vectors theory relation.
1. Introduction
The development of new intellectual technologies is closely related to the disclosure of the properties of the human cognitive system. Unlike computers, with classic architecture, a person determines the next action in accordance with the internal motivation, the current situation and goals, most of them aimed at the knowledge of the world. Researchers in the field of massively parallel, neurocomputing and artificial intelligence focus their efforts on the search for the best hardware and software solutions. At the same time they have to deal with a number of architectural issues. For example, how to ensure the synchronization of a large number of processors, how to ensure the identification and accumulation of knowledge, what structure should have a memory for establishing links between description the situation, knowledge and methods of solving problems, how to be processed and put into storage structure a new information order that the system can be trained, increasing the level of its intellect. The most radical solution to this problem is to develop a fundamentally new "thinking" architectures of intelligent computers with developed artificial intelligence in which the programming of applications comes down to education and knowledge of the external world into the internal structure of these computers. One of these architectures is an active, associative neural structure - neural-like growing network implemented in hardware.
© Yashchenko V.A., 2015
ISSN 1028-9763. Математичш машини i системи, 2015, № 4
2. Multidimensional neural-like growing networks
A. Neural-like growing networks
Neural-like growing networks (n-GN) are formally defined as S = (R, A, D, P, M, N).
Here R = {r.}, i = 1,n , A = {at}, i = 1 ,k , D = {dt}, i = 1, e , P = {P}, i = 1, k N = h, where P is the excitation threshold of node a ; P = f (m) > P° (P is the minimum allowed excitation threshold) given that the set of arcs D entering the node ai, is assigned the set of weights M = {mt}, i = 1, w , where mi, may take both positive and negative values [1-6].
Neural-like growing networks are a dynamic structure that changes depending on the value and the time the information gets to the receptors, as well as on the previous state of the network. The information about the objects is presented as the ensembles of excited nodes and the connections between them. Memorization of objects and situations descriptions is accompanied by the addition of new nodes and arcs to mi the network when a group of receptors and neural-like elements enter into a state of excitement. B. Basic Operations of building n-GN
In the complex structures of multiply neuro-like GN interconnection between network elements are most conveniently expressed in terms of relations. Consideration of any information network is not merely a numeration elements entering into its composition, but also the determination of possible kinds connections and interactions between them. For this purpose, a formalized apparatus of the theory relationship is usually applied. To describe the linkages between the two elements (x, y), x £ X, y £ Y, applies the concept of binary relations R(x,y), which is
represented as a set of ordered pairs (x, y) of elements x £ X, y £ Y, defined on the set R [7, 8].
In theory, multiply neural-like growing networks are considered binary relations, which are given a set of vertices or neural elements {al, a2,..., a„}, where ai. Boolean vector of finite
dimension, {a , ak}, a plurality of pairs of these elements. A pair a., ak in the subset, R only
when the vector a is in relation to R with the element a .
i k
Consider the basic properties of pairs of vectors, based on the conjunction operation applied by the components of the vectors, i.e. a. xak = (a^ Ab^, a^ Ab(2),...,a^ Ab^ ), here x
- operation "vector" conjunction, A - a conjunction.
The basic properties of conjunctive pairs of vectors, such as the following: 1. a x c = a , 2. a x c # a , 3. a x c = c ,
4. a x c # c , 5. a x c = 0, 6. a x c # 0.
Combinations of the basic properties of pairs of vectors give eight mutually exclusive relationships:
a r 1 c ° (a x c = a ) n (a x c = c ) n (a x c # 0); a r2 c ° (a x c ^ a ) n (a x c ^ c ) n (a x c = 0); a r 3 c ° (a x c ^ a ) n (a x c ^ c ) n (a x c ^ 0); a R4 c ° (a x c ^ a ) n (a x c = c ) n (a x c ^ 0);
a R 5 c ° (a x c = a ) n (a x c * c ) n (a x c * 0); a R 6 c ° (a x c = a ) n (a x c * c ) n (a x c = 0); a R 7 c ° (a x c * a ) n (a x c = c ) n (a x c = 0); a rs c ° (a x c = a ) n (a x c = c ) n (a x c = 0); here n - logical AND.
Obviously, the relationship R6, R7, RS is trivial, because in each of them, one or both vectors are zero. Based on the analysis of the basic properties of vector pairs formulate the following statements:
Statement 1. On the set of pairs of vectors a, a e A, you can define five major, mutually exclusive relationship R1, R2, R3, R4, R5.
aRl a ° " a , a e A:(a x a = a )n(a x a = a )n
i i+1 . . ,
( a.x a * 0 );
i i+1 i i i+1 i+1
i i+1
a R2a = "a t, a i+1e A : (a t x a t+1 * a i) n (a ix a i+1 * a i+1) n (a ix i+1 = 0 );
a R3 a a , a e A: ( < x a * a/ )n( a x < * < )n
i i+1 i i+1 i i i+1 i+1
( a x < * 0);
1 i+1
a R4 a a , a e A : (<2 x ¿J * ¿J )n(<2 x a = <2 )n
i i+1 • ; ■1 ; • ;■1 ; ■1
( a x a * 0 ); i i+1
a R5 a a , a e A : ( a x ¿J = ¿J )n( a x ¿J * ¿J )n
i i+1
(a x < * 0 ).
i+1 i i i+1 i+1
i i+1 i i i+1 i+1
i i+1
Here a. X ai+1 - the conjunction of vectors a. u a+1, n - logical AND.
Obviously, the relationship R6, R7, Rs is trivial, because in each of them, one or both vectors are zero. On the basis of assertions 1 are determined by the following basic operations for constructing n-GN.
If a pair of vectors ( < 1, < k) is located in relation to R1, R2, R3, R4, R5, respectively, the operation Qj1, Qj2, Qj3, Qj4, or Qj5, is performed, which consist in the construction of vectors the pair (a , a ) on of three vectors (a ,a ,a ) and are defined as follows:
Q (a , a) = a ak, a^^, a:= a, ak := 0, ak+1:= 0;
02, — — ' . ,—1 3k —k+1 /1 /1 3k —k —k+1 „
(a , a ) = (a , a , a ), a := a , a := a , a := 0;
(a, a)=(a, a, a ), a := (a x a x a) u c, a := (a x a x a) u c, a := a x a ;
04, — —' 1 — k — k+1 — k — k —1 —1 — k — 1 — — k+1 „ (a, a)=(a, a , a ), a := a, a := (a x a x a) u c , a := 0;
05, — —\ 1 — k —k+1 —1 —1 — k —1 — k —k — —k+1 „ (a,a)=(a,a ,a ), a := a, a := (a xa xa) uc ,ari := 0:
here u - disjunction vectors, applied to the components of the vectors.
C. The basic attitude of complete vectors consisting of a combination relations sub-vectors
The input vector neural-like growing network of representing a description of the concepts or situations outside world usually has a greater dimension. For example, an image size of 640x480 pixels in the network represented by the vector 307200 bytes in size. Therefore, for the hardware implementation of n-PC input (complete), the vector must be divided into subvector.
Determination 1. Full vector is a vector consisting of subvectors.
If each vector a. of the set A={al ,a2,...,an} determined by the aggregate by a subvectors , i.e., a=(a,a, -, a), denote: a a plurality of sub-vectors by (full) vectors a £ A; by ur - a sub-
i i i
set of subvectors, which are given a binary relation r.
7 j 7 j+i
Statement 2. On the set of pairs of subvectors a , a £ a can be identified eight mutually exclusive relationship r1, r2, r3, r4, r5, r6, r7, r8.
arlc a, a e A: (a x a = a)n( a x a = a )n( a x a *0),
7 j 7 j+i
here a xa - the conjunction of vectors — + H a+1, n - logical AND;
ar2c —a +1 e A: (—xaj+1 * aj;naxaj+1 * aj+1)^a xaj+1=0>;
a r3c —,—+1 e A: (a1 x —j+1 * —j) n a x * aM) n a x —+1 * 0);
— «/I — W —j —j+1 A • ✓ —j —j+1 — j, ^ —j —j+1 —j+1, ✓ —j —j+1
ar 4c a , a e A . (a x a * a ) n (a x a = a ) n (a x a * 0); ar5c a , a e A. (a x a = a )n(a x a * a )n(a x a *0);
—wi— ^ —j —j+1 A • / —j —j+1 —j > / —j —j+1 —j+K / —j —j+1n >
ar6ca , a e A. (a x a = a )n(a x a * a )n(a x a = 0);
— ,„n— w —j —j+1 A- , —j —j+1 — K , —j —j+1 —j+\ , —j —j+1^ ar7c a , a e A: (a x a * a )n(a x a = a )n(a x a = 0);
— 0 — —j —j+1 A • —j —j+1 — j —j —j+1 —j+1 —j —j+1
ar8c a, a e A: (a x a = a )n( a x a = a )n( a x a = 0).
In this case, all eight relations used, since a full vector will be zero only in the case of the vanishing of all its subvectors.
Formulate statements that define the relationship on the set of vectors of full relations subvector that make them. These statements are obviously follows from the definition of equality (inequality) vectors:
Determination 2. The full vectors are equal if equal all their corresponding subvectors in particular equal to their respective components (bits), i.e.
a = b : a = b « "js : —js) = b(s).
Determination 3. The full vectors are not equal if there is at least one pair of corresponding unequal subvectors, in particular, subvectors not match at least in one a discharge, i.e.
a *b ^$g:ag*bg ^ $g,k : ags)*b8(s)
Statement 3. In accordance with the definition 2 full vectors a and b are in relation to R1 if and
^j 7j
only if all subvectors a , b are in the relation r1, i.e.
a R1 b ^ V: a rl b .
Statement 4. In accordance with the definition 2 full vectors a and b are in relation to R2 if and
7J 7J
only if all subvectors a , b are in the relation r2, i.e.
< R2b « V : ar2 b .
Statement 5. In accordance with the definition 2 full vectors a and b are in relation to R3 if and
7 j 7 j
only if all subvectors a , b are in the relation r3, i.e.
a R3b V : ar3 b ■
7 —*
Statement 6. In accordance with the definition 2 full vectors a and b are in relation to R4 if and
7 J 7 J
only if all subvectors a , b are in the relation r4, i.e.
< R4 b ^ V : a r4 b .
7 —*
Statement 7. In accordance with the definition 2 full vectors a and b are in relation to R5 if and
7 J 7 J
only if all subvectors a , b are in the relation r5, i.e.
< R5 b V : a r5 b .
7 —*
Statement s. In accordance with the definition 2 full vectors a and b are in relation to R6 if and
7 J 7 J
only if all subvectors a , b are in the relation r6, i.e.
< R6 b ^ V : a r6 b .
7 —*
Statement 9. In accordance with the definition 2 full vectors a and b are in relation to R7 if and
7 J 7 J
only if all subvectors a , b are in the relation r7, i.e.
a R7 b V: a r7 b '
7 —>
Statement 10. In accordance with the definition 2 full vectors a and b are in relation to R8 if
7 J 7 J
and only if all subvectors a , b are in the relation r8, i.e.
a R8 b V.: ar8 b ■
7 j — j
In the case where the subvectors a E a full vectors a E A represent different combi-
7 j
nations of relations r1, r2, r3, r4, r5, r6, r7, r8, the ratio of the full of the vectors a E A is not so obvious and require evidence.
7
Theorem 1. Let on the set of subvectors a vectors a. e A given binary relation R=r1 ur2, and ar1 ^ 0 and ar2 . Then, on the set of vectors A is given by a binary relation R3, i.e, to
v a, a e A : a R3 a .
Substantiation. Since ar1 ^ 0, then at least one pair of respective subvectors ag, ag of vec-
7 7 7 g 7 g
tors a1, a2 £ A fair treatment R1, where al x a2 ^ 0, and since ar2 ^ 0, then at least one pair of subvectors a j, a 2 of vectors a1, a 2 £ A fair treatment R2, and where al x a 2 ^ al, al x a2 ^ a2. This means, by the above definition of 3 that al x a2 ^ al, al x a2 ^ a2, ax x a2 ^ 0, i.e. a 1 R3 a2 for Va 1, a2 £ A, as required.
7
Theorem 2. Let on the set of subvectors a vectors a £ A given binary relation R=r1 ur3, and ar1 ^ 0 and ar3 ^ 0. Then, on the set of vectors A is given by a binary relation R3, i.e, to
vaaeA:a R3 a..
7 g 7 g
Substantiation. Since ar1 ^ 0, then at least one pair of respective subvectors ax, a2 of vectors
7 7 7g 7g
a1, a2 £ A fair treatment R1, where ax xa2 ^ 0, and since ar3 ^ 0, then at least one pair of
subvectors al, a 2 of vectors aj, a 2 £ A fair treatment R3, and where al x a 2 ^ al,
al x a2 ^ a2, al x a2 ^ 0 . This means, by the above definition of 3 that al x a2 ^ al,
a1 x a 2 ^ a 2, a 1 x a 2 ^ 0, i.e. a 1 R3 a2 for Va1, a 2 £ A, as required. Similarly possible formulate the following assertion.
Statement 11. The relationship of R3 on the set of complete vectors a1 , a2 £ A defined by the
union of relationship is (r2ur3) or (r4ur3), or (r5ur3), or (r6ur3), or (r7ur3), or (r8ur3)
7 j
on subvector a £ a, here conjunctive properties relationship r3 absorb properties relationship r2, r4, r5, r6, r7 and r8.
Also it can be shown that the relationship of R3 on the set of full vectors al, a2 £ A given association relationship (r4u r5) or (r4u r6), or (r5u r7), or (r2u r5), or (r2u r4) on the subvec-
7j
tor a £ a .
7
Theorem 2. Let on the set of subvectors a vectors ai £ A given binary relation R=r2 ur6, and ar2 ^ 0 and ar6 ^ 0. Then, on the set of vectors A is given by a binary relation R2, i.e., to
va,a e a :a ri a.
7 g 7 g
Substantiation. Since ar2 ^ 0, then at least one pair of respective subvectors ax, a2 of vectors
7 7 7 g 7 g 7 g 7 g 7 g 7 g 7g 7g
a1, a2 £ A fair treatment R2, where, a 1 x a2 ^ a 1 , a 1 x a2 ^ a2, at x a2 ^ 0 and since
ar6 ^ 0, then at least one pair of subvectors al, a2 of vectors al, a2 £ A fair treatment R6, —► k —^ k —^ k —^ k —^ k —^ k —^ k —^ k and where a1 x a2 = a1, a1 x a2 ^ a1 and a1 x a2 ^ 0 . This means, by the above definition
3, that al x a2 ^ al, al x a2 ^ a2, al x a2 ^ 0, i.e. a 1 R2 a2 for al, a2 £ A, as required.
Similarly possible formulate the following assertion.
Statement 12. The relationship of R2 on the set of complete vectors a1, a2 E A defined by the union of relationship is (r2ur7) or (réurî), on subvector at E a.
Theorem 4. Let on the set of subvectors a vectors a. EA given binary relation R= r1 ur4, and ar1 ^ 0 and ar4 ^ 0 . Then, on the set of vectors A is given by a binary relation R4, i.e.
va>aeA:a R4 ai.
7 g 7 g
Substantiation. Since ar1 ^ 0, then at least one pair of respective subvectors a1, a2 of vectors
77 7 g 7 g 7 g 7 g 7 g 7 g 7 g 7 g
a1, a 2 E A fair treatment R1, where, a1 x a 2 = a1 , a 1 x a 2 = a 2, a1 x a 2 ^ 0 and since
7 k 7 k 7 7
ar4 ^0, then at least one pair of subvectors a 1, a2 of vectors a1, a2 E A fair treatment R4,
7k 7k 7 k 7k 7k 7 k 7k 7k
and where, a1 x a 2 ^ a1, a1 x a 2 = a1 and a1 x a 2 ^ 0 .
This means, by the above definition 3, that a1 x a2 ^ a1, a1 x a2 = a2, a 1 x a2 ^ 0, i.e.
ax R4 a2 for Va 1, a2 E A, as required. Similarly possible formulate the following assertion.
Statement 13. The relationship of R4 on the set of complete vectors a1, a2 E A defined by the
7 j
union of relationship is (r1ur7) or (r4ur7), on subvector a E a.
7
Theorem 5. Let on the set of subvectors a vectors at E A given binary relation R=r1 ur5, and ar1 ^ 0 and ar5 ^0. Then, on the set of vectors A is given by a binary relation R5, i.e.
7 7 ' 7 7 (
V a, a E A : aR5a .
7 g 7 g
Substantiation. Since ar1 ^ 0, then at least one pair of respective subvectors a 1, a2 of vectors
7 7 7g 7g 7g 7g 7g 7g 7 g 7 g
a1, a2 E A fair treatment R1, where a1 x a2 = a1 , a1 x a2 = a2, a1 x a2 ^ 0, and since
ar5 ^ 0, then at least one pair of subvectors a 1, a2 of vectors a1, a2 E A fair treatment R5,
7 k 7 k 7k 7 k 7 k 7k 7k 7k
and where a 1 x a2 = a1, a 1 x a2 ^ a2 and a1 x a2 ^ 0 . This means, by the above definition
7 7 7 7
3, that i.e. a 1 R5 a2 for V a1, a2 E A, as required. Similarly possible formulate the following assertion.
Statement 14. The relationship of R5 on the set of complete vectors a1, a2 E A defined by the
7j
union of relationship is (r1ur6) or (r5ur6), on subvector a E a .
7
Theorem 6. Let on the set of subvectors a vectors a. E A given binary relation R=r1 ur2ur3, and ar1 ^ 0, ar2 ^ 0 and ar3 ^ 0. Then, on the set of vectors A is given by a binary relation
7 7 ' 7 7 (
R3, i.e. V a, a E A : a R3 a .
Substantiation. Since ar1 ^ 0 and ar2 ^ 0, then at least one pair of respective subvectors
7 g 7 g 7 k 7 k 77
a 1, a2 and a 1, a2 of vectors a1, a2 £ A fair treatments r1 and r2, as well as the relationship r'=(r1 ur2) in accordance with theorem 1 given the relationship r3 and as ar3 ^ 0, then at least
7 n 7 n
for one pair of corresponding subvectors al, a2 to the fair treatment r3, and the definition 1 of
the ratio of r"=(r3u r3) sets r3, then on the set of complete vectors a1, a2 £ A ratio R=r'u
r''=(r1 ur2) ur3 coincides with r3, i.e. al R3 a2 for al, a2 £ A, as required.
Theorem 7. Let on the set of subvectors a vectors ai £ A given binary relation R=r1 ur2ur4,
and ar1 ^ 0, ar2 ^ 0 and ar4 ^ 0 . Then, on the set of vectors A is given by a binary relation
7 7' 7 7 (
R3, i.e. V a, a £ A : aR3 a .
Substantiation. Since ar1 ^ 0 and ar2 ^ 0, then at least one pair of respective subvectors
7 g 7 g 7k 7k 7 7
a 1, a2 and aj, a2 of vectors a1, a2 £ A fair treatments r1 and r2, as well as the relationship r'=(r1 ur2) in accordance with theorem 1 given the relationship r3 and as ar4 ^ 0, then at
7 n 7 n
least for one pair of corresponding subvectors a1 , a2 to the fair treatment r4, and the definition
13 of the ratio of r"=(r3u r4) sets r3, then on the set of complete vectors al, a2 £A ratio
R=r'ur"=(r1 ur2) ur4 coincides with r3, i.e. a 1 R3 a2 for V a 1, a2 £ A , as required. Similarly possible formulate the following assertion.
Statement 17. The relationship of R3 on the set of complete vectors al, a2 £ A defined by the
union of relationship is (r1u r2u r5), or ..., (r2u r3u r4), or ..., ( r5u r6u r7) on subvector
7 j
a £ a.
Theorem 8. Let on the set of subvectors a vectors at £ A given binary relation R=r1 ur2ur3u
r4, and ar1 ^ 0, ar2 ^ 0, ar3 ^ 0 and ar4 ^ 0 . Then, on the set of vectors A is given by a
7 7' 7 7 (
binary relation R3, i.e. V a, a £ A : a R3 a .
Substantiation. Since ar1 ^ 0, ar2 ^ 0, and ar3 ^ 0, then at least one pair of respective
7 g 7 g 7n 7n 7k 7k 7 7
subvectors a a 2, a a 2 and a a 2 of vectors al, a 2 £ A fair treatments r1, r2 and r3, as well as the relationship r'=(r1 ur2ur3) in accordance with theorem 5 given the relationship r3
7 n 7 n
and as ar4 ^ 0, then at least for one pair of corresponding subvectors al, a2 to the fair treatment r4, and the definition 13 of the ratio of r"=(r3ur4) sets r3, then on the set of complete vectors a1, a2 £ A ratio R=r'ur''=(r1 ur2ur3) (ur3ur4) coincides with R3, i.e. a 1 R3 a2 for
77
V al, a2 £ A, as required.
Similarly possible formulate the following assertion.
Statement 18. The relationship of R3 on the set of complete vectors a., a2 E A defined by the union of relationship is (r1ur2ur3ur5),..., (r5ur6ur7ur8) on subvector a E a .
7
Theorem 9. Let on the set of subvectors a vectors at E A given binary relation R=r1 ur2ur3u r4ur5, and ar1 ^ 0, ar2 ^ 0, ar3 ^ 0, ar4 ^ 0 and ar5 ^ 0. Then, on the set of vectors
7 7' 7 7 (
A is given by a binary relation R3, i.e. V a, a E A : a R3 a .
Substantiation. Since ar1 ^ 0, ar2 ^ 0, ar3 ^ 0 and ar4 ^ 0, then at least one pair of re-
7 g 7 g 7 n 7 n 7 m 7 m 7 k 7 k 7 7
spective subvectors a., a2, aa2, a. , a2 and aa2 of vectors a., a2 E A fair treatments r1, r2, r3 and r4, as well as the relationship r'=(r1 ur2ur3ur4) in accordance with theorem 7 given the relationship r3 and as ar5 ^ 0, then at least for one pair of corresponding
7 n 7 n
subvectors a., a2 to the fair treatment r5, and the definition 13 of the ratio of r''=(r3ur5) sets r3, then on the set of complete vectors a., a2 EA ratio R=r'ur''= (r1 u r2u r3)u (r4u r5)
7 7 7 7
coincides with R3, i.e. a. R3 a2 for V a., a2 E A , as required. Similarly possible formulate the following assertion.
Statement 19. The relationship of R3 on the set of complete vectors a., a2 E A defined by the union of relationship is (r1u r2u r3u r5u r6), . . . , (r4ur5u r6u r7u rS), on subvector
7 J
a E a.
Theorem 10. Let on the set of subvectors a vectors a1 E A given binary relation R=r1 ur2u r3ur4ur5ur6, and ar1 ^0, ar2 ^0, ar3 ^0, ar4 ^0, ar5 ^0 and ar6 ^0. Then,
7 7 ' 7 7
on the set of vectors A is given by a binary relation R3, i.e. V a, a E A : a R3 a . Substantiation. Since ar1 ^ 0, ar2 ^ 0, ar3 ^ 0, ar4 ^ 0 and ar5 ^ 0, then at least one
—g — g 7 n 7 n 7 m 7 m 7 s 7 s 7 k 7 k
pair of respective subvectors aa 2, aa 2, a. , a 2 aa 2, and aa 2 of vectors
a., a2 E A fair treatments r1, r2, r3, r4 and r5, as well as the relationship r'=(r1 ur2ur3u r4ur5) in accordance with theorem 9 given the relationship r3 and as ar6 ^ 0, then at least for
7 n 7 n
one pair of corresponding subvectors a., a2 to the fair treatment r6, and the definition 13 of the ratio of r''=(r3ur6) sets r3, then on the set of complete vectors a., a2 E A ratio R=r'ur''= r1
7 7 7 7
ur2ur3ur4ur5 ur6 coincides with R3, i.e. a. R3 a2 for V a., a2 E A , as required. Similarly possible formulate the following assertion.
Statement 20. The relationship of R3 on the set of complete vectors a., a2 E A defined by the
union of relationship is (r1ur2ur3ur4ur5ur7),..., (r3 ur4ur5ur6ur7ur8) on subvec-
7 J
tor a E a.
Theorem 11. Let on the set of subvectors a vectors a1 E A given binary relation R=r1 ur2u r3ur4ur5ur6ur7, and ar1 ^0, ar2 ^0, ar3 ^0, ar4 ^0, ar5 ^0, ar6 ^0 and
ar7 ^ 0. Then, on the set of vectors A is given by a binary relation R3, i.e.
7 7' 7 7 (
V a, a £ A : a R3 a .
Substantiation. Since ar1 ^ 0, ar2 ^ 0, ar3 ^ 0, ar4 ^ 0, ar5 ^ 0 and ar6 ^ 0, then at
7 g 7 g 7 n 7 n 7 m 7 m 7 s 7 S 7 rf 7 rf 7 k 7 k
least one pair of respective subvectors al, a2, al, a2, a 1 , a 2 , aj, a2 aj, a2 and aj, a2
of vectors aj, a2 £ A fair treatments r1, r2, r3, r4, r5 and r6, as well as the relationship r'=(r1 u r2u r3u r4u r5u r6) in accordance with theorem 10 given the relationship r3 and as
7 n 7 n
ar7 ^ 0 ,then at least for one pair of corresponding subvectors al, a2 to the fair treatment r7, and the definition 13 of the ratio of r''=(r3u r7) sets r3, then on the set of complete vectors
aj, a2 £ A ratio R=r'ur''=r1 ur2ur3ur4ur5 ur6 ur7 coincides with R3, i.e. a 1 R3 a2
77
for V al, a2 £ A, as required.
Similarly possible formulate the following assertion.
Statement 21. The relationship of R3 on the set of complete vectors al, a2 £ A defined by the union of relationship is (r1ur2ur3ur4ur5ur6ur8), ... ,(r1ur2 ur3ur4ur5ur7ur8) on subvector a £ a.
7 j
Theorem 12. Let on the set of subvectors a vectors a £ A given binary relation R=r1 ur2u r3ur4ur5ur6ur7,ur8 and ar1 ^ 0, ar2 ^0, ar3 ^0, ar4 ^0, ar5 ^0, ar6 ^0, ar7 ^ 0and ar8 ^0. Then, on the set of vectors A is given by a binary relation R3, i.e.
7 7' 7 7 (
V a, a £ A : a R3 a .
Substantiation. Since ar1 ^0, ar2 ^ 0, ar3 ^0, ar4 ^0, ar5 ^0, ar6 ^0, and
7 g 7 g 7n 7n 7m 7m 7 s 7s
ar7 ^ 0, then at least one pair of respective subvectors a 1, a2, a 1, a2, a 1 , a2 , a 1, a 2
a 1, a2, al, a2 and al, a2 of vectors al, a2 £ A fair treatments r1, r2, r3, r4,r5, r6 and r7, as well as the relationship r'=(r1 ur2ur3ur4ur5ur6ur7) in accordance with theorem 11 given the relationship r3 and as ar8 ^0, then at least for one pair of corresponding sub-vectors
n 7 n 2
a 1, a2 to the fair treatment r7, and the definition 13 of the ratio of r''=(r3ur7) sets r3, then on the set of complete vectors a1 , a2 £A ratio R=r'ur''=r1 ur2ur3ur4ur5 ur6 ur7ur8
7 7 7 7
coincides with R3, i.e. a 1 R3 a2 for V al, a2 £ A , as required.
On the basis of theorems and assertions are defined basic attitude of full vectors, given
7j
union relationship subvectors. On fig. 1 shows the basic relationship of full of vectors a £ A by combining relations subvectors a.
(rl i_/r2) v (r5 c/r4) v (r2 wr6) v (rl c/r7) = R3
Fig. 1. Basic relationship of full of vectors by combining relations subvectors
C. Matrix representation multiply neural-like growing networks
According to the theory of graphs [7, 8], the conversion performed on the matrix correspond structural change graphs, and in application of the theory of multiply neural-like growing networks of relevant transformation of the structure of these networks.
In the theory of multiply neural-like growing networks with the help of matrices the topological structure of network is represented.
Due to the fact that the multiply neural-like growing networks are dynamic structures, which change (growing) as a result of receipt of new information on the receptor field, the matrix n-PC are also converted in the process of analysis and storage of information.
Line numbers of such matrix, are numbers of neural-like set elements A = {ai}, where
i e I = {1,2,3,...,k}.
Line matrix consists of a combination of the vectors M and N , where M - the vector representing description of the object, the signs of which are arranged from left to right in accordance with the numbering of the receptors from the set R = {rl,r2,...,rn}, and N - vector which elements are numbered from left to right in the order of numbering of vertices of A, i.e.
M ={ n/i e H }, N ={ k/i e U }, here
0, if the sign of the object, corresponding i the receptor missing,
1, if the sign of the object, corresponding i the receptor there is;
0, if the vector ai corresponding j line of the matrix, has no connection with the new vector,
1, if the vector ai corresponding j line of the matrix, has connection with the new vector.
In order to form a matrix of n-GN it is installed connection coefficients hr > n1. Let the external information coming on receptor field, represented by a Wr = {r? }, i £ Ir, j £ Jr . For
all pairs of vectors a,a £ Vr, where Vr is the set of row vectors of length k receptor area, introduce mutually exclusive relationship Rr i, for the receptor zone.
7 n 1 7' w 7 j 7 j+1 A . 7 j 7 j+j 7 j 7 j 7 j+1 7 j+j 7 j 7 j+1
aR 1 a a, a e A .(a x a = a )n(a.x a = a n(a x a *0),
i i i i i i i i
7 j 7j+1 — j 7 j + 1
here at xa_ - the conjunction of vectors at and at , H - a logical AND;
7 rt 7 tJ 7j+j A . 7j 7j+1 7j 7j 7j+1 7j+l 7j 7j+1
aR 2a *"a, a e A :(a x a ■*■ a )n(a x a ■*■ a )n( a x a =0);
r i i i i i i i i i i
7 7' 7j 7j+1 7j 7j+1 7j 7 j 7j+1 7 j+1 7j 7 j+1
aR 3a a , a e A :(a x a ■*■ a )n(a x a ■*■ a )u(a x a *0);
r i i i i i i i i i i
7 7 ' 7 j 7 j +1 7 j 7 j +1 7 j 7 j 7 j +1 7 j+1 7 j 7 j +1
aR 4a a , a e A :(a x a ■*■ a )n(a x a = a )u(a x a *0);
r i i i i i i i i i i
7 7' 7 j 7 j+1 7j 7j+1 7 j 7 j 7 j+1 7 j+1 7 j 7 j+1
aR 5a a, a e A :(a x a = a)n(a.x a * a )n(a.x a *0).
i i i i i i i i
7 1 7 2 73 7k
A. Let there a set of vectors there is a„, a„, a„,..., a (.
ri j ri j ri '
B. Next is checks in which of the relations Rr1, Rr2, Rr3, Rr4, Rr5 is a pair of vectors a, a of a
/ — 1 7 k \ ¡7 2 7. \ /7 3 7k \ /7 k -1 7 k \
sets of pairs of receptor areas (ari, ar. ), (an, an ),(ari, a r. ),..., (a, , a, ), where kranges from
2 to k + g, here g-a number of new vectors.
/71 — M
If the couple of vectors (a,, a, ), is in respect of the Rr1, Rr2, Rr3, R4, Rr5, then the operations are performed respectively Qj1, Qrj2, Qrj3, Qrj4 or Qrj5:
01 77' 71 7 k 7 k +1 71 71 7 k 7 k+1 at a2 7
(a , a ) = (a , a , a ), a := a , a := 0, a := 0, m := be m':= b,,
r1 ri ri ri ri ri ri ri k k k k
7 2
„a,
-0 - a ^0 = _ a ^
a i k a72 k
P 7, = / (ma), P 7 2 = / (ma);
02 7 7' 71 7 k 7 k 71 71 7 k 7 k 7 k +1 a 7
( a , a ) = ( a , a , a ), a := a , a := a , a := 0, m':= b,
r1 r r r r r r r r k k
ma:= b,, P 7 = /( ma'), P1 = /( ma);
2
a, k a,
O (a , a) = (a , a , a ), a := (a x a x a ) u c , a := (a x a x a ) u c
r1 ri ri ri ri ri ri ri rj ri ri ri ri rj
71 7k 71,k 7+1k
7k+1 71 7k a, i a, 7 a, ... \ r>0 a ,
a := a x a , m, = b,, m':= b,, m := /(P7M), P7k+j=/(m, ),
k k k k
0 ' 1 ' 1 0 -k -7k
P_>=/(ma', ma), P7k=/(ma', mala, k c a, k c
a, a,
4 7 7' 71 7k 7k+1 71 71 7k 71 7k 7k 7k+1 a1
O (a, a)=( a, a, a ), a :=( a x a x a)u c, a := a, a :=0, m := b,
r1 r r r r r r r rj r r r k k
7k 0 0 7k 0 71 71
ma := b,, ma :=/( PtJ Pt. =/( ma ), P^=/(ma', ma);
k k c a, a, k a, k c
05 7 7 71 7k 7.+1 71 71 7k 71 7k 7k 7K+1 a 1
(a, a)=(a , a , a ), a := a , a := (a x a x a )u c , a := 0, m := b,
r1 r r r r r r r r r rj r k k
7k 0 0 t' 0
m := b, ma' :=/( Pt,), Pt=/(ma:\ Pt.=/(nt, ma") .
k k c a, a, k a, k c
2 3 4 5 —1 —k
Operations Qr1, Qr1 , Qri , Qri or Qr1 are valid if the hr > n1, otherwise, if ari ^ ari
then
1111 1 1
a=a, a = a, a+>-=o, mal -= brt, mat -= brt, P =f(ma), P =f(mb'
" n n n n k k k k a k a
-1 - k 7 k k+1 i / 1 \ if a„ = a,, then a„ := ° a^ := 0, m^ := bk, Pj = f (ma ).
k = <
1, if the operation was performed Qrj1,
2, if the operation was performed Qr12, Qr14, Qr15,
3, if the operation was performed Qrj .
/-2 - M
If the couple of vectors I ari, ar. ), is in respect of the Rr1, Rr2, Rr3, Rr4, Rr5, then the operations are performed respectively Qrj1, Qrj2, Qrj, Qrj4 or Qr]5:
1 ^ ^ ' ^ 2 ^ k ^ k +1 ^ 2 ^ 2 ^ k ^ k + 1 ^ 1
01 — — ' — 2 — k — k +1 — 2 — 2 — k — k + 1 a j
( a , a ) = ( a , a , a ), a := a , a := o, a := o, m ' := b
r 2 r ' r' r' r' r' r' r' k i
— 2 — 1 — 2
m a' := b , P 0_1 = f ( ma-), P \2 = f ( m a');
k k a' k a ' k
02 — — ' — 2 — k — k — 2 — 2 — k — k — k+1 a i a" 7
( a , a ) = ( a , a , a ), a := a , a := a , a := o, m ':= b,, m ':= b,
r 2 riririririririri k k k k
P —1 = f( mp, P — 2 = f( m"');
kk
0— —' —2 —k —k+1 —2 —2 —k —2 —k —2 —k —k —k+1 —2 —k
(a, a) =(a, a, a ), a :=( a x a x a )è c , a : =(a x a x a )è c , a := a x a
2 r r r r r r r r r r r r r r r r —1 —k —k —1 —k —k n?:= b, nf:= b, nf:=f( Pj, P\=f( mf), P— =f( nt, nfl
k c a a k a k c
04 —»■ —► —»2 ——»k+1 —»2 —»2 ——»2 ———»k+1 a J
( a, a)=( a , a , a ), a :=( a x a x a )è c , a := a , a := o, m = b,
r2 r r r r r r r rj r r r k k
—k —k o o —k o —1 —1
ma' := b,, ma' :=f( P—k), P—k=f( ma' ), P—1 = f( ma', ma');
k k c a—k ak k a1 k c
0— —' —2 —k —k+1 —2 —2 —k —2 —k — —k+1 a 7 a 1
(a, a)=( a, a, a ) a := a, a =(a x a x a )u c, a :=o, m = h, m' = b
2 r r r nnnnnnnn k k k k
m :=f(p—+1), P—k+1=f(nl■,k), P1=f(m5', m^), P!=f(m, nll).
-k+1 - k+1 Here for Qr24, Qr25 , ari 0 , if as a result of perform operations Qr14, Qr15, ari 0
-k+1
and if as a result of perform operations Qr13, Qr14, ari 0 .
The above operations are performed, if hr > n1, otherwise, if it is ari ^ ari, then
2 k 1 2 — 2 — k
a2 := a2, a := a , ma := b„, m a-:= b t, P01 = /(ma ), P\ = /(ma ) and if ari = ari,
' a,
ri ri ri k k k k a k a k
— 2 — 2 — k a 2 1 o
a J= aa„:= o, maa := bt, P 2 = f( m?').
3 - k
Further, if the couple of vectors (ari = ari ), is in respect of the Rr1, Rr4, Rr5,
then the operations are performed respectively Qr/, Qry2, Qrj3, Qrj4 orQrj5: etc. while a plurality of
¡->1 7 k \ /7 2 /7 3 -k \ /^k-1 - k \
pairs will not become exhausted ¡an.,ar. ),lan,an. ),lari,ari ),...,\ari ,ar. ).
Thus, descriptions of concepts, objects, conditions or situations are formed in a matrix that contains information about these concepts, objects, conditions or situations and relationships between them, pointing to interdependence of their submission.
Fig. 2 depicts the determination of the ratio of two full vectors a 15 a2 consisting of sub-
vectors
7 1 7 2 7 3 7 4 7 5 7 6 7 7 7 8
a i ' a i 'a 1 'a 1 'a 1 'a 1 'a 1 'a 1 '
7 1 7 2 7 3 7 4 7 5 7 6 7 7 7 8
a 2 a 2 a 2 a 2 a 2 a 2 a 2 a 2 .
a, [ a2 [
a! a, a? a! aI a, aï
ii ii ii ii ii ii ii
Ü2 a22 ai at Ü2 at at at
1 II II II II II II II
ri V2 T5 Ï4 Y2 V6 ri r-
/•,; « r2 /V
rt, ft,
Fig. 2. Defining of vectors relationships a 1, a 2
Binary relationships between pairs of subvectors aj, aj are determined simultaneously.
The first pair of subvectors is in relation r1. The second pair of subvectors is in relation r2. The third pair of subvectors is in relation r5. The fourth pair of subvectors is in relation r4.
The fifth pair of subvectors is in relation r2. The sixth pair of subvectors is in relation to r6. Seventh pair of subvectors is in relation r1. The eighth pair of subvectors is in relation to r7.
All relationship of subvectors are determined parallel in one time interval. Then, in accordance with the ratio of the base vectors defined by combining their subvectors (Fig. 1) are determined in what respect are the full vectors a 1, a 2 .
The first and second pair of subvectors is in the relation r3. The third and fourth pair of subvectors is in the relation r3. The fifth and sixth pair of subvectors is in the relation r2. The seventh and eighth pair of subvectors is in the relation r4. The fifth and sixth, seventh and eighth pairs of subvectors are in the relation r3. In general, full vectors are in the relation R3. In accordance with a relation of R3 are given operations for converting network.
£ a¿ iii al al k at
1 1 ! i i 1 II ! • It [ ]
al a' a; a; a! Sr at
1 1 f 1 \ If '1 ... i =1
a't at ii: ai ai aî
1 1 i 1 1 □1 B *** [ ]
ai «Í al a¡ a', al M
1 1 1 1 1 11 1 ... 1 1
in ai al al a} Sí
! Il II II II II 1 ... 1 1
(ti ai ¡i; •is al a: Ü:
The input redor
The knwitdgt bast
Fig. 3. Here shows the definition of the relationship of the input full vector with the full vectors of the knowledge base
In Fig. 3 shows the definition of the relationship of the input full vector with the full vectors of the knowledge base. The input vector describes the concepts, objects, conditions, situa-
tions, or images of the outside world. For example, the input vector describes the image of the human face, and the base knowledge contains descriptions of entities stored earlier. Relationship vectors are determined as described above. As a result, either the input vector is written into the knowledge base (in knowledge base there is no such image) or not written (such image is available in the knowledge base). Wherein the defining relationships of all vectors of the knowledge base with input vector is carried out simultaneously in a single time interval. Thus, with increasing of the knowledge base, execution time of operations remains unchanged. A relative speed of the information processing with increasing the knowledge base increases.
This property of n-GN at a hardware representation in matrix form (Fig. 4) allows increasing the volume of information processed freely (due to the modular structure) without loss of time of its processing (due to the massive parallelism of operations performing).
3. Conclusion
Matrix representation multiply neural-like growing networks in the form of a matrix allow us to solve the problem of hardware implementation of intelligent systems using these networks as a knowledge base. Due to the fact that n-GN is a dynamic structure that changes (grows) as a result of receipt of new information on the receptor field, their physical implementation is difficult due to the organization of dynamically changing connections between network elements.
In matrix representation, multiply n-GN, input information into the network is the process of redistribution of pointers (links) between existing and emerging elements of the matrix.
As a result of this process the object is included in class to which it belongs, or a new class of objects.
Forming the matrix associative links between descriptions of the objects on their common characteristics is set up automatically. Description of the object or class of objects localizes in certain parts of the network that can effectively perform different operations of associative search.
Minimization of the presentation of information in the matrix n-GN is carried out by compression of the information at every level, as well as due to the fact that the same combination of attributes of multiple objects represented by.
The relative speed of information processing in the n-GN with the increase in its volume increases.
Fig. 4.This is hardware representation in matrix form
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