Possible proof relating to slow synthesis.

[gmc_nxtman]

I may have found a way to synthesise any slow salvo using a set, finite amount of gliders.

One could synthesise a device that counts in quaternary. Each '0' could represent an elbow push operation, each '1' could be a pull, each '2' could fire a glider of one color, and '3' could fire a glider of the opposite color. That would mean that every possible sequence of push, pull, fire1, and fire2 operations would eventually appear. Then add a toggleable gate that allows the gliders to escape. Synthesize the base device using n gliders, and two more gliders to turn the device on and off, and some more to destroy it.

Would that be sufficient to prove that any slow salvo could be synthesized using only a finite amount of gliders?

[dvgrn]

I may have found a way to synthesise any slow salvo using a set, finite amount of gliders.

One could synthesise a device that counts in quaternary. Each '0' could represent an elbow push operation, each '1' could be a pull, each '2' could fire a glider of one color, and '3' could fire a glider of the opposite color. That would mean that every possible sequence of push, pull, fire1, and fire2 operations would eventually appear. Then add a toggleable gate that allows the gliders to escape. Synthesize the base device using n gliders, and two more gliders to turn the device on and off, and some more to destroy it.

Would that be sufficient to prove that any slow salvo could be synthesized using only a finite amount of gliders?

Absolutely!

The corollary seems to be that there's a maximum number of gliders N needed to construct any glider-constructible pattern, no matter how complicated -- and we could probably show by construction that N is no more than four digits long.

-- It's kind of an odd definition of "glider recipe", though, with all but two of the gliders always exactly the same, and the last two so far away from the others that it would require a special simulator to actually see the construction happening -- Golly HashLife wouldn't be anywhere near good enough.

It's especially weird that a recipe for just one half-baked knightship would have the same cost in gliders as the recipe for a fleet of a million Gemini spaceships.

EDIT: I'm not sure if your design -- count to N, turn on output gate, construct, turn off, destroy -- will turn out to be any cheaper to build than the current fixed-cost constructor design, which encodes an arbitrarily long recipe with the position of a single block. Seems like the two methods are fairly comparable, especially in the sense that both would produce construction recipes far too big and slow to be simulated by Golly, at least without some serious sneaky shortcuts.