Still life (cellular automaton)

In Conway's Game of Life and other cellular automata, a still life is a pattern that does not change from one generation to the next. The term comes from the art world where a still life painting or photograph depicts an inanimate scene. In cellular automata, a still life can be thought of as an oscillator with unit period.[1]

Classification

Pseudo still life
Strict still life

A pseudo still life consists of two or more adjacent islands (connected components) which can be partitioned (either individually or as sets) into non-interacting subparts, which are also still lifes. This compares with a strict still life, which may not be partitioned in this way. A strict still life may have only a single island, or it may have multiple islands that depend on one another for stability, and thus cannot be decomposed. The distinction between the two is not always obvious, as a strict still life may have multiple connected components all of which are needed for its stability. However, it is possible to determine whether a still life pattern is a strict still life or a pseudo still life in polynomial time by searching for cycles in an associated skew-symmetric graph.[2]

Examples

There are many naturally occurring still lifes in Conway's Game of Life. A random initial pattern will leave behind a great deal of debris, containing small oscillators and a large variety of still lifes.

The most common still life (i.e. that most likely to be generated from a random initial state) is the block.[3] A pair of blocks placed side-by-side (or bi-block) is the simplest pseudo still life. Blocks are used as components in many complex devices, an example being the Gosper glider gun.

Block
Bi-block

The second most common still life is the hive (or beehive).[3] Hives are frequently created in (non-interacting) sets of four, in a formation known as a honey farm.

Hive
Honey farm

The third most common still life is the loaf.[3] Loaves are often found together in a pairing known as a bi-loaf. Bi-loaves themselves are often created in a further (non-interacting) pairing known as a bakery. Two bakeries can extremely rarely form next to each other, forming a set of four loaves known as a tetraloaf alongside two more bi-loafs.

Loaf
Bi-loaf
Bakery

A tub consists of four live cells placed in a diamond shape around a central dead cell. Placing an extra live cell diagonally to the central cell gives another still life, known as a boat. Placing a further live cell on the opposite side gives yet another still life, known as a ship. A tub, a boat or a ship can be extended by adding a pair of live cells, to give a barge, a long-boat or a long-ship respectively. This extension can be repeated indefinitely, to give arbitrarily large structures.

From left: tub, barge, long-barge, etc...
From left: boat, long-boat, etc...
From left: ship, long-ship, etc...

A pair of boats can be combined to give another still life known as the boat tie (a pun on bow tie, which it superficially resembles). Similarly, a pair of ships can be combined into a ship tie.

Boat tie
Ship tie

Eaters

Fish-hook (eater1)
eater2

Still lifes can be used to modify or destroy other objects. A still life is called an eater when it can be used to absorb some other pattern (often a glider, spaceship, or the debris from a more complicated reaction) and returns to its original state after the collision. Many examples exist, with the most notable being the fish-hook (Also known as eater 1), which is capable of absorbing several types of spaceship. A similar device is the 'reflector', which alters the direction of an incoming spaceship. Oscillators with similar properties may also be called eaters or reflectors, but are more difficult to apply as they must be synchronized to the pattern they modify. Still life eaters and reflectors, on the other hand, work correctly regardless of the timing of the pattern they modify, as long as successive reactions occur with enough separation in time to allow the eater or reflector to recover its original shape.

Enumeration

The number of strict and pseudo still lifes in Conway's Game of Life existing for a given number of live cells has been documented up to a value of 34 (sequences A019473 and A056613 respectively in the On-Line Encyclopedia of Integer Sequences (OEIS).[4][5]

Live cells Strict still lifes Pseudo still lifes Examples[1]
100
200
300
420Block, tub
510Boat
650Barge, beehive, carrier, ship, snake
740Fishhook, loaf, long boat, python
891Canoe, mango, long barge, pond
9101Hat, integral sign
10257Block on table, boat-tie, loop
114616
1212155Ship-tie[citation needed]
13240110
14619279Bi-loaf[citation needed]
151,353620
163,2861,645
177,7734,067
1819,04410,843
1945,75927,250Eater 2[citation needed]
20112,24370,637
21273,188179,011
22672,172462,086
231,646,1471,184,882
244,051,7323,069,135
259,971,3777,906,676
2624,619,30720,463,274
2760,823,00852,816,265
28150,613,157136,655,095
29373,188,952353,198,379
30926,068,847914,075,620
312,299,616,6372,364,815,358
325,716,948,6836,123,084,116
3314,223,867,29815,851,861,075
3435,422,864,10441,058,173,683

Density

19x19 maximum-density still life in Conway's game of life
20x20 maximum-density still life in Conway's game of life

The problem of fitting an n×n region with a maximally dense still life has attracted attention as a test case for constraint programming.[6][7][8][9][10] In the limit of an infinitely large grid, no more than half of the cells in the plane can be live.[11] For finite square grids, greater densities can be achieved. For instance, the maximum density still life within an 8×8 square is a regular grid of nine blocks, with density 36/64 = 0.5625.[6] Optimal solutions are known for squares of all sizes.[12] Yorke-Smith provides a listing of known finite maximum-density patterns.[13]

References

  1. "Still Life - from Eric Weisstein's Treasure Trove of Life C.A." Retrieved 2009-01-24.
  2. Cook, Matthew (2003). "Still life theory". New Constructions in Cellular Automata. Santa Fe Institute Studies in the Sciences of Complexity, Oxford University Press. pp. 93–118.
  3. Achim Flammenkamp. "Top 100 of Game-of-Life Ash Objects". Retrieved 2008-11-05.
  4. Number of stable n-celled patterns ("still lifes") in Conway's game of Life (sequence A019473 in the OEIS).
  5. Number of n-celled pseudo-still-lifes in Conway's game of Life (sequence A056613 in the OEIS).
  6. Bosch, R. A. (1999). "Integer programming and Conway's game of Life". SIAM Review. 41 (3): 594–604. Bibcode:1999SIAMR..41..594B. doi:10.1137/S0036144598338252..
  7. Bosch, R. A. (2000). "Maximum density stable patterns in variants of Conway's game of Life". Operations Research Letters. 27 (1): 7–11. doi:10.1016/S0167-6377(00)00016-X..
  8. Smith, Barbara M. (2002). "A dual graph translation of a problem in 'Life'". Principles and Practice of Constraint Programming - CP 2002. Lecture Notes in Computer Science. Vol. 2470. Springer-Verlag. pp. 89–94. doi:10.1007/3-540-46135-3_27..
  9. Bosch, Robert; Trick, Michael (2004). "Constraint programming and hybrid formulations for three Life designs". Annals of Operations Research. 130 (1–4): 41–56. doi:10.1023/B:ANOR.0000032569.86938.2f. S2CID 27359250..
  10. Cheng, Kenil C. K.; Yap, Roland H. C. (2006). "Applying ad-hoc global constraints with the case constraint to still-life". Constraints. 11 (2–3): 91–114. doi:10.1007/s10601-006-8058-9. S2CID 8241518..
  11. Elkies, Noam D. (1998). "The still life density problem and its generalizations". Voronoi's Impact on Modern Science, Book I. Proc. Inst. Math. Nat. Acad. Sci. Ukraine, vol. 21. pp. 228–253. arXiv:math.CO/9905194.
  12. Chu, Geoffrey; Stuckey, Peter J. (2012-06-01). "A complete solution to the Maximum Density Still Life Problem". Artificial Intelligence. 184–185: 1–16. doi:10.1016/j.artint.2012.02.001.
  13. Neil Yorke-Smith. "Maximum Density Still Life". Artificial Intelligence Center. SRI International.

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