X-Rule’s Precursor is also Logically Universal
We re-examine the isotropic Precursor-Rule (of the anisotropic X-Rule) and show that it is also logically universal. The Precursor-Rule was selected from a sample of biased cellular automata rules classified by input-entropy. These biases followed most “Life-Like” constraints — in particular isotropy, but not simple birth/survival logic. The Precursor-Rule was chosen for its spontaneously emergent mobile and stable patterns, gliders and eaters/reflectors, but glider-guns, originally absent, have recently been discovered, as well as other complex structures from the Game-of-Life lexicon. We demonstrate these newly discovered structures, and build the logical gates required for universality in the logical sense.
keywords: universality, cellular automata, glider-gun, logical gates.
Since the publication of Conway’s Game-of-Life, many rules have been found with, to a degree, similarly interesting behavior. Most of these rules are Game-of-Life variants and “Life-Like” in that they follow a simple birth/survival logic based on the total of 1s in the outer neighborhood, which for Life is defined as birth=3, survival=2 or 3 (B3S23). The variants are useful to study the nature and context of the Game-of-Life, to underline why the Game-of-Life itself is so special, and why the birth/survival scheme is able in some cases to produce gliders, glider-guns, logic gates, and universal computation.
To generalise these questions, another approach is to consider rules without the birth/survival scheme, to study their characteristics, and thus to enrich the landscape that makes universal computation possible in binary 2D cellular automata with a Moore neighborhood. Rules have been found that do not follow simple birth/survival but are nevertheless candidates for universality. To mention two examples, the isotropic R-Rule discovered by Sapin and the anisotropic X-Rule discovered by the authors of this paper. The Precursor-Rule, defined in figures 4 to 7, belongs to this latter class of cellular automata, not following birth/survival, but still isotropic, where all rotations/flips of a given Moore neighborhood map to the same output.
Gliders and stable “eaters” emerge spontaneously in the Game-of-Life, but a glider-gun was originally absent and only subsequently discovered by Gosper[5, 1]. In a curious imitation of this order of events, glider-guns in the Precursor-Rule have only recently been discovered. Thanks to these glider-guns, its possible to build the logical gates for negation, conjunction and disjunction and satisfy the third of Conway’s three conditions for universality to demonstrate universal computation in the logical sense.
The Precursor-Rule was selected from a sample of biased rules classified by the input-entropy method[11, 12], giving the scatter-plot in figure 2. These biases followed “Life-Like” constraints though not simple birth/survival logic, to the extent that the rules are binary, with a Moore neighborhood, and in particular that they are isotropic, and where the parameter, the density of 1s in the look-up table, is similar to the Game-of-Life where . Excluding the chaotic sector of the sample (the most heavily populated) a short list of 71 rules with spontaneously emergent gliders and eaters (also called eaters/reflectors) were selected from the ordered sector which has low entropy variability.
The Precursor-Rule itself was selected from this short list, firstly because it featured two spontaneously emergent glider types, moving orthogonally (Gc, figure 10) and diagonally (Ga, figure 8), and secondly because it was possible to construct oscillating behavior where glider Gc was made to bounce between stable reflectors (figures 22, 23). This became the basis for the design of the glider-guns in the anisotropic X-rule, a close mutant of its isotropic precursor. Isotropic behaviour, where gliders and any other dynamical mechanisms operate equivalently in any direction, has arguably an advantage over anisotropy in that it simplifies and makes the design of the mechanisms more flexible.
Glider-guns are the key components for logical gates and thus universality. However, at the time we were unable to discover or construct glider-guns in the Precursor-Rule. Lately, with the collaboration of members of the ConwayLife forum , glider-guns have now been created for gliders Gc and Ga (figure 1). In addition, the forum contributed a plethora of complex structures from the Game-of-Life lexicon, including other glider-guns, oscillators, ships, puffer-trains, rakes, and breeders, which enrich the Precursor-Rule’s behavior and complexity. There is another orthogonal glider (Gb, figure 9) less likely to emerge spontaneously because of its more complicated phases, but as yet a glider-gun for Gb has not been discovered.
The Precursor-Rule has a number of glider-guns, any of which can be used to build logical gates, however we have chosen to use the basic glider-gun in figure 1(b) to demonstrate logical universality, using analogous methods to Conway and the X-Rule.
The paper is organised into the following further sections, (2) the Precursor-Rule definition, (3) a description of gliders, eaters, and collisions, (4) the basic glider-guns for gliders Gc and Ga, (5) logical universality by logical gates using glider-gun GGa, (6) a review of alternative glider-guns and other dynamical structures discovered to date, and (7) the concluding remarks.
2 The Precursor-Rule definition
Figures 4 to 7 define the
Precursor-Rule in four ways; the rule-table, the rule-table expanded
to show all 512 neighborhoods, as a 102-bit isotropic
rule-table111Ongoing investigation shows that a small but
significant proportion of rule-table outputs are quasi-neutral
(wildcards) — their mutations have little or no effect on most
glider-guns featured in this paper, making the Precursor-Rule
part of a cluster of very similar rules., and in terms of
birth/survival where a simple logic is not evident.
3 Gliders, Eaters, and Collisions
A glider is a special kind of oscillator, a mobile pattern that recovers its form but in a displaced position, thus moving at a given velocity. A rule with the ability to support a glider, together with a stable eater/reflector, and a diversity of interactions between gliders and eaters, provides the first hint of potential universality.
From a typical chaotic initial condition as in figure 3, and evolution subject to the Precursor-Rule, its easy to detect the spontaneous emergence of two eater types, and (and their spins/flips), and two glider types, glider Ga (Figure 8) and glider Gc (Figure 10). A combined glider G2a, two Ga gliders joined together with a one cell overlap, can also emerge (figure 11). Glider Gb (Figure 9) is not detected immediately but with a more patient search it can be found. Figures 12 to 18 describe some of the collision results between Ga and Gc gliders, and between these gliders and eaters. Similar experiments could include G2a gliders and also the oscillators in sections 6.1 and 6.2, to provide a more thorough collision catalog.
|—————- Ga —————-|
|————————– Gc ————————–|
|————————– Gc ————————–|
3.1 Gliders colliding with gliders
The outcomes of collisions between gliders are very diverse, depending on the phase, angle, and point of impact, and include the destruction of either or both gliders, a bounce, or a transformation to different or combined glider types. A residual pattern of eaters/reflectors may also be created.
The speed of a glider (or other periodic mobile structure) relative to the speed of light , is measured by the number of squares advanced within its period. In general, orthogonal gliders based on Gc advance 2 squares in a period of 4 giving a speed of , whereas diagonal gliders based on Ga advance 1 square (on both axes) in a period of 4 giving an speed of .
3.2 Gliders colliding with eaters/reflectors
Stable structures emerge spontaneously in the Precursor-Rule which may destroy and/or reflect colliding gliders. The two basic eaters/reflectors (also known as “still life”) are isolated patterns consisting of 3 cells in an “L” shape, and two adjoining cells, giving the following with all rotations/flips:
The eaters/reflectors may themselves be destroyed or transformed in the collision, and the glider may be destroyed, bounce, and transform to a different or combined glider. As with collisions between gliders, the outcomes of collisions between a glider and an eater/reflector are very diverse, depending on the phase, angle, and point of impact.
4 Basic glider-guns
Although a diversity of interactions between gliders and eaters provides the first hint of potential universality, the essential ingredient is a glider-gun, a dynamic structure that ejects gliders periodically into space. A glider-gun can also be seen as an oscillator that adds to its form periodically to shed gliders. In some rules a glider-gun may emerge spontaneously[8, 13], but not in the Game-of-Life, the X-Rule, or the Precursor-Rule — in these cases the glider-gun is a complex structure with a negligible probability of emerging from a random pattern — it has to be found, discovered or somehow constructed.
Gosper found the game-of-Life glider-gun[5, 1]. The anisotropic X-Rule gliders-guns were constructed from reflecting/bouncing oscillators in its isotropic precursor by Gómez, the search for a glider-gun in the Precursor-Rule itself having been abandoned at that time. However, since the publication of  and its announcement on the ConwayLife forum, a member, Arie Paap, discovered the first two glider-guns in the Precursor-Rule — GG2a shooting the G2a glider (two Ga’s combined) shown in figure 26, followed by the “basic” GGc in figure 1(a). A number of other Gc and Ga glider-guns were later announced in the forum, described in section 6.3, together with a diversity of other complex structures. However, the “basic” glider-gun that we apply to demonstrate logical gates is GGa — latterly constructed by Gómez by colliding two GGc glider-streams head on (figure 1(b)). For a while this was the smallest Ga glider-gun, but a comparably compact gun with double the period has lately been found by colliding two GGc glider-streams at 90 (figure 45).
In the next section the basic glider-gun GGa will be harnessed to demonstrate the logical gates, NOT, AND, and OR, to show that the Precursor-Rule is logically universal.
5 Logical Universality
Traditionally the proof for universality in cellular automata is based on the Turing Machine or an equivalent mechanism, but in another approach by Conway, a cellular automata is universal in the full sense if it is capable of the following,
Data storage or memory.
Data transmission requiring wires and an internal clock.
Data processing requiring a universal set of logic gates NOT, AND, and OR, to satisfy negation, conjunction and disjunction.
This paper is confined to proving condition 3 only, for universality in the logical sense. To demonstrate universality in the full sense as for the Game-of-Life, it would be necessary to also prove conditions 1 and 2, or to prove universality in terms of the Turing Machine, as was done by Randall for the Game-of-Life.
5.1 Logical Gates
Logical universality in the Precursor-Rule, as in the Game-of-Life, is based on Post’s Functional Completeness Theorem (FCT). This theorem guarantees that it is possible to construct a conjunctive (or disjunctive) normal form formula using only the logical gates NOT, AND and OR.
Using a specific right-angle collision, two Ga gliders can self-destruct leaving no residue. Applying this between GGa glider-gun streams, and a Ga glider/gap sequence with the correct spacing and phases representing a “string” of information, its possible to build logical gates. Gates NOT, AND and OR are illustrated in figures 19 to 21. Note that the AND and OR gates include intermediate NOT and NOR gates, explained in the captions.
Gaps in a string are indicated by grey circles, dynamic trails=10 are included, and eaters are positioned to eventually stop gliders.
5.1.1 Logical Gate NOT
The logical gate NOT (10 and 01), also called
an “inverter”, requires one GGa glider-gun interacting with
a string of Ga gliders/gaps, illustrated as before/after
snapshots in figure 19.
5.1.2 Logical gate AND (also NOR)
The logical gate AND (111, else0),
also a NOR gate (0+01, else0),
requires one GGa glider-gun interacting with
two input strings of Ga gliders/gaps, illustrated as before/after
snapshots in figure 20. Note that gate AND contains gate NOT.
5.1.3 Logical Gate OR
The logical gate OR (0+00, else1) requires two GGa glider-guns interacting with two input strings of Ga gliders/gaps, illustrated as before/after snapshots in figure 21. Note that gate OR contains both NOT and AND/NOR gates.
6 Precursor-Rule Universe
As well as the gliders, eaters, and collisions in section 3, the basic glider-guns in section 4, and the logical gates built from some of these components in section 5, the Precursor-Rule is capable of an astonishing diversity of dynamical behaviour. This section gives examples, starting with oscillators, then various complex dynamical structures — glider-guns, space ships, puffer-trains, rakes, and breeders, named from the Game-of-Life lexicon, and discovered by members of the ConwayLife forum. These kinds of structures may eventually provide the components for universality in the full sense discussed in section 5.
6.1 Variable length/period oscillators
6.2 Other oscillators
6.3 Other glider-guns
The first two glider-guns in the Precursor-Rule (discovered by ), opened the floodgates for further discovery. The second was the basic GGc (figure 1(a)) already discussed. The first “QuadGG2a” shoots G2a, double Ga gliders (figure 26) and is significant because it provides the building blocks for meta-glider-guns made from interacting simpler glider-guns, including the Ga glider-gun discovered by, shown in figure 27(b).
Figure 27 shows 4 examples of meta-glider-guns – they need to be seen in action to appreciate their extraordinary dynamics.
Figure 27(d) demonstrates a multi-glider-gun in that it shoots more than one glider type. In figure 28 we show another multi-glider-gun, also a meta-glider-gun because it is constructed from 4 interacting GGc glider-gun sub-units, discovered by . Sending different glider types simultaneously from the same gun is arguably novel in relation to the Game-of-Life. Of course, any of the glider streams can be blocked by strategically positioned eaters.
6.3.3 Variable period Gc glider-guns
A system of variable period glider-guns, GGcV, for glider Gc was created by . Built from his complex reflecting/bouncing oscillators (cRBOs), the system is demonstrated with the smallest cRBO (gap/period of 28/41 in figure 24). Two cRBOs are positioned as in figure 29(a), with the distance between their centers =85 cells. A Gc glider pointing West is introduced, and its interaction brushing past the pulsating cRBO creates a new Gc glider moving East as in figure 29(b), which interacts with the second cRBO repeating the cycle.
The result is glider-gun GGcV shooting Gc gliders East and West. The whole structure has a period of 328 time-steps. The distance between cRBO centers can be adjusted by modular amounts to increase the period. To date the following have been demonstrated, = 85/328, 167/656, 249/984, 331/1312. The series continues +82/+328. Similar structures can also be built with the larger cRBOs in figure 24.
6.4 Spaceships and puffer-trains
A spaceship is a mobile periodic pattern larger than a simple glider — a puffer-train is similar but leaves debris in its wake. The combined Ga gliders in figure 11 could be classified as spaceships. Figures 30 to 33 show examples of spaceships and puffer-trains, some suggested at the ConwayLife forum, and relating to combinations of either Ga or Gc gliders, as well as other patterns. The period and speed are indicated.
A rake is a mobile periodic pattern that
sheds a succession of gliders in its wake,
including Ga or Gc gliders, or a combination of both, a sort of mobile glider-gun.
In this sense a rake is an adapted spaceship, or a puffer-train if debris is
also left in the wake.
A rake can also be a sub-component in building a “meta-rake”.
Several rakes of various type and complexity have been discovered
at the ConwayLife forum.
Figures 35 to 41 provide details.
Whereas a glider-gun or a rake ejects a stream of gliders, a breeder is a pattern that ejects a stream of glider-guns, rakes, or puffer-trains, in various combinations. Breeders are said to exhibit unbounded quadratic growth by creating multiple copies of a second object, each of which creates multiple copies of a third object. The significance of breeders lies in their high level of complexity demonstrating open ended pattern evolution, as well as providing further components for computation and memory. Breeders have been constructed in the Precursor-Rule at the ConwayLife forum. In figures 42 to 44, we show four examples of breeders ejecting rakes.
7 Concluding remarks
The X-Rule’s isotropic precursor was the original cellular automaton that we studied in our search for universal computation, but at the time we found it necessary to modify the rule to find glider-guns — we were then able to demonstrate logical universality in the anisotropic X-rule. After announcing the X-rule and its precursor on ConwayLife, members of the forum applied their considerable know-how in Game-of-Life pattern search to discover glider-guns in the isotropic Precursor-Rule. As well as glider-guns, many other important complex dynamical mechanisms have been constructed and we have presented a selection, which incidentally shows the power of diversified search in a dedicated community.
Armed with glider-guns, we were able to construct the logical gates and demonstrate logical universality in the Precursor-Rule, which like the Game-of-Life exhibits an extraordinary diversity of dynamics, but according to a rule not based on birth/survival logic. The results documented in this paper are an initial exploration — further interesting dynamics can be discovered possibly including memory functions required for universality in the Turing sense. The dynamics are open-ended and impossible to pin down within a sufficiently large space-time.
Although the Game-of-Life has accumulated a vast compendium of behaviour, it can be argued that the Precursor-Rule has a more diverse range of basic gliders and glider-guns, providing a richer diversity of the fundamental particles from which more complex structures can be built.
The Precursor-Rule belongs to the ordered zone in rule-space within the input-entropy scatter-plot (figure 2), the zone with low values of entropy variability and mean entropy. It seems that the ingredients that enable logical universality — gliders, eaters, and crucially glider-guns — are more likely to occur in this zone, rather than in the “complex” zone with high entropy variability, were activity tends to overwhelm stability. Although 2D binary cellular automata rules supporting glider-guns are exceedingly rare, it appears nonetheless that many such rules are to be found in this zone, which begs the question, what are the underlying principles for the existence of glider-guns?
7.1 SansDomino rule-space
We have recently become aware of “SansDomino” rule-space where rules can support gliders similar to Ga and Gc gliders in the Precursor-Rule, with analogous glider-guns shooting these gliders. These rules are based on modified birth/survival (B2/S13 and B2/S14) with the exception that an adjoining pair of 1s in the outer neighborhood outputs zero. An example of such a modified B2/S14 glider-gun is shown in figure 46. It will be important to investigate the relationship between the Precursor-Rule and ‘SansDomino” rules.
Experiments were done with Discrete Dynamics Lab [14, 15], Mathematica and Golly. The Precursor-Rule was found during a collaboration at June workshops in 2013 and 2014 at the DDLab Complex Systems Institute in Ariege, France, and also at the Universidad Autónoma de Zacatecas, México, and in London, UK. Later patterns were discovered during interactions with the ConwayLife forum where many people made important contributions. J.M. Gómez Soto also acknowledges his residency at the DDLab Complex Systems Institute, and financial support from the Research Council of Zacatecas (COZCyT).
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