Yes, I’m a big fan of the Entropic Uncertainty Principle. One thing to note about it is that the definition of entropy only uses the measure space structure of the reals, whereas the definition of variance also uses the metric on the reals as well. So Heisenberg’s principle uses *more* structure to say *less* stuff. And it’s not like the extra structure is merely redundant either. You can say useful stuff using the metric structure, like Hardy’s Uncertainty Principle. So Heisenberg’s version is taking useful information and then just throwing it away.

I’d almost support teaching the Entropic Uncertainty Principle instead of Heisenberg’s to students first learning quantum theory. But unfortunately its proof is much harder. And students are generally more familiar with variance than entropy.

With regards to Eliezer’s original point, the distibutions |f|^2 and |g|^2 don’t actually describe our uncertainty about anything. We have perfect knowledge of the wavefunction; there is no uncertainty. I suppose you could say that H(|f|^2) and H(|g|^2) quantify the uncertainty you would have if you were to measure the position and momentum (in Eliezer’s point of view this would be indexical uncertainty about which world we were in), although you can’t perform both of these measurements on the same particle.

Some related discussions: 1. https://www.conwaylife.com/forums/viewtopic.php?f=2&t=979 2. https://www.conwaylife.com/forums/viewtopic.php?f=7&t=2877 3. https://www.conwaylife.com/forums/viewtopic.php?p=86140#p86140

My own thoughts.

Patterns in GoL are generally not robust. Typically changing anything will cause the whole pattern to disintegrate in a catastrophic explosion and revert to the usual ‘ash’ of randomly placed small still lifes and oscillators along with some escaping gliders.

The pattern Eater 2 can eat gliders along 4 adjacent lanes.

The Highway Robber can eat gliders travelling along a lane right at the edge of the pattern, such that gliders on the next lane pass by unaffected. So one can use several staggered highway robbers to make a wall which eats any gliders coming at it from a given direction along multiple adjacent lanes. The wall will be very oblique and will fail if two gliders come in on the same lane too close together.

The block is robust to deleting any one of its live cells, but is not robust to placing single live cells next to it.

The maximum speed at which GoL patterns can propagate into empty space is 1 cell every 2 generations, measured in the L_1 norm. Spaceships which travel at this speed limit (such as the glider, XWSS, and Sir Robin) are therefore robust to things happening behind them, in the sense that nothing can catch up with them.

It’s long been hypothesised that it should be possible to make a pattern which can eat any single incoming glider. My method for doing this would be to design a wall around the pattern which is designed to fall apart in a predictable way whenever it is hit by a glider. This collapse would then trigger construction machinery on the interior of the pattern that rebuilds the wall. The trick would be to make sure that the collapse of the wall didn’t emit any escaping gliders and whose debris didn’t depend on where the glider hit it. That way the construction machinery would have a reliable blank slate on which to rebuild.

If one did have a pattern with the above property that it could eat any glider that hit it, one could then arrange several copies of this pattern in a ring around any other pattern to make it safe from any single glider. Of course such a pattern would not be safe against other perturbations, and the recovery time would be so slow that it would not even be safe against two gliders a million generations apart.

It’s an open problem whether there exists a pattern that recovers if any single cell is toggled.

I think the most promising approach is the recently discovered methods in this thread. These methods are designed to clear large areas of the random ash that Life tends to evolve into. One could use these methods to create a machine that clears the area around itself and then builds copies of itself into the cleared space. As this repeated it would spread copies of itself across the grid. The replicators could build walls of random ash between themselves and their children, so that if one of them explodes the explosion does not spread to all copies. If one of these copies hit something it couldn’t deal with, it would explode (hopefully also destroying the obstruction) and then be replaced by a new child of the replicators behind it. Thus such a pattern would be truly robust. If one wanted the pattern to be robust and not spread, one could make every copy keep track of its coordinates relative to the first copy, and not replicate if it was outside a certain distance. I think this would produce what you desire: a bounded pattern that is robust to many of the contexts it could be placed in. However, there are many details still to be worked out. The main problem is that the above cleaning methods are not guaranteed to work on every arrangement of ash. So the question is whether they can clear a large enough area before they hit something that makes them explode. We only need each replicator to succeed often enough that their population growth rate is positive.