## Searching algorithms

For general discussion about Conway's Game of Life.

### Re: Searching algorithms

simsim314 wrote:I think I have the basic idea how to construct pretty fast integer converter.

For the parity reader, one need to send gliders to collide with SL and leave the SL but will shoot two gliders back. Now one can use this simple fact: in the following gliders position if the distance is even the left reaction will shoot back a glider, if odd the right one... we can shoot glider salvo accordingly to fix the remaining debris.

I haven't thought through all the details, but I think I understand the basic idea here. The metric you're trying to optimize is the number of ticks that it takes to return parity information from a faraway block.

I think that for this particular project, the parity read time is not a very useful quantity to optimize.

The original plan was to simply pull the encoded recipe block back to zero once and feeding the number of pulls, P, into a binary counter. Calcyman's suggestion was to use two push-and-pull SBM registers, and pull the blocks back log2(P) times and use the parity readings after each pull -- which is slower, but you can throw away the binary counter, which is another kind of simplicity. Calcyman's design will almost certainly reduce the value of N, the maximum number of gliders needed to construct anything glider-constructible.

This new proposal appears to add a nightmare tangle of complex circuitry, just to reduce the number of ticks it takes to read a sequence of parities from a given block position. If we're using this time-to-integer converter for the stated purpose of running a universal constructor to glider-construct larger objects, all that circuitry has to be constructed with gliders, which will hugely increase N.

Whatever N turns out to be, there will be no point in using this mechanism to build anything that can be constructed with fewer than N gliders. I'll do some proper estimates later today, but it seems clear that for an encoding of a slow-salvo construction of more than N synchronized gliders -- a Gemini spaceship, say, or very likely an HBK -- the block will be placed a frankly silly distance away as Kazyan says.

Frankly-silly distances are far enough away that in an actual pattern, Golly won't even be able to send a glider to interact with the block in a reasonable amount of time, at least not without a new specialized simulation algorithm. For example, if we could somehow encode the 38,380-glider recipe for an HBK using just one U.C. operation per resulting glider -- an underestimate by several orders of magnitude -- then the block would have to start at a distance of around 2^76760 cells from the U.C.

dvgrn
Moderator

Posts: 5753
Joined: May 17th, 2009, 11:00 pm

### Re: Searching algorithms

As more practical implementation of this approach, we could use static tape instead of remotely located block.

We have all that's needed for this purpose. The real life UC will work as follows:

0. Place a sequence of blocks (instead of single remotely located block) encoding binary operations using parity.
1. Read block parity of the closest block.
2. Shoot a sequence that will move the block out of the way for the next block.
3. Adjust the internal state according to the parity, for slow salvo.
4. Repeat 1.
5. At the end:
- for guns: move back all the blocks.
- for spaceships objects: move the block sequence into the new lane.

NOTE It's also possible to have some "workers" that move the block sequence few lanes toward base, thus increasing the replication speed in cost of design complexity.

simsim314

Posts: 1696
Joined: February 10th, 2014, 1:27 pm

### Re: Searching algorithms

I made a block to binary converter based on 18 p1920 guns. If the least significant bit of the number is 1 then the next glider will leave towards the NE otherwise to the SE. In the end the contraption gets stuck in the loop corresponding to x == 0 with gliders leaving towards the SE every 1920 generations.

The pattern is based on the following reactions that allow detection of the cases x == 0, x == 1 and x > 1:

x = 174, y = 93, rule = B3/S232bo71bo72bo$3b2o70b2o71b2o$2b2o70b2o71b2o5$10bo71bo72bo$11bo71bo72bo$9b3o69b3o70b3o3$15bo71bo72bo$13bobo69bobo70bobo$14b2o70b2o71b2o4$172b2o$98b2o72b2o$25b2o71b2o$25b2o62$bo71bo72bo$b2o70b2o71b2o$obo69bobo70bobo5$16b3o69b3o70b3o$18bo71bo72bo$17bo71bo72bo!

The 18 shotguns break down as: 5 for block detection, 4 for salvo selection, 2 for block copy, 4 (=2x2) for block pull and 3 for block push.

The block in the example below is at distance 341 from "home". 341 has binary representation 101010101 and so the gliders alternate between NE and SE until the final loop is reached.

x = 1163, y = 1041, rule = 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chris_c

Posts: 895
Joined: June 28th, 2014, 7:15 am

### Re: Searching algorithms

simsim314 wrote:As more practical implementation of this approach, we could use static tape instead of remotely located block.

OK, now I'm definitely a little confused. Seems as if the topic was originally a universal universal constructor with a fixed number of cells:

simsim314 wrote:EDIT2 This is pretty amazing - constant cells universal constructor (meaning one can encode caterpillar using around few K cells).

But as soon as you're encoding recipes on a static tape, you've lost the constant-cells property.

Also, once you allow a static tape, why not allow multiple object types, for higher-density information storage?

Then it doesn't seem like a very big step to have a static tape that's really a freeze-dried salvo... which brings back the possibility of encoding relative timings directly on the tape.

dvgrn
Moderator

Posts: 5753
Joined: May 17th, 2009, 11:00 pm

### Re: Searching algorithms

An interesting question for the U.C. stage of the fixed-number-of-gliders Constructor of Everything Constructible:

To my knowledge there has never been an optimized design for a single-arm universal constructor, where the thing to be minimized is the number of different glider locations in the combined PUSH/PULL/WHITE/BLACK glider salvos.

The closest approach was Paul Chapman's prototype Spartan universal constructor -- but we really didn't spend any time searching for the best combination of gliders. It was a prototype, so we just took the first thing that worked and ran with it. Here's what it looks like:

Send green gliders: PUSH
Send green & white gliders: PULL
Send green & yellow gliders: G0 (i.e., fire glider of parity 0, on black checkerboard squares)
Send green & red gliders: G1 (parity 1, a.k.a. white checkerboared squares)

Code: Select all
x = 38, y = 27, rule = LifeHistory23.2E$22.E.E$24.E7.2A$31.A.A2.2C$33.A2.2C$21.D$21.2D$20.D.D$26.C$26.2C3.A$25.C.C3.2A$30.A.A2$16.D$16.2D$15.D.D9$3E$2.E$.E![[ VIEWONLY THUMBNAIL ]] The PUSH and PULL combination with just three gliders seems pretty lucky, but the others are just what could be found by hand with a small amount of effort. Seems as if there's sure to be something more efficient out there: a set of K gliders, where some subset does a PUSH(n), another subset does a PULL(m) with m relatively prime to n, another subset that does any PUSH or PULL operation plus a G0 glider, and a last subset that does any (the same or different) PUSH or PULL plus a G1 glider in the same direction. A solution with K=6 gliders seems like a sure thing; K=5 seems like a fairly good bet, and K=4 is theoretically possible but I wouldn't expect to be that lucky. One obvious place to look is sets of gliders starting with a two-glider DEC3 consisting of two block pulls. The second of the two gliders retains a degree of freedom, so then it's a matter of picking a useful placement for that glider, and looking for third and fourth glider locations that allow subset salvos that produce clean sideways gliders. Kazyan pointed out that a universal constructor is most likely still universal even if it can only fire P1 monochromatic salvos. If we can prove that, then we can get away with only PUSH, PULL, and G0 (or G1) operations, and we could use a blinker as an elbow instead of a block. This opens up another pair of gliders to start a search from, again with a degree of freedom to play with -- this time a PULL2: x = 10, y = 9, rule = LifeHistory7.3C4$6.3C$8.C$.2C4.C$C.C$2.C!

With only three operations needed instead of four, K=4 starts looking a lot more likely, and K=3 becomes the unbelievably unlikely possibility.

In general, the lower a value of K we can get, the less circuitry we'll need to build a complete universal constructor connected to a sliding-block memory, and the lower N will turn out to be for the N-glider maximum needed for any glider construction.

dvgrn
Moderator

Posts: 5753
Joined: May 17th, 2009, 11:00 pm

### Re: Searching algorithms

If it doesn't matter how efficient the salvo encoding is, then it seems like we don't need a push1 and a pull1 to access every glider lane possible, only a relatively prime pair of distances. So couldn't we open up a new degree of freedom in our search by using a version of the trivial pull3, along with whatever non-multiple-of-three push is most convenient?
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

A for awesome

Posts: 1862
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

### Re: Searching algorithms

A for awesome wrote:If it doesn't matter how efficient the salvo encoding is, then it seems like we don't need a push1 and a pull1 to access every glider lane possible, only a relatively prime pair of distances. So couldn't we open up a new degree of freedom in our search by using a version of the trivial pull3, along with whatever non-multiple-of-three push is most convenient?

Exactly. Or if we try using a blinker elbow, a version of the trivial PULL2. That's a lot of options to search through, especially if we only need one clean glider-output reaction where it doesn't matter how the elbow moves.

(My messages must be getting too high on the TL;DR scale again. I'm doing better with this post...)

dvgrn
Moderator

Posts: 5753
Joined: May 17th, 2009, 11:00 pm

### Re: Searching algorithms

Actually it was fairly easy to make a base 3 converter instead of binary. After that I plonked down some fairly brain-dead p1920 circuitry and that should be enough to make a block to P1 monochromatic salvo converter (EDIT: I mean slow salvo of course).

In the example below the block is 140 steps from home and the output sequence is Pull, Fire, Push, Pull, Fire, Push, Push, Push, Push, .... since 140 = 0000000012012 in base 3.

x = 2448, y = 2077, rule = 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I'm sure this monstrosity can be optimised a lot more.

EDIT2: I did eventually optimise by removing some guns and using cheaper p690 technology. This new pattern is probably glider constructible in around 1,000 gliders. Note that the operations have been swapped so that now we have PUSH = 0, PULL = 1 and FIRE = 2.

x = 1235, y = 944, rule = 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Last edited by chris_c on December 9th, 2015, 11:54 am, edited 1 time in total. chris_c Posts: 895 Joined: June 28th, 2014, 7:15 am ### Re: Searching algorithms chris_c wrote:Actually it was fairly easy to make a base 3 converter instead of binary Wow! Congrats on the so elegant design for the limited cells UC! This is truly simple design, and amazing execution. I've written a small script, that converts slow salvo operations into UC design. import golly as g block = g.parse("2o$2o!")UC = 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311bo$1310bobo$1310b2o2$1411b2o$1319b2o90b2o$1319b2o$1418bob2o$\1418b2obo$1319b2o$1318b3o$1308b2o5bob2o17bo$1308b2o5bo2bo16bobo$1315bo\b2o16bo2bo$1318b3o17bo$1319b2o14bo2bo5bo2bob2obo2bo$1336b3o4b2o2bo4bo\2b2o63bobo$1335b2ob2o4bo2bob2obo2bo64bo3bo$1334b5o85bo$1334b4o82bo4bo\4b2o$1334bo89bo5b2o$1420bo3bo$1420bobo$1322b2o3b2o2$1323bo3bo14b3o9b2o\$1324b3o15bo9bo2bo7bo$1324b3o10bo5bo7bo7b2o3bo$1335b4o12bo6bo5bo$1329b\2o3bobob2o11bo7b5o47b2obob2o$1322b2o5b2o2bo2bob3o11bo2bo$1323bo10bobob\2o14b2o55bo5bo$1320b3o12b4o$1320bo16bo74b2ob2o$1414bo2$1375b2o8bo$\1375b2o6b3o$1382bo$1367b2o13b2o30b2o$1368bo45b2o$1368bobo$1369b2o$\1374b2o$1374b2o4$1371bo21bo8b2o$1370bobo20b3o6b2o$1370b2o24bo$1395b2o\13b2o$1410bo$1379b2o27bobo$1379b2o27b2o2$1401b2o$1378b2o8bo12b2o$1378b\2o6b3o$1385bo$1370b2o13b2o20bo$1371bo34bobo$1371bobo33b2o$1372b2o2$\1379b2o17b2o$1379b2o17b2o3$1374bo$1373bobo$1373b2o$1534bo$1532bobo$\1382b2o149b2o$1382b2o2$1533bo$1380b2o149bobo5bo$1380b2o150b2o6b2o37bob\o$1371b2o3bo6b2o13b2o139b2o38b2o$1371bobo3bo5b3o11bobo180bo$1372b5o6b\2o12bo$1373b3o4b2o128b2o$1380b2o16bo2bo5bo2bob2obo2bo92bo$1401bo5b4ob\2ob4o92bobo$1400b2o5bo2bob2obo2bo93b2o$1388bo$1386b2ob2o4bo$1396b2o$\1385bo5bo4b2o4b2o$1397bo4bobo$1385b2obob2o3b2o5bo108b2o$1511b2o4b2o$\1415b2o100b2o$1415bobo$1405bo4b2o4b3o7b2o$1404bobob2o2bo4b3o6b2o$1392b\2o9bo3bob3o4b3o97b2o$1385b2o5b2o9bo3bob2o4bobo94b2o2b2o$1386bo16bo3b2o\6b2o94bobo$1383b3o18bobo104bo12b2o$1383bo21bo104b2o11bobo$1523bo$1522b\2o5$1524b2o$1523bobo$1523bo$1522b2o7$1532b2o$1532b2o$1583bo$1539bob2o\38bobo482bo$1539b2obo39b2o480b2o$2065b2o7$1541bo$1539bobo$1537b2o$\1537b2o12b2o$1537b2o12b2o$1539bobo$1541bo3$1534b3o$1534b3o$1533bo3bo2$\1532b2o3b2o7$1496b2o8bo$1496b2o6b3o$1503bo$1488b2o13b2o30b2o$1489bo45b\2o$1489bobo$1490b2o$1495b2o$1495b2o4$1492bo21bo8b2o$1491bobo20b3o6b2o$\1491b2o24bo$1516b2o13b2o$1531bo$1500b2o27bobo$1500b2o27b2o$1518b2o4b2o\$1519b2o3b2o$1499b2o8bo8bo$1499b2o6b3o$1506bo$1491b2o13b2o20bo$1492bo\34bobo$1492bobo33b2o$1493b2o$1498b2o$1498b2o19b2o$1519b2o3$1495bo$\1494bobo$1494b2o3$1503b2o$1503b2o3$1503b2o$1502b3o10bo$1492b2o5bob2o$\1492b2o5bo2bo14b3o$1499bob2o14b2o$1502b3o11b2o$1503b2o6b3o2bo11bo2bob\2obo2bo$1510bo16b2o2bo4bo2b2o$1519b2o7bo2bob2obo2bo$1519bobo$1519bo4$\1506b2o3b2o2$1507bo3bo14b3o9b2o$1508b3o15bo9bo2bo7bo$1508b3o10bo5bo7bo\7b2o3bo$1519b4o12bo6bo5bo$1513b2o3bobob2o11bo7b5o$1506b2o5b2o2bo2bob3o\11bo2bo$1507bo10bobob2o14b2o$1504b3o12b4o$1504bo16bo13$1651b2o$1652bo$\1652bobo$1653b2o6$1652b2o$1652b2o4b2o$1658b2o4$1657b2o$1653b2o2b2o$\1652bobo$1652bo12b2o$1651b2o11bobo$1664bo$1663b2o5$1665b2o$1664bobo$\1664bo$1663b2o7$1673b2o$1673b2o2$1680bob2o$1680b2obo8$1681b2o$1681bobo\$1676b2o6bo$1675bo2bo2bo2bo7b2o$1676b2o6bo7b2o$1681bobo$1676bo4b2o$\1675bobo$1675bobo$1676bo3$1673b2obob2o$1673bo5bo$1674bo3bo$1675b3o5$\1637b2o8bo$1637b2o6b3o$1644bo$1629b2o13b2o30b2o$1630bo45b2o$1630bobo$\1631b2o2$1638b2o$1638b2o3$1633bo21bo8b2o$1632bobo20b3o6b2o$1632b2o24bo\$1657b2o13b2o$1672bo$1641b2o27bobo$1641b2o27b2o2$1663b2o$1640b2o8bo12b\2o$1640b2o6b3o$1647bo$1632b2o13b2o20bo$1633bo34bobo$1633bobo33b2o$\1634b2o2$1641b2o17b2o$1641b2o17b2o3$1636bo$1635bobo$1635b2o3$1644b2o$\1644b2o2$1657b3o$1641bo15bo$1640bobo15bo$1633b2o3b2o3bo$1633b2o3b2o3bo\$1638b2o3bo29b4o$1640bobo29b6o$1641bo17bo11b8o$1658b2o10b2o6b2o$1658bo\bo10b8o$1672b6o$1673b4o2$1649b3o$1648bo3bo$1647bo5bo12b2o$1647b2obob2o\11b2o$1667bo9bo$1675bobo$1663b2o9bobo11b2o$1663b3o7bo2bo11b2o$1654b2o\9b2obo5bobo$1647b2o5bo5bo4bo2bo6bobo$1648bo10bo5b2obo8bo$1645b3o7bo3bo\3b3o$1645bo11bo5b2o115$1995b2o$1995b2o40$1976b2o$1976b2o!\")operations = g.getstring("Please enter slow salvo oeprations left to righ \nPush = 0, Shoot = 1, Pull = 2", "110120")decimal = 0for i in xrange(0, len(operations)):   decimal *= 3   decimal += int(operations[i])g.new("")g.putcells(UC)g.putcells(block, 630 + decimal, 211 - decimal)g.fit()

@dvgrn - Regarding my suggestion about static tape, I wasn't talking about limited cells UC. I was talking about the option to use this same technology, to design executable patterns, obviously with unbounded size. We could for example "Giminize" Chris's design.

simsim314

Posts: 1696
Joined: February 10th, 2014, 1:27 pm

### Re: Searching algorithms

simsim314 wrote:@dvgrn - Regarding my suggestion about static tape, I wasn't talking about limited cells UC. I was talking about the option to use this same technology, to design executable patterns, obviously with unbounded size. We could for example "Giminize" Chris's design.

There's a static tape in Golly's Patterns/Life/Signal-Circuitry/constructor-memory-tape.rle. Paul Chapman and I discussed designs back and forth for several years, but never really found anything that was worth building. It's so much less efficient to encode recipes in digital form on a static tape, that the overall size of the design had a tendency to explode.

For a Geminization of Chris's decoder, presumably we'd have to end up with a copy of the static tape some distance away, where the copy of the circuitry could act on it. Gemini's glider-stream data moves to the new location pretty much automatically, but with a static tape it seems like we'd need a lot of extra circuitry to make the copy.

It was partly the fact that the Gemini design simply threw away three-quarters of the circuitry in constructor-memory-tape.rle, that made it such a huge leap forward... I'm not so sure it's a good idea to go back, at least unless we can do something really new like a 2D memory array, or callable subroutines. The limited-cells UC doesn't need anything like this -- add more circuitry of any kind, and it will just increase N, with no practical good effects.

An interesting trick along the static-tape line might create a new ultra-slow knightship, though. If we found a tape-reading reaction that produced a glider and a new block at (2,1) -- let's say (if a block codes for a 1 and empty space codes for a 0) -- then the tape could tell the U.C. to build slow salvo-seeds, or synchronized-salvo seeds for that matter, that would shoot down the U.C. and rebuild it at a (2,1) offset. I''d certainly love to see a pattern like that in operation!

Some diamond-shaped Geminoid spaceship designs could use a similar trick, and the amount of circuitry that.would have to be destroyed and rebuilt is actually quite small... but different at each of the four corners of the diamond, so probably each corner would read a different quarter of the construction recipe, maybe using chains of three semi-Snarks.

dvgrn
Moderator

Posts: 5753
Joined: May 17th, 2009, 11:00 pm

### Re: Searching algorithms

I have an idea for a nonexhaustive but faster version of the WLS/JLS algorithm. It works essentially the same, but throws away branches that appear unlikely to be completed, and discounts trivial stator variants or disconnected solutions. It is likely to be significantly faster with things like finding high-period patterns (because such searches contain many unpromising branches that waste a lot of time) or finding low-period ships where small examples are already known (because once the small example is found, the search wastes time trying to add cells way behind it that do not interact with it in any way). I am not sure how to measure a criterion for determining which branches are promising, but I think with some modification there would be some way to implement this. However, I don't know how much of a performance boost would be gained.

In case you're wondering, I basically took everything I disliked about JLS and figured out a possible way to improve it. This was not really any special insight; I prune searches manually sometimes and I figured it might be easier to have an automated way to do that.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

A for awesome

Posts: 1862
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

### Re: Searching algorithms

A for awesome wrote:I have an idea for a nonexhaustive but faster version of the WLS/JLS algorithm. It works essentially the same, but throws away branches that appear unlikely to be completed, and discounts trivial stator variants or disconnected solutions.

The variant of this idea that I've been trying to get a handle on for years, is a hash-enabled WLS/JLS. The key would be successful partitioning of the problem: if there were a way to monitor the search and recognize that a lot of searches were being duplicated because Section A was actually completely independent of Section B, then it would be possible to run a Section A search just once, and a Section B search just once.

Then you could simply output the result that any Section A solution could be combined with any Section B solution, and safely skip a huge chunk of the search space.

Roughly speaking, this could theoretically speed things up from O(n^2) to O(2n), I suppose, which would be a Really Really Big Improvement (sorry for the technical math jargon). The problem is that there are ridiculous numbers of possible Sections A and B, and I don't see how to record them all in a hash table, or to check in a reasonable amount of time whether JLS/WLS has entered the kind of brain-dead loop that it would be so nice to avoid.

So far, it seems as if a very clever helper algorithm would be needed to do this: HLPR, let's call it, short for Human-Like Pattern Recognition. If you can specify that algorithm in a way that can be converted into code, you'll definitely have something! Unfortunately it seems to be a really tough problem.

Another nice trick might be a human-assisted Blind Watchmaker-type utility, that would run small snippets of different possible searches, and display the partial results for a human to review and pick the ones that look the most promising. Maybe that would make it easier to zero in on the interesting parts of a search space -- with enough practice, anyway.

EDIT: Added some related notes in another thread about SAT solvers, a set of algorithms developed mostly just in the last ten years that have been impressively successful when applied to Conway's Life search problems.

dvgrn
Moderator

Posts: 5753
Joined: May 17th, 2009, 11:00 pm

### Re: Searching algorithms

A for awesome wrote:I have an idea for a nonexhaustive but faster version of the WLS/JLS algorithm. It works essentially the same, but throws away branches that appear unlikely to be completed, and discounts trivial stator variants or disconnected solutions.

I have been musing over a version of JLS which could discard disconnected solutions for quite some time, particularly after watching many high period searches in B2/S0 for speed c ships/puffers become hopelessly bogged down by umpteen permutations of non-interacting moons. I believe the constraint system in JLS is well suited to implementing such a restriction of the search, but I haven't invested enough time to gain the required familiarity with JLS' code. My idea to implement this constraint is to determine the number of connected blobs in the current partial solution and backtrack when the number of blobs is larger than the allowed value. A connected blob includes all On cells in a group wherever there is less than a two cell gap between them in any one time step, or two On cells are neighbours in adjacent time steps (where first and last time step are considered adjacent whenever a periodic solution is sought). If any cell in a blob is within two cells of an unknown cell then that blob is not yet complete. All incomplete blobs are collectively counted as a single blob. I believe this constraint could be tested at the completion of each layer and whilst it would add some computational overhead it could make many searches which are currently intractable without manual pruning possible to run to completion.
The latest version of the 5S Project contains over 196,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1209
Joined: August 9th, 2013, 12:45 am

### Re: Searching algorithms

wildmyron wrote:
A for awesome wrote:I have an idea for a nonexhaustive but faster version of the WLS/JLS algorithm. It works essentially the same, but throws away branches that appear unlikely to be completed, and discounts trivial stator variants or disconnected solutions.

I have been musing over a version of JLS which could discard disconnected solutions for quite some time, particularly after watching many high period searches in B2/S0 for speed c ships/puffers become hopelessly bogged down by umpteen permutations of non-interacting moons. I believe the constraint system in JLS is well suited to implementing such a restriction of the search, but I haven't invested enough time to gain the required familiarity with JLS' code. My idea to implement this constraint is to determine the number of connected blobs in the current partial solution and backtrack when the number of blobs is larger than the allowed value. A connected blob includes all On cells in a group wherever there is less than a two cell gap between them in any one time step, or two On cells are neighbours in adjacent time steps (where first and last time step are considered adjacent whenever a periodic solution is sought). If any cell in a blob is within two cells of an unknown cell then that blob is not yet complete. All incomplete blobs are collectively counted as a single blob. I believe this constraint could be tested at the completion of each layer and whilst it would add some computational overhead it could make many searches which are currently intractable without manual pruning possible to run to completion.

I was thinking it might be possible instead to not allow changing cells from a known solution that cannot interact with the aforementioned solution. This approach would successfully find all solutions, including supported ones, while eliminating most disjoint ones.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

A for awesome

Posts: 1862
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

### Re: Searching algorithms

I am considering the idea of a tile-based search algorithm for spaceships (and possibly oscillators, but that would be less useful). In order to fully explain it, I will have to define a few things first:

• Tile: A square subsection of the search area, extends in time and space. They can be skewed in time (shifted 1 square after n gens, possibly multiple times) if the displacement in one period of the ships desired is greater than 1.

Corner tile: A tile with two adjacent constrained edges.

Edge tile: A tile with one constrained edge.

.
• Constrained edge: An edge of a tile that is required to behave as if the neighboring tile (and the diagonally adjacent tiles) is filled with vacuum or a repetitive agar.
• Solve: To find a set of states for a tile that match the search criteria (with some deterministic sequential algorithm that I haven't determined yet), including all constraints imposed by constrained edges and neighboring tiles.
• Complete an edge: To solve a tile on the edge of the search as a corner tile.

The algorithm is as follows:
Find all solutions that fit within one tile (I.E. have 4 constrained edges). If there are more than a limit that is input as a search parameter, then you are done (in fact, if the amount of solutions found ever exceeds the limit, the search is over). Next, solve Tile 1 with 3 constrained edges, if possible. If successful, try to complete partial with the tile adjacent to the unconstrained edge. Enumerate all solutions of this type (if there are any). Then, remove the constraint on the edge opposite the original tile, and try to solve. For every solution, go to the next tile and try to solve it with 3, then 2 constrained edges as before. When the number of tiles exceeds the dimension of the search, do not solve the final tile with only 2 constrained edges. When this phase is done, return to Tile 1 and try to solve with 2 adjacent constrained edges. If there are no solutions, than spaceships of this type are impossible. If there are solutions, go to one of the tiles adjacent to tile #1 and try to solve that with one constrained edge, a continuation of Tile 1's constrained edge. If unsuccessful, try the next solution of tile 1. Then, repeat this procedure with the other tile adjacent to Tile 1. Next, solve tiles in diagonal rows across the search area. If at any time there is no solution to a tile, return to the tiles adjacent to them and try different solutions for them. Always remember unsolvable combinations of tiles, and never try to reuse them. If you can solve an edge tile as a corner tile, try that first (only try this if you are at the limits of the search bounding area). This restricts the bounding area of the search and forces all tiles in the same orthogonal row as that one to have a continuation of that tile's constrained edge. If both edges have been completed and you are at the corner of the search area, solve the last tile as a corner tile and enumerate all solutions as complete solutions to the search, afterwards returning to the previous tiles to try different combinations. If you exhaust all possible states for tile 1, the search is complete.

I apologize for the long and unoptimized algorithm; I could not find any way to improve it without making it impossible to explain (there may be huge errors or gaps in my explanation that I haven't found; I haven't checked). I still have some things that need to be worked out, such as whether the tile size is input or automatically derived from the other search parameters, or exactly how to solve the tiles or how to skew them in time to move with the ship. I do think this search algorithm could have significant advantages over conventional algorithms, especially in large searches wherein normal algorithms would just keep trying the same futile combinations of cells in one corner of the search every time they changed something somewhere else. Somebody check my math, but I think that this could lead to improvements in runtime on the order of O(n^(2^d)), where d is the smaller dimension (in tiles) of the search.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

A for awesome

Posts: 1862
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

### Re: Searching algorithms

dvgrn wrote:Frankly-silly distances are far enough away that in an actual pattern, Golly won't even be able to send a glider to interact with the block in a reasonable amount of time, at least not without a new specialized simulation algorithm. For example, if we could somehow encode the 38,380-glider recipe for an HBK using just one U.C. operation per resulting glider -- an underestimate by several orders of magnitude -- then the block would have to start at a distance of around 2^76760 cells from the U.C.

I think you underestimate the power of HashLife -- 2^76760 cells seems comfortably within reach. After all, the quadtree will look something like:

(((([...], 0, 0, 0), 0, 0, 0), 0, 0, 0), 0, 0, (0, 0, 0, (0, 0, 0, (0, 0, 0, [...]))))

and take roughly 2 * 76760 nodes to store -- that's only a few megabytes of memory, and you're not going to be generating many new nodes when you run the behemoth.

I haven't tested this, but one proof-of-concept would be to run Caber Tosser in Golly's hlife algorithm with 'Hyperspeed' enabled and leave it running overnight.
What do you do with ill crystallographers? Take them to the mono-clinic!

calcyman

Posts: 2072
Joined: June 1st, 2009, 4:32 pm

### Re: Searching algorithms

I recently came up with an extremely speculative idea for an oscillator and/or spaceship searching method. It was inspired by a simple observation I made a while back: For any synchronous two-dimensional cellular automaton (call it rule O), there is at least one anisotropic 3-dimensional automaton (call it rule S) with the same number of states as rule O for which stable patterns correspond exactly to the full histories of patterns in rule O. This is a relatively simple transformation to make, and it basically crystallizes the time dimension in rule O into a spatial dimension in rule S. If rule S is run on a grid periodic with period n in its third dimension and offset by (i,j), a stable pattern (a wick in non-periodic rule S) corresponds to a (i,j)c/n spaceship in rule O. In the degenerate case of i and j being zero, it corresponds to a period-n oscillator.
This is simply a restatement of the initial problem of finding oscillators and spaceships in rule O when rules O and S are defined as above. However, by adding one or more extra states to rule S, the problem's behavior changes. In particular, if given any rule O, a rule S can be constructed with more states than rule O so that the following conditions can be met, random soups in rule S on a periodic grid could evolve into stable configurations with regularity, corresponding to rule O oscillators or spaceships.

These conditions are:
• Random soups in periodic rule S do not tend to die out overwhelmingly.
• The extra rule S state does not strongly invade and permeate patterns composed of normal states.
• The extra rule S state or states cannot "cap" an incomplete pattern consisting of normal states, i.e. no stable patterns can exist that contain the extra state or states.
• Rule S is not explosive on the required periodic grid.
• Rule S oscillators are uncommon or impossible.

If such a rule O/rule S pair were found, all that would remain would be to create some apgsearch/gsearch/cs-like program that generates starting patterns in periodic rule S and evolves them until a stable pattern or stable patterns are formed, and then to output the rule O equivalent pattern. I could see this as useful to generate small spaceships of a particular speed, although I am not sure enough about the dynamics of three-dimensional CAs, especially anisotropic ones, to say for sure what the distribution of spaceships generated would be.

This is, of course, mostly speculative, and I do not have any working example of a rule O/rule S pair that meets the conditions. I don't consider it likely that a valid rule S even exists for any given rule O that would be at all useful (not trivially equivalent to the action of an existing search program), let alone that such a rule could be reliably generated. But I do believe that the payoff if this approach truly works would be great enough that it's worth looking into.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

A for awesome

Posts: 1862
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

### Re: Searching algorithms

Genetic algorithms are total failure for GOL. On the other hand statistical based algorithms are good direction which wasn't explored well enough yet.

I'm currently doing a master's in CS in this subject.

As and when things progress to a postable state I'll keep you updated.
JohanB

Posts: 16
Joined: June 10th, 2016, 11:59 pm

### Re: Searching algorithms

What about a variant of the lifesrc algorithm that divides the cell array in half along the axis which is being searched, first trying to solve one half and then the other? Something like this:
x = 50, y = 20, rule = LifeHistory50A$A48BA$A48BA$A48BA$A48BA$A48BA$A48BA$A48BA$A48BA$A48BA$A48DA$A48DA$A48DA$A48DA$A48DA$A48DA$A48DA$A48DA$A48DA$50A! Instead of trying to run a width-18 asymmetric search on the entire space, it runs a width-9 search on the blue cells and then runs width-9 searches on the red cells for all of the other solutions. In JLS (I'm not familiar with lifesrc itself or WLS) this could be achieved by a modification to allow a custom search ordering. Would this approach actually result in significant speedup for wide searches? My instinct says it would, but I'm not entirely sure. x₁=ηx V ⃰_η=c²√(Λη) K=(Λu²)/2 Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt) $$x_1=\eta x$$ $$V^*_\eta=c^2\sqrt{\Lambda\eta}$$ $$K=\frac{\Lambda u^2}2$$ $$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$ http://conwaylife.com/wiki/A_for_all Aidan F. Pierce A for awesome Posts: 1862 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 ### Re: Searching algorithms A for awesome wrote:What about a variant of the lifesrc algorithm that divides the cell array in half along the axis which is being searched, first trying to solve one half and then the other? Something like this: x = 50, y = 20, rule = LifeHistory50A$A48BA$A48BA$A48BA$A48BA$A48BA$A48BA$A48BA$A48BA$A48BA$A48DA$A48DA$A48DA$A48DA$A48DA$A48DA$A48DA$A48DA$A48DA$50A!

Instead of trying to run a width-18 asymmetric search on the entire space, it runs a width-9 search on the blue cells and then runs width-9 searches on the red cells for all of the other solutions. In JLS (I'm not familiar with lifesrc itself or WLS) this could be achieved by a modification to allow a custom search ordering.

Would this approach actually result in significant speedup for wide searches? My instinct says it would, but I'm not entirely sure.

I don't believe you could get any benefit from this kind of cell ordering because there is very little constraint on the solutions for the first half of the search. Consider a search which might find the following c/3 ship with a width of 12 (a contrived example, but illustrative):
x = 37, y = 12, rule = LifeHistory4.2A4.5D5.A4.A.A$10.5D4.A.A2.2A.A.2A3.A.A$2.2A2.A3.5D4.A3.A8.A2.A$.A8.5D4.9A4.A2.2A$2A4.3A.5D5.A8.2A$.A4.3A.5D2.A4.7A$.3A8.A4.A4.A5.A$2.A6.2A.A3.A.A4.4A3.2A.A$4.A4.A2.3A9.A5.A.A$2.2A4.2A.4A2.A6.2A$2.2A5.A2.A3.A8.A\$4.A5.2A14.A!

The search would be set to find partial solutions in the top six rows before searching the bottom half. I presume you would require an On cell in the top half of the first row. Because the bottom half of the rows aren't completed there are many partial results for the top half which would need to be searched through before there is any hope of completing the bottom half. As the ship above shows, the lack of constraints means the search could not exclude partials with unconnected segments and would therefore attempt completing a combinatorial number of partials for the top half where there might not be any hope of completing the partial based on the top half of the first segment. I apologise for being a bit unclear in my explanation - hopefully you can see what I mean. If you try such a search in JLS or WLS with the bottom half of rows set to "Don't Care" I think it will be immediately apparent what I mean. I suppose you might somewhat remedy this if you require all the cells in the top half to be in a connected segment, at the cost of not finding ships such as the example above. Even with that I think that the benefit from only searching half the rows will be lost to the widening of the search space by having unconstrained rows - especially if you allow the ship to be reasonably long. This alludes to another problem caused by a need to predetermine the maximum length of the ship - too short and there won't be a solution, too long and the duration of the search will become prohibitive.
The latest version of the 5S Project contains over 196,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.
wildmyron

Posts: 1209
Joined: August 9th, 2013, 12:45 am

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