sorrge wrote:I'm skeptical about the idea of filling the space with something. An unbounded chain reaction in this field cannot be excluded, I think. At least I don't see how a bound can be proven.
Oh, I certainly agree. Let's say we look at random pairs of gliders striking the left edge of thousand-cell-wide sparse random fields of blocks, and measure the maximum distance D that the resulting reaction travels rightward before dying out.
If you run enough experiments with enough different random fields, eventually a reaction will get all the way across the thousand-cell-wide "minefield". There's very likely a nice clean power law that you can work out from a few million experiments, that will give you a very good estimate of how many glider pairs you'll have to throw at the block field before there's a 50% chance that a chain reaction will make it across a width-N minefield, for any width N and any specific random density and content of the minefield.
If the probabilities get exponentially smaller as the minefield gets wider, as I'm claiming, then you can pick a value of N that gives a probability of permeability that's safely lower than any chosen finite value P ... but it will never go quite to zero.
Silly Thought Experiment
Now, as we start looking at really ridiculously low values of P, we start having to deal with the inconvenient fact that a glider can perfectly well hit a patch of random debris and accidentally produce a working glider gun.
To date,
Catagolue has generated 200 trillion objects from random asymmetric soup, and this has never happened. But on the other hand it's fairly trivial to set up some random-looking blocks that are really secretly a seed for a gun -- a
period-156 multi-gun, maybe. The most likely configuration may still be so improbable that we'll never see anything like this come out of an asymmetric soup in Catagolue, but in theory it's bound to happen eventually.
This might cause trouble with that nice clean power law I mentioned, depending on the intricate details of how different period glider streams interact with whatever is in the minefield.
Maybe even more painful, there are quadratic-growth patterns that could randomly get started on the inside edge of your far-apart layers, that start up a series of glider-producing switch engines. Again, they won't be a problem if you just don't leave any empty space between your layers (!).
Even (Much) More Ridiculous...
But probably we're just not being imaginative enough. There's a non-zero probability that a couple of gliders (or whatever) could hit a minefield and transform it into a glider loop that happens to repeatedly shoot a single-channel glider stream off to the side, that happens to be a construction recipe for a glider gun, spaceship gun, or whatever.
Now you have not just one stream of spaceships attacking your structure, but an increasing number of them as time goes on... and the guns could be constructed with some sparse block shielding, so they'd be less likely to be destroyed immediately by random return gliders from whatever their gliders hit... and the original single-channel constructor that's causing all this trouble would be around a corner out of the way of all the excitement, so it might well keep causing trouble for quite a long time.
If your calculations don't take this possibility into account -- along with an unlimited number of even less likely but correspondingly more evil mechanisms -- then you won't know
for sure that the sum of your hypothetical series actually diverges. In other words: there's an awful lot of really weird stuff out there
sorrge wrote:In your other post you wrote that an active repair mechanism is more vulnerable to attack than a passive barrier. Let's consider a defense against gliders. The active defense can be broken only if another attack comes while the damage from the previous attack is not yet repaired. But the passive defense is always broken on the second attack...
That's not quite accurate. It's theoretically possible that a second glider could cause an unpredictable chain reaction. But much more often the first glider just hits and destroys a single block, or moves it by (2,1), and in all those cases a sufficiently thick sparse-block passive defense is just about as good after the strike as before.
The high odds of cleanly absorbing a glider, with no interesting fireworks at all, are what make the passive defense about as good as we can get. Again, as soon as you start worrying about other spaceship types, all bets are off and a whole different analysis has to be done.