Max-Plus Generalization of Conway's Game of Life
aa r X i v : . [ n li n . C G ] S e p Max-Plus Generalization of Conway’s Game ofLife
Kotaro Sakata, Yuta Tanaka and Daisuke TakahashiDepartment of Pure and Applied Mathematics, Waseda University,3-4-1, Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan
Abstract
We propose a max-plus equation which includes Conway’s Game ofLife (GoL) as a special case. There are some special solutions to theequation which include and unify those to GoL. Moreover, the multi-value extension of GoL is derived from the equation and the behavior ofsolutions is discussed.
Keywords: Cellular Automaton; Conway’s Game of Life; Max-Plus Equation
Conway’s Game of Life (GoL) is a binary cellular automaton (CA) and ex-presses a kind of population ecology[1, 2]. It is an evolution game using atwo-dimensional orthogonal grid of cells and each cell has either of two states,alive or dead. The evolution rule for the discrete generation is defined as follows.1. Birth: If there are just 3 live cells in the Moore neighborhood of a deadcell, the dead cell changes to the live cell at the next generation.2. Survival: If there are 2 or 3 live cells in the Moore neighborhood of a livecell, it is alive at the next generation.3. Death: Otherwise, the cell is dead at the next generation.Let us assume values of two states, 1 for alive and 0 for dead. If u nij denotesthe value at ( i, j ) cell of the generation n , the above evolution rule can betranscribed into the evolution equation, u n +1 ij = ( u nij , s nij ) = (0 , ,
2) or (1 , , (1)where s nij = u ni − j − + u nij − + u ni +1 j − + u ni − j + u ni +1 j + u ni − j +1 + u nij +1 + u ni +1 j +1 .Specific solutions to GoL have been vastly searched and listed. There are varioustypes of evolution of solutions and they are analogous to activities of life.In this paper, we propose an extended model of GoL using max-plus opera-tion. Max-plus operation is based on max-plus algebra which is a commutativesemiring defined by addition ‘max’ and multiplication ‘+’[3]. It is used for thedescription of discrete event systems and utilized to analyze the dynamics with1ax-plus linear system theory based on the Perron-Frobenius theory[4]. It isalso obtained by ultradiscretizing the difference equations through the limitingprocedure, lim ε → +0 ε log( e A/ε + e B/ε ) = max(
A, B ) . (2)Tokihiro et. al. found that the binary CA called ‘box and ball system’ isobtained by ultradiscretizing the discrete soliton equation through the abovelimit and showed that the solutions to the box and ball system giving solitoninteractions among groups of balls can be derived by the same limit of multi-soliton solutions to the discrete equation[5, 6].In the above context, we can consider that max-plus expression proposesnovel viewpoint and mathematical tools to pure digital systems like CA. Wepropose a max-plus equation with a continuous dependent variable in whichGoL is embedded as a special case. There exist real-valued exact solutions tothe equation and they include and unify those to GoL. Contents of this paper areas follows. In Section 2, the max-plus equation extended from GoL is proposed.In Section 3, special solutions to the max-plus equation and their relations tothose to GoL are shown. In Section 4, we show the multi-value CA obtainedfrom the max-plus equation and discuss the behavior of solutions. In Section 5,we give concluding remarks. Let us consider the following evolution equation using operators max, + and − . u n +1 ij = F ( u nij , s nij ) , (3)where i and j are integer space coordinates, n integer time, s nij the sum of eight u ’s in the Moore neighborhood, s nij = u ni − j − + u nij − + u ni +1 j − + u ni − j + u ni +1 j + u ni − j +1 + u nij +1 + u ni +1 j +1 , and F ( u, s ) is defined by F ( u, s ) = max(0 , u + s − − max(0 , u + s − − max(0 , s −
3) + max(0 , s − . If 0 ≤ u ≤
1, we can easily show 0 ≤ F ( u, s ) ≤
1. Figure 1 shows the graphs of F (0 , s ), F (0 . , s ) and F (1 , s ).Consider the initial value problem for equation (3) and assume n = 0 is aninitial time. If we set the initial data u ij to satisfy 0 ≤ u ij ≤ i and j ,then any u nij ( n >
0) also since 0 ≤ F ( u, s ) ≤
1. Moreover, if we assume u nij at acertain n for any i and j takes either of the values 0 and 1, s nij is one of the nineinteger values from 0 to 8. Then the value of RHS of equation (3) is also 0 or 1considering the graphs of F (0 , s ) and F (1 , s ). Therefore, the value of solution u nij to equation (3) can be closed in the binary set { , } if the initial data u ij is.Then the evolution equation (3) becomes equivalent to equation (1) consideringthe profiles of F (0 , s ) and F (1 , s ). Thus equation (3) includes the rule of GoL asa special case. We call the evolution system defined by equation (3) ‘MaxLife’2 s F (a) F (0 , s ) s F (b) F (0 . , s ) s F (c) F (1 , s )Figure 1: Graphs of F ( u, s ).in this meaning and discuss the behavior of its real-valued solutions closed inthe range of [0 ,
1] relating them to binary solutions closed in the range of { , } which are also solutions to GoL. In this section, we show the special solutions to MaxLife. Since it is difficultto solve equation (3) in a systematic way, we assume dimensions, symmetryand period of solutions within the background u = 0. Solutions shown below isconfined in 4 × × region (‘Block’) The first example is a static solution confined in 2 × . . . . . .. . . c d . . .. . . a b . . .. . . . . . , where a , b , c and d are all real constants from 0 to 1. The region outside theshown is u = 0. Substituting the above solution to equation (3), we obtain oneof equations, a = max(0 , a + b + c + d − − max(0 , a + b + c + d − − max(0 , b + c + d −
3) + max(0 , b + c + d − . Since a , b , c , d ∈ [0 , a = max(0 , a + b + c + d − − max(0 , a + b + c + d − a = b = c = d, a = max(0 , a − − max(0 , a − . Solving this condition, we have a = b = c = d = 0 or 23 or 1 . The above values give a trivial solution ( a = 0), a non-integer solution ( a = 2 / a = 1). Solutions from this subsection are shown schematically as figures without proof.The four colored cells shown in Figure 2 are used to denote the values of u where a is any constant from 0 to 1. The ‘blinker’ type of solution is shown in Figure 3. u = 0 1 a − a Figure 2: Four colored cells denoting the values of u .This solution is periodic with period 2. The double arrow ‘ ↔ ’ means the stateat next time of the left (right) state is the right (left). Since the ‘blinker’ of GoL ↔ Figure 3: ‘Blinker’ type of solution.is obtained in the case of a = 0 and 1, the solution in Figure 3 includes twoconfigurations of ‘blinker’ rotated 90 degree to each other as shown in Figure 4. ↔ ↔ a = 0 a = 1Figure 4: ‘Blinker’ of GoL.4 ↔ (a) ‘clock’ ( a = 0), ‘block’ ( a = 1) (b) ‘clock’ ( a = 0), ‘snake’ ( a = 1) ↔ ↔ (c) ‘clock’ ( a = 0), ‘barge’ ( a = 1) (d) ‘clock’ ( a = 0), ‘pond’ ( a = 1) ↔ (e) ‘clock’ ( a = 0), ‘clock’ ( a = 1)Figure 5: Solutions to equation (3) with period 2 giving ‘clock’ for a = 0. For a = 1, they give a static solution ((a)–(d)) or ‘clock’ rotated by 90 degree ((e)). The next examples shown in Figure 5 are 5 types of solutions giving the ‘clock’and another of GoL in a special case. All solutions give ‘clock’ for a = 0. For a = 1, static solution of GoL or another ‘clock’ rotated by 90 degree are given.The solutions of GoL described here are shown in Figure 6. Note that ‘clock’ isperiodic with period 2 and the other solutions are all static. ↔ ‘clock’‘block’ ‘snake’ ‘barge’ ‘pond’Figure 6: Solutions to GoL included in Figure 5 The next group of solutions include ‘toad’ of GoL for a = 0. Figure 7 shows thesolutions and Figure 8 shows ‘toad’. 5 ↔ (a) ‘toad’ ( a = 0), ‘block’ ( a = 1) (b) ‘toad’ ( a = 0), ‘barge’ ( a = 1) ↔ ↔ (c) ‘toad’ ( a = 0), ‘pond’ ( a = 1) (d) ‘toad’ ( a = 0), ‘snake’ ( a = 1) ↔ ↔ (e) ‘toad’ ( a = 0), ‘snake’ ( a = 1) (f) ‘toad’ ( a = 0), ‘snake’ ( a = 1) ↔ ↔ (g) ‘toad’ ( a = 0), ‘snake’ ( a = 1) (h) ‘toad’ ( a = 0), ‘clock’ ( a = 1) ↔ ↔ (i) ‘toad’ ( a = 0), ‘clock’ ( a = 1) (j) ‘toad’ ( a = 0), ‘toad’ ( a = 1) ↔ ↔ (k) ‘toad’ ( a = 0), ‘toad’ ( a = 1) (l) ‘toad’ ( a = 0), ‘toad’ ( a = 1) ↔ ↔ (m) ‘toad’ ( a = 0), ‘toad’ ( a = 1) (n) ‘toad’ ( a = 0), ‘toad’ ( a = 1)Figure 7: Solutions to equation (3) with period 2 giving ‘toad’ for a = 0. For a = 1, they give (a)-(g) a static solution, (h)-(i) ‘clock’, and (j)-(n) anotherconfiguration of ‘toad’. 6 ‘toad’Figure 8: ‘Toad’ of GoL. There are solutions giving a moving pattern of GoL. One of the simplest so-lutions is called ‘glider’ shown in Figure 9. Figure 10 shows the real-valued → → → →
Figure 9: ‘Glider’ of GoL.solution and it coincides with that of Figure 9 if a = 1 and gives another gliderof different time phase if a = 0. → → → → Figure 10: Solution to equation (3) giving ‘glider’ with different time phase for a = 0 and 1. There are other variations of solution obtained by rotating or reflecting thosedescribed in Subsections 3.3 and 3.4. We can derive a general solution unifyingall such solutions. Assume a periodic and symmetric solution with period 2 andwith a point symmetry confined in 4 × u u u u u u u u u u
00 0 u u ↔ u u u u u u u u u u
00 0 u u i (1 ≤ i ≤
20) as follows. u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a ,u = a + a + a + a + a + a + a + a + a + a , (4)where 0 ≤ a i ≤ i and a + a + · · · + a = 1. If we set a i = 1 and a j = 0( j = i ), one of the solutions reported in Subsections 3.3 and 3.4 or its reflectionor rotation is obtained. Note that 20 parameters are redundant and we canreduce them to 6 through the transformation of parameters though the aboveexpression is convenient to give a special solution. Figure 11 shows examples ofsolution obtained by equation (4). Figure 11 (a), (b) and (c) show solutions for a = 0 .
25, 0.5 and 0.75 respectively where other a i ’s are randomly given. Wecan observe that the ‘toad’ solution to GoL emerges as a approaches 1. ↔ ↔ (a) a = 0 .
25 (b) a = 0 . ↔ ↔ (c) a = 0 .
75 (d) a = 1Figure 11: Examples of solution by equation (4). The range of u in equation (3) can be closed to the finite set { , /N, /N, . . . , ( N − /N, } for a positive integer N . Then equation (3) becomes( N + 1)-value CA. The case N = 1 is the original GoL. Figure 12 shows anexample of evolution for 2, 3 and 10-value cases ( N = 1, 2, 9) with periodicboundary condition from random initial data. Since it is difficult to evaluate8 = 0 n = 200 n = 400(a) 2-value case n = 0 n = 200 n = 400(b) 3-value case n = 0 n = 200 n = 400(c) 10-value caseFigure 12: Evolution from random initial data for equation (3) as multi-valueCA. 9he behavior of general solution quantitatively, we describe our observation fromnumerical computation as shown in Figure 12. Solution of 2-value case (GoL)tends to change drastically as time proceeds and often results in the steadystate with separated static and periodic patterns. In contrast to 2-value case,solution of 3 or more value case rarely results in the steady state and continuesto evolve with connected non-zero domains interacting one another.There are various basic static or periodic solutions confined in a finite regionfor multi-value case. Some of them can be obtained by choosing the parametersof solutions reported in Section 3. For example, if a is set to 1 / /
3, 4-value case. Moreover, if the dimensions of region for u = 0 and theperiod are assumed, all solutions can be searched numerically. For example,there are 40 static solutions and 23 periodic solutions with period 2 for 3-valuecase. Solutions to GoL are included in them and 13 of 40 static solutions and3 of 23 periodic solutions are constructed only from 0 and 1. The number ofsteady basic solutions for 3-value case are much larger than 2-value case, andit suggests the persistence of evolution of non-zero area for multi-value case asshown in Figure 12. We proposed the max-plus equation (3) as the difference equation on a real-valued state variable. It includes GoL as a special case if the state value isrestricted to 0 and 1. It has special solutions including a free parameter and theyunify those to GoL by special choice of parameter. Though various solutionsto GoL have been reported independently, their relations are suggested throughthis unification. Among such solutions, we obtained a solution (4) includingmany parameters unifying various solutions to GoL. However, a systematic wayto derive general solutions has not been found yet. It is one of future problemsto propose a way to solve equation (3) as we solve the differential equations.Max-plus equation can be obtained from difference equation using an expo-nential type of transformation of variables with a limiting parameter. Considerthe following difference equation, U n +1 ij = C (1 + δ U nij S nij )(1 + δ S nij )(1 + δ U nij S nij )(1 + δ S nij ) , (5)where S nij = U ni − j − U nij − U ni +1 j − U ni − j U ni +1 j U ni − j +1 U nij +1 U ni +1 j +1 , and C = (1 + δ ) (1 + δ )(1 + δ ) . If we use the transformation including a parameter ε , U nij = e u nij /ε , S nij = e s nij /ε , δ = e − /ε , equation (3) is obtained from equation (5) by the limit ε → +0. Note that thelimit formula (2) is used in the derivation.10xample of evolution of solution to equation (5) is shown in Figure 13 for ε = 0 .
1. The background value is U = e /ε = 1 and randomly chosen cells areset to U = e /ε for initial data. The initial data changes drastically at n = 1,some patterns survive and evolve from n = 2 to 16, and they merge and extendto the whole area from n = 32 to 128. This extended pattern is evolving untilat least n = 10000 and the range of U is always preserved about from 1 to e /ε . Though some stable static patterns confined in a finite area are foundnumerically, exact solutions giving static, periodic or moving patterns have notyet been found. It is another future problem to find exact solutions and todiscuss the relation between solutions to equation (3) and to equation (5). n = 0 1 24 8 1632 64 128Figure 13: Example of evolution of solution to equation (5) for ε = 0 . References [1] M. Gardner, “The fantastic combinations of John Conway’s newsolitaire game ‘life’”, Scientific American , 1970 pp.120-123.doi:10.1038/scientificamerican1070-120.[2] S. Wolfram, “A New Kind of Science”, Wolfram Media Inc, 2002.113] S. Gaubert and Max Plus, “Methods and applications of (max,+) linearalgebra”, Lecture Notes in Computer Science, , Springer, Berlin, 1997pp.261-282. doi:10.1007/BFb0023465.[4] F. Baccelli, G. Cohen, G. J. Olsder and J. P. Quadrat, “Synchronizationand Linearity: An Algebra for Discrete Event Systems” (Wiley Series inProbability and Statistics), John Wiley & Son, 1992.[5] T. Tokihiro, D. Takahashi, J. Matsukidaira and J. Satsuma, “FromSoliton Equations to Integrable Cellular Automata through a Lim-iting Procedure”, Physical Review Letters , 1996 pp.3247-3250.doi:10.1103/PhysRevLett.76.3247.[6] D. Takahashi and J. Satsuma, “A Soliton Cellular Automaton”,Journal of the Physical Society of Japan59