A Study of the O rigin of Comp lex ity in the Science of Comp lex ity

2002 10 10 : 100026788 (2002) 1020066206 1 2, (1., 510090; 2., 519070) : Λ, Λ,, Λ :, Λ, 20 90 (CA S), Λ CA S Λ,, Λ, Λ : ; ; : N 94 : A A Study of the O rigin of Comp lex ity in the Science of Comp lex ity ZHAN G Q i2ren 1, L IN Fu2yong 2 ( 1. Guangdong U n iversity of T echno logy, Guangzhou 510090, Ch ina; 2. Zhuhai Co llege, J inan U n iversity, Zhuhai 519070, Ch ina) Abstract: T he science of comp lexity has been one of the mo st concerned scientific fieldṡ In the science of comp lexity, one of the mo st p rim ary issues is about w hat is the o rigin of comp lexity. T h is paper aim s to m athem atically give its so lutions in the context of general system s studieṡ It p roves that fo r any comp lex system in nature and society, its comp lexity is caused and dom inated by the relationsh ip cycle over the so2called fundam ental levelh d of its h ierarchy. O n the o ther hand, it has obtained rich fruits in the study of physical, chem ical, and b io logical system s, and has started on related question s in eco2 nom ics as developm ent on the comp lex adap tive system s (CA S) to the effect that a new fo rce suddenly cam e to the fo re in Comp lexity Science in the 90s. How ever, the developm ent of the CA S concep t in the context of o rganizational and m anagem ent research is still at its earliest stageṡ T he advances in system s theo ry, m athem atics and computer science on w h ich the above p rogress rests should be useful also to m anagem ent science. R esearch on the evo lution o r reengineering fo r business o rganization should focus ultim ately on develop ing the adap tive enterp rise models em bodied in useful softw are packageṡ W e p re2 sent th is paper in m emo ry of P rof. Xu Guozh i and w ith the hope to carry fo rw ard h is unfinished w isheṡ Key words: the science of comp lexity; comp lex system; theo rem of comp lexity o rigin 1 (T he Science of Com p lex ity), Λ,, Λ, : 2002203221 : (0S010S2) : (1960- ),,,,,,

10 67 [ 1-4,, ] Λ (H all fo r w o rk shop of m etasyn thetic en2 gineering) [ 5-7 ] ; 1991 1,,,,, ; 1994 9 ; 1997 1 68 ; 1999 3 Λ Λ1996, 85, [ 3 ], Λ, 2000 45 68 [ 1-2 ], Λ,,,,, Λ 2 1984, (L o s A lam ou s N ational L ab. ) Geo rge A. Cow an,, W. B rian A rthu r J. H. Ho lland J. Casti G. L eonard,, K. J. A rrow Ph ilip A nderson (San ta Fe In stitu te), Λ 10,,, (A u tom ata N etw o rk, AN ) (Genetic A lgo rithm, GA ) Λ,, (Com p lex ity) Λ,, Λ,, Λ 1994, J. H. Ho lland (Com p lex A dap tive System - CA S),, Λ Λ 1998 3 W ilsonville 5 125, : : Λ Λ 2000 8, 9 Λ,,, (Evo lu tion of Com p lex Structu re and Fo rm ) ; ( D ynam ics of Evo lu tionary Search) ; ( M o lecu lar and Genetic In sigh ts In to Evo2 lu tion) ; ( Evo lu tionary, D evelopm en tal, and Eco logical A spects of B iodiversity) ; ( Social and Cu ltu ral Evo lu tion) [ 8-12 ] Λ Λ,, Λ F. A. Do ria, :W e go from com p lex ity to perp lex ity ( ) [ 13 ] Λ,,, Λ,, Λ 3 [ 5 ], Z (n) E (S ), [14 ] E (S ), S B, Z (n) t H d R z d (t) = {R ij (t) gf ij (si (t), R ij (t), sj (t) ) = 0; 1Φ i, j Φ n; nε 2} : R z d (t) Υ1 (S ), S B E (S )

68 2002 10 4 [ 14-22 ], : E (S ), S B, Z (n) t H d R z d (t) S z (t) H z (t), 1,, H d e (p ) Z (n), sp E (S ) e (p i) Z (n) (i= 1, 2,, k; kε 0) sp i ( 2 ), sp (t) = <p (S (t), sp 1 (t),, sp i (t),, sp k (t) ), 7 1 (S, R (t), R z d (t) ) = 0 7 2 (S, R z d (t), S z (t) ) = 0 7 3 (S, R z d (t), H z (t) ) = 0, S B t E (S ) ; R (t) t E (S ) Z (n), H d 5,. E (S ), S B, Z (n) t H d R z d (t) = {R ij (t) gf ij (si (t), R ij (t), sj (t) ) = 0; 1Φ i, j Φ n; nε 2},, Ηd = 0,, R ij (t) R z d (t), R ij (t) = <ij (S ) S z (t) = Υ2 (S ) H z (t) = Υ3 (S ) 1 Z (n) 2 e (p ), S B t E (S ) ; R (t) t E (S ) Z (n) E (S ) S B, Z (n) t H d R z d (t), Ηd= 0 : k H k (k = d, d - 1,, 2, 1; i= 1, 2,, nk) ek (i) Z (n), 3, sk (i) : sk (i) = <k (i) (S ) ; k = d, d - 1,, 2, 1; i = 1, 2,, n k Ηd= 0, H k (k= d, d - 1,, 2, 1; i= 1, 2,, nk) ek (i) 4, (x, y ) ek (i) (x = 1, 2,, m ; y = 1, 2,, bx; b1+ b2+ + bm Φ nk),, E 3 (S 3 ) = E (S ) {ek (l) gk = k- 1, k- 2,, 2, 1; l= 1, 2,, nk }. g., 3 H k ek (i), sk (i) S 3 ek (i) ek (j) (1Φ j Φ nk; j i) sk (j), sk (i) = <k (i) (S 3 (t), {sk (j) g1 Φ j Φ nk; j i}) sk (i) : = <k (i) (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }, {sk (j) g1 Φ j Φ nk; j i}), i = 1, 2,, n k

10 69 sk (i) = <k (i) (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }, i = 1, 2,, n k} : i) x = 1, s1, y = <1, y (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }), y = 1, 2,, b1 3 ( ) Z (n) 4 H k ek (i) ii) x = 2, s2, y = <2, y (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }, {s1, q gq = 1, 2,, b1}) = <2, y (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }) s2, y = <2, y (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }), y = 1, 2,, b2 iii) x = x, sx, y sx, y = <x, y (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }), y = 1, 2,, bx, x = x + 1, sx + 1, y sx + 1, y = <x + 1, y (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }, {sp, qgp = x, x - 1,, 2, 1; q = 1, 2,, bp }) = <x + 1, y (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }) sx + 1, y = <x + 1, y (S, {sk (l) gk = k - 1, k - 2,, 2, 1; l = 1, 2,, n k }), y = 1, 2,, bx + 1 i), ii) iii), H k ek (i), sk (i) : sk (i) = <k (i) (S, {sk (l) gk = k - 1, k - 2,, 2, 1, l = 1, 2,, n k }), i = 1, 2,, n k g. k= d, H d ed (i), sd (i) sd (i) = <d (i) (S, {sk (l) gk = d - 1, d - 2,, 2, 1, l = 1, 2,, n k }), i = 1, 2,, n d., H d ed (i), sd (i) sd (i) = <d (i) (S, {sk (l) gk = d - 1, d - 2,, 2, 1; l = 1, 2,, n k }, g, {sd (j) g1 Φ j Φ nd; j i}), i = 1, 2,, n d sd (i) = <d (i) (S, {sk (l) gk = d - 1, d - 2,, 2, 1; l = 1, 2,, n k }), i = 1, 2,, n d (1) g. k= d - 1, H k ek (i), sk (i) sk (i) = <k (i) (S, {sk (l) gk = k - 1,, 2, 1; l = 1, 2,, n k }), i = 1, 2,, n k

70 2002 10 sd- 1 (i) = <d- 1 (i) (S, {sk (l) gk = d - 2, d - 3,, 2, 1; l = 1, 2,, n k }, i = 1, 2,, n d- 1. R i, j (t) R z k+ 1 (t) - R z k (t) = {R i, j (t) gf ij (sk+ 1 (i), R i, j (t), sk+ 1 (j) ) = 0; 1Φ i, j Φ nk+ 1}, R i, j (t) = f ij (sk+ 1 (i), sk+ 1 (j) ) = f ij (S, {sk (l) gk = k, k - 1,, 2, 1; l = 1, 2,, n k }) R i, j (t) = f ij (S, {sk (l) gk = k, k - 1,, 2, 1; l = 1, 2,, n k }) H k ek (i), (M - 1), sk (i) sk (i) = <k (i) (S, {sk (l) gk = k - 1,, 2, 1; l = 1, 2,, n k }, {sk (j) gj = 1, 2,, n k; j i}), i = 1, 2,, n k, g, H k ek (i), sk (i) sk (i) = <k (i) (S, {sk (l) gk = k - 1,, 2, 1; l = 1, 2,, n k }), i = 1, 2,, n k sd- 1 (i) = <d- 1 (i) (S, {sk (l) gk = d - 2, d - 3,, 2, 1; l = 1, 2,, n k }), i = 1, 2,, n d- 1 (2) g. k= d - 2, d - 3,, 2, 1, g, sd- 2 (i) = <d- 2 (i) (S, {sk (l) gk = d - 3,, 2, 1; l = 1, 2,, n k }), i = 1, 2,, n d- 2 (3) sd- 3 (i) = <d- 3 (i) (S, {sk (l) gk = d - 4,, 2, 1; l = 1, 2,, n k }), i = 1, 2,, n d- 3 (4) g s2 (i) = <2 (i) (S, {s1 (i) gi = 1, 2,, n 1), i = 1, 2,, n 2 (5) s1 (i) = <1 (i) (S ), i = 1, 2,, n 1 (6) (1) - (6), k H k (k= d, d - 1,, 2, 1; i= 1, 2,, nk) ek (i) Z (n), sk (i) sk (i) = <k (i) (S ), k = 1, 2,, d; i = 1, 2,, n k, R i, j (t) R z d (t), R i, j (t) = f i, j (si (t), sj (t) ) = f i, j (<i (S (t) ), <j (S (t) ) ) = <i, j (S (t) ) R i, j (t) = <i, j (S (t) ), 1 Φ i, j Φ n,, (M - 2) (M - 3), S z (t) = <2 (S (t) ) H z (t) = <3 (S (t) ), Z (n),, H d,,,,,, 6 1999 8,,,,, ( ) ( ) ( ), [ 23 ] [ 24 ]

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