Minimal Glider-Gun in a 2D Cellular Automaton
MMinimal Glider-Gun in a 2D Cellular Automaton
Jos´e Manuel G´omez Soto ∗ Universidad Aut´onoma de Zacatecas.Unidad Acad´emica de Matem´aticas. Zacatecas, Zac. M´exico.
Andrew Wuensche † Discrete Dynamics Lab.
September 11, 2017
Abstract
To understand the underlying principles of self-organisation and compu-tation in cellular automata, it would be helpful to find the simplest formof the essential ingredients, glider-guns and eaters, because then the dy-namics would be easier to interpret. Such minimal components emergespontaneously in the newly discovered Sayab-rule, a binary 2D cellular au-tomaton with a Moore neighborhood and isotropic dynamics. The Sayab-rule has the smallest glider-gun reported to date, consisting of just fourlive cells at its minimal phases. We show that the Sayab-rule can imple-ment complex dynamical interactions and the gates required for logicaluniversality. keywords: universality, cellular automata, glider-gun, logical gates.
The study of 2D cellular automata (CA) with complex properties has progressedover time in a kind of regression from the complicated to the simple. Justto mention a few key moments in CA history, the original CA was von Neu-mann’s with 29 states designed to model self-reproduction, and by extension –universality[18]. Codd simplified von Neumann’s CA to 8 states[5], and Bankssimplified it further to 3 and 4 states[2, 1]. In modelling self-reproduction itsalso worth mentioning Langton’s “Loops”[12] with 8 states, which was simpli-fied by Byl to 6 states[4]. These 2D CA all featured the 5-cell “von Neumann”neighborhood.Another line of research was based on the larger 9 × ∗ [email protected], http://matematicas.reduaz.mx/ ∼ jmgomez † a r X i v : . [ n li n . C G ] S e p igure 1: Left : One of the Sayab-rule’s minimal glider-gun patterns, of 4 live cells.
Right : the glider-gun GG1 in action shooting two diagonal glider streams with afrequency of 20 time-steps and glider spacing of 5 cells. Each glider streams isstopped by an eater. Because the system is isotropic, any orientation of the glider-gun is equally valid. Green dynamic trail are set to 10 time-steps.
Note: Green dynamic trails mark any change on a zero (white) cell within the last 10 time-steps, giving a glider a green trailing wake. 10 time-steps is the setting in all subsequentfigures with green dynamic trails. xxxxxxFigure 2:
The Sayab-rule glider-gun attractor cycle[19] with a period of 20 time-steps composed of two phases, where opposite glider-gun patterns are flipped. Thedirection of time is clockwize. A small patch was isolated around a glider-gun bytwo close eaters.
Left : A detail of a patch with a minimal glider-gun (green denoteschange) alongside the same pattern on the attractor cycle. on state in their minimum phase. Here wepresent a much smaller glider-gun which emerges spontaneously in the newlydiscovered Sayab-rule, named after the Mayan-Yucatec word for a spring (ofrunning water).The Sayab-rule is a binary 2D CA with a Moore neighborhood and isotropicdynamics. Though analogous to the game-of-Life and the recently discoveredPrecursor-rule, the Sayab-rule has the smallest glider-gun reported to date, con-sisting of just four live cells at its minimal phase, as well as eaters and otheressential ingredients. We show that the Sayab-rule can implement a diversityof complex dynamical structures and the logical gates required for logical uni-versality , and supports analogous complex structures from the Game-of-Lifelexicon — still lives, eaters, oscillators and spaceships.The paper is organised into the following further sections, (2) the Sayab-Ruledefinition, (3) the Sayab-Rule’s gliders-guns, eaters, collisions, and other com-plex structures, (4) logical universality by logical gates, and (5) the concludingremarks. The Sayab-Rule is found in the ordered region of the input-entropy scatter-plot[20] close to the Precursor Rule[11], and from the same sample and short-list[10, 11]. The input-entropy criteria in this sample followed “Life-Like” con-straints (but not birth/survival logic) to the extent that the rules are binary,isotropic, with a Moore neighborhood, and with the λ parameter[13], the den- We designate a CA “logically universal” if its possible build the logical gates NOT, AND,and OR, to satisfy negation, conjunction and disjunction. “Universal computation” as inthe Game-of-Life requires additional functions[15, 3], memory registers, auxiliary storage andother components. λ = 0 . = 512 neighbor-hood outputs to just 102 effective outputs[17], from which just 29 “symmetryclasses” map to 1 (figure 4).Figure 3: Top
The Sayab rule-table based on to all 512 neighborhoods, and
Below expanded to show each neighborhood pattern. 131 black neighborhoods map to 1,381 blue neighborhoods map to 0. Because the rule is isotropic, only 102 symmetryclasses are significant, as described in figure 4
From the game-of-Life lexicon, we borrow the various names for characteristicpatterns or objects, including glider-guns, gliders, eaters, still-lives, oscillators,and space-ships. A glider is a periodic mobile pattern that recovers its shape butat a displaced position, making it move at a given velocity, sometimes referredto as a mobile particle. A glider is usually identified as moving on the diagonal,whereas an orthogonal “glider” is called a space-ship. A glider-gun is a periodicpattern in a fixed location that sends, shoots, or sheds, gliders into space atregular intervals.In the Sayab-rule, the spontaneous emergence of its basic glider-gun, as wellas isolated gliders, is highly probable from a sufficiently large random initialstate because the four glider patterns are very simple and likely to occur oremerge by chance – likewise, the smallest glider-gun patterns. Simple still-livesand oscillators (which may act as eaters which destroy gliders but remain active)4igure 4:
The Sayab-rule’s 29 isotropic neighborhood symmetry classes that mapto 1 (the remaining 73 symmetry classes map to 0, making 102 in total). Each classis identified by the smallest decimal equivalent of the class, where the 3 × patternis taken as a string in the order — for example, the pattern is the string001110111 representing the symmetry class 119. The class numbers are coloreddepending on the value of the central cell to distinguish birth (blue) from survival(red), but no clear “Life-like” birth/survival logic is discernible. are also likely to occur or emerge from random patterns. The basic glider-gunis also probable in subsequent evolution because it can result from the collisionof two gliders, or a glider and an oscillator, though the glider-gun can also bedestroyed by incoming gliders and other interactions. ← —————— Ga —————– → Figure 5:
The 4 phases of the Sayab-rule glider Ga, moving NE with speed c /4,where c is the “speed of light”, in this case, for a Moore neighborhood, c equalsone cell per time-step, diagonally or orthogonally. Figure 6:
Examples of still-lives. p =2 ———- p =2 ——– p =4 —— p =4 —— p =4 ——— p =9Figure 7: Sayab rule oscillators with the periods indicated.Figure 8: A typical evolution emerging after 108 time-steps from a 50x50 30%density random zone. Two stable glider-guns have emerged, together with othergliders, still-lives and oscilators.
Figure 9:
The glider-gun core for 10 successive time-steps — in the next next 10time-steps the same reversed patterns are repeated, to make the period 20 attractorcycle (figure 2). The pattern sequence is from left to right. Any of these patternsare the seeds of a glider-gun, with the smallest, 4 live cells, being the most probableto occur in a random pattern.
6s can be seen in its attractor[19] (figure 2), the Sayab-rule’s basic glider-gun GG1 (figure 1) has a core that varies between just 4 and 11 live cells duringits cycle of twenty time-steps, which is composed of two equivalent phases of 10time-steps. After 10 time-steps the core patters are reversed. In figures 2 and 9the core and its twin 45 ◦ glider streams face towards the North, but the glider-gun can be oriented to face in any of 4 directions. The glider-gun shoots glidersat 20 time-steps intervals with a speed is c /4, and a glider takes 20 time-steps totraverse 5 (diagonal) cells, which is also the spacing of gliders in a glider stream.This spacing can be doubled (without limit) by combining the basic glider-gunsinto compound glider-guns (figures 16 and 17).In the Sayab rule, there are many possible outcomes resulting from collisionsbetween two (or more) gliders, and between gliders and still-lives or oscillators.These have been examined experimentally but not exhaustively.The outcomesdepend on the precise timing and points of impact, and can result in the destruc-tion, survival, or modification of the various colliding objects. For the purposesof this paper we highlight some significant collision outcomes.Eaters that are able to stop a stream of gliders, are a necessary componentin the computation machinery. They can be derived from still-lives or oscillators(figure 10). The glider-gun itself can be the outcome of a collision between aglider and an oscillator (figures 11), or between two gliders (figure 12).(a) (b)Figure 10: Collisions between a glider and an eater, (a) derived from a still-life,and (b) from an oscillator. (a) (b)Figure 11: (a) three different collisions between a glider with an oscillator create aglider-gun (b) shown after 43 time-steps. (a) two gliders colliding at 90 ◦ create a glider-gun (b) shown after 48time-steps. A particular but not infrequent collision situation can arise between a streamof gliders and an oscillator which results in a retrograde stable pattern movingbackwards, a sort of footprint. This eventually destroys the originating glider-gun as illustrated in figure 13.(a) (b) (c)Figure 13:
Glider-gun stream (a) collides with an oscillator resuting in a retrogradestable pattern (b) moving backwards that eventualy destroys the glider-gun (c).
A small slow moving space-ship (an orthogonal glider) can result from a col-lision between a glider and an oscillator, as shown in figure 14. The spaceshipthat emerges has a frequency of 12 and speed of c /12, so it takes 12 time-stepsto advance one cell. Larger space-ships with various frequencies are shown infigure 15. (a) (b)Figure 14: (a) a glider collides with an oscillator creating a slow moving space-ship(b) shown after 25 time-steps. The 12 phases of the space-ship are shown. Figure 15:
Six large space-ships moving North with speed c /2. Periods, from leftto right, are 2, 2, 2, 4, 4, 4. (a) two pairs of gliders, each pair colliding at 90 ◦ , form a pre-image ofGG2. (b) the compound glider-gun GG2 shown after 138 time-steps, shoots gliderswith a frequency of 40 time-steps and glider spacing is 10 cells. Figure 17:
The compound glider-gun GG4 shoots gliders with a frequency of 80time-steps and glider spacing is 20 cells.
A compound glider-gun (GG2) can be built from two interlocking GG1glider-guns. GG2 shoots two glider streams in opposite directions with a fre-quency of 40 time-steps and a glider spacing is 10 cells (twice GG1). Thedynamics depend on glider streams colliding at 90 ◦ resulting in the destructionof one glider-stream, and alternate gliders in the other glider-stream. Collisionsleave behind a sacrificial “eater” which destroys one of the next pair of incominggliders.Two GG2 glider-guns can be combined into a larger compound glider-gun(GG4, figure 17) where analogous collisions result in doubling the GG2 fre-quency and spacing, so the GG4 glider-stream has a frequency of 80 time-stepsand spacing of 20 cells. This doubling of glider-stream frequency and spacingwith greater compound glider-guns can be continued without limit.9 Logical Universality and Logical Gates
Post’s Functional Completeness Theorem[14, 8] established that it is possible tomake a disjuntive (or conjuctive) normal form formula using the logical gatesNOT, AND and OR. Conway applies this as his 3rd condition for a cellularautomata to be universal in the full sense. The three conditions, applied to thegame-of-Life[3], state that the system must be capable of the following:1. Data storage or memory.2. Data transmission requiring wires and an internal clock.3. Data processing requiring a universal set of logic gates NOT, AND, andOR, to satisfy negation, conjunction and disjunction.This section is confined to demonstrating the logical gates, so Conway’scondition 3, for universality in the logical sense. To demonstrate universality inConway’s full sense it would be necessary to also prove conditions 1 and 2.We propose that the basic existential ingredients for constructing logicalgates, and thus logical universality, are as follows:1. A glider-gun or “pulse generator”, that sends a stream of gilders intospace (figures 1 and 2).2. An eater, based on a still-life or oscillator, that destroys an incoming gliderand survives the collision, so can stop a glider stream (figure 10).3. Complete self-destruction when two gliders collide at an angle. Any debrismust quickly dissipate, and the gap between gliders must be sufficient soas not to interfere with the next glider collision (figure 18).These ingredients exist in Sayab-rule dynamics, where collision outcomesdepend on the precise timing and point of impact. Interacting GG1 glider-gun streams with glider/gap sequences with the correct spacing and phasesrepresenting a “string” of data, we present examples of the logical gates NOT,AND and OR, in figures 19, 20 and 21. Gaps in a string are indicated by greycircles, and dynamic trails of 10 time-steps are included. Any input strings canbe substituted for those shown. Eaters are positioned to eventually stop gliders.Figure 18: Two gliders colliding at 90 ◦ self-destruct. 5 consecutive time-steps areshown. This is a key collision in making logical gates. Head-on collisions also selfdestruct, but are not as useful in this context. Alternatively, full universality could be proved in terms of the Turing Machine, as wasdone by Randall[15]. Gliders are not listed separately because they are implicit in the glider-gun. npu t A N O T - A Figure 19:
An example of the NOT gate: ( ¬ , → →
1) or inverter, whichtransforms a stream of data to its compliment, represented by gliders and gaps. The5-bit input string A (11001) moving SE interacts with a GG1 glider-stream movingNE, resulting in NOT-A (00110) moving NE, shown after 94 time-steps. i npu t A i npu t B A - A ND - B A - N O R - B Figure 20:
An example of the AND gate (1 ∧ →
1, else →
0) making a conjunc-tion between two streams of data, represented by gliders and gaps. The 5-bit inputstrings A (11001) and B (10101) both moving SE interact with a GG1 glider-streammoving NE, resulting in A-AND-B (10001) moving SE shown after 174 time-steps.The dynamics making this AND gate first makes an intermediate NOT-A string00110 (as in figure 19) which then interacts with input string B to simultane-ously produce both the A-AND-B string moving SE described above, and also theA-NOR-B string 00010 moving NE. npu t A i npu t B A - O R - BA - A ND - B Figure 21:
An example of the OR gate (1 ∨ →
1, else →
0) which makes adisjuntion between two stream of data represented by two streams of gliders andgaps. The 5-bit input strings A (11001) and B (10101) both moving SE interactwith two GG1 glider-streams, the lower GG1 shooting NE, and subsequently with anupper GG1 shooting SE, finally resulting in the A-OR-B string (11101) moving SEshown after 232 time-steps. The dynamics first makes an intermediate NOT-A string00110 (as in figure 19), which then interacts with string B to simultaneously produceboth the AND string (10001, which appears in the figure) and an intermediate A-NOR-B string 00010 — this is inverted by the upper glider-gun stream to makeNOT(A-NOR-B) which is the same as the A-OR-B string (11101).
The Sayab-rule’s glider-gun is the smallest reported to date in 2D CA, consistingof just four live cells at its minimal phases. From this glider-gun and otherartefacts it is possible to build the logical gates NOT, AND and OR requiredfor logical universality, which are constructed by collision dynamics dependingon precise timing and points of impact. Furthermore, the fact that the glider-guncan result from a collision between two gliders, or between a glider and a simpleoscillator, opens up possibilities for making complex dynamical structures.Three basic existential ingredients are proposed for constructing logical gates,12o summarise: a glider-gun, an eater, and self-destruction when two gliders col-lide at an angle. Rules with these ingredients are certainly elusive; in previouswork[20, 10, 11] we described how they can nevertheless be found. These meth-ods and the frequency of such rules in rule-space requires further research. Therules occur as families of genetically related rules — this aspect in itself requiresinvestigation — for example, variants of the Sayab-rule make up a family withrelated behaviour.Finally, the minimal size of the Sayab-rule’s glider-gun is significant be-cause it should make it easier to interpret its dynamical machinery, employingDe Bruijn diagrams and other mathematical and computational tools. Such fur-ther research holds the promise of understanding how glider-guns and relatedartefacts can exist, and so reveal the underlying principles of self-organisationin CA, and by extension in nature itself.
Experiments were done with Discrete Dynamics Lab [24, 25], Mathematica andGolly. The Sayab-Rule was found during a collaboration at June workshops in2017 at the DDLab Complex Systems Institute in Ariege, France, and also at theUniversidad Aut´onoma de Zacatecas, M´exico, and in London, UK. J. M. G´omezSoto acknowledges his residency at the DDLab Complex Systems Institute, andfinancial support from the Research Council of Zacatecas (COZCyT).
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