Isotropic non-totalistic Life-like cellular automaton

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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.


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
Neighborhood 0.png Neighborhood 8.png
Neighborhood 1c.png Neighborhood 2c.png Neighborhood 3c.png Neighborhood 4c.png Neighborhood 5c.png Neighborhood 6c.png Neighborhood 7c.png
Neighborhood 1e.png Neighborhood 2e.png Neighborhood 3e.png Neighborhood 4e.png Neighborhood 5e.png Neighborhood 6e.png Neighborhood 7e.png
Neighborhood 2k.png Neighborhood 3k.png Neighborhood 4k.png Neighborhood 5k.png Neighborhood 6k.png
Neighborhood 2a.png Neighborhood 3a.png Neighborhood 4a.png Neighborhood 5a.png Neighborhood 6a.png
i Neighborhood 2i.png Neighborhood 3i.png Neighborhood 4i.png Neighborhood 5i.png Neighborhood 6i.png
n Neighborhood 2n.png Neighborhood 3n.png Neighborhood 4n.png Neighborhood 5n.png Neighborhood 6n.png
y Neighborhood 3y.png Neighborhood 4y.png Neighborhood 5y.png
q Neighborhood 3q.png Neighborhood 4q.png Neighborhood 5q.png
j Neighborhood 3j.png Neighborhood 4j.png Neighborhood 5j.png
r Neighborhood 3r.png Neighborhood 4r.png Neighborhood 5r.png
t Neighborhood 4t.png
w Neighborhood 4w.png
z Neighborhood 4z.png

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
Outer4 a.png Outer4 c.png Outer4 d.png Outer4 e.png Outer4 f.png Outer4 g.png Outer4 i.png Outer4 j.png
k l m n o p q r
Outer4 k.png Outer4 l.png Outer4 m.png Outer4 n.png Outer4 o.png Outer4 p.png Outer4 q.png Outer4 r.png

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
Hexagonal neighborhood 0.png Hexagonal neighborhood 1.png Hexagonal neighborhood 5.png Hexagonal neighborhood 6.png
Hexagonal neighborhood 2o.png Hexagonal neighborhood 3o.png Hexagonal neighborhood 4o.png
Hexagonal neighborhood 2m.png Hexagonal neighborhood 3m.png Hexagonal neighborhood 4m.png
Hexagonal neighborhood 2p.png Hexagonal neighborhood 3p.png Hexagonal neighborhood 4p.png

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
  1. For n amount of states, the number of unique transitions is given by the doubly triangular numbers: A002817
  2. The rulespace can be simulated via MAP strings, or, less preferably, ruletables, however no direct support for the notation exists.
  3. Based on the Moore neighbourhood, with the four edge cells and/or the four corner cells moved away from the origin a defined distance.
  4. The range-2 "far corners" and range-3 "far edges" cases are supported, but the general case of exploded Moore neighbourhoods are not.
  5. A "circle" formed of four three-cell lines at distance 2 from the center
  6. Four "pre-blocks" with the gaps facing the center
  7. Four "pre-blocks" with the gaps facing outwards
  8. Range-1 Moore, with added range-2 corners
  9. 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.
  10. 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


  2. muzik (February 9, 2019). Range-2 von Neumann isotropic non-totalistic rulespace (discussion thread) at the forums
  3. 3.0 3.1 AforAmpere (February 23, 2019). Re: Range-2 von Neumann isotropic non-totalistic rulespace (discussion thread) at the forums
  4. bubblegum (August 26, 2020). Re: Range-2 von Neumann isotropic non-totalistic rulespace (discussion thread) at the forums
  5. Lemon41625 (November 29, 2020). Re: CAViewer - A Cellular Automaton Simulator written in Java (discussion thread) at the forums
  7. 7.0 7.1
  8. muzik (January 19, 2020). Re: 3-state range-1 outer-totalistic rulespace (discussion thread) at the forums
  9. Paul Callahan (December 3, 1997). "Experiments with a somewhat "Life-like" hexagonal CA (long)". Retrieved on September 29, 2017.
  10. "ExtendedCallahanHexagonal.gif". forums. Retrieved on July 22, 2017.
  11. Aidan F. Pierce (December 17, 2015). "Re: Hacking apgsearch". forums. Retrieved on June 12, 2016.
  12. Adam P. Goucher (January 21, 2016). "Re: apgsearch v2.2". forums. Retrieved on June 12, 2016.
  13. Adam P. Goucher (September 10, 2017). Re: apgsearch v4.2 (discussion thread) at the forums
  14. Adam P. Goucher (December 1, 2018). "Re: Non-totalistic hex rules". forums. Retrieved on December 1, 2018.
  15. Lemon41625 (June 19, 2020). Re: Range-2 von Neumann isotropic non-totalistic rulespace (discussion thread) at the forums

External links