## The Omniperiodicity Problem

For discussion of specific patterns or specific families of patterns, both newly-discovered and well-known.

### The Omniperiodicity Problem

I looked through 11 pages of this forum, and was surprised to see no thread devoted to finding all oscillator periods, so I thought I'd make one.

What oscillator periods are known:

•All periods under 43, except for 19, 23, 34, 38, and 41

•All periods above 43, using snark loops

What oscillator types are known:

Billiard table configurations, oscillators in which the rotor is encased in the stator. The definition of "encased" is not always clear.

Examples of billiard table oscillators:

`x = 58, y = 13, rule = B3/S234bob2o42b2o\$4b2obo26b2o14b2o\$34b2o\$4b3o12b2ob2o26b4o\$2obobobo10bobobobo9b4o11bo4bob2o\$2obo3bo10bo3bobo6bobo4bo10bobo2bob2o\$3bo3bob2o4b2obo3bob2o5b2obo3bo7b2obobo2bo\$3bo3bob2o5bobo3bo11bo3bob2o4b2obo2bobo\$4b3o9bobobobo11bo4bobo8b4o\$17b2ob2o13b4o\$3bob2o45b2o\$3b2obo30b2o13b2o\$37b2o!`

Hassler oscillators, oscillators in which an unstable object is perturbed by other oscillator(s) (or still life[s]). The perturbed object need not necessarily be unstable, but the perturbers have to affect it in some way.

Examples of hassler oscillators:

`x = 69, y = 15, rule = B3/S2338b2o\$2o12b2o23bo21bo4bo\$bo12bo24bobo9b2o8b6o\$bobo8bobo5b3o10b3o4bobo8bo9bo4bo\$2b2o8b2o7b3o8b3o14bobo\$49b2o\$5b6o14b6o12b2o\$43b2ob2o13b6o\$2b2o8b2o7b3o8b3o11b2o12bo6bo\$bobo8bobo5b3o10b3o4b2o17bo8bo\$bo12bo24bobo18bo6bo\$2o12b2o23bo8bobo10b6o\$38b2o9bobo\$51bo\$51b2o!`

Shuttle oscillators, oscillators usually of an even period in which an active object goes back and forth, typically reflecting off some sort of stabilization at the ends. Some oscillators are a trivial type of shuttle without stabilizations, like the Tumbler.

Examples of shuttle oscillators:

`x = 180, y = 39, rule = B3/S2379b2o2bo38bo2b2o\$79bo2bobo36bobo2bo\$80bobobo36bobobo\$79b2obob2o34b2obob2o\$78bo3bo2bo11bo22bo2bo3bo\$77bob2o2b3o10bobo21b3o2b2obo\$77bo4b4o10bobo14b2o5b4o4bo\$78b3o2b3o11bo2b2o2b2o4b2obo6b3o2b3o\$52b2o28bo2bo14bo3b2o3bo10bo2bo\$22bo28bobo22b5obob2o14b2o7bo10b2obob5o\$20bobo28bo24bo2bo2bobo24b3o9bobo2bo2bo\$2o3bo7b2o4bobo23bo5b3o13bo13bo2bo36bo2bo\$2ob2o8b2o3bo2bo21b3o21b3o12b2o38b2o\$4bobo12bobo20bo27bo\$5bo14bobo9b2o9b3o21b3o12b2o\$5b3o2b2o10bo9bobo10bo5b3o13bo13bo2bo62bo4bo\$10b2o22bo16bo24bo2bo2bobo24b3o33b2ob4ob2o\$34b2o15bobo22b5obob2o14b2o7bo11bo25bo4bo\$52b2o28bo2bo14bo3b2o3bo9b3o39bo\$78b3o2b3o11bo2b2o2b2o4b2obo4bo43b2o8bo4bo\$77bo4b4o10bobo14b2o2bobo41b2o7b2ob4ob2o\$77bob2o2b3o10bobo19bo53bo4bo\$78bo3bo2bo11bo\$79b2obob2o34bo\$80bobobo32bobobo\$79bo2bobo32b2o2bo\$79b2o2bo37b2o\$111b3o\$110b5o\$109bob3obo\$107b3o5b3o\$106bo3bo3bo3bo\$107b4obob3obo\$111b3o3bo\$106bob2obobobo\$106b2obo2bo2b2o\$110bo2bobo\$111b2o2bo\$115b2o!`

Other oscillators, any kind of oscillator that does not meet any of the above definitions.

What do you think the missing oscillator periods could be?

Examples:

I think that a p19 oscillator could be....

•A pre-pulsar shuttle with pulsars bouncing back and forth
•A loop of gliders reflecting between p19 dependent reflectors of some sort
•An extensible pattern based on a stabilization of the several known p19 wicks
•An unstable object being perturbed at p19 by a group of still lifes
•A billiard table configuration

I think that a p34 oscillator could be....

•Two p17s interacting to hassle something like a toad
•A pre-pulsar shuttle based on (for example) clocks or killer toads

What to post in this thread:

•Challening the definitions posted above
•Ideas for how to make an oscillator of period n
•Oscillators that break a record of being the smallest of that period
•Solutions to the problem

gmc_nxtman

Posts: 1147
Joined: May 26th, 2015, 7:20 pm

### Re: The Omniperiodicity Problem

There is also somewhat high probability that the first known oscillators of yet unknown periods are going to be loops of signals, when any 90-degree signal turner is found.
Ivan Fomichev

codeholic
Moderator

Posts: 1141
Joined: September 13th, 2011, 8:23 am
Location: Hamburg, Germany

### Re: The Omniperiodicity Problem

codeholic wrote:There is also somewhat high probability that the first known oscillators of yet unknown periods are going to be loops of signals, when any 90-degree signal turner is found.

Agreed. Specifically, I'd bet a nickel that the omniperiodicity problem will eventually be solved by someone finding a 90-degree 2c/3 signal elbow, with a recovery time under 19 ticks.

The 2c/3 signal itself is so simple that it can be repeated every three ticks (?!) so there's some room to maneuver here.

I don't think that a SAT-solver approach is likely to produce any definitive results yet -- too many cells will be involved to do an exhaustive search. Maybe things will be different if Moore's Law has a few more decades to work its magic.

If I'm understanding the Garden of Eden #6 paper correctly, the search limit for current technology seems to be around 30-40 cells. That doesn't sound too different from the size of search that lifesrc/WLS/JLS might be able to manage, counting all unknown cells in all phases, and efficiently discarding unworkable combinations.

If someone can figure out how to state the elbow problem in a way that requires less than 40 unknown cells to be tested in combination, then there might be a chance of running a distributed search to test every possible workable elbow in that search space.

Another idea: we already have a perfectly good 2c/3 elbow, but it doubles the signal so the elbow can only be used once:

`#C another piece of circuitry overdue for replacement#C   (with Snarks and syringes, mostly)x = 316, y = 425, rule = B3/S23140b2o\$140b3o\$139bob2o\$139b3o\$140bo11\$136bobo5bobo2bo\$136b2o6b2o3bobo\$137bo7bo3b2o5\$133bo2bo\$131bobo2bobo\$132b2o2b2o3\$129b2o\$129bo2bob2o\$130b3ob2o2\$130b6o\$129bo6bo\$129b5o2bo\$126bo7bobobo\$126b6o2bo2b2o\$132bobo\$124b6o2bob2o14bo4b2o\$123bo6bobo16b2o3b2o\$123b5o2bobo16bobo4bo\$120bo7bob2o\$120b6o2bo\$126bobo\$118b6o2bob2o22b2o\$117bo6bobo25bobo\$117b5o2bobo25bo\$114bo7bob2o\$114b6o2bo\$120bobo\$112b6o2bob2o\$111bo6bobo\$111b5o2bobo\$108bo7bob2o\$108b6o2bo\$114bobo\$106b6o2bob2o\$105bo6bobo\$105b5o2bobo\$102bo7bob2o\$102b6o2bo\$108bobo\$100b6o2bob2o\$99bo6bobo\$99b5o2bobo\$96bo7bob2o\$96b6o2bo\$102bobo\$94b6o2bob2o\$93bo6bobo\$93b5o2bobo\$90bo7bob2o\$90b6o2bo\$96bobo\$88b6o2bob2o\$87bo6bobo\$87b5o2bobo\$84bo7bob2o\$84b6o2bo\$90bobo\$82b6o2bob2o\$81bo6bobo\$81b5o2bobo\$78bo7bob2o\$78b6o2bo\$84bobo\$76b6o2bob2o\$75bo6bobo\$75b5o2bobo\$72bo7bob2o\$72b6o2bo\$78bobo\$70b6o2bob2o\$69bo6bobo\$69b5o2bobo\$66bo7bob2o\$66b6o2bo\$72bobo\$64b6o2bob2o\$63bo6bobo\$63b5o2bobo\$60bo7bob2o\$60b6o2bo\$66bobo\$58b6o2bob2o\$57bo6bobo\$57b5o2bobo\$54bo7bob2o\$54b6o2bo\$60bobo\$52b6o2bob2o\$51bo6bobo\$51b5o2bobo\$48bo7bob2o\$48b6o2bo\$54bobo\$46b6o2bob2o\$45bo6bobo\$45b5o2bobo\$42bo7bob2o\$42b6o2bo\$48bobo\$40b6o2bob2o\$39bo6bobo\$39b5o2bobo\$36bo7bob2o\$36b6o2bo\$42bobo\$34b6o2bob2o\$33bo6bobo\$33b5o2bobo\$30bo7bob2o\$30b6o2bo\$36bobo\$28b6o2bob2o\$27bo6bobo\$27b5o2bobo\$24bo7bob2o\$24b6o2bo\$30bobo\$22b6o2bob2o\$21bo6bobo\$21b5o2bobo\$18bo7bob2o\$18b6o2bo\$24bobo\$16b6o2bob2o280bob2o\$10b2o3bo6bobo283b2obo\$9bo2bo2b5o2bobo\$8bob3o7bob2o282b5o\$4b2obobo3b5o2bo263b2o20bo4bo2b2o\$5bobo3bo6bobo264bo23bo2bo2bo\$5bobo2b6o2bob2o263bobo21b2obobo\$3bobobobo6bobo267b2o18bo5bob2o\$2bob2o2bob4o2bobo286bobo4bo\$2bo3bobobo3bob2o287bo2bo2b2o\$2ob2obobo3bobo291b2o\$bobo2bob4obob3o257b2o\$o2bobo7bo3bo258bo\$b3o2b8o262bobo\$4bobo270b2o\$3b2obo2b7o\$2bo2b2obo7bo278b2o\$2b2o4bo2b6o278b2o\$8bobo\$7b2obo2b6o\$10bobo6bo\$10bobo2b5o\$11b2obo7bo\$14bo2b6o\$14bobo\$13b2obo2b6o\$16bobo6bo\$16bobo2b5o\$17b2obo7bo\$20bo2b6o257bo\$20bobo262bobo\$19b2obo2b6o254bobo\$22bobo6bo254bo\$22bobo2b5o\$23b2obo7bo\$26bo2b6o\$26bobo\$25b2obo2b6o114b2o\$28bobo6bo106b2o5b2o\$28bobo2b5o106b2o\$29b2obo7bo140bo\$32bo2b6o138b3o\$32bobo111b2o17b2o11bo\$31b2obo2b6o103b2o17bo12b2o\$34bobo6bo96b2o21bobo20bo\$34bobo2b5o96b2o21b2o19b3o\$35b2obo7bo136bo\$38bo2b6o136b2o\$38bobo\$37b2obo2b6o\$40bobo6bo136b2o9b2o26b2o7b2o\$40bobo2b5o136bo11bo25bobo7b2o\$41b2obo7bo131bobo11bobo21b3obobo\$44bo2b6o131b2o13b2o20bo5b2o39bo\$44bobo174b2o45b3o\$43b2obo2b6o216bo14bo\$46bobo6bo135bo78b2o12b3o\$46bobo2b5o81bo51b3o91bo\$47b2obo7bo78b3o48bo94b2o\$50bo2b6o81bo47b2o\$50bobo86b2o\$49b2obo2b6o221b2o\$52bobo6bo201b2o17b2o\$52bobo2b5o201b2o\$53b2obo7bo\$56bo2b6o\$56bobo98b2o\$55b2obo2b6o90bo\$58bobo6bo87bobo42b2o\$58bobo2b5o87b2o43b2o11b2o51b2o\$59b2obo7bo142bo53bo\$62bo2b6o109b2o32b3o48bo\$62bobo73b2o40bo35bo48b2o\$61b2obo2b6o65b2o15b2o24b3o85b2o\$64bobo6bo81bobo25bo12b2o38b2o32bo\$64bobo2b5o83bo38bo39bo30b3o\$65b2obo7bo80b2o38b3o37b3o27bo\$68bo2b6o122bo39bo\$68bobo\$67b2obo2b6o\$70bobo6bo58b2o\$70bobo2b5o58bo\$71b2obo7bo53bobo\$74bo2b6o53b2o\$74bobo\$73b2obo2b6o\$76bobo6bo\$76bobo2b5o67b2obo\$77b2obo7bo64bob2o\$80bo2b6o\$80bobo63b2o\$79b2obo2b6o55b2o122b2o\$82bobo6bo177bobo\$82bobo2b5o177bo29b2o\$83b2obo7bo173b2o29bobo\$86bo2b6o206bo\$86bobo185b2o25b2o\$85b2obo2b6o176bobo4b2o\$88bobo6bo38b2o135bo7bo\$88bobo2b5o39bo134b2o4b3o\$89b2obo7bo36bobo138bo\$92bo2b6o37b2o146b2o\$92bobo190bobo\$91b2obo2b6o182bo\$94bobo6bo180b2o\$94bobo2b5o53bo\$95b2obo7bo48b3o\$98bo2b6o47bo\$98bobo53b2o\$97b2obo2b6o193b2o\$100bobo6bo168bo23bobo\$100bobo2b5o49b2o117b3o23bo\$101b2obo7bo46b2o120bo22b2o\$104bo2b6o167b2o14b2o\$104bobo189b2o\$103b2obo2b6o54b2o\$106bobo6bo53bo\$106bobo2b5o51bobo\$107b2obo7bo48b2o\$110bo2b6o11b2o\$110bobo16bobo\$109b2obo2b6o8bo\$112bobo6bo6b2o\$112bobo2b5o\$113b2obo7bo15b2o\$116bo2b6o15b2o\$116bobo137b2o7b2o30b2o\$115b2obo2b6o61b2o66b2o7bobo29bobo\$118bobo6bo60b2o45bo27bobob3o29bo\$118bobo2b5o107b3o25b2o5bo28b2o\$119b2obo7bo107bo30b2o\$122bo2b6o55b2o49b2o\$122bobo20b2o39b2o\$121b2obo2b6o11bobo26b2o43bo\$124bobo6bo10bo29bo43b3o\$124bobo2b5o9b2o29bobo44bo\$125b2obo7bo38b2o43b2o11b2o\$128bo2b6o34b2o60b2o\$128bobo40b2o\$127b2obo2b6o\$130bobo6bob2o\$130bobo2b5ob2o\$131b2obo138b2o21b2o\$134bo2b6o19b2o108bobo21b2o\$134bobo5bo19bo109bo17b2o\$133b2obo2b3o18bobo21b2o85b2o17b2o\$136bobo5b2o14b2o22b2o\$136bo2bo4b2o93b2o\$137b2o9bo46b2o39b2o2bo2b2o47b2o\$144b2o2b3o43bo2bo38b2obo3bo41b2o5b2o\$144b2o5bo42bobo23b2o17bobobo10b2o29b2o\$150b2o43bo20bo3b2o14b2obob2o12bo\$215bobo12b2o4bo2bo12b3o\$214bobo13b2o6b2o12bo\$163b2o49bo\$162bobo48b2o\$163bo\$177b2o\$177b2o4\$143b2o\$143b2o5\$145bo\$115b2o28b3o\$108b2o5b2o31bo92bo\$108b2o37b2o90b3o11bo\$143b2o93bo14b3o\$143bo94b2o16bo14bo\$110b2o33bo109b2o12b3o\$110b2o32b2o69bo52bo\$104b2o109b3o50b2o\$104b2o112bo\$217b2o\$267b2o\$248b2o17b2o\$141b2o105b2o\$141b2o17b2o\$160b2o3\$161b2o\$161bo89b2o\$148b2o12b3o41bo23b2o20bo\$131b2o16bo14bo40bobo22bo19bo\$131bo14b3o57bo14b2o8b3o16b2o\$124bo7b3o11bo74bo11bo20b2o\$122b3o9bo87b3o30bo\$121bo102bo27b3o\$121b2o129bo\$113b2o93b2o\$113b2o92bobo\$207bo\$206b2o62b2o\$222b2obo44bobo\$222bob2o46bo\$272b2o\$130b2o83b2o\$130bo84b2o\$128bobo\$128b2o3\$262b2o\$262b2o2\$253b2obo\$253bob2o2\$132b2o\$132bobo\$134bo91b2o44b2o\$134b2o90bo44bobo\$124b2o98bobo44bo\$124b2o98b2o44b2o\$105bob2o\$105b2obo2\$114b2o135b2o\$114b2o136bo\$252bobo\$253b2o15b2o\$210b2o58b2o\$210bo2b2o\$211b2obo\$212bo40b2o\$124b2o86bobo4b2o31bobo\$124bo88b2o5bo31bo\$122bobo94bo31b2o\$122b2o68b2o25b2o\$166bo25bo\$154bo11b3o21bobo\$152b3o14bo20b2o\$136bo14bo16b2o\$136b3o12b2o70b2o44b2o\$139bo58b2o24bo44bo\$100b2o36b2o59bo13b2o6b3o46b3o\$101bo97bobo11b2o6bo50bo\$101bobo96b2o\$99b2ob2o35b2o\$98bobo38b2o17b2o\$98bobo57b2o\$97b2ob2o20b2o\$97bo24bo\$98bob2o18bobo108b2o\$96bobob2o18b2o109b2o\$96b2o\$155b2o32b2o\$155bo20b2o11b2o\$120b2o35bo19bo67b2o21b2o\$120bobo33b2o16b3o67bobo21b2o\$122bo29b2o20bo53b2o14bo17b2o\$122b3o27bo75bo14b2o17b2o\$125bo27b3o37b2o14b2o3b2o13bo\$124b2o29bo38bo15bo3bo13b2o\$191b3o13b3o5b3o46b2o\$191bo15bo9bo39b2o5b2o\$257b2o4\$114b2o\$114b2o2\$105b2o\$106bo\$103b3o45b2o\$103bo47b2o\$111b2o32b2o\$111bo33b2o\$112bo\$111b2o\$147b2o\$140b2o5b2o\$140b2o!`

The elbow's repeat rate is 15 ticks, plenty good enough to solve the remaining oscillator periods, so maybe it's worth looking for a component that reduces the double signal back to a singleton signal. I don't know whether thorough searches have already been done on this. (?)

dvgrn
Moderator

Posts: 5703
Joined: May 17th, 2009, 11:00 pm

### Re: The Omniperiodicity Problem

What are the known signals besides the two 2c/3 (single and double ones)?
Ivan Fomichev

codeholic
Moderator

Posts: 1141
Joined: September 13th, 2011, 8:23 am
Location: Hamburg, Germany

### Re: The Omniperiodicity Problem

codeholic wrote:What are the known signals besides the two 2c/3 (single and double ones)?

There's Dean Hickerson's 5c/9 signal wire from 11 April 1997, which can be connected to a 2c/3 wire. I've quoted a sample signal injector at the end. The link will take you to a larger collection, but that's maybe not a likely line of research to get even as high as p19.

Dean Hickerson wrote:
`#C A signal moves along a diagonal wire with speed 5c/9 and period 18/2.#C When it reaches the end it fizzles out.  Successive signals must be#C at least 8 gens apart.#C Dean Hickerson, 4/11/97x = 52, y = 504bob2o\$4b2o2bo\$7bo2bo\$2b5ob2obo2bo\$bo2bo3bo2b4o\$bob2obobobo6bo\$2obob4obo2b5o\$3bo6bobo5b2o\$2obob4obo2bob2obobo\$o2bobo2bob2obobobo2bo\$2b2o2bo2bo3bobo4bob2o\$4b2o4b4ob2o2b2o2bo\$4bo3bobo6bo3bo\$5b4obob5ob3o3bo\$9bobo4bobo2b4o\$7bo3bo2bo3bobo6bo\$7b2o2bobob4obo2b5o\$10b2obo6bobo5b2o\$13bob4obo2bob2obobo\$13bobo2bob2obobobo2bo\$12b2o2bo2bo3bobo4bob2o\$14b2o4b4ob2o2b2o2bo\$14bo3bobo6bo3bo\$15b4obob5ob3o3bo\$19bobo4bobo2b4o\$17bo3bo2bo3bobo6bo\$17b2o2bobob4obo2b5o\$20b2obo6bobo5b2o\$23bob4obo2bob2obobo\$23bobo2bob2obobobo2bo\$22b2o2bo2bo3bobo4bob2o\$24b2o4b4ob2o2b2o2bo\$24bo3bobo6bo3bo\$25b4obob5ob3o3bo\$29bobo4bobo2b4o\$27bo3bo2bo3bobo6bo\$27b2o2bobob4obo2b5o\$30b2obo6bobo5b2o\$33bob4obo2bob2obo2bo\$33bobo2bob2obobobo2b2o\$32b2o2bo2bo3bobo\$34b2o4b4ob2o\$34bo3bobo6bo\$35b4obob5obo\$39bobo4bobo\$37bo3bo2bo3b2o\$37b2o2bobob3o2bo\$40b2obo5bo\$44bob3o\$45b2o!`

This showed up starting with a 2c/3 wire. Here's the connection,
also showing a different way for the 5c/9 to fizzle out:

`#C Two 2c/3 signals are converted to 5c/9 signals and then fizzle out.#C The original signals are 6 gens apart, but after conversion they're#C 9 gens apart.  (Don't try this twice in a row.)#C Dean Hickerson, 4/11/97x = 61, y = 566bo2bo\$4b6o\$3bo\$3bobob5o\$2obobo6bo\$2obobo2b5o\$4b2obo7bo\$7bobob5o\$7bobo\$6b2obo2b6o\$9bobo6bo\$9bobo2b5o\$10b2obo7b2o\$13bo2b6o\$13bobo7bo\$12b2obo2b5obo2b2o\$15bobo6bo3bo\$15bobo2b4ob3o3bo\$14b2obobo3bobo2b4o\$18bo2bo3bobo6bo\$19b2ob4obo2b5o\$20bo6bobo5b2o\$20bob4obo2bob2obobo\$17b2obobo2bob2obobobo2bo\$17bob2o2bo2bo3bobo4bob2o\$21b2o4b4ob2o2b2o2bo\$21bo3bobo6bo3bo\$22b4obob5ob3o3bo\$26bobo4bobo2b4o\$24bo3bo2bo3bobo6bo\$24b2o2bobob4obo2b5o\$27b2obo6bobo5b2o\$30bob4obo2bob2obobo\$30bobo2bob2obobobo2bo\$29b2o2bo2bo3bobo4bob2o\$31b2o4b4ob2o2b2o2bo\$31bo3bobo6bo3bo\$32b4obob5ob3o3bo\$36bobo4bobo2b4o\$34bo3bo2bo3bobo6bo\$34b2o2bobob4obo2b5o\$37b2obo6bobo5b2o\$40bob4obo2bob2obobob2o\$40bobo2bob2obobobo2bob2o\$39b2o2bo2bo3bobo4bo\$41b2o4b4ob2o2b2o\$41bo3bobo6bo\$42b4obob5ob3o\$46bobo4bobo2bo\$44bo3bo2bo3bobobo\$44b2o2bobob4obobo\$47b2obo6bob2o\$50bob4obo\$47b2obobo2bobob2o\$47b2obobo2bobob2o\$51bo4bo!`

For a short time I thought I'd found a way for the 5c/9 signal to
turn a corner and become a 2c/3. But, like the glider-activated
2c/3 that I sent earlier, it only works once:

`#C 5c/9 signal turns a corner, becoming 2c/3.  But the corner is#C damaged.#C Dean Hickerson, 4/11/97x = 63, y = 404bob2o\$4b2o2bo\$7bo2bo\$2b5ob2obo2bo\$bo2bo3bo2b4o\$bob2obobobo6bo\$2obob4obo2b5o38b2o\$3bo6bobo5b2o35bo2bo2b2o\$2obob4obo2bob2obobo34bobobo2bo\$o2bobo2bob2obobobo2bo31b2obobo2b2o\$2b2o2bo2bo3bobo4bob2o29bobobo\$4b2o4b4ob2o2b2o2bo29bobo2b4o\$4bo3bobo6bo3bo28b2obobo5bo\$5b4obob5ob3o3bo24bobobo2b3o\$9bobo4bobo2b4o24bobobo4bo\$7bo3bo2bo3bobo6bo18b2obobo2b4o\$7b2o2bobob4obo2b5o19bobobo\$10b2obo6bobo5b2o3b2o12bobo2b4o\$13bob4obo2bob2obobobobo9b2obobo5bo\$13bobo2bob2obobobo2bobo10bobobo2b3o\$12b2o2bo2bo3bobo4bob2o9bobobo4bo\$14b2o4b4ob2o2b2o4bo4b2obobo2b4o\$14bo3bobo6bo3b4obo4bobobo\$15b4obob5ob3o5bo4bobo2b4o\$19bobo4bobo2b6ob2obobo5bo\$17bo3bo2bo3bobo6bobobo2b3o\$17b2o2bobob4obo2b2o2bobobo4bo\$20b2obo6bobo2bobobo2b4o\$23bob4obo2bobobobo\$23bobo2bob2obobobo2b4o\$20bob2o2bo2bo3bobobo5bo\$20b2o2b2o4b4obo2b3o\$23bo2b4o5bo4bo\$24bo5b4ob5o\$25b3o2bo2bo\$27bo3bo2b6o\$28b3o5bo2bo2\$28b2obo\$28bob2o!`

[2 June 1997:]
A p11 can inject signals into a 5c/9 track. Here are 2 versions, showing
both fizzle reactions that I know about:

`#C p11 oscillators inject signals into p18/2, speed 5c/9 diagonal tracksx = 116, y = 557b2o57b2o\$3b2obo2bo52b2obo2bo\$2bobob2obo51bobob2obo\$bo2bo4bob2o47bo2bo4bob2o\$bobob5o2b3obob2o40bobob5o2b3obob2o\$2obo6bo4b2ob2o39b2obo6bo4b2ob2o\$3bob2ob2ob3o48bob2ob2ob3o\$3bo3bobobo2b6o42bo3bobobo2b6o\$4b2obo3bobo6bo42b2obo3bobo6bo\$6bob4obo2b5o44bob4obo2b5o\$6bo6bobo5b2o42bo6bobo5b2o\$5b2ob4obo2bob2obobo40b2ob4obo2bob2obobo\$6bobo2bob2obobobo2bo41bobo2bob2obobobo2bo\$6bo2bo2bo3bobo4bob2o38bo2bo2bo3bobo4bob2o\$7b2o4b4ob2o2b2o2bo39b2o4b4ob2o2b2o2bo\$11bobo6bo3bo45bobo6bo3bo\$7b5obob5ob3o3bo38b5obob5ob3o3bo\$7bo4bobo4bobo2b4o38bo4bobo4bobo2b4o\$10bo3bo2bo3bobo6bo38bo3bo2bo3bobo6bo\$10b2o2bobob4obo2b5o38b2o2bobob4obo2b5o\$13b2obo6bobo5b2o39b2obo6bobo5b2o\$16bob4obo2bob2obobo41bob4obo2bob2obobo\$16bobo2bob2obobobo2bo41bobo2bob2obobobo2bo\$15b2o2bo2bo3bobo4bob2o37b2o2bo2bo3bobo4bob2o\$17b2o4b4ob2o2b2o2bo39b2o4b4ob2o2b2o2bo\$17bo3bobo6bo3bo41bo3bobo6bo3bo\$18b4obob5ob3o3bo39b4obob5ob3o3bo\$22bobo4bobo2b4o43bobo4bobo2b4o\$20bo3bo2bo3bobo6bo38bo3bo2bo3bobo6bo\$20b2o2bobob4obo2b5o38b2o2bobob4obo2b5o\$23b2obo6bobo5b2o39b2obo6bobo5b2o\$26bob4obo2bob2obobo41bob4obo2bob2obobo\$26bobo2bob2obobobo2bo41bobo2bob2obobobo2bo\$25b2o2bo2bo3bobo4bob2o37b2o2bo2bo3bobo4bob2o\$27b2o4b4ob2o2b2o2bo39b2o4b4ob2o2b2o2bo\$27bo3bobo6bo3bo41bo3bobo6bo3bo\$28b4obob5ob3o3bo39b4obob5ob3o3bo\$32bobo4bobo2b4o43bobo4bobo2b4o\$30bo3bo2bo3bobo6bo38bo3bo2bo3bobo6bo\$30b2o2bobob4obo2b5o38b2o2bobob4obo2b5o\$33b2obo6bobo5b2o39b2obo6bobo5b2o\$36bob4obo2bob2obo2bo40bob4obo2bob2obobob2o\$36bobo2bob2obobobo2b2o40bobo2bob2obobobo2bob2o\$35b2o2bo2bo3bobo45b2o2bo2bo3bobo4bo\$37b2o4b4ob2o46b2o4b4ob2o2b2o\$37bo3bobo6bo45bo3bobo6bo\$38b4obob5obo45b4obob5ob3o\$42bobo4bobo49bobo4bobo2bo\$40bo3bo2bo3b2o46bo3bo2bo3bobobo\$40b2o2bobob3o2bo45b2o2bobob4obobo\$43b2obo5bo49b2obo6bob2o\$47bob3o53bob4obo\$48b2o52b2obobo2bobob2o\$102b2obobo2bobob2o\$106bo4bo!`

Then there's a c/2 by Hartmut Holzwart, I believe:

`x = 60, y = 60, rule = B3/S23bo\$obo\$bobo\$2bobo\$3bobo\$5bo\$7bo\$3bo3bo\$7b2o\$4b3o3bo\$8b3o\$9b2o\$12bo\$12bobo2\$13bobo\$14bobo\$15bobo2bo\$16bo5bo\$17b2o3bo\$22bo\$19b3o\$21bobo\$23b2o\$22b3o2\$25b2o\$28bo\$26bo2bo\$28bobo\$29bobo\$30bobo\$31bobo\$32bobo\$33bobo\$34bobo\$35bobo\$36bobo\$37bobo\$38bobo\$39bobo\$40bobo\$41bobo\$42bobo\$43bobo\$44bobo\$45bobo\$46bobo\$47bobo\$48bobo\$49bobo\$50bobo\$51bobo\$52bobo\$53bobo\$54bobo\$55bobo\$56bobo\$57bobo\$58bo!`

-- and Jason Summers' orthogonal lightspeed beehive wire, I suppose, though the wire drifts and has to be moved back into place with a series of later signals... and Gabriel Nivasch's diagonal lightspeed signals, which are complicated enough that it seems unlikely that there will ever be a recipe to generate them.

(Might be missing a few. Anyone have other contributions?)

dvgrn
Moderator

Posts: 5703
Joined: May 17th, 2009, 11:00 pm

### Re: The Omniperiodicity Problem

Dave Greene wrote:(Might be missing a few. Anyone have other contributions?)

There exists this simple orthogonal lightspeed signal:

`x = 26, y = 11, rule = B3/S23bo2bo2bo2bo2bo2bo2bo2bo\$b24o\$25bo\$b24o\$o4bo\$b4o4b16o\$6bo18bo\$b24o\$o\$b24o\$3bo2bo2bo2bo2bo2bo2bo2bo!`

Now that we have collected so many of them, it seems a good idea to invent some sort of nomenclature for signal turners, like the one for Hershel conduits (or does there already exist one?).
There are 10 types of people in the world: those who understand binary and those who don't.

Alexey_Nigin

Posts: 323
Joined: August 4th, 2014, 12:33 pm
Location: Ann Arbor, MI

### Re: The Omniperiodicity Problem

Alexey_Nigin wrote: it seems a good idea to invent some sort of nomenclature for signal turners
Wish if we had any to name them... There are currently only three converters that I know of.
Best wishes to you, Scorbie

Scorbie

Posts: 1379
Joined: December 7th, 2013, 1:05 am

### Re: The Omniperiodicity Problem

I was wondering if it would actually be possible to convert an orthogonal lightspeed signal to another signal (or even try turning it around a corner)... the simple one Alexey Nigin mentioned below looks the easiest to form (due to size, though some larger signals have the symmetry advantage), but I'm not aware of any way to even absorb it. I went and played around with some of the ones that can be absorbed, though, and a couple of them are kinda interesting:

`#C Maybe a drifter search could do something with it (not necessarily this form exactly)x = 45, y = 21, rule = B3/S2332b2o\$32bobo\$34bo\$34b2o\$36bo\$bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2b2obo\$b34o2bo\$36b2ob2o\$b35o2bobo2b2o\$o5bo29bobobobobo\$b4o4b28o2bo2bo\$4bo2bo29b2o3b2o\$b4o4b28o\$o4bo31b2o\$b36o2bo\$38bobo\$b38obo\$bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo5b2o\$37b3o\$36bo2bo\$36b2o!`

`#C Orthogonal lightspeed signal turns 135 degrees to become a 2c/3 signal - but one important cell is turned OFF in the process.x = 42, y = 26, rule = B3/S233bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo\$3b33o\$36bo\$b4ob2obo2b25o3b2o\$o4bo3b2o26b2o2bo\$b6o2bo2b25o2b2o\$37bobo\$b35obobo\$bo2bo2bo2bo2bo2bo2bo2bo12bob2o\$25b8o2bo\$25bo7bobob2o\$26b5o2bobobo\$20b2obo7bobobobo\$21bob6o2bobob2o\$20bo8bobobo\$21b6o2bobobo\$27bobob2o\$19b6o2bobo\$18bo6bobobo\$18b5o2bobob2o\$23bobobo\$18b3o2bobobo\$18bo2bobob2o\$19bobobo\$20bo2bob2o\$23b2obo!`
Sphenocorona

Posts: 480
Joined: April 9th, 2013, 11:03 pm

### Re: The Omniperiodicity Problem

Sphenocorona wrote:I was wondering if it would actually be possible to convert an orthogonal lightspeed signal to another signal (or even try turning it around a corner)...

To prove omniperiodicity it probably makes the most sense to look for a fast direct 90-degree turn where the input and output signals are the same -- or a 180-degree signal turner would be fine if it leaves space for two wires next to each other, but that seems a little less likely.

Otherwise the problem doesn't really get any smaller. If we could fix the ortho-lightspeed to 2c/3 signal turner, then to complete a loop we'd still need a 2c/3 to ortho-lightspeed turner, at minimum, or a chain of other signal converters that could be combined to produce that.

I'll be really enormously impressed and surprised if anyone comes up with a working X-to-ortho-lightspeed signal converter any time soon -- let's say, before someone figures out how to program the question into an actual working 1000-qubit quantum computer. Actually I doubt that even quantum computing will be enough of a boost. This is just a ridiculously big search space.

How Big Is Ridiculously Big?

I did some very rough estimating. If anyone wants some slightly more specific crackpot calculations, I can post the rest of my notes...

Bigger Signal Object Means Bigger Search Space

Creating an ortho-lightspeed signal involves figuring out how to adjust the states of at least 18 bits. Really it might be more like 9 bits for the leading p1 part and another 18 for the trailing p2 part, but let's just call it 18 cells.

We have to start from some kind of large still life, give it an input signal, get that specific ortho-lightspeed output signal going, and then have the still life return to its exact original state. Even if the input is just a one-bit spark somewhere nearby, quite a few more cells will be needed to guide that signal into the stabilized end of an ortho-lightspeed wire.

Longer Recovery Also Means Bigger Search Space

And even if that one-bit spark propagates at lightspeed, it's going to take at least 6 ticks for that signal to build a full-length ortho-lightspeed signal in the beginning of the wire. After that we can stop worrying -- the wire will recover naturally.

-- But we already have on the order of 6*18 = 108 cell states to worry about, not counting stator cells around the edges. (Yes, it's possible to do a lot of hand-waving and imagine vaguely that we could get away with less, but really it will probably be more.)

Summary: Brute Force Has Its Limits

Anywhere above 2^40 cases is pretty difficult -- that's a trillion, which is apgnano/Catagolue territory -- and we're talking a lot more than 2^40 sets of 2^40 cases. So we need an algorithm clever enough to remove a factor of something like 2^70 from this search problem...!

My theory is that if we had the tools needed to solve the ortho-lightspeed elbow problem, we would also be able to solve the 2c/3 elbow problem, about a million times more easily. A 2c/3 elbow -- or even just a double-signal to single-signal converter -- is all we really need to prove omniperiodicity.

dvgrn
Moderator

Posts: 5703
Joined: May 17th, 2009, 11:00 pm

### Re: The Omniperiodicity Problem

This is close to a 180 signal turner:

`x = 26, y = 11, rule = B3/S23bo2bo2bo2bo2bo2bo2bo2bo\$b24o\$25bo\$b25o\$o4bo\$5o4b17o\$6bo18bo\$25o\$o\$b24o\$3bo2bo2bo2bo2bo2bo2bo2bo!`

Keep in mind, that there are an infinity of lightspeed signals, very few of which could be potentially be turned.

These even simpler lightspeed signal might also be turnable:

`x = 29, y = 29, rule = B3/S233bo2bo2bo2bo2bo2bo2bo2bo2bo\$b27o\$o\$b27o\$28bo\$b2o2b24o\$ob2o\$b2o2b24o\$28bo\$b27o\$o\$b27o\$3bo2bo2bo2bo2bo2bo2bo2bo2bo4\$3bo2bo2bo2bo2bo2bo2bo2bo2bo\$b27o\$o\$b27o\$28bo\$b3o2b23o\$o2bo\$b3o2b23o\$28bo\$b27o\$o\$b27o\$3bo2bo2bo2bo2bo2bo2bo2bo2bo!`

gmc_nxtman

Posts: 1147
Joined: May 26th, 2015, 7:20 pm

### Re: The Omniperiodicity Problem

dvgrn wrote:Anywhere above 2^40 cases is pretty difficult -- that's a trillion, which is apgnano/Catagolue territory -- and we're talking a lot more than 2^40 sets of 2^40 cases. So we need an algorithm clever enough to remove a factor of something like 2^70 from this search problem...!

No such algorithm can exist. We are better off getting supercomputer time and running dr or Bellman to find stabilisers for what partial results we have. It may be easier to disprove omniperiodicity.
Princess of Science, Parcly Taxel

Freywa

Posts: 556
Joined: June 23rd, 2011, 3:20 am
Location: Singapore

### Re: The Omniperiodicity Problem

gmc_nxtman wrote:This is close to a 180 signal turner:

`x = 26, y = 11, rule = B3/S23bo2bo2bo2bo2bo2bo2bo2bo\$b24o\$25bo\$b25o\$o4bo\$5o4b17o\$6bo18bo\$25o\$o\$b24o\$3bo2bo2bo2bo2bo2bo2bo2bo!`

Not a potentially useful 180-degree turner for the omniperiodicity problem, though, is it? To make a p[19|23|34|38|41] oscillator we'd have to be able to add several signals to one conduit, so a 180-degree turner would have to be offset so the return signal could travel back on a separate wire.

Freywa wrote:We are better off getting supercomputer time and running dr or Bellman to find stabilisers for what partial results we have.

Yeah, I've been really curious to see what a high-performing computing cluster could do with this problem. I had access to a system with several thousand cores for a while last year, but sadly had too many other things to do to spend much time playing around with it.

I did figure out that the cluster was just about powerful enough to brute-force the discovery of the loafer, and every other spaceship of any speed that's the same size or smaller: there are "only" 2^53 arrangements of cells in a triangle 10 cells on a side, with the corner cells removed... a few thousand modified copies of apgnano running on a cluster could classify every possible object inside that snub triangle, so we'd get a lot of loafers and who knows what else besides.

At 2000 soups per minute per core a completely brute-force method would take about ten years on 2000 cores (!) That could be cut down a lot more with symmetries and so on... and most of those bit patterns aren't very challenging soups, after all. If we were really just looking for spaceships instead of cataloguing everything, we could cut off several orders of magnitude I would think.

But I don't know whether a modified dr or a special-purpose Bellman would be a better choice for hunting for a 2c/3 elbow, or a 2c/3 period doubler/pulse divider.

Freywa wrote:It may be easier to disprove omniperiodicity.

I hope not, since it's clearly impossible to prove the non-existence of those five oscillators. As far as I can see, any or all of those periods could perfectly well pop out of a RandomAgar or symmetrical Catagolue soup search tomorrow.

You'd have to show mathematically that at least one of those periods could not possibly be attained by a B3/S23 pattern of any size. A proof of non-omniperiodicity would have to give good reasons why no 2c/3 elbow can possibly exist -- along with an infinite number of other possible objects, like 5c/9 elbows.

Even if you somehow knew that no 2c/3 elbow exists inside a 20x20 bounding box, the jury would still be out on 20x21...!

dvgrn
Moderator

Posts: 5703
Joined: May 17th, 2009, 11:00 pm

### Re: The Omniperiodicity Problem

dvgrn wrote:At 2000 soups per minute per core a completely brute-force method would take about ten years on 2000 cores (!) That could be cut down a lot more with symmetries and so on... and most of those bit patterns aren't very challenging soups, after all. If we were really just looking for spaceships instead of cataloguing everything, we could cut off several orders of magnitude I would think.
Yep... I think modifying code to something similar to gsearch would make it faster, I think. (Or maybe one can use unmodified gsearch... Depending on whether gsearch's life iterator is as fast as apgnano's.)
Best wishes to you, Scorbie

Scorbie

Posts: 1379
Joined: December 7th, 2013, 1:05 am

### Re: The Omniperiodicity Problem

Freywa wrote:It may be easier to disprove omniperiodicity.

Finding a proof of lack of omniperiodicity isn't completely impossible, but going this route would be incredibly, incredibly hard, far harder than searching for the existence of them in the first place, as well as still requiring a ton of this sort of searching anyway - by which point you may have already found what you've been attempting to prove impossible.
Sphenocorona

Posts: 480
Joined: April 9th, 2013, 11:03 pm

### Re: The Omniperiodicity Problem

Sphenocorona wrote:Finding a proof of lack of omniperiodicity isn't completely impossible

There is a systematic way to convert any Pi^0_2 statement to a cellular automaton, such that the cellular automaton is omniperiodic if and only if the Pi^0_2 statement is true. Also, asking whether a cellular automaton is omniperiodic is easily expressed as a Pi^0_2 statement, so these two problems are genuinely equivalent.

Fortunately, for B3/S23 there are only finitely many periods for which the existence of oscillators is unknown, so this is only a Sigma^0_1 statement. Resolving an arbitrary Sigma^0_1 statement is equivalent to solving the halting problem for an arbitrary Turing machine. This is still undecidable, but not nearly as undecidable as resolving an arbitrary Pi^0_2 statement -- so you have more of a chance of victory.

The notation is explained in https://en.wikipedia.org/wiki/Arithmetical_hierarchy if you haven't encountered it before.
What do you do with ill crystallographers? Take them to the mono-clinic!

calcyman

Posts: 2054
Joined: June 1st, 2009, 4:32 pm

### Re: The Omniperiodicity Problem

calcyman wrote:Fortunately, for B3/S23 there are only finitely many periods for which the existence of oscillators is unknown, so this is only a Sigma^0_1 statement. Resolving an arbitrary Sigma^0_1 statement is equivalent to solving the halting problem for an arbitrary Turing machine. This is still undecidable, but not nearly as undecidable as resolving an arbitrary Pi^0_2 statement -- so you have more of a chance of victory.

I'm not so theoretically-minded, I guess. One infinitesimal chance of victory looks about the same as another. @Calcyman, it sounds like you're comparing "one chance in aleph_0" to "one chance in aleph_1" -- metaphorically speaking, of course.

In other words, B3/S23 is so clearly omniperiodic that it seems like a waste of time to even think about trying to prove the opposite. Any such proof would exceedingly complex, headache-inducingly subtle, and ridiculously long... and it's a safe bet it would contain at least one error somewhere.

Let's have another look at how close we are to proving omniperiodicity right now:

`x = 137, y = 84, rule = LifeHistory15.2A\$14.3A\$14.2A.A\$15.3A\$16.A11\$7.A2.A.A5.A.A\$5.A.A3.2A6.2A84.2A\$6.2A3.A7.A85.2A\$109.A\$105.5A\$104.A\$104.6A\$20.A2.A84.A.A.2A\$18.A.A2.A.A76.7A.A.2A\$19.2A2.2A71.2A3.A6.A.A\$95.A2.A2.5A2.A.A\$94.A.3A7.A.2A\$26.2A62.2A.A.A3.5A2.A\$21.2A.A2.A63.A.A3.A6.A.A\$21.2A.3A64.A.A2.6A2.A.2A\$89.A.A.A.A6.A.A\$21.6A61.A.2A2.A.4A2.A.A\$20.A6.A60.A3.A.A.A3.A.2A\$20.A2.5A58.2A.2A.A.A3.A.A\$18.A.A.A7.A56.A.A2.A.4A.A.3A\$18.2A2.A2.6A55.A2.A.A7.A3.A\$22.A.A62.3A2.8A\$2A4.A14.2A.A2.6A57.A.A\$.2A3.2A16.A.A6.A55.2A.A2.7A\$A4.A.A16.A.A2.5A54.A2.2A.A7.A\$25.2A.A7.A51.2A4.A2.6A\$28.A2.6A57.A.A\$28.A.A62.2A.A2.6A\$3.2A22.2A.A2.6A57.A.A6.A\$2.A.A25.A.A6.A56.A.A2.5A\$4.A25.A.A2.5A57.2A.A7.A\$31.2A.A7.A57.A2.6A\$34.A2.6A57.A.A\$34.A.A62.2A.A2.6A\$33.2A.A2.6A57.A.A6.A.2A\$36.A.A6.A.2A53.A.A2.5A.2A\$36.A.A2.5A.2A54.2A.A\$37.2A.A65.A2.6A19.2A\$40.A2.6A19.2A36.A.A5.A19.A\$40.A.A5.A19.A36.2A.A2.3A18.A.A\$39.2A.A2.3A18.A.A39.A.A5.2A14.2A\$42.A.A5.2A14.2A40.A2.A4.2A\$42.A2.A4.2A57.2A9.A\$43.2A9.A61.2A2.3A\$50.2A2.3A59.2A5.A\$50.2A5.A64.2A\$56.2A2\$135.2A\$69.2A63.A.A\$68.A.A64.A\$69.A5\$115.2A10.D2.2D\$49.2A10.D2.2D49.2A11.3D\$49.2A11.3D64.D\$63.D2\$124.C.2C\$58.C.2C60.3C.2C\$56.3C.2C59.C\$55.C66.3C.2C\$56.3C.2C62.C.C\$58.C.C63.C.C\$58.C.C64.C\$59.C!`

We already have ways to get a 2c/3 signal started (though it's certainly not repeatable at p19 yet).

The same wire can carry either a single or a double signal, and we already have a way to turn the corner while doubling the signal.

We even already have a mechanism that can accept either a single or a double signal, and turn either one into the exact same spark.

It's clearly not easy to turn that spark, or one of its variants, back into a single 2c/3 signal again. But 'dr' searches have only been done up to a certain number of active cells, with recovery within a certain number of ticks.

Every time Moore's Law lets us bump up the search limits a little bit, we find more interesting stuff. Here's a stamp collection of fizzles from 2008, to give a sense of the range of possibilities. (@Calcyman, this should look suspiciously familiar to you, but I'm not sure everyone else here has a copy handy...!)

`#C Top-left section: 2c/3 and corresponding TL fizzles.#C Bottom-left section: various reactions that can't be categorised#C anywhere else.#C Right section: Up to 7 different variations for each reaction.#C Each row#C corresponds to a different key reaction, and each column#C corresponds to a different input.#C#C A) Small TL eater#C B) Medium TL eater#C C) Large TL eater#C D) Perpendicular TL eater (with split)#C E) Small 2c/3 fizzle#C F) Medium 2c/3 fizzle#C G) Large 2c/3 fizzle#C#C Adam P. Goucher, 23 November 2008x = 578, y = 336, rule = 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The LifeViewer is pretty good for looking at these -- space bar and 'b' keyboard shortcuts highly recommended. Look at how far some of those signals can travel before they fizzle out -- including across the 2c/3 wire-end boundary! And most of those paths, not counting initial traffic-light stages, could easily accept another signal within 19 ticks.

All someone has to do to prove omniperiodicity is find a way to adapt any one of these, or any similar traveling signal, to allow them to be chained linearly at any angle, and also to turn a 90-degree corner. That's an awful lot of possibilities that would all have to be ruled out somehow before a proof of the nonexistence of p19 oscillators would be within reach.

dvgrn
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Joined: May 17th, 2009, 11:00 pm