## lifeplets - enumerating small connected still lifes

For scripts to aid with computation or simulation in cellular automata.

### lifeplets - enumerating small connected still lifes

So there was a recent question on MathOverflow by Sebastien Palcoux about polyplets (i.e. connected patterns of cells) that vanish in one step in CGoL, and specifically about enumerating all of them with up to N cells.

Given that such vanishing polyplets in CGoL are equivalent to connected still lifes in the "dual" rule B3/S0145678, I figured this should be a simple task for standard Life pattern finding programs, but none of the programs I tried actually did quite what I wanted (specifically, doing an exhaustive search for all matching patterns up to N cells, and not wasting time on disconnected constellations), so I decided to write my own.

My answer on MathOverflow has some more detail on the search algorithm, but basically it's a straightforward row-by-row depth first search on an N+2 cell cylindrical lattice (which is just wide enough that the two sides of a connected N cell pattern cannot possibly wrap around and interact). The clever bit is that the code keeps track of the connected components of the partial pattern, and backtracks as soon as any component becomes fully surrounded by dead cells, or if the minimum number of extra live cells needed to connect all the components exceeds the number still available.

The code (and some output) is here on GitHub, in case anyone is curious. The current code is written in Python, and fairly ugly and inefficient Python at that, and should really be regarded more as a proof of concept than as an efficient search tool. I'd expect that just porting the code to C or C++ and using more efficient data structures ought to speed it up at least 10-fold. Since it's a DFS, it could also be fairly easily parallelized, and I can already think of several possible improvements to the pruning scheme (such as recursively propagating forced cell states into yet-to-be-searched cells, while keeping track of the cell count limit) that should hopefully also cut down the runtime by a significant factor.

Nonetheless, I did manage to produce this collection of all one-step vanishing polyplets in CGoL with up to 20 cells using just the current Python code:

`x = 158, y = 450, rule = 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Also, as a simple test, here are all the connected still lifes in CGoL with up to 12 cells, generated using the same code:

`x = 157, y = 181, rule = B3/S232o4bo\$2o3bobo\$6bo7\$2o\$obo\$bo7\$2o4b2obo3b2o4bo\$obo3bob2o2bo2bo2bobo\$b2o10b2o4bobo\$20bo7\$2o5b2o5b2o6b2o\$obo4bobo4bobobo2bo2bo\$bobo5bo7b2o2bobo\$2bo6b2o11bo7\$2o6b2o6b2o6b2o6b2o6b2o7b2o6b2o6bo\$obo5bobo5bo2b2o4bo7bo6bobo5bo2bo4bo2bo4bobo\$bobo7bo5b2obo4bob2o4bobo7bobo2bo2bo5bo2bo4bobo\$2b2o8bo13bobo5bobo7b2o3b2o7b2o6bobo\$11b2o22bo31bo7\$2ob2o4b2o2b2o3b2o7b2o7b2o7b2o7b2o7b2o7b2o7bob2o\$bobo5bobo2bo3bobo6bobo6bobo7bo7bobo7bo2b2o3bobo6b2o2bo\$bobo8b2o5bobo6bobo7bo7bob2o7bo6bobobo6bo8bobo\$2bo17bobo7bo7bobo6bobo8bo6bo10bobo6bo\$21bo8b2o7b2o7bo10bo17b2o\$58b2o7\$2ob2o5b2ob2obo3b2o8b2o3b2o3b2o8b2o8b2o8b2o8b2o8b2o8b2o8b2o8b2o8b2o8b2o8b2o\$bobo6bob2ob2o3bobo7bobo2bo4bobo7bobo7bobo7bo2b2o5bo2bo6bo3b2o4bo10bo2bo6bo9bo9bo8bobo2b2o\$o2bo17bobo9bobo5bo2b2o6bo9bo8b2o2bo5b2obo6bo2bo6bo2b2o5bobobo5bob2o6bobo7bobo9bo2bo\$b2o19bobo9bo7b2obo6bob2o6bobo9b2o8bo7bobo7bo2bo6bobo7bo2bo6bobo7bobo9b2o\$23b2o28bobo7bobo18b2o7bo9b2o8bo9b2o8bobo8bo\$64bo69bo9b2o3\$2o8b2o9b2o8bo9bo9bo\$obo7bobo7bo2bo6bobo7bobo7bobo2b2o\$3bo9bo6bobobo6b2o8bobo7bo2bobo\$4bo9bo6bobo9b2o7bobo7b2o\$5bo9bobo4bo10bobo7bobo\$6bo9b2o16bo9bo\$5b2o7\$2ob2o6b2ob2o6b2ob2o6b2ob2o6b2obo7b2o2b2o5b2o2b2o5b2o2bo6b2o9b2o9b2o9b2o9b2o9b2o\$bobo8bobobo5bob2obo5bob2obobo3bob2o7bobo2bo5bo2bo2bo4bo2bobo5bobob2o5bobo8bobo8bobo8bobo8bobo\$o2bo8bo2bo11bo11b2o7b2o7bobo7b2o2b2o5b2o2bo7bobo7b2o9bobo8bobo8bobo9bo\$obo10b2o12b2o19bobo6b2o23b2o6bobo9b2o8bobo8bobo9bo9b3o\$bo47bo41bo10bobo8bobo9bo9bobo10bo\$103bo10bo10b2o9b2o9b2o3\$2o9b2o9b2o9b2o9b2o4bo4b2o9b2o9b2o9b2o9b2o9b2o9b2o9b2o9b2o\$obo9bo2b2o6bo2bo7bo9bobo2bobo3bobo2b2o4bobo8bo2b2obo4bo2b2o6bo4bo5bo10bo10bo11bo2bo\$2bo9bobobo6bobobo6bob2o9bo2bo5bo2bobo6bo2b2o5bobob2o5b2o2bo6bo2bobo5b3ob2o5bo3b2o5bo10bobobo\$2bob2o7bobo8bo2bo7bo2bo9b2o7b2o9bobobo6bo12bobo6bo2bo8b2obo6bo2bo7bo2b2o6bo2bo\$3bobo8bo10b2o9bobo30bo23bo8b2o20bobo8bo2bo9b2o\$4bo32bo87bo10b2o3\$2o9b2o9b2o9b2o9b2o9bob2o7bob2o7bobo8bobo9b2o9bo\$bo9bobo3b2o3bobo8bobo8bobo8b2o2bo6b2o2bo6b2obo7b2obo7bo2bo7bobo\$bob2o9bo2bo7bo2b2o6bo10bo10bobo8bo10bo10bo7bobobo7bobo\$2bo2bo9bobo8bo2bo7bo10bo10bobo8b3o7bob2o7bobo6bo2bo9bo\$4bobo9bo10b2o9bo10bo10bo11bo8bobo8bobo6b2o10bobo\$5bo33bo10bobo40bo20bobo\$40bo10b2o62bo\$39b2o7\$2ob2o7b2ob2o7b2obo8b2obo8b2obo8b2o2b2o6b2o2b2o6b2o2bo7b2o2bobo5b2o2bo7b2o3b2o5b2o3b2o5b2o3b2o\$ob2obo6bob2obo6bob2o8bob2o8bob2o8bobo2bo6bo2bo2bo5bo2bobo6bo2bob2o5bo2bobo6bobo3bo5bo4bo7bo3bo\$6bo11bobo7b2o10b2o10b2obo6bobo8b2o2bobo5bobobo7bobo9bobo2bo7bo2bo7b3obo7bobobo\$7bo11b2o7bo11bobo9bob2o5bobo14bo7bobo9b2o10bo2b2o7bobo10b2o9bobo\$6b2o21bo11b2o19bo24bo35bo23bo\$28b2o3\$2o3b2o5b2o3bo6b2o3bo6b2o3bo6b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o\$bo2bo2bo5bo2bobo5bobobobo6bo2bobo5bobob2o6bobob2o6bobo9bobo9bobo9bobo9bo2b2o7bo2b2o8bo3bo\$bobo2b2o5bobobo9b2o2bo5bobo2bo7bobo9bobo8b2o10bobo10bo11bo10b2o2bo7bobobo7bob3o\$2bo11bobo13b2o6bo3b2o5bo2bo9bo2bo9b2o9bobo8bob2o8bob2o10bobo8bo2bo8bo\$15bo34b2o11b2o10bobo9bobo8bo2bo8bo2bo9b2o10b2o10bo\$76b2o10b2o9b2o11b2o32b2o3\$2o10b2o4bo5b2o4b2o4b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o\$bo11bo2b3o5bobo2bobo4bobo9bobo9bobo9bobo9bobo9bobo9bobo9bobo9bobo9bobo\$o12bobo9bo2bo8bobo9bobo9bobo9bo2b2o7bo2bo9bo2bo8bo11bo11bo11bo\$2o12bobo9b2o10bo2b2o8bo11bo10b2o2bo7b2obo8bobobo7b3o9bob2o8bob2o8bobo\$2bob2o9bo23b2obo8bob2o8bobo11b2o10bo9bobo11bo9bo2bo8bo2bo8bobo\$2b2obo46bobo9bobo22b2o9bo11bobo9b2o11b2o9bobo\$65bo47bo35bo3\$2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o\$obo9bo2b2o7bo4b2o5bo4b2o5bo11bo12bo2b2o7bo11bo11bo11bo11bo11bo\$2bo10bobobo7b3o2bo6bo2bobo6b3o9b3o9bobo2bo5bo12bob2o8bob2o8bobo9bobo9bobo\$2bobo9bo2bo9bobo8bo2bo9b3o9bo10bo2bo7b3ob2o7b2obo8bo2bo8b2o10bobo9bobo\$3bobo11b2o9bo10b2o13bo8bob2o8b2o10b2obo10bo9bo2bo9b2o9bobo9bobo\$5bo47b2o9bobo34b2o9b2o10bobo9bobo10bo\$5b2o118bo11bo11b2o3\$2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o10b2o\$bo10bobob2o6bobobo7bobo3b2o4bo2b2ob2o4bo5b2o4bo11bo11bo12bo2b2o7bo10bobob2o6bobo4bo\$bobo11b2obo8b2o9bo2bobo5b2ob2obo5b3obobo5bo4bo6bo11bo11bobo2bo6bob2ob2o7b2obobo6bo2bobo\$2bobo13bo10b2o7bobo22b2o9bo2bobo6bo3b2o6bo11bo2bobo6bob2obo11b2o7bo2bo\$3bo14b2o9bobo7bo35bo2bo8bo2bo8bo2b2o10bo34b2o\$4b3o23bo45b2o10bobo9bo2bo\$6bo82bo11b2o3\$2o10b2o10b2o10b2o10bob2o8bob2o8bob2o8bobo9bobo9bobo9bo12b2o10b2o\$obo9bobo9bobo9bobo9b2obo8b2ob3o6b2o2bo7b2obob2o5b2obo8b2obo8b3o9bobo9bobo\$3bo3b2o6bo11bo11bo11bob2o11bo8bo2b2o7bobo9bo11bo2b2o7bo2b2o4bobob2o6bo\$4bo2bo8bo2b2o7bo11bo10b2obo10bobo8b2obo7bobo9bob2o8bobobo6bo2bobo5bobobo7bo2b2o\$5bobo9bo2bo8bo11bo24bo21bo11bobo9bo10b2o9bo11bo2bo\$6bo11b2o10bo11bo58bo45b2o\$31bo11bobo\$32bo11b2o\$31b2o3\$b2o10b2o10b2o10bo11bo11bo11bo11bo11bo11bo11bo11bo12b2o\$o2bo8bobo9bo2bo2b2o4bobob2o6bobo2b2o5bobo2b2o5bobo9bobo9bobo9bobo9bobo9bobo10bo2bo\$b2obo8bo10b2o2bo2bo4bobobo8b2o3bo6bobo2bo6b2o10bobob2o6bobo9bobo2b2o5bobo9bo2b2o6bob2obo\$3bobo8b3o12b2o6bo2bo10b3o9bobo9b2o10bobo8bobo9bo2bobo7bo2b2o6b2o2bo6bo2bo\$3bobo11bo20b2o11bo11b2o10bobo9bobo9bobo9b2o10bobobo9bobo6b2o\$4bo11bobo57bobo9bo11bobo21bo13bo\$17bo59bo23bo3\$2bo\$bobo\$bo2bo\$2o3bo\$2b3o\$2bo!`

(I'm pretty sure these collections should be exhaustive, but there's always the possibility that I might still have some subtle bug in my code that none of my own tests have caught. If anyone else can confirm these results, that would be appreciated. Note that these results deliberately exclude patterns with more than one island, such as the aircraft carrier, and so are not directly comparable with conventional still life counts.)

Any feedback is, of course, welcome.

vyznev

Posts: 27
Joined: April 23rd, 2016, 4:08 am

### Re: lifeplets - enumerating small connected still lifes

vyznev wrote:(I'm pretty sure these collections should be exhaustive, but there's always the possibility that I might still have some subtle bug in my code that none of my own tests have caught. If anyone else can confirm these results, that would be appreciated...)

It would take me a little while to write and run an independent check that you haven't missed any polyplets, but there certainly aren't any obvious ones missing at the lower bit counts.

As a quick spot check, I looked at Catagolue's 12-bit still life collection and confirmed that 29 of the 121 still lifes in the list contain two or more islands -- which leaves your remaining count of 92 connected 12-bit still lifes.

dvgrn
Moderator

Posts: 5705
Joined: May 17th, 2009, 11:00 pm