XRule’s Precursor is also Logically Universal
Abstract
We reexamine the isotropic PrecursorRule (of the anisotropic XRule[6]) and show that it is also logically universal. The PrecursorRule was selected from a sample of biased cellular automata rules classified by inputentropy[11]. These biases followed most “LifeLike” constraints — in particular isotropy, but not simple birth/survival logic. The PrecursorRule was chosen for its spontaneously emergent mobile and stable patterns, gliders and eaters/reflectors, but gliderguns, originally absent, have recently been discovered, as well as other complex structures from the GameofLife lexicon. We demonstrate these newly discovered structures, and build the logical gates required for universality in the logical sense.
keywords: universality, cellular automata, glidergun, logical gates.
1 Introduction
Since the publication of Conway’s GameofLife[5], many rules have been found with, to a degree, similarly interesting behavior[2]. Most of these rules are GameofLife variants and “LifeLike” 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 GameofLife, to underline why the GameofLife itself is so special, and why the birth/survival scheme is able in some cases to produce gliders, gliderguns, 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 RRule discovered by Sapin[8] and the anisotropic XRule[6] discovered by the authors of this paper. The PrecursorRule, 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 GameofLife, but a glidergun was originally absent and only subsequently discovered by Gosper[5, 1]. In a curious imitation of this order of events, gliderguns in the PrecursorRule have only recently been discovered. Thanks to these gliderguns, its possible to build the logical gates for negation, conjunction and disjunction and satisfy the third of Conway’s three conditions for universality[1] to demonstrate universal computation in the logical sense.
The PrecursorRule was selected from a sample of biased rules classified by the inputentropy method[11, 12], giving the scatterplot in figure 2. These biases followed “LifeLike” 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 lookup table, is similar to the GameofLife 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 PrecursorRule 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 gliderguns in the anisotropic Xrule[6], 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.
Gliderguns are the key components for logical gates and thus universality. However, at the time we were unable to discover or construct gliderguns in the PrecursorRule. Lately, with the collaboration of members of the ConwayLife forum [19], gliderguns have now been created for gliders Gc and Ga (figure 1). In addition, the forum contributed a plethora of complex structures from the GameofLife lexicon, including other gliderguns, oscillators, ships, puffertrains, rakes, and breeders, which enrich the PrecursorRule’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 glidergun for Gb has not been discovered.
The PrecursorRule has a number of gliderguns, any of which can be used to build logical gates, however we have chosen to use the basic glidergun in figure 1(b) to demonstrate logical universality, using analogous methods to Conway[1] and the XRule[6].
The paper is organised into the following further sections, (2) the PrecursorRule definition, (3) a description of gliders, eaters, and collisions, (4) the basic gliderguns for gliders Gc and Ga, (5) logical universality by logical gates using glidergun GGa, (6) a review of alternative gliderguns and other dynamical structures discovered to date, and (7) the concluding remarks.
2 The PrecursorRule definition
Figures 4 to 7 define the
PrecursorRule in four ways; the ruletable, the ruletable expanded
to show all 512 neighborhoods, as a 102bit isotropic
ruletable^{1}^{1}1Ongoing investigation shows that a small but
significant proportion of ruletable outputs are quasineutral
(wildcards) — their mutations have little or no effect on most
gliderguns featured in this paper, making the PrecursorRule
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 PrecursorRule, 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 —————  
    
1  2  3  4  5 
————————– Gc ————————–  

    
1  2  3  4  5 
————————– Gc ————————–  

    
1  2  3  4  5 
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 PrecursorRule 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 gliderguns
Although a diversity of interactions between gliders and eaters provides the first hint of potential universality, the essential ingredient is a glidergun, a dynamic structure that ejects gliders periodically into space. A glidergun can also be seen as an oscillator that adds to its form periodically to shed gliders. In some rules a glidergun may emerge spontaneously[8, 13], but not in the GameofLife, the XRule, or the PrecursorRule — in these cases the glidergun 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 gameofLife glidergun[5, 1]. The anisotropic XRule glidersguns were constructed from reflecting/bouncing oscillators in its isotropic precursor by Gómez[6], the search for a glidergun in the PrecursorRule itself having been abandoned at that time. However, since the publication of [6] and its announcement on the ConwayLife forum[19], a member, Arie Paap[26], discovered the first two gliderguns in the PrecursorRule — 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 gliderguns were later announced in the forum, described in section 6.3, together with a diversity of other complex structures. However, the “basic” glidergun that we apply to demonstrate logical gates is GGa — latterly constructed by Gómez[21] by colliding two GGc gliderstreams head on (figure 1(b)). For a while this was the smallest Ga glidergun, but a comparably compact gun with double the period has lately been found[17] by colliding two GGc gliderstreams at 90 (figure 45).
In the next section the basic glidergun GGa will be harnessed to demonstrate the logical gates, NOT, AND, and OR, to show that the PrecursorRule 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[1], 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 GameofLife, 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[7] for the GameofLife.
5.1 Logical Gates
Logical universality in the PrecursorRule, as in the GameofLife, is based on Post’s Functional Completeness Theorem (FCT)[4]. 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 rightangle collision, two Ga gliders can selfdestruct leaving no residue. Applying this between GGa glidergun 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[3], 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 glidergun 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 glidergun 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 gliderguns 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 PrecursorRule Universe
As well as the gliders, eaters, and collisions in section 3, the basic gliderguns in section 4, and the logical gates built from some of these components in section 5, the PrecursorRule is capable of an astonishing diversity of dynamical behaviour. This section gives examples, starting with oscillators, then various complex dynamical structures — gliderguns, space ships, puffertrains, rakes, and breeders, named from the GameofLife lexicon, and discovered by members of the ConwayLife forum[19]. 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 gliderguns
The first two gliderguns in the PrecursorRule (discovered by [26]), 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 metagliderguns made from interacting simpler gliderguns, including the Ga glidergun discovered by[27], shown in figure 27(b).
6.3.1 Metagliderguns
Figure 27 shows 4 examples of metagliderguns – they need to be seen in action to appreciate their extraordinary dynamics.
6.3.2 Multigliderguns
Figure 27(d) demonstrates a multiglidergun in that it shoots more than one glider type. In figure 28 we show another multiglidergun, also a metaglidergun because it is constructed from 4 interacting GGc glidergun subunits, discovered by [21]. Sending different glider types simultaneously from the same gun is arguably novel in relation to the GameofLife. Of course, any of the glider streams can be blocked by strategically positioned eaters.
6.3.3 Variable period Gc gliderguns
A system of variable period gliderguns, GGcV, for glider Gc was created by [26]. 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 glidergun GGcV shooting Gc gliders East and West. The whole structure has a period of 328 timesteps. 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 puffertrains
A spaceship is a mobile periodic pattern larger than a simple glider — a puffertrain 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 puffertrains, 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.
6.5 Rakes
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 glidergun.
In this sense a rake is an adapted spaceship, or a puffertrain if debris is
also left in the wake.
A rake can also be a subcomponent in building a “metarake”.
Several rakes of various type and complexity have been discovered
at the ConwayLife forum[19].
Figures 35 to 41 provide details.
6.6 Breeders
Whereas a glidergun or a rake ejects a stream of gliders, a breeder is a pattern that ejects a stream of gliderguns, rakes, or puffertrains, 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[18]. 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 PrecursorRule at the ConwayLife forum[19]. In figures 42 to 44, we show four examples of breeders ejecting rakes.
7 Concluding remarks
The XRule’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 gliderguns — we were then able to demonstrate logical universality in the anisotropic Xrule[6]. After announcing the Xrule and its precursor on ConwayLife[19], members of the forum applied their considerable knowhow in GameofLife pattern search to discover gliderguns in the isotropic PrecursorRule. As well as gliderguns, 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 gliderguns, we were able to construct the logical gates and demonstrate logical universality in the PrecursorRule, which like the GameofLife 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 openended and impossible to pin down within a sufficiently large spacetime.
Although the GameofLife has accumulated a vast compendium of behaviour, it can be argued that the PrecursorRule has a more diverse range of basic gliders and gliderguns, providing a richer diversity of the fundamental particles from which more complex structures can be built.
The PrecursorRule belongs to the ordered zone in rulespace within the inputentropy scatterplot (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 gliderguns — 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 gliderguns 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 gliderguns?
7.1 SansDomino rulespace
We have recently become aware of “SansDomino” rulespace[25] where rules can support gliders similar to Ga and Gc gliders in the PrecursorRule, with analogous gliderguns 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 glidergun is shown in figure 46. It will be important to investigate the relationship between the PrecursorRule and ‘SansDomino” rules.
8 Acknowledgements
Experiments were done with Discrete Dynamics Lab [14, 15], Mathematica and Golly. The PrecursorRule 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[19] 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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