x(t + 1) = W x(t) w j = w j W w = : 1 x x x , 1 (cellular automata) (1) :

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1 (t = 0, 1,,...) t t x 1, x, x 3 x 1 x ( 1) (+1) A, B A B -1 x x 1 1 x3 1: B A 1 W = , (1) W j w j j t 3 (x 1 (t), x (t), x 3 (t)) T = x(t) t + 1 1

2 x(t + 1) = W x(t) w j = w j W w = : 1 x x x , 1 (cellular automata) (1) : 1 ( 0.1 ) x t x x x t+1 x x (x 1, x, x 3 ) (000),(011),(101) (fxed pont) (001) (110) (lmt cycle) (010),(100),(111) (001)

3 : 1 (basn) ( ) (attractor) : 1 (111) (000), (110) (000) (110) 3

4 x = (1, 1, 1, 0) T y = (1, 0, 0) T x T y = W x W 4 x y x1 x x3 x4 y1 y y3 4: ( m x (), y ()), = 1... m j k w jk = m =1 x () j y() k, () (1, 0, 0, 0) (0, 0, 1) (0, 1, 0, 0) (0, 0, 1, 0) (0, 1, 0) (0, 1, 1) (0, 0, 0, 1) (1, 0, 0) (3) ( ) W = , (4) 4

5 m m (1978) 3 (1979) ( ) n s = (s 1, s,..., s n ) s 1, 0, m () s 1, 0, +1 w ±1 0 ŝ w ŝ = φ (φ (w) s), (5) φ +1, x > 0 φ(x) = 0, x = 0, (6) 1, x < 0 ( ) ( ) 5

6 4 (Hopfeld & Tank, 1985) 1 1. w j = w j w j = 0. n x = 0, 1 ( = 1,,..., n) n n t(= 0, 1,,...) u (t), x (t) t + 1 x (t + 1) w j θ (t) 1, f u (t) > 0 x (t + 1) = x (t), f u (t) = 0 0, f u (t) 0 u (t) = (7) w j x j (t) θ (t) (8) j x j (t) w j x j (t) θ (t) E = 1 =1 w j x x j + θ x, (9) (7) (8) (9) k 1 6

7 E = 1 w j x x j + θ x 1 x k k j k j k w kj x j 1 x k w k x + θ k x k j t t+1 k (10) x k (t) x k (t + 1) (11) x k = x k (t + 1) x k (t) 1-1 x k E k k ( E k = 1 n ) w kj x j + w k x x k + θ k x k =1 = w kj + w jk =1 x j x k + θ k x k (1) w j = w j E k = w kj x j θ k x k (13) u k E k = u k x k (14) (7) x k > 0 x k 0 1 u k > 0 E k < 0 x k < 0 x k 1 0 u k < 0 E k < 0 x k = 0 E k = 0 E k 0 (15) (Travelng salesman problem:tsp) 7

8 N N A,B,C,D,E 5 TSP 5 BACED A B C D E N! X x X X Y d XY 1. ( 1 ) x X x Xj = 0 (16) X j. ( 1 ) x X x Y = 0 (17) X Y X 3. ( 1 ) x X = N (18) X ( ) 1 X d XY x X (x Y, 1 + x Y,+1 ) (19) Y X 8

9 E = A x X x Xj X j + B x X x Y ( X Y X ) + C x X N X + D d XY x X (x Y, 1 + x Y,+1 ) X Y X A = B = 500, C = 00, D = 500, N = 15 (0) E = 1 w X,Y j x X x Y j + θ X x X (1) X Y j X w X,Y j = Aδ XY (1 δ j ) Bδ j (1 δ XY ) C Dd XY (δ j,+1 + δ j, 1 ) θ X = CN δ j { 1, f = j, δ j = 0, f j () (3) programng ụ dt = u τ + nw j x j θ (4) u, t u u = τ + nw j x j θ (5) u u (t + 1) = u (t) + u (6) 4.. 0, 1 1, 1 9

10 P s x s = (x s 1, x s,..., x s n) (s = 1,,..., P ) s 0 x s E s = 1 wjx s s x s j (7) =1 E s x x j ws j 3 w j s = xs xs j E s = 1 wjx s s x s j = 1 =1 (x s ) ( x s j) (8) =1 ( ) w j = P wj s = s=1 P x s x s j (9) s=1 n P 15 % generalzed nverse matrx Moore Penrose ( ) 1 Z + = Z T Z Z T (30) [ ξ 1, ξ,..., ξ p] N P X W ( ) 1 W = XX + = X X T X X T (31) W 7 px 0, 1 n w j = (x s 1) xs j 1 3 s=1 x -1 1 x 10

11 5 4 A/D 4 x 0 x 1 x x 3 (0 or 1) A/D a E = 1 = 1 ( a ) x + 1 =0 =0 j =0 x (1 x ) =0 ( +j ) x x j (3) ( 1 + a ) x + 1 a =0 0 3 j j =0 w j = 0 I E = 1 =0 j =0 w j x x j (3) (33) x I (33) =0 w j = (+j) ( j) (34) I = ( 1) + a (35) Lyapunov functon (3) de dt = dx (+j) x dt j ( 1) + a (36) =0 j =0 0 (36) de dt du dt 11

12 [ ] du dt = j =0 de dt = (+j) x j ( 1) + a ( = 0, 1,..., 3) (37) =0 dx dt x = f(u ) (38) du dt = f (u ) de dt =0 ( ) du (39) dt f(u ) < 0 (1978).. :. (1979).. :. Hopfeld, J. J. & Tank, D. W. (1985). Neural computaton of decsons n optmzaton problems. Bologcal Cybernetcs, 5,

9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x

2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin