Totalistic Life-like cellular automaton

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Totalistic Life-like cellular automaton can refer to two related but distinct classes of cellular automata.

Definition

In a more narrow sense, a Life-like cellular automaton is said to be totalistic / inner-totalistic[1] when the new state of a (live or dead) cell in the next generation can be expressed as a function of the total number of live cells in its neighborhood, counting the cell itself.

There are precisely 210 = 1024 different totalistic CAs, compared to 218 = 262,144 outer-totalistic CAs.[note 1]

For a given Life-like rule to be totalistic, the following condition is necessary and sufficient: for all 1 ≤ n ≤ 8, a live cell survives with n - 1 neighbors iff a dead cell gets born with n neighbors. For example, the rule given by the rulestring B3/S2 is totalistic; any cell will be alive in the next generation if it has exactly three live cells (including the cell itself) in its neighborhood, and dead otherwise. Conway's Game of Life (B3/S23), on the other hand, is not totalistic: a live cell with three neighbors will survive to the next generation, but a dead cell with four neighbors will not get born.

In the program Mirek's Cellebration, totalistic cellular automata fall into a family called Vote for Life, which is named after the rule Vote (B5678/S45678).

As a synonym for "outer-totalistic"

In common parlance, "totalistic" is also often (but incorrectly) used as a synonym for "outer-totalistic" / "semi-totalistic", meaning that the new state of a cell is a function of both the total number of live cells surrounding the cell, and the state of the cell itself. The rest of this article uses the more narrow definition.

The two definitions differ in that in the second case, the transition function may afford special consideration to the state of the cell itself. For example, the following two configurations may evolve differently in an outer-totalistic CA, but must be treated the same by a totalistic CA:

INT Moore R1 2a.png INT Moore R1 3i dead.png

Notation

Totalistic CAs are described by totalistic rulestrings: strings of digits specifying the live cell counts which will cause a given cell to be alive in the next generation. For example, Fredkin (sometimes referred to as "Replicator 2") has the totalistic rulestring 13579; the equivalent rulestring in B/S notation is B1357/S02468.

Etymology

The word "totalistic" stems from the fact that evolution in a totalistic CA depends only on the total number of live cells in a given cell's neighborhood; similarly, in outer-totalistic CAs, evolution depends on the total number of outer cells, rather than their specific alignment. The word "semi-totalistic" expresses that an outer-totalistic CA, while not necessarily totalistic, is not entirely non-totalistic either: the transition function still depends on some certain total number of live cells.

Generalizations

Totalistic and outer-totalistic CAs can be generalized in several straightforward ways: taking into account the relative (but not absolute) alignment of live cells in a cell's neighborhood yields non-totalistic (isotropic) CAs, while also considering the absolute alignment yields non-isotropic CAs.

They can also be generalized to larger neighbourhoods. apgsearch v4.63 and above are able to search any higher-range outer-totalistic cellular automaton up to a range of 5.[2] LifeViewer and Golly can simulate these from ranges 1 to 500, on various neighbourhoods. A subset of HROT rules is the Larger than Life rulespace, in which birth and survival conditions must all be subsequent.

Outer-totalistic rules can also be generalised to higher state counts; a notation for 3-state rules was proposed in January 2020.[3]

See also

Notes

  1. Although there 262,144 different outer-totalistic Life-like CAs, some of these are essentially the same in a natural sense; the number of essentially different rules is 131,328. See Black/white reversal and Self-complementary for more.

References

  1. Dave Greene (September 18, 2017). Re: Thread for basic questions (discussion thread) at the ConwayLife.com forums
  2. Adam P. Goucher (November 6, 2018). Re: apgsearch v4.0 (discussion thread) at the ConwayLife.com forums
  3. Connor Steppie (January 8, 2020). 3-state range-1 outer-totalistic rulespace (discussion thread) at the ConwayLife.com forums

External links