Isotropic non-totalistic Life-like cellular automaton
An isotropic non-totalistic Life-like cellular automaton is a generalization of the concept of a Life-like cellular automaton in which transitions take into account not only the total number of live neighbors of a cell, but also the relative configuration of those neighbors.
Isotropic non-totalistic rules are described using several different classes of notation.
Contents
Rulespaces
Moore neighbourhood
Isotropic non-totalistic rules are described using Hensel notation, an extension of B/S notation developed by Alan Hensel additionally describing allowed or forbidden configurations. Each digit in the rule's birth and survival conditions is followed by an optional suffix, with each allowed configuration described by a specific letter; a minus sign may be used to forbid configurations rather than allow them. If no configurations are specified, all are considered to be allowed, as in the totalistic case. This notation is not used by non-isotropic Life-like cellular automata.
For instance, B2-a/S12 (the Just Friends rule) indicates that a dead cell will be born with 2 neighbors, except when they are adjacent (indicated by the "-a"), and that a live cell will survive with 1 or 2 neighbors in any configuration. This exclusion of the "B2a" transition prevents the rule from exploding in a similar manner as Seeds.
This notation has the following symmetry: For any letter x and number n≠4, nx is defined if and only (8-n)x is defined and moreover (8-n)x is the complement (change live cells to dead and dead cells to live; ignore the center cell) of nx.
Alongside the convential Moore neighbourhood, this notation can also be used to describe transitions on the 8-cell "exploded" Moore neighbourhoods,^{[1]} such as the "Far Corners" and "Far Edges" neighbourhoods supported by CAViewer.
The following table describes all possible neighborhood configurations for the Moore neighbourhood of range 1:
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|---|
— (no letter) |
|||||||||
c (corner) |
|||||||||
e (edge) |
|||||||||
k (knight) |
|||||||||
a (adjacent) |
|||||||||
i | |||||||||
n | |||||||||
y | |||||||||
q | |||||||||
j | |||||||||
r | |||||||||
t | |||||||||
w | |||||||||
z |
Rules using the von Neumann neighbourhood can be simulated via isotropic non-totalistic rules on the Moore neighbourhood; for example, B1/SV becomes B1e2ak3inqy4ny5e/S.
Range-2 von Neumann neighbourhood
There are 618 different transitions possible in the range-2 von Neumann neighbourhood. Four notations have been proposed as of February 2018.^{[2]}
The currently accepted notation is the Feb 24 notation by AForAmpere and MiloJacquet.^{[3]}
For each transition, use the transition corresponding to the configuration of the center 8 cells, aligned with the canonical directions in the table above. Then take the outside 4 cells, and depending on the configurations in the table below, add that letter.
a | c | d | e | f | g | i | j | |
---|---|---|---|---|---|---|---|---|
— |
k | l | m | n | o | p | q | r | |
---|---|---|---|---|---|---|---|---|
— |
For transitions that are the same under reflections or rotations, the canonical transition is the lowest alphabetically.
'x' is used after a totalistic number. For example, 'B3x2ic1ei5x-3kr/S0x8x'.
bubblegum has also proposed to use the 'v' character to represent outer totalistic transitions for the inner 8 cells.^{[4]}
As of 27 November 2020, the only cellular automaton simulation software natively supports such rules is CAViewer.^{[5]}
An earlier proposed notation for range-2 von Neumann isotropic non-totalistic rules is based on Hensel notation.^{[3]} It never entered common use.
Range-2 cross neighbourhood
A proposal for this neighbourhood was mistakenly created in 2017,^{[6]} and a second notation was created in mid-2020.^{[7]}
Range-2 knight neighbourhood
A notation for this was proposed in mid-2020.^{[7]}
3-state Moore neighbourhood
A notation for 3-state range-1 rules is underway.^{[8]}
Hexagonal neighbourhood
- Main article: Hexagonal neighbourhood
It is possible to define isotropic non-totalistic Life-like CAs on a hexagonal grid as well. The following table describes all possible neighborhood configurations for the hexagonal neighbourhood, using notation due to Paul Callahan;^{[9]}^{[10]} the names ortho, meta and para were chosen in analogy to arene substitution patterns in aromatic chemistry:
0 | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|
— (no letter) |
|||||||
o (ortho) |
|||||||
m (meta) |
|||||||
p (para) |
Golly does not support isotropic non-totalistic hexagonal rules using this syntax, so they must instead be simulated using either rule tables or MAP strings. LifeViewer and lifelib support them natively.
Table of isotropic non-totalistic rulespaces
The number of unique transitions for higher ranges tends to be extremely large, and the development of notations for such rules is generally infeasible. The following table documents notable examples:
Common name | Rulespace definitions | Transitions | Notated? | Major software support | |||||
---|---|---|---|---|---|---|---|---|---|
Grid | Range | States | Neighbourhood | Golly | LifeViewer | lifelib | |||
Block cellular automata | {4,4} | 1/2 | 2 | Moore | 6 | Yes | (Unnecessary) | ||
- | {4,4} | 1 | 2 | von Neumann | 6^{[n 1]} | Trivial | (Unnecessary) | ||
Non-totalistic hexagonal | {6,3} | 1 | 2 | hexagonal | 13 | Yes | No^{[n 2]} | Yes | Yes |
Knight INT | {4,4} | 2 | 2 | knight | 43 | Yes | No | No | No |
Isotropic non-totalistic | {4,4} | 1 | 2 | Moore | 51 | Yes | Yes | Yes | Yes |
Exploded | {4,4} | ≥1 | 2 | variable^{[n 3]} | 51 | Yes | No | No | Some^{[n 4]} |
Cross INT | {4,4} | 2 | 2 | cross | 55 | Yes | No | No | No |
- | {4,4} | 2 | 2 | nonstandard^{[n 5]} | 570 | No | No | No | No |
- | {4,4} | 2 | 2 | nonstandard^{[n 6]} | 570 | No | No | No | No |
- | {4,4} | 2 | 2 | nonstandard^{[n 7]} | 570 | No | No | No | No |
R2 von Neumann INT | {4,4} | 2 | 2 | von Neumann | 618 | Yes | No | No | Yes |
- | {4,4} | 2 | 2 | nonstandard^{[n 8]} | 618 | No | No | No | No |
3-state knight INT | {4,4} | 2 | 3 | knight | 873 | No | No | No | No |
3-state INT | {4,4} | 1 | 3 | Moore | 954 | WIP | No^{[n 9]} | No^{[n 9]} | No^{[n 9]} |
3-state cross INT | {4,4} | 2 | 3 | cross | 1035 | No | No | No | No |
4-state knight INT | {4,4} | 2 | 4 | knight | 8356 | No | No | No | No |
4-state INT | {4,4} | 1 | 4 | Moore | 8740 | No | No^{[n 9]} | No^{[n 9]} | No^{[n 9]} |
4-state cross INT | {4,4} | 2 | 4 | cross | 9316 | No | No | No | No |
R2 hexagonal INT | {6,3} | 2 | 2 | hexagonal | 22668 | No | No | No | No |
3-state R2 vN INT | {4,4} | 2 | 3 | von Neumann | 68715 | No | No | No | No |
R2 Moore-without-corners | {4,4} | 2 | 2 | circular | 132744 | No | No | No | No |
3D non-totalistic | {4,3,4} | 1 | 2 | Moore | 1426144^{[n 10]} | No | No | No | No |
R2 Moore INT | {4,4} | 2 | 2 | Moore | 2105872 | No | No | No | No |
- ↑ For n amount of states, the number of unique transitions is given by the doubly triangular numbers: A002817
- ↑ The rulespace can be simulated via MAP strings, or, less preferably, ruletables, however no direct support for the notation exists.
- ↑ Based on the Moore neighbourhood, with the four edge cells and/or the four corner cells moved away from the origin a defined distance.
- ↑ The range-2 "far corners" and range-3 "far edges" cases are supported, but the general case of exploded Moore neighbourhoods are not.
- ↑ A "circle" formed of four three-cell lines at distance 2 from the center
- ↑ Four "pre-blocks" with the gaps facing the center
- ↑ Four "pre-blocks" with the gaps facing outwards
- ↑ Range-1 Moore, with added range-2 corners
- ↑ ^{9.0} ^{9.1} ^{9.2} ^{9.3} ^{9.4} ^{9.5} The rulespace can be simulated via ruletables, however no direct support for the notation exists.
- ↑ Upper bound; calculations by Milo Jacquet returned this number of transitions, however calculations by wildmyron returned 1426132
Soup-searching non-totalistic rules
Adam P. Goucher's apgsearch was modified to support isotropic non-totalistic rules by Aidan F. Pierce on December 17, 2015.^{[11]} This hacked version was later modified in late January 2016 to be able to upload the search results to Catagolue.^{[12]} However, apgsearch did not gain native support for these rules until v4.2, released on September 10, 2017, which can search isotropic non-totalistic rules without B0.^{[13]} v4.66 and above also support the searching of isotropic hexagonal neighborhood rules.^{[14]} Range 2 von Neumann isotropic rules can also be search via the means of a ruletable using a custom neighbourhood.^{[15]}
See also
- Totalistic Life-like cellular automaton
- Non-isotropic Life-like cellular automaton
- Generations
- Larger than Life
- Weighted Life
References
- ↑ https://www.conwaylife.com/forums/viewtopic.php?f=11&t=4763#p128801
- ↑ muzik (February 9, 2019). Range-2 von Neumann isotropic non-totalistic rulespace (discussion thread) at the ConwayLife.com forums
- ↑ ^{3.0} ^{3.1} AforAmpere (February 23, 2019). Re: Range-2 von Neumann isotropic non-totalistic rulespace (discussion thread) at the ConwayLife.com forums
- ↑ bubblegum (August 26, 2020). Re: Range-2 von Neumann isotropic non-totalistic rulespace (discussion thread) at the ConwayLife.com forums
- ↑ Lemon41625 (November 29, 2020). Re: CAViewer - A Cellular Automaton Simulator written in Java (discussion thread) at the ConwayLife.com forums
- ↑ https://www.conwaylife.com/forums/viewtopic.php?f=7&t=2576&p=50367#p50386
- ↑ ^{7.0} ^{7.1} https://www.conwaylife.com/forums/viewtopic.php?f=3&t=1622&p=101003#p101003
- ↑ muzik (January 19, 2020). Re: 3-state range-1 outer-totalistic rulespace (discussion thread) at the ConwayLife.com forums
- ↑ Paul Callahan (December 3, 1997). "Experiments with a somewhat "Life-like" hexagonal CA (long)". Retrieved on September 29, 2017.
- ↑ "ExtendedCallahanHexagonal.gif". ConwayLife.com forums. Retrieved on July 22, 2017.
- ↑ Aidan F. Pierce (December 17, 2015). "Re: Hacking apgsearch". ConwayLife.com forums. Retrieved on June 12, 2016.
- ↑ Adam P. Goucher (January 21, 2016). "Re: apgsearch v2.2". ConwayLife.com forums. Retrieved on June 12, 2016.
- ↑ Adam P. Goucher (September 10, 2017). Re: apgsearch v4.2 (discussion thread) at the ConwayLife.com forums
- ↑ Adam P. Goucher (December 1, 2018). "Re: Non-totalistic hex rules". ConwayLife.com forums. Retrieved on December 1, 2018.
- ↑ Lemon41625 (June 19, 2020). Re: Range-2 von Neumann isotropic non-totalistic rulespace (discussion thread) at the ConwayLife.com forums
External links
- Alan Hensel. "Table of non-totalistic neighborhoods". Retrieved on 2016-06-12.
- Alan Hensel. "Rule notation". Retrieved on 2016-06-12. (note that the table on this page describes an earlier version of Hensel notation that has fallen into disuse)
- Non-totalistic Rules - notations, projects, & discussion (discussion thread) at the ConwayLife.com forums