Self-complementary: Difference between revisions
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A | A two-state [[cellular automaton]] is said to be '''self-complementary''' if it remains unchanged under [[black/white reversal]], i.e. if a [[pattern]] composed of live [[cell]]s in a dead [[universe]] behaves the same as an equivalent pattern composed of dead cells in a live universe. | ||
A self-complementary rule's birth conditions completely determine its survival conditions, and vice versa, but no other constraints exist; as such there are 2<sup>9</sup> = 512 distinct self-complementary [[ | A self-complementary rule's birth conditions completely determine its survival conditions, and vice versa, but no other constraints exist; as such there are 2<sup>9</sup> = 512 distinct self-complementary [[Life-like cellular automaton|Life-like rules]]. | ||
Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is [[strobing]]; in the latter case, there exists an [[equivalent strobing rule]]. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are [[black/white reversal|inverted]] in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior. | Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is [[strobing]]; in the latter case, there exists an [[equivalent strobing rule]]. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are [[black/white reversal|inverted]] in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior. | ||
Revision as of 19:38, 4 September 2022
A two-state cellular automaton is said to be self-complementary if it remains unchanged under black/white reversal, i.e. if a pattern composed of live cells in a dead universe behaves the same as an equivalent pattern composed of dead cells in a live universe.
A self-complementary rule's birth conditions completely determine its survival conditions, and vice versa, but no other constraints exist; as such there are 29 = 512 distinct self-complementary Life-like rules.
Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is strobing; in the latter case, there exists an equivalent strobing rule. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are inverted in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior.
Examples
The most well-known and well-investigated self-complementary rule is Day & Night (B3678/S34678); its equivalent strobing version is B01245/S0125.
Generalizations
Self-complementary rules exhibiting complex behavior are known to exist in higher-range outer-totalistic rules, where spaceships, still lifes, oscillators, replicators, rakes and guns inhabit[1]. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary 1D cellular automata with higher ranges and custom neighborhoods[2].
It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1. Yoel Matveyev's multistate cyclic rules are also a generalization of self-complementarity, where every pattern behaves exactly the same, if all live cells of a certain state are cyclically changed to another live state[3]. Classic self-complementary rules can be emulated by multistate cyclic rules. For example, Matveyev's rule Triple-DN has 2 live states, which act exactly like Day and Night by themselves, as well as when a pattern in one live state evolves on the background of the other live state.
References
- ↑ Yoel Matveyev (November 23, 2021). Re: Larger than Life (discussion thread) at the ConwayLife.com forums
- ↑ Yoel Matveyev (July 28, 2021). Re: Larger than Life (discussion thread) at the ConwayLife.com forums
- ↑ Yoel Matveyev (July 24, 2020). Re: Gluonic (discussion thread) at the ConwayLife.com forums