Strobing rule

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A two-state rule is said to be strobing iff neither of the two cellstates is quiescent. For example, a strobing Life-like cellular automaton contains B0 (dead cells get born if they have zero live neighbors) but not S8 (live cells do not survive if they have eight live neighbors). In strobing rules, the entire universe will "light up" (get born) and die again in successive alternate generations, creating a visual "strobing" effect.[note 1]

Strobing duals of self-complementary rules

Self-complementary outer-totalistic rules (i.e. Life-like cellular automata that are invariant under black/white reversal) necessarily include precisely one of B0 and S8. Applying the transformation below to a self-complementary rule with S8 but not B0 yields a rule with B0 but not S8, i.e. a strobing rule in the above sense. This rule is called the equivalent strobing rule or strobing dual of the original rule.

Applying the "strobing rule" transformation a second time yields the original rule again, so each (non-strobing) self-complementary rule and its equivalent strobing rule form a pair. As there are exactly 29 = 512 self-complementary outer-totalistic rules, and each equivalent strobing rule is itself self-complementary, there are exactly 28 = 256 such pairs.

There are instead 251 self-complementary isotropic non-totalistic rules, with equivalence classes of quartets formed by the strobing dual and checkerboard dual.

Patterns evolve identically in the strobing and non-strobing versions of a self-complementary rule, except that in the strobing version, every other generation is inverted.

Determining the equivalent strobing rule of a self-complementary rule

A given self-complementary rule's strobing dual is determined in the same way as its black/white reversal, except that its original B and S conditions are not negated.

That is to say, to determine the strobing version of a given rule:

  1. Subtract each condition in B and S from 8, yielding B′ and S′.
  2. The equivalent strobing rule of B/S is S′/B′.

For self-complementary rules, this can be reduced to the following, easier method:

  1. Negate each condition in B and S, yielding B′ and S′.
  2. The equivalent strobing rule of B/S is B′/S′.

Using the latter algorithm, it is easy to see that the rule integers of a rule and its strobing equivalent's always add up to 262143 (218 - 1), which is also the number of possible range-1 outer-totalistic rules (minus one).

Example

Applying the first method to the rule B356/S014678, we have:

  1. B = 356; S = 014678
  2. B′ = 235; S′ = 012478

Or, using the second method:

  1. B = 356; S = 014678
  2. B′ = 012478; S′ = 235

Therefore, the equivalent strobing rule of B356/S014678 is B012478/S235.

Note that similarly to black/white reversal, when converting isotropic non-totalistic rules written in Hensel notation, B4 and S4 conditions are not only subtracted from 8 (which would map them to themselves), but must be considered on a per-letter basis. After the negation, a mapping (M) is enacted that swaps elements of the pairs (4c,4e), (4i,4t), (4n,4r), (4y,4j), and (4q,4w) with their partners, but preserves 4k, 4a, and 4z.

Equivalent alternating rules of strobing rules

Some programs, such as Golly, don't simulate strobing rules directly, but alternate between two equivalent non-strobing ones on even and odd generations.[1] For even generations, invert all conditions (ie. the strobing rule followed by a cellwise NOT), and for odd ones, subtract conditions' outer-totals from 8 (applying map M to B4 and S4 ones), and swap the birth and survival ones (ie. the black/white reversal of the strobing rule).

If followed on a non-strobing rule, this will (unusefully) return an alternating pair of equivalent strobing rules.

Notes

  1. This effect may not actually be rendered by software supporting such rules, e.g. Golly.

See also

References