On weak universality of three-dimensional Larger than Life cellular automaton
OOn weak universality of three-dimensionalLarger than Life cellular automaton
Katsunobu Imai, Kyosuke Oroji, and Tomohiro Kubota
Hiroshima University, Hiroshima, Japan [email protected]
Abstract.
Larger than Life cellular automaton (LtL) is a class of cellu-lar automata and is a generalization of the game of Life by extending itsneighborhood radius. We have studied the three-dimensional extensionof LtL. In this paper, we show a radius-4 three-dimensional LtL rule isa candidate for weakly universal one.
Keywords: cellular automata, Larger than Life, three-dimension, weakuniversality
A class of two-dimensional cellular automata,
Larger than Life (LtL) [6], is anatural generalization of the game of Life [4] by extending the radius of itsneighborhood. A lot of interesting patters such as bugs are found in LtL [7].A bug is a kind of glider or spaceship patterns found in the game of Life [9,4].An radius-5 LtL rule, Bosco [6] has high computing efficiency employing thefunction of bugs and the other coherent structures [8].As far as the game of Life, there are several studies about three-dimensionalextensions [1,2]. Although there are many combination of the ranges of birth andsurvival, Bays found some better candidates as the three-dimensional “Life” andfound several glider patterns [1,3]. Although the computational universality of(two-dimensional) Life was shown more than 35 years ago [4], there is no relatedstudy of universality of three-dimensional Life, to the best of our knowledge.We have studied LtL rules in three-dimension. We showed a methodology todesign period-1 bugs and constructed several period-1 bugs in the cases of radiifrom 3 to 7 and we also found several stable patterns and blinker patterns [10].At the time, we also tried to find some functional patterns such as turning a bugor copying a bug by colliding a bug to blinkers or stable blocks. Because theyare thought to be inevitable for embedding a computing function to a cellularspace. But the trial was not fruitful. Because there was two reasons: (1) thenumber of found bugs was too small, (2) period-1 bugs are too ‘solid’ to recoverthe shapes after some reactions cased by the collisions of another bug, stableblock or blinker.In the process of constructing these bugs and the simulation of collisions ofthem, we also found a period-10 bug in the case of radius 4 [10]. We noticed that a r X i v : . [ n li n . C G ] M a r he bug has a kind of stabilizing property to recover the original shape of the bugeven starting from a pattern which is not any pattern of the bug. Unfortunatelythe bug did not have any useful function, we have been trying to find bugs withbit longer periods.We have finally found a radius-4 three-dimensional LtL rule as a weaklyuniversal candidate, i.e, starting from a periodic infinite initial configuration, itis possible to simulate a universal machine (the rule 110 cellular automaton canbe simulated [5]). Larger than Life (LtL) [6] L is a natural generalization of the game of Life anddefined by L = ( r, β , β , δ , δ ), where r is the radius of its neighborhood, [ β , β ]is the range of the total number of live neighborhood cells for giving birth to thecentral focus cell (0 → δ , δ ] is the range for survival of the central focuscell (1 → r + 1,thus the neighborhood includes (2 r + 1) cells. Note that the game of Life canbe denoted by (1 , , , , bug . A bug is a pattern which crawls the cellular space and the same patternrepeatedly appears at a certain displaced position d = ( d , d ) after a certain period τ . Fig. 1.
Family of bugs τ ; ( d , d ) supported by Bosco’s rule (5 , , , ,
58) [6].
LtL can be also defined in three-dimension in the same way, except thenumber of neighborhood cells is (2 r + 1) . It is extremely large, for example, 729in the case of radius-4. A displacement vector of a bug needs to be also modifiedto three-dimension. , , , , In this section we show that the three-dimensional LtL rule, L = (4 , , , , τ B = 13 , d B = (0 , , t +1 t +2 t +3 t +4 t B0 B1 B2 B3 B4B5 B6 B7 B8 B9B10 B11 B12 t +6 t +5 t +7 t +8 t +9 t +11 t +10 t +12 t +13 Fig. 2.
The evolution of a bug B in a period.
Fig. 3.
The cross-sectional view of a bug B0.
We also found two blinkers P and Q. P is a period-4 blinker. Fig. 4 shows theevolution of the patterns in a period. Fig. 5 shows the cross-sectional patternP2 of the blinker P in Fig. 4.Q is a period-14 blinker. Fig. 6 shows the evolution of the patterns in a period.In contrast to the blinker P, the blinker Q is asymmetric in the structure. Q i andQ i +7 (0 < i <
6) are symmetric and both sequences are alternatingly appeared.Fig. 7 shows the cross-sectional pattern of Q1 of the blinker Q in Fig. 6.A collision of a bug B to a blinker Q, with a special offset and phase of aperiod, realizes a turning module of the bug B. Fig. 8 shows a process of theright turning. An input bug B0 changes the direction. Place B0 at (0 , ,
0) andQ1 at (4 , ,
6) as an initial configuration, after 34 steps, a right-turned B0 isgenerated at 14 , , +1 t +2 t +3 t +4 t P0 P1 P2 P3
Fig. 4.
The evolution of period-4 brinker P in a period.
Fig. 5.
The cross-sectional view of P2. t +1 t +2 t +3 t +4 t Q0 Q1 Q2 Q3 Q4Q5 Q6 Q7 Q8 Q9Q10 Q11 Q12 Q13 t +6 t +5 t +7 t +8 t +9 t +14 t +11 t +10 t +12 t +13 Fig. 6.
The evolution of period-14 brinker Q in a period. ig. 7.
The cross-sectional view of Q1. output pattern is almost recovered but it takes several more steps to convergethe exact shape of the bug B. t +7 t +8 t +9 t t +11 t +6 t +10 t +12 t +13 t +14 t +16 t +15 t +17 t +18 t +34B0 B0Q1 Fig. 8.
A process of turning a bug. Q1B0
15 12
B0B0
14 (generated)(z-offset: +6)(input)
Fig. 9.
A layout of turning configuration in xy -plane and the relative positions of inputand output bugs. A duplicator of a bug B can be realized by a collision of B to two blinkers,Q, with an offset and a phase. Fig. 10 shows a duplicating process of a bug B.A bug B0 as an input, two B0 patterns are generated as outputs. As in Fig.11, B , ,
0) and two Q1 are placed at ( ± , ,
1) as an initial configuration,two B ± , ,
1) after 44 time steps.Note that these collisions are destructive and each function is only used once. +10 t +11 t +12 t +13 t t +15 t +14 t +16 t +17 t +18 t +20 t +19 t +21 t +22 t +44B0 B0B0Q1Q1 Fig. 10.
A process of duplicating a bug. P2 P2B0
13 88
B0B0 (generated) (generated)(z-offset: +1)(input) B0 Fig. 11.
A layout of duplicate configuration in xy -plane and the relative positions ofinput and output bugs. We shows that a NOT gate module and an AND gate module can be realizedby combining a duplicator and a turning module.Fig. 12 is a NOT gate module. A duplicator is used as a kind of “clocking”module and generate a bug. An input signal to the NOT gate is encoded bythe existence of a bug. If the input is 0 then the generated bug is used as theoutput. If the input is 1 then the associated input bug B0 causes a collision tothe generated bug. Almost all types of collisions of two bugs finally annihilateall live cells produced by the collision. Thus there is no output, i.e., 0.Fig. 13 shows an AND gate module. It can be realized by the similar way.Only in the case that two bugs are placed as inputs 1 and 1, the input2 bugsurvives as its output.
The position of input and output bugs of a duplicator (a turning module) differsin z-coordinate. The value of the offset of a duplicator and turning module are1 and 6 respectively. Each module flipped upside down is also effective, i.e., a nput: 0 Output: 1 Output: 0Input: 1
Fig. 12.
NOT gate modules (input 0 and 1).
Input1: 1 Output: 1Input2: 1
Fig. 13.
An AND gate module (input 1 and 1). flipped duplicator is also a duplicator, and a flipped (right) turning module canbe used as a (left) turning module.The positions of duplicated bugs by the pattern in Fig. 11 is ( ± , ,
1) andthe difference of their phase of period from the input bug is −
5. Because anotherB0 pattern appear at 44-step later, but the original bug must be B0 pattern againin 39-step later. The change of the phase of period causes a serious problem forcombining the pattern of gate modules. Because we need to change the phaseof blinkers P and Q depending on the number of cascading modules. But thedifference of the phase of period between input and output bugs of turningmodule pattern (Fig 9) is “fortunately” 5 (= 39 −
12 14
12 1488 B0
13 8 8 B0 Fig. 14.
A synchronized duplicator. B0
13 8 8 B0 Input Output
Fig. 15.
A synchronized NOT gate.
In this paper, we show that an LtL, (4 , , , , cknowledgments. This work is supported by the JSPS KAKENHI GrantNumbers JP26330016, JP17K00015.
References
1. Bays, C., Candidates for the game of Life in three dimensions,
Complex Systems , ,373-400 (1987).2. Bays, C., A new game of three-dimensional Life, Complex Systems , , 15-18 (1991).3. Bays, C., Further notes on the game of three-dimensional Life, Complex Systems , , 67-73 (1994).4. Berlekamp, E.R., Conway, J.H., Guy, R.K., Winning ways for your mathematicalplays, vol. 2, Academic Press (1982).5. Cook, M.: Universality in elementary cellular automata, Complex Systems , (1)1–40 (2004).6. Evans, K.M.: Larger than Life: Digital creatures in a family of two-dimensional cel-lular automata, Discrete Mathematics and Theoretical Computer Science Proceedings
AA (DM-CCG) 177–192 (2001).7. Evans, K.M.: Larger than Life: threshold-range scaling of Life’s coherent structure,
Physica D , 43–67 (2001).8. Evans, K.M.: Is Bosco’s rule universal?, Proceedings of Machines, Computations, andUniversality , Saint Petersburg,
Lecture Notes in Computer Science , Springer,188–199 (2004).9. Gardner, M.: Mathematical games – the fantastic combinations of John Conway’snew solitare game, Life,
Scientific American , 120–123 (1970).10. Imai, K., Imai, K., Masamori, Y., Iwamoto, C., and Morita, K.: On designinggliders in three-dimensional Larger than Life cellular automata,
Natural Computing:4th International Workshop on Natural Computing , Himeji,
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