BioSystems, 26 (1992) 135-138 135 Elsevier Scientific Publishers Ireland Ltd.
Self-organization toward criticality in the Game of Life
Keisuke Ito and Yukio-Pegio Gunji Department of Earth Sciences, Faculty of Science, Kobe University, Nada, Kobe 657 (Japan) (Received November 12th, 1991)
Life seems to be at the border between order and chaos. The Game of Life, which is a cellular automaton to mimic life, also lies at the transition between ordered and chaotic structures. Kauffman recently suggested that the organizations at the edge of chaos may be the characteristic target of selection for systems able to coordinate complex tasks and adapt. In this paper, we present the idea of perpetual disequilibration proposed by Gunji and others as a general principle governing self-organization of complex systems towards the critical state lying at the border of order and chaos. The rule for the Game of Life has the minimum degree of pel~petual disequilibrium among 2 TM rules of the class to which it belongs.
Keywards: Life game; Cellular automata; Self-organization; Critical state; Evolution.
Phenomena governed by power laws are wide- biological systems, in which the propagation ly known in nature as 1If noise and the fractal velocity of information is slow (Matsuno, 1989), structure. Bak et al. (1988) proposed the idea of is undeterminable. Let us briefly explain this self-organized criticality as a general principle using the description of a cellular automaton to explain such phenomena. A canonical exam- system (Fig. 1). Suppose that the state of the cell ple of self-organi~'ed criticality is a pile of sand. is 0 when the cell does not contain material A or It was proved that self-organized criticality of 1 when it contains A. The propagation velocity the sand-pile model is a direct consequence of of information v0 is one cell-distance per one the local conservation law (Hwa and Karder, time-step, while the moving velocity of material 1989). Bak et al. (1989) argued that the Game of A vp is two cell-distance per one time-step, be- Life (Gardner, .1970) is a new type of self- ing supposed to be faster than v0. The central organized criticality, having no local conserva- cell x can know the state of the nearest two cells tion law. A question arises why the particular only, and cannot know at time t whether rule of the Game of Life is selected. It has been material A is present in the next cells or not. conjectured by several researchers that complex Nevertheless, the next state of cell x is deter- systems which are able to change their internal mined to be 0 if the system is in the state of Fig. structure or relationship tend to self-organize la, or to be 1 ff it is in the state of Fig. lb. The toward criticality, and some supporting evi- next state is determined only a posteriori. When dence has been presented (Kauffman, 1991; Kauffman and Johnsen, 1991; Langton, 1990; McIntosh, 1990). We need to find the general a b principle to explain why. o,,o o o!o olo o o!e Gunji and Kon-no (1991) and Gunji and ~t2 Nakamura (199].) consider that the behavior of t÷l 0 • Fig, 1. A cellular automaton in which the observation veloci- Correspondence to: K. Ito. ty v0 is less than the particle velocity Vp.
0303-2647/92/$05.00 © 1992 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland 136 we study this kind of system, we have been tak- A(t) f(0) . A(t+0 fm. A(t+2) ing the probabilistic approach. We collect results, estimate the probability, and then pre- dict the future using the probability. The valuable information that material A has come B(i)~ S(i)~ or not at the next time-step is not used directly in such a probabilistic approach. We present an Fig. 2. The procedure of perpetual disequilibration. Sub- alternative, stronger approach. scripts: i = 0, 1,...,n for all possible dynamical equations, and k = 0,1 for all possible time-reversed functions. The time evolution of the above-mentioned Superscripts: h = 1 ..... m for all possible values of one-to- system cannot be described by a deterministic many type functions. equation. We assume a deterministic equation f as a tentative one for the time evolution from time t to t + 1. A set of possible equations which the system may obey are denoted by ~i); i = 0,1,...,n]. In order to use the information it by g(1), and let J~l) be the time-forward func- obtained at time t + 1, we define the time- tion corresponding to g(1). The system will be reversed function off in the following forms. best described, when J~l) is used at the next For the cellular automaton of Fig. 1, we have time-step so that the discrepancy caused by the in general temporal use ofj~0) at time t may be minimized. As long as the system is not reversible, the dis- s(x;t + 1) = f(o(s(x - 1;t),s(x;t),s(x + 1;t)) (1) crepancy persists perpetually and the Hamming distance remains positive. When the system has where s is the state of cell. Two kinds of time- the capability to evolve like living organisms, reversed function ofj~i) are defined as the process of minimizing the discrepancy by changing the operational function 2~) is nothing s(x;t) = g(Oo(S(x;t + 1),s(x- 1;t),s(x + 1;t)) (2a) but the evolution dynamics of the system to modify its structure and function. Accordingly, and the principle of perpetual disequilibration is the base of the rule dynamics of evolution. We apply s(x + 1;t) = g(,)l(s(x;t + 1),s(x- 1;t),s(x;t)) (2b) the principle to explain why the system evolves to select the rule of the Game of Life among Since the system is irreversible, g(i)0 and g(~)l are many other possible ones. not unique but are one-to-many (m) type func- The Game of Life is a kind of two-dimensional tions. Each of them is denoted by g(~ h cellular automaton. The rule is expressed by (i= 0,...,n;k--- 0,1;h = 1 .... ,m). The procedure of perpetual disequilibration is s(ij;t + 1) = f(s(i,j;t), sum(i,j; t)) (3) explained in Fig. 2. Starting from the configura- tion A(t) of the system at time t, the configura- where s is the state of cell being either 0 or 1, tion A(t + 1) is obtained through the operation and sum is the total sum of the state values of of Y~0). Interesting results arise from the follow- 8-nearest neighbors of the cell at (ij). The rule ing backward operation. All kinds of time- of the Game of Life is one of the possible 2 is reversed functions g(~)kh operate on the con- rules, and is described particularly as figuration A(t + 1) and corresponding con- figurations B(0kh are obtained. In order to most f(O,sum) = 1, only when sum = 3, otherwise effectively use the information obtained at f(O,sum) = 0 t + 1 a posteriori, we take the following pro- cedure. Select g(~gkh that minimizes the Hamm- f(1,sum) = 1, only when sum = 2 or 3, other- ing distance between A(t) and B(,~ h, and denote wisef(1,sum) = 0 137
Two types of the time-reversed function of fare and defined as d' = do' + wold1' (7c) s(ij;t) = goh(s(i,j;t + 1), sum(i,j;t)) (4a) The average is taken for all steady-state con- sum(i,j;t) = glh(s(i,j;t + 1),s(i,j;t)) (4b) figurations. The measures of PD for the set of rules, in- Both go and gl are; one-to-many (m) type func- cluding the Game of Life, are computed, using a tions, and each ils denoted by go h and gl h two-dimensional lattice of 16 × 16. The rule (h = 1,...,m). Let the configuration of the number RN is given by system at t be S(t). We get S(t + 1), operating f on S(t). We then reverse the operation, RN = ~{29 × s(i,3) + 2 sum(ia)} × f(s,sum) operating gkh(k = 0,1) on S(t + 1), and get Sk h. The measure of perpetual disequilibrium (PD) is (8) a function of the Hamming distance between S(t) and Sk h. The zeroth order measure at time Starting from a random configuration with an t is defined by the minimum of them as equal proportion of 0 and 1, we iterate using rule f over 200 steps to obtain a steady state, then compute do(S(t)) and dl(S(t)) operatingf and gk h do(S(t)) = min(~'lls(i,j;t)- Soh(i,j;t)l) (5a) repeatedly. We repeat this procedure 50 times, i,j starting from different initial configurations. We then obtain do, do', dl, and dl', taking the d~(S(t)) = min ( ~'~lsum(i,j;t)- suml~(i,j;t)l ) minimum or the average of do(S(t)) and dl(S(t)), i,j respectively. (5b) Because it is hard to compute for all 2 is rules, we computed only for 28 = 256 rules which are where Sk h and sumk h are s and sum, respective- close to the rule of the Game of Life. Namely, ly, in the configuration Sk h. The minimum is f(s,sum) is fixed to be zero when sum is 0 or taken over all possible h of gk h which is one-to- greater than 4. The results are shown in Fig. 3. many type. The zeroth order measure of PD dk The 256 rules are arranged in increasing order for a pair of functionsfand gk is then defined by of the zeroth order measure d of PD. For rules the minimum of dk(S(t)) over an infinitely long of which d is equal to zero, the rules are arrang- time. Finally, the zeroth order measure d of PD ed in decreasing order of the first order measure for the transition rule f is defined as the sum of d' of PD. In general, rules with d > 0 belong to do and d I as the class III cellular automata by Wolfram's classification (Wolfram, 1984) with a few excep- d = do + wold1 (6)
Here, w01 is taken to be 1/8, because the max- imum of Is(ij;t) - Soh(ij;t)l is 1, while the max- imum of Isum(ij;t) - sumlh(ij;t)l is 8. dtl j// Furthermore, we ,define the first order measure f.~f of PD by the average of the Hamming distance as S
Fig. 3. The zeroth order and the first order measures of per- do' =