Glider synthesis
|
Glider synthesis (or glider construction) is the construction of an object by means of collisions of gliders and glider-constructible spaceships. It is generally assumed that the gliders should be arranged so that they could come from infinity - that is, gliders should not have had to pass through one another to achieve the initial arrangement (or else it is considered “not fully functional”). LWSSes, MWSSes and HWSSes can also be used in syntheses; these spaceships can themselves be easily synthesized from gliders at any point along their trajectory, so this conversion is often left as an implicit step.
Features of syntheses
Four main characterizing features of a synthesis are the geometry, construction time, glider cost, and number of stages.
The geometry is the number of directions of incoming gliders:
- four-directional: gliders collide from all four directions
- three-directional: gliders collide from all directions but one
- two-directional; further divisible in head-on and 90° syntheses. All two-glider syntheses are necessarily two-directional.
- unidirectional, which assumes the initial presence of a target (usually a still life or an oscillator) to be hit with gliders. Such syntheses are also known as slow or synchronized salvo syntheses.
Since gliders are themselves glider-constructible, any multidirectional synthesis can be technically downgraded to a fewer-directional one, usually at the cost of increasing the construction time, cost, and/or number of stages needed for the synthesis. More challenging is finding a two- or three-directional synthesis for a particular object where few or no parts of the synthesis reactions extend outside the final pattern's bounding box in a particular direction. This is especially important for the synthesis of temporary bait objects, which will need to be placed sometimes quite close to other components without perturbing them. For especially tight locations, sometimes it will be useful to construct an LWSS (or another standard c/2 spaceship) some distance away from the synthesis nexus and let that collide with a glider in the final stages; this allows synthesis at a 45° angle, rather than a 90° angle as required for synthesis by gliders from separate directions.
The construction time is simply the number of generations it takes to complete a synthesis. For multi-stage syntheses, each stage has its own construction time.
The number of stages is a count of how many separate operations a synthesis can be divided into, with pauses of arbitrary length between the stages. Often a particular synthesis operation cannot be achieved by a direct collision of gliders, and a synthesis procedure instead requires first synthesizing a number of bait objects, and then hitting these with gliders to produce the final result.
The cost is the number of gliders expended over the course of the synthesis. Similar to the construction time, it can be defined also for individual synthesis stages. A *WSS is considered to cost 3 gliders. The discovery of the reverse caber tosser in 2018 proved that there is a universal constant upper bound on the cost to synthesise any synthesisable object; currently, the best known upper bound is 15 gliders.
Of particular interest is slow salvo synthesis: unidirectional synthesis where every stage has a glider cost of one. Perhaps surprisingly, anything that is glider synthesizable is also slow salvo synthesizable, a result that crucially depends on the existence of universal construction arms as well as one-time turners and splitters.
An incremental synthesis is a synthesis with multiple stages. The final step (final stage, activation step/stage) of an incremental synthesis is the step that converts a previously constructed stationary object or constellation into the target object. Finding the final step of a synthesis is often a nontrivial, complicated process.[1][2] After the final step may come one or more cleanup steps to remove stray objects that are not part of the intended target.
Still life syntheses
In the 1990s, glider syntheses for all still lifes and known oscillators with at most 14 cells were found by David Buckingham. Almost all of these were successfully reduced to a synthesis cost of less than 1 glider per ON cell, or "1 glider per bit".[3]
A collaborative effort ending in May 2014 completed glider syntheses of all still lifes with 17 or fewer cells.[4][5][6] Lengthy projects to complete syntheses for all still lifes up to specific populations were completed for 18 bits in October 2019,[7] 19 bits in February 2020,[8] 20 bits in March 2021,[9] 21 bits in November 2022,[10] and 22 bits in August 2024.[11] Later optimization projects reduced the maximum cost of construction for 15-bit,[12][13] 16-bit,[14][15] and 17-bit[16] still lifes to less than one glider per bit, in November 2016, May 2017, and September 2019 respectively.
In September 2020, the 17-glider reverse caber-tosser proved that all synthesizable still lifes could theoretically be constructed with no more than one glider per bit.[17]
The following table displays statistics about the costs (excluding RCT constructions) for strict and pseudo still lifes with up to 21 cells as of November 15, 2022.[18]
Live cells | Strict still lifes | Pseudo still lifes | Quasi still lifes | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Count ( A019473) |
Min. cost | Avg. cost | Max. cost | Count ( A056613) |
Min. cost | Max. cost | Count ( A330283) |
Min. cost | Max. cost | |
4 | 2 | 2 | 2.500 | 3 | 0 | – | – | 0 | – | – |
5 | 1 | 2 | 2.000 | 2 | 0 | – | – | 0 | – | – |
6 | 5 | 2 | 3.200 | 4 | 0 | – | – | 0 | – | – |
7 | 4 | 2 | 2.750 | 4 | 0 | – | – | 0 | – | – |
8 | 9 | 2 | 3.556 | 4 | 1 | 2 | 2 | 6 | 3 | 4 |
9 | 10 | 3 | 4.000 | 5 | 1 | 3 | 3 | 13 | 3 | 4 |
10 | 25 | 4 | 4.360 | 5 | 7 | 3 | 5 | 57 | 3 | ? |
11 | 46 | 4 | 4.543 | 5 | 16 | 3 | 6 | 141 | 3 | ? |
12 | 121 | 4 | 4.983 | 7 | 55 | 3 | 9 | 465 | 3 | ? |
13 | 240 | 4 | 5.383 | 8 | 110 | 4 | 9 | 1,224 | 3 | ? |
14 | 619 | 3 | 5.975 | 9 | 279 | 3 | 10 | 3,956 | 3 | ? |
15 | 1,353 | 4 | 6.823 | 10 | 620 | 4 | 12 | 11,599 | 4 | ? |
16 | 3,286 | 3 | 7.755 | 13 | 1,645 | 4 | 16[n 1][n 2] | 36,538 | 3 | ? |
17 | 7,773 | 4 | 9.015 | 15 | 4,067 | 4 | 18[n 1][n 3] | 107,415 | 3 | ? |
18 | 19,044 | 4 | 10.339 | 24[n 1][n 4] | 10,843 | 4 | 29[n 1][n 5] | 327,250 | 3 | ? |
19 | 45,759 | 4 | 11.632 | 46[n 1][n 6] | 27,250 | 4 | 31[n 1][n 7] | 972,040 | 3 | ? |
20 | 112,243 | 4 | 13.066 | 70[n 1][n 8] | 70,637 | 4 | 40[n 1][n 9] | 2,957,488 | 4 | ? |
21 | 273,188 | 5 | 14.734 | 96[n 1][n 10] | 179,011 | [n 11] | 8,879,327 | 3 | ? | |
22 | 672,172 | 4 | 16.604 | 158[n 1][n 12] | 462,086 | 26,943,317 | ? | ? |
In January 2022, work by Ilkka Törmä and Ville Salo demonstrated a still life with 306 cells that is impossible to synthesize with gliders, reduced by 400spartans to 278 on September 4, 2023, 236 on March 12, 2024 and 184 on September 1, 2024, meaning there is a threshold 22 < p ≤ 184 such that not all still lifes with ≥ p cells are constructible. The lower bound can be improved with a relaxing of the condition: no self-forcing patch exists within 8 × 8, so a still life would require at least 32 cells to contain one.[19]
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 15 gliders using the reverse caber-tosser
- ↑ All but 2 16-bit pseudo still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
- ↑ All but 2 17-bit pseudo still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
- ↑ All but 84 18-bit strict still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
- ↑ All but 33 18-bit pseudo still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
- ↑ All but 1054 19-bit strict still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
- ↑ All but 51 19-bit pseudo still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
- ↑ All but 5527 20-bit strict still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
- ↑ All but 258 20-bit pseudo still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
- ↑ All but 22,035 21-bit strict still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
- ↑ Only 82941 of 179011 21-bit pseudo still lifes are currently tabulated on Catagolue.
- ↑ All but 86,237 22-bit strict still lifes can be constructed with strictly less than one glider per bit as of September 1, 2024.
Oscillator syntheses
The following is a table of the smallest oscillators with synthesis statistics:
Min. population | Period-2 oscillators | Period-3 oscillators | Period-4 oscillators | ||||||
---|---|---|---|---|---|---|---|---|---|
Count ( A056614) |
Min. cost | Max. cost | Count | Min. cost | Max. cost | Count | Min. cost | Max. cost | |
3 | 1 | 2 | 2 | 0 | - | - | 0 | - | - |
6 | 3 | 3 | 4 | 0 | - | - | 0 | - | - |
8 | 1 | 4 | 4 | 0 | - | - | 0 | - | - |
9 | 1 | 5 | 5 | 0 | - | - | 0 | - | - |
10 | 1 | 5 | 5 | 0 | - | - | 0 | - | - |
11 | 1 | 8 | 8 | 0 | - | - | 0 | - | - |
12 | 6 | 5 | 7 | 1 | 5 | 5 | 2 | 5 | 5 |
13 | 3 | 5 | 11 | 1 | 5 | 5 | 0 | - | - |
14 | 20 | 5 | 30 | 1 | 6 | 6 | 0 | - | - |
15 | 29 | 5 | 14 | 0 | - | - | 0 | - | - |
16 | 98 | 4 | 56 | 7 | 5 | 8 | 0 | - | - |
17 | 199 | 7(?) | Incomplete | 8 | 5 | 13 | 0 | - | - |
18 | 484 | 4(?) | Incomplete | 39 | 5 | 12 | 3 | ? | ? |
19 | 1083 | ? | Incomplete | 66 | ? | ? | 7 | ? | ? |
20 | 2722 | ? | Incomplete | 213 | ? | ? | 23 | ? | ? |
21 | 6596 | ? | Incomplete | 420 | ? | ? | 34 | ? | ? |
Min. population | Period-5 oscillators | Period-6 oscillators | Higher periods | ||||||
---|---|---|---|---|---|---|---|---|---|
Count | Min. cost | Max. cost | Count | Min. cost | Max. cost | Count | Min. cost | Max. cost | |
12 | 0 | - | - | 0 | - | - | 2 | 3 | 4 |
15 | 1 | 9 | 9 | 0 | - | - | 1 | 5 | 5 |
16 | 1 | 5 | 5 | 1 | 5 | 5 | 1 | 5 | 5 |
17 | 2 | ? | ? | 0 | - | - | ? | - | - |
18 | 4 | ? | ? | 4 | ? | ? | ? | - | - |
19 | 8 | ? | ? | 0 | - | - | ? | - | - |
20 | 16 | ? | ? | 11 | ? | ? | ? | - | - |
21 | 42 | ? | ? | 15 | ? | ? | ? | - | - |
Spaceship syntheses
Perhaps the most interesting glider syntheses are those of spaceships, because these can be used to create corresponding guns and rakes. Many of the c/2 spaceships that are based on standard spaceships have been synthesized, mostly by Mark Niemiec. In June 1998, Stephen Silver found syntheses for some of the Corderships (although it was not until July 1999 that Jason Summers used this to build a Cordership gun). Many larger Corderships also have known glider syntheses, and others could easily be generated using the same techniques. In general, larger Corderships have declined in importance after the discovery of four-, three- and two-engine versions.
In May 2000, Noam Elkies suggested that 60P5H2V0, a 2c/5 spaceship found by Tim Coe in May 1996, might be a candidate for glider synthesis. Initial attempts to construct a synthesis for this spaceship got fairly close, but it was only in March 2003 that Summers and Elkies managed to find a way to perform the crucial last step. Summers then used the new synthesis to build a c/2 forward rake for the 2c/5 spaceship; this was the first example in Life of a rake which fires spaceships that travel in the same direction as the rake but more slowly.
After the loafer was discovered and synthesized in 2013, a number of new spaceship syntheses were found during a short period of time in late 2014 and early 2015, including the dart, crab, 25P3H1V0.2, 30P5H2V0, x66, and weekender. Most of this was due to the work of Martin Grant.
Current research is ongoing for one-sided spaceship synthesis of XWSS. This allows for easier construction of convoys and flotillas, which are important components of helices, in turn important in several classes of engineered spaceships.
As of June 2023, the smallest spaceships with no known syntheses are as follows:
- Unnamed tagalong 23P4H2V0 supported by a LWSS: 32 cells
- 32P4H2V0: 32 cells
- Unnamed tagalong 23P4H2V0 supported by a MWSS top spark: 34 cells
- Unnamed tagalong 23P4H2V0 supported by a MWSS back spark: 34 cells
- 34P5H2V0: 34 cells
- 35P4H1V1: 35 cells
- Unnamed 7WSS sidecar: 36 cells
- Canada goose: 36 cells
- The two ways for Wainwright's tagalong to follow a B29: 36 cells
Other syntheses of note
A 3-glider synthesis of a pentadecathlon was found in April 1997 by Heinrich Koenig, which came as a surprise because it was widely assumed that such a small synthesis would already be known.
Along similar lines, a 3-glider synthesis of the bi-pond was discovered in June 2014 by Bob Shemyakin[25], a 3-glider synthesis of a messy glider-producing switch engine was found in October 2014 by Michael Simkin[26], and a 3-glider synthesis of a clean switch engine was discovered in March 2017 by Luka Okanishi.[27]
On March 27, 2022, dani found that two copies of Simkin's GPSE synthesis could be combined to produce a record-breaking 6-glider synthesis[28] of a breeder which had been discovered a week earlier.[29]
2-glider syntheses
- Main article: 2-glider collision
There are 71 distinct 2-glider collisions, of which 28 produce nothing, six produce a block, five produce a honey farm, three produce a B-heptomino, three produce a pi-heptomino, three produce a blinker, three produce a traffic light, two produce a glider, two produce a pond, two produce a loaf and a blinker, one produces a boat, one produces a beehive, one produces a loaf, one produces an eater 1, one produces lumps of muck, one produces a teardrop, one produces an interchange, one produces a traffic light and a glider, one produces an octomino, one produces a bi-block, one produces four blocks, one produces two blocks, one produces a blinker, loaf, tub and block, and one produces the so-called two-glider mess, a methuselah stabilizing after 530 generations and consisting of four gliders, eight blinkers (including a traffic light), four blocks, a beehive and a ship.
All 71 such syntheses can be seen below in a pattern put together by Jason Summers on January 29, 2005:
All 71 distinct 2-glider collisions, arranged by what they synthesize (click above to open LifeViewer) RLE: here Plaintext: here |
See also
- Patterns constructible by a given number of gliders (category)
- Salvo
- Tutorials/Glider syntheses
- Shinjuku
- Glider destruction
References
- ↑ Dave Greene (August 14, 2022). Re: Suggested LifeWiki edits (discussion thread) at the ConwayLife.com forums
- ↑ Carson Cheng (April 6, 2023). Re: Small Spaceship Syntheses (discussion thread) at the ConwayLife.com forums
- ↑ Mark D. Niemiec (June 20, 2015). Re: 4 glider syntheses (discussion thread) at the ConwayLife.com forums
- ↑ Constructions Known for All Still Lifes up to 17 Bits at Game of Life News. Posted by Dave Greene on May 23, 2014.
- ↑ Martin Grant (January 6, 2014). 17-bit SL Syntheses (100% Complete!) (discussion thread) at the ConwayLife.com forums
- ↑ Martin Grant (May 17, 2014). Re: 17-bit SL Syntheses (discussion thread) at the ConwayLife.com forums
- ↑ Ian07 (October 9, 2019). Re: 18-bit SL Syntheses (100% Complete!) (discussion thread) at the ConwayLife.com forums
- ↑ Martin Grant (February 8, 2020). Re: 19-bit still life syntheses (discussion thread) at the ConwayLife.com forums
- ↑ Martin Grant (March 12, 2021). Re: 20-bit still life syntheses (discussion thread) at the ConwayLife.com forums
- ↑ May13 (November 14, 2022). Re: 21-bit still life syntheses (discussion thread) at the ConwayLife.com forums
- ↑ May13 (August 19, 2024). Re: 22-bit still life syntheses (discussion thread) at the ConwayLife.com forums
- ↑ BlinkerSpawn (October 27, 2016). 15 in 15: Efficient 15-bit Synthesis Project (DONE!) (discussion thread) at the ConwayLife.com forums
- ↑ Martin Grant (November 19, 2016). Re: 15 in 15: Efficient 15-bit Synthesis Project (2 SLs remain) (discussion thread) at the ConwayLife.com forums
- ↑ Bob Shemyakin (December 20, 2016). 16 in 16: Efficient 16-bit Synthesis Project (discussion thread) at the ConwayLife.com forums
- ↑ Goldtiger997 (May 24, 2017). Re: 15 in 15: Efficient 15-bit Synthesis Project (2 SLs remain) (discussion thread) at the ConwayLife.com forums
- ↑ Tanner Jacobi (September 9, 2019). Re: 17 in 17: Efficient 17-bit synthesis project (discussion thread) at the ConwayLife.com forums
- ↑ Adam P. Goucher (September 19, 2020). Re: Binary slow salvos (discussion thread) at the ConwayLife.com forums
- ↑ Adam P. Goucher. "Syntheses". Catagolue. Retrieved on November 15, 2022.
- ↑ mu.sicman (400spartans) (September 2, 2024). Message in #cgol on the Conwaylife Lounge Discord server
- ↑ Goldtiger997 (March 8, 2021). Re: Small Spaceship Syntheses (discussion thread) at the ConwayLife.com forums
- ↑ Goldtiger997 (April 10, 2021). Re: Small Spaceship Syntheses (discussion thread) at the ConwayLife.com forums
- ↑ Mark Niemiec (January 14, 2022). Re: Small Spaceship Syntheses (discussion thread) at the ConwayLife.com forums
- ↑ https://conwaylife.com/forums/viewtopic.php?f=2&t=1557&start=625#p154625
- ↑ https://conwaylife.com/forums/viewtopic.php?f=2&t=1557&start=625#p154645
- ↑ Bob Shemyakin (June 16, 2014). 4 glider syntheses (discussion thread) at the ConwayLife.com forums
- ↑ Michael Simkin (October 24, 2014). Re: Making switch-engines (discussion thread) at the ConwayLife.com forums
- ↑ Luka Okanishi (March 12, 2017). Re: Thread For Your Accidental Discoveries (discussion thread) at the ConwayLife.com forums
- ↑ dani (March 27, 2022). Re: Small Quadratic Growth Patterns (discussion thread) at the ConwayLife.com forums
- ↑ dani (March 20, 2022). Re: Small Quadratic Growth Patterns (discussion thread) at the ConwayLife.com forums
External links
- Catagolue synthesis database
- Mark Niemiec's glider synthesis database at Mark D. Niemiec's Life Page
- Dean Hickerson's Life page with four pages of glider syntheses
- Martin Grant's synthesis component database, version 3
- Heinrich Koenig's component catalog on pentadecathlon.com
- Glider synthesis at the Life Lexicon
Forum threads
- Still Life Synthesis Thread (discussion thread) at the ConwayLife.com forums
- Synthesising Oscillators (discussion thread) at the ConwayLife.com forums
- Small Spaceship Syntheses (discussion thread) at the ConwayLife.com forums
- Soup-based syntheses (discussion thread) at the ConwayLife.com forums
- Randomly enumerating glider syntheses (discussion thread) at the ConwayLife.com forums
- 4 glider syntheses (discussion thread) at the ConwayLife.com forums
- Synthesis components (discussion thread) at the ConwayLife.com forums