1 : ( ) ( ) ( ) ( ) ( ) etc (SCA)
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1 START: 17th Symp. Auto. Decentr. Sys., Jan. 28, 2005 Symplectic cellular automata as a test-bed for research on the emergence of natural systems
2 1 : ( ) ( ) ( ) ( ) ( ) etc (SCA)
3 2 SCA 2.0 CA ( ) E.g. ERCA [4]-[12] [14, 15] ( ) SCA CA SCA
4 2.1 [ ] R E.g. K = Z/(2) n- (x 1,..., x n ), x i {0, 1}. (K n ) A E.g. A K n- K[x 1,..., x n ] a 1,...,a N Z + b k x a 1 1 x a N N, b k {0, 1}.
5 [ ] a, b A, α, β R A 1 v : A A v(αa + βb) = αv(a) + βv(b) v(ab) = av(b) + v(a)b A A- Der(A) Der(A) [a, b] = ab ba R α i, β i R, u, v, w Der(A) [ α i u i, β j u j ] = α i β j [u i, u j ], [u, u] = 0 ( [v, u] = [u, v]), [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0. E.g. F i K = Z/(2) i F i x i Der(A).
6 [ ] 1 R A Ω 1 (A) = {0} 0 R, Ω 0 (A) = R Ω 1 (A) = {da a A} a, b A, α R d(αa) = αda, d(a + b) = da + db, d(ab) = da b + a db, a db = db a, da db = db da, da da = 0 d : Ω n (A) Ω n+1 (A) : Ω m (A) Ω n (A) Ω m+n (A) (m, n 0) Ω 2 (A), Ω 3 (A),... Ω(A) = Ω p (A) p=0 A M Ω(A)
7 [ ] 2 A = R[x 1,..., x n ] Der(A) { x i } n i=1 Ω1 (A) {dx i } n i=1 A p- ω v ι v : Ω p (A) Ω p 1 (A) 1. ω Ω 0 (A) = R v = 0 ι v (ω) = ω = dx i 1 dx ip v = x j ( 1) k 1 dx i 1 dx i k 1 dx i k+1 dx i p (j = i k ) ι v (ω) = 0 (j i k for k {1... p}) 3. ι v (ω) i.e., ι v+w (ω) = ι v (ω)+ι w (ω), ι v (ω+µ) = ι v (ω)+ι v (µ). 4. ι av (ω) = ι v (aω) = aι v (ω) (a A). da Ω 1 (A) v Der(A) da, v ι v (da) = v(a) ω Ω p (A) v 1... v p ι v p( (ι v1 (ω))) R ω(v 1,..., v p )
8 [ ] X F X 2 3 A R- ω Ω 2 (A) α : Der(A) Ω 1 (A)(α : v α(v)) α(v)(w) ω(w, v) α : Der(A) Ω 1 (A) A- 2- ω A 4 A F F v F v F α 1 (df ) ( α(v F ) = df )
9 5 {, } : A A A A 2 F G {F, G} v F G = dg, v F = dg(v F ) = ω(v F, v G ). 1 A ω {, } : A A A A α i, β i R, F, G, H A { α i F i, β j F j } = α i β j {F i, F j }, {F, G} = {G, F }, {F G, H} = {F, H}G + F {G, H}, {F, {G, H}} + {G, {H, F }} + {H, {F, G}} = 0 R
10 2 F, G A v {F,G} = [v F, v G ]. F v F
11 2.2 - [ - ] M q M T M (q, q) T M L : T M R L M q(t) S T L(q, q)dt 0 S δs δs = = = T 0 T 0 T 0 δl(q, q)dt T L(q, q) L(q, q) δqdt + δ qdt q 0 q ( L(q, q) d ) L(q, q) δqdt = 0 q dt q S (δs = 0) -
12 [ - ] ( R ) A A ( ) L A A {0,..., T } H A (T +1) = A 0 A T (A 0 = = A T = A) S S T 1 i=0 L i (L i 1 0 }{{} L 1 T H). i & (i+1) th A Ω(A) p Ω p (A A) = Ω q (A) Ω p q (A) q=0 A A Ω(A A)
13 Ω(H) 1- Ω 1 (H) = Ω 1 (A 1 A T ) = T A 0 A 1 Ω 1 (A i ) A T. i=0 i p i : Ω 1 (H) Ω 1 (H) d i p i d H d = T i=0 d i 0 T H δ T 1 i=1 d i. S δs = T 1 (d i L i + d i L i 1 ) = 0. (1) i=1
14 ϕ : A 2 A 0 A 1 (pull-back) ϕ 0, 1 q 0, q 1 2 q 2 = ϕ(q 0, q 1 ) f A 2 f(q 2 ) = f( ϕ(q 0, q 1 )) = (f ϕ)(q 0, q 1 ) ϕ(f) f ϕ ϕ Φ : A 1 A 2 A 0 A 1 Φ : a 1 1 a 1 a ϕ(a) (2) ϕ Φ - Φ (d 1 L 1 ) + d 1 L 0 = 0. (3) Φ : Ω 1 (A 1 A 2 ) Ω 1 (A 0 A 1 ) Φ Ω 1 (A 0 A 1 )
15 Φ A A Φ : A 1 A 2 A 1 A 2 - Φ (d 1 L) + d 2 L = 0, (4) L = L 1 A 1 A 2 dl = d 1 L + d 2 L E.g. d 1 L 1 = d 1 L(q 1, q 2 ) = ( L(q 1, q 2 )/ q 1 ) dq 1 Ω 1 (A 1 A 2 ) q 2 = ϕ(q 0, q 1 ) Φ d 1 L 1 = ( L(q 1, ϕ(q 0, q 1 ))/ q 1 ) dq 1 Ω 1 (A 0 A 1 ) ( L(q1, q 2 ) + L(q ) 0, q 1 ) dq 1 = 0 q 1 q 1 dq 1 -
16 2.3 [ ] M L : T M R T M h s : M M s R L (q, q) T M L(h s (q), h s ( q)) = L(q, q) 1- L I : T M R M q I I(q, q) = L dh s (q) q ds. s=0
17 [ ] A v A A D D 1 v + v 1, i.e., D(a b) a v(b) + v(a) b 6 DL = (1 v + v 1)L = 0 v L A A 3 ( ) Φ - v L F v (v 1)L F v A A Φ Φ(F v ) = F v 4 L Der(A A)
18 A A A A 7 L 2- ω Ω 2 (A A) ω dd 1 L = dd 2 L ω ω 5 Φ - ω Φ Φ ω = ω
19 ω Der(A A) Ω 1 (A A) α : v ω(, v) k- 6 ω v L v F v = (v 1)L L A A
20 2.5 SCA SCA (3) (4) - 1 {1,..., N} i σ i R R N L N L({σi t, σt+1 i } N i=1 ) = L({σj t, σt+1 j } j N (i) ) (5) i=1 N (i) i N L L (3) 1 SCA {j i N (j)} L({σ t k, σt+1 k } k N (j) ) σ t i + L({σt 1 k, σk t } k N (j)) σi t SCA = 0. (6)
21 3 SCA ERCA Takesue 1 CA(ERCA) [4]-[8],[12] E.g. etc SCA 3.1 ERCA (σ 1,..., σ N ) = {0, 1} N σ t+1 i = f(σ t i 1, σt i, σt i+1 ) + σt 1 i mod 2.
22 f : {0, 1} 3 {0, 1} 3 f(σ 1, σ 0, σ 1 )2 σ 1+2σ 0 +4σ 1 σ 1,σ 0,σ 1 {0,1} R E.g. f(s 1, s 0, s 1 ) = σ 1 + σ 1 90R. C : {0, 1} 2N R C(σ1 t... σn; t σ t σ t 1 N ) = C(σt σ t+1 N ; σt 1... σn) t N C ERCA c : {0, 1} 2(α+1) R 0 α < N N C(σ1 t... σt N ; σt σ t 1 ) = N i=1 c(σ t i... σt i+α ; σt 1 i... σ t 1 i+α ) C α
23 3.2 SCA ERCA R 2 Z/(2) Z/(2) 1 i N (i) = {i 1, i, i + 1} L({σj t, σt+1 } j N (i) ) j L({σ t j, σt+1 j } j N (i) ) = σ t+1 i σ t i V (σt i 1, σt i, σt i+1 ) (7) SCA (6) σ t+1 i = +1 j= 1 σ t i V (σ t i+j 1, σt i+j, σt i+j+1 ) σt 1 i. (8) V R[σ i 1, σ i, σ i+1 ] ERCA 256 ω dd 2 L = i dσt+1 i dσ t i
24 3.3 ERCA SCA (i) (8) 3 V R[σ i 1, σ i, σ i ] V E.g. 90R V σ i 1 σ i ( 1 2 (σ i σ i 1 ) 2 ) σi 1 2 σ i + σ i 1 σi 2. (ii) V (7) (5) (iii) 6((1 v + v 1)L = 0) α Der(R[σ 1,..., σ α ]) 3 E.g. V = σ i 1 σ i L v = N i=1 σ i+1 + σ σ i i σ i+1 F v (v 1)L = N i=1 σt+1 i+1 σt i + σt+1 i σi+1 t. v V = σi 1 2 σ i + σ i 1 σi 2
25 SCA 4, 6 R[σ 1,..., σ N ] [4]-[8],[12] SCA
26 3.4 SCA (i) (ii). SCA (i) (ii) SCA R E.g. R = Z/(p), p > 2 L
27 4 E.g. [18, 19] SCA End
28 References [1] K. Morita: Computation-universal models of two-dimensional 16-state reversible cellular automata, IEICE Trans. Inf. & Syst., E75-D-1, 141/147 (1992) [2] H.H. Chou, J.A. Reggia: Emergence of self-replicating structures in a cellular automata space, Physica D, 110, 252/276 (1997) [3] C. Salzberg, A. Antony, H. Sayama: Evolutionary dynamics of cellular automata-based self-replicators in hostile environment, BioSystems, 78, 119/134 (2004) [4] S. Takesue: Reversible cellular automata and statistical mechanics, Phys. Rev. Lett., 59, 22, 2499/2502 (1987) [5] S. Takesue: Ergodic properties and thermodynamic behavior of elementary reversible cellular automata. I. basic properties, J. Stat. Phys., 56, 371/402 (1989) [6] S. Takesue: Relaxation properties of elementary reversible cellular automata, Physica D, 45, 278/284 (1990) [7] S. Takesue: Fourie s law and the Green-Kubo formula in a cellular-automaton model, Phys. Rev. Lett., 64, 3, 252/255 (1990) [8] S. Takesue: Boltzmann-type equations for elementary reversible cellular automata, Physica D, 103, 190/200 (1997) [9] G.Y. Vichniac: Simulating physics with cellular automata, Physica D, 10, 96/116 (1984)
29 [10] T. Toffoli and N. Margolus: Invertible cellular automata: A review, Physica D, 45, 229/253 (1990) [11] R.M. D Souza and N.H. Margolus: Thermodynamically reversible generalization of diffusion limited aggregation, Phys. Rev. E, 61, 1, 264/274 (1999) [12] T. Hattori, S. Takesue: Additive conserved quantities in discrete-time lattice dynamical systems, Physica D, 49, 295/332 (1991) [13] K. Kaneko: Symplectic cellular automata, Physics Letters A, 129-1, 9/16 (1988) [14] J.C. Baez and J.W. Gilliam: An algebraic approach to discrete mechanics, Lett. Math. Phys., 31, 3, 205/212 (1994) [15] J.W. Gilliam: Lagrangian and symplectic techniques in discrete mechanics, Ph.D. dissertation, Univ. California Riverside (1996) [16] (2004 [17] 7 (2000 [18] T. Nozawa, Y. Miyake: Description of systematicity intrinsic to the dynamics of complex-system models, Chaos, Sol. & Frac., 14-7, 1095/1115 (2002) [19] (2002)
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