Information Encoding by Using Cellular Automata

Transcription

1 Vol. 44 No. SIG 7(TOM 8) May 2003 CA CA CA 1 CA CA Information Encoding by Using Cellular Automata Yoshihisa Fukuhara and Yoshiyasu Takefuji In this paper, we propose a new information encoding method by using cellular automata (CA). In this approach, plural CA systems work simultaneously and input values are encoded into small bits by simple deterministic process. The proposed system cannot work under all situation depend on their transition rules and input values. We studied when the proposed system works correctly from various kinds of views. 1. CA 1) CA CA CA CA Berlekamp 2) Wolfram 3),4) CA 4 Wolfram IV CA Langton 5) SFC Takefuji Laboratory, Keio Research Institute at SFC Faculty of Environmental Information, Keio University Mitchell 10) CA CA CNN 7),8) CA Mitchell CA EvCA 9),10) CA CA 5) 1 110

2 Vol. 44 No. SIG 7(TOM 8) 111 CA CA CA 2. CA 2.1 CA CA CA 2 3 CA (1) 8 k n k kn 2 3 CA 2 23 = 256 f a (1) (2) a t i,j t i j a t+1 i,j = f(a t i,j x,..., a t i,j,..., a t i,j+x) (1) x = n 1 (2) 2 CA CA Fig. 1 1 CA An example of the information encoder by using CA. 2.2 CA CA

3 112 May 2003 (3) Sj t m CA t j S a (1) f a (4) j CA j j CA { f(x) = S t j = f(a t 1,j,a t 2,j,...a t i,j) (3) 0 a t 1,j = a t 2,j... = a t i,j 1 (4) (1) CA 2 3 (2) (3) (4) (5) (3) (4) (6) 4 CA CA 3. Langton 5) λ (5) λ λ = kn n q (5) k n k n n q 1 λ λ k n CA k kn

4 Vol. 44 No. SIG 7(TOM 8) G Fig. 2 Calculation of parameter G. G G (6) (7) (2) G = k n x qc pq C p (6) 3 Fig. 3 λ 2 5 CA C = { p=1 q= x 0 1 (7) p 1 q C pq p C p p n 1 (1) G =0+( 1) =0 2 G CA 0 CA G λ G 2 CA CA CA ,000 λ G 2 5 CA Fig. 4 CA λ % 10% 4 G λ G CA

5 114 May 2003 Fig CA Fig CA Fig CA Fig CA CA λ =0.5 G G =0 5 6 G =0 λ =0.3 G =0 λ =0.5 4 G λ G λ =0.3 G = CA λ =0.5 λ G λ /k

6 Vol. 44 No. SIG 7(TOM 8) >G> 1 Table 1 Effective parameters for the encoder. λ G bit 0.34 (0.66) bit 0.34 (0.66) bit (0.875) bit bit Fig Fig Fig Fig. 12 λ 2 3 CA Encode ability of CA (2 states 3 neighbors). G 2 3 CA Encode ability of CA (2 states 3 neighbors). λ 3 3 CA Encode ability of CA (3 states 3 neighbors). G 3 3 CA Encode ability of CA (3 states 3 neighbors). 4) λ = 1.0 1/k λ 0 1 λ λ =0.3 CA IV 5) 13 CA (a) (b) (d) II (c) III (c) 2 3 λ =0.5 II CA CA 2 CA CA

7 116 May 2003 (a) (b) (c) (d) 13 CA (a) (b) (c) (d) Fig. 13 Encode ability of CA: (a) 2 states 5 neighbors with 4 patterns to 2 bit, (b) 2 states 5 neighbors with 5 patterns to 3 bit, (c) 2 states 3 neighbors with 4 patterns to 2 bit, (d) 3 states 3 neighbors with 6 patterns to 2 bit. II III 5),6),11) Mitchell 10) III CA 12) 5.3 G G =0 G 1 1 G = G = (1) (2) λ (3) II λ G 2 (1) λ =1 1/k G =0 (2) λ G 6. CA

8 Vol. 44 No. SIG 7(TOM 8) 117 CA 1 CA Langton λ G 7. G CA CA 13) 1) von Neumann, J.: Theory of self-reproducing automata, edited and completed by A. Burks, University of Illinois Press (1966). 2) Berlekamp, E., Conway, J. and Guy, R.: Winning Ways for your Mathematical Plays, Academic Press (1982). 3) Wolfram, S.: Statistical mechanics of cellular automata, Rev. Mod. Phys., Vol.55, pp (1983). 4) Wolfram, S.: Universality and Complexity in Cellular Automata, Physica D, Vol.10, pp.1 35 (1984). 5) Langton, C.G.: Computation at the Edge of Chaos: Phase Transitions and Emergent Computation, Physica D, Vol.42, pp (1990). 6) Packard, N.H.: Adaptation toward the edge of chaos, Dynamic Patterns in Complex Systems, pp , Singapore, World Scientific (1988). 7) Chua, L.O. and Yang, L: Cellular neural networks: Theory, IEEE Trans. Circuits and Syst., Vol.35, No.10, pp (1988). 8) Chua, L.O. and Yang, L.: Cellular neural networks: Applications, IEEE Trans. Circuits and Syst., Vol.35, No.10, pp (1988). 9) Mitchell, M., et al.: Revisiting the edge of chaos: evolving cellular automata to perform computations, Complex Systems, Vol.7, pp (1993). 10) Mitchell, M., Crutchfield, J.P. and Hraber, P.T.: Dynamics, Computation and the Edge of Chaos : A Re-Examination, Cowan, G., Pines, D. and Melzner, D. (Eds), Complexity: Metaphors, Models and Reality, Santa Fe Institute Stuides in the Sciences of Complexity, Vol.19. Addison-Wesley (1993). 11) Kauffman, S.A.: The origin of order, Oxford University Press (1993). 12) Suzudo, T.: Crystallization of two-dimensional cellular automata, Complexity International, Vol.6 (1999). 13) Mori, T., et al.: Edge of Chaos in Rulechanging Cellular Automata, Physica D, Vol.116, pp (1998). ( ) ( ) ( ) SFC NTT 1998 Dr. Yoshiyasu Takefuji: professor of Keio University from 1992, was on the faculty of Case Western Reserve Univ. since 1988, Univ. of South Caroline ( ), Univ. of South Florida ( ). Research: neural computing and hyperspectral computing. Awards: NSF-RIA in 1989, distinguished service from IEEE Trans. on NN in 1992, best paper of IPSJ in 1980 and that of IFAC at AIRTC 98, TEPCO and KAST in 1993, Takayanagi award in 1995, KDD award in 1997, NTT tele-education courseware award in Government advisor: NCC of Philippines, VITTI of Vietnam, Jordan CTTISC, Thailand, Srilanka, Hong Kong, Multimedia Univ. in Malaysia. Published: 22 books and more than 200 papers.