There is a notation used to describe these automata. It is written in the form S/B. S (for survival) is a list of all the numbers of states that cause an ON cell to remain ON. B (for birth) is a list of all the numbers of states that cause an OFF cell to turn on. If 0 is in the list, then blank regions of the universe will turn on in one generation.
As an example, the Seeds rule is described as /2. Thus every ON cell dies in every generation, since the survival list is empty. All OFF cells that had exactly two adjacent ON cells then turn on.
This class of cellular automata is named for the Game of Life (23/3), the most famous cellular automaton. Many different terms are used to describe this class. It is common to refer to it as the "Life family" or to simply use phrases like "similar to Life".
There are far too many possible Life-like rules to list them all here. The following table combines notable rules compiled as part of Mirek's Cellebration with the rules mentioned by Wolfram and some additional named rules.
|Rule||Name||Description or source|
|/2||Seeds||Chaotic growth; all patterns are phoenixes. Has spaceships|
|012345678/1||Wolfram Fig. 7(e)|
|012345678/3||Flakes, Life without Death||Ladder-like patterns can be used to simulate arbitrary Boolean circuits|
|012345678/378||Wolfram Fig. 9(a)|
|01356/13456||Wolfram Fig. 7(d)|
|018/018||Wolfram Fig. 13(c); class 2|
|0238/123567||Wolfram Fig. 13(f); class 3|
|03456/34||Wolfram Fig. 7(g)|
|045/0578||Wolfram Fig. 7(i)|
|0468/236||Wolfram Figs. 7(a), 13(g); class 3|
|1/1||Gnarl||Investigated by Kellie Evans; forms interesting patterns starting even from such simple seeds as a single live cell|
|12345/3||Maze||Forms maze-like designs|
|12456/0578||Wolfram Fig. 7(h)|
|125/36||2x2||If a pattern is composed of 2x2 blocks, it will continue to evolve in the same form. Has many oscillators and spaceships|
|135/135||Wolfram Fig. 13(h); class 3|
|1357/1357||Replicator||Edward Fredkin's replicating automaton: every pattern is eventually replaced by multiple copies of itself|
|1358/357||Amoeba||Well balanced between life and death; forms patterns with chaotic interiors and wildly vacillating boundaries. Has spaceships|
|23/3||Life||Highly complex behavior|
|23/36||HighLife||Similar to Life but with a small self-replicating pattern|
|234/3||Wolfram Figs. 9(b), 13(b); Class 2. Has spaceships Stable growths.|
|2345/45678||Walled Cities||Forms centers of activity separated by walls|
|2346/367||Wolfram Fig. 9(c). Has spaceships|
|235678/3678||Stains||Patterns quickly stabilize, curiously different from nearby rules|
|235678/378||Coagulations||Patterns tend to expand forever in contrast to the nearby rule Stains|
|238/357||Pseudo life||Pattern evolution resembles Life but few patterns from Life work in this rule because the glider is unstable.|
|245/368||Move||Random patterns tend to stabilize, but has many naturally occurring and engineered spaceships|
|27/257||Wolfram Fig. 7(b); has spaceships|
|34/34||34 Life||Was initially thought to be a stable alternative to Life, until computer simulation found that larger patterns tend to explode. Has many small oscillators and spaceships|
|34678/3678||Day & Night||Symmetric under on-off reversal. Has engineered patterns with highly complex behavior|
|4567/345||Assimilation||Forms permanent diamond shaped patterns with partially filled interiors|
|45678/137||Wolfram Fig. 7(f)|
|45678/3||Coral||Patterns grow slowly forming coral-like textures|
|5/345||Long life||Studied by Andrew Trevorrow, has very high period oscillators|
|5678/35678||Diamoeba||Forms large diamonds with chaotically oscillating boundaries, first studied by Dean Hickerson. Gravner and Griffeath posed the existence of quadratic growth patterns as an open problem, later solved by Hickerson|
Note that any automaton of the above form that contains the element /1 (e.g. 78/17, or 34/145) will always be explosive for any finite pattern: at any step, consider the cell (x,y) that has minimum x-coordinate among cells that are on, and among such cells the one with minimum y-coordinate. Then the cell (x-1,y-1) must have exactly one neighbor, and will become on in the next step. Similarly, the pattern must grow at each step in each of the four diagonal directions. Thus, any nonempty starting pattern leads to explosive growth.