From skimming the paper, "continuity" seems to be a deceptively strong assumption here, and the claim "this is a physical system so it's continuous" is doing a lot of heavy lifting. Unless I'm misunderstanding the example in the first paragraph, "move towards point 1 at a constant pace" can't be represented as a continuous A_t(x) for t above some finite threshold.
Consider the following decision: If x < 0.5 go left at a constant speed till the hay is reached, else go right at a constant speed until the hay is reached.
Then A(t,x) is clearly not continuous in x, and we can easily bound the time required to make a decision. The nuance here is that we have to somehow be able to distinguish 0.5 - eps from 0.5 for very small epsilon.
Edit: on further thought, suppose we had a device which measured reasonably well. More precisely it tells us x < 0.5 if x is actually <= 0.5 - c, it tells us x >= 0.5 if x > 0.5 + c, and tells us it is unsure otherwise. We do not know c, but it is deterministic (and hopefully reasonably small). Then we can decide to go left if it tells us x < 0.5, and right if it tells us unsure or that x >= 0.5.
> The nuance here is that we have to somehow be able to distinguish 0.5 - eps from 0.5 for very small epsilon.
If you ignore the problem then the problem indeed goes away. The need for distinguishing very small epsilon exists because of the continuity assumption, and because of the continuity assumption you can't really solve it either.
> Then we can decide to go left if it tells us x < 0.5, and right if it tells us unsure or that x >= 0.5.
Now you just moved the problem to deciding at which point you are unsure. As long as there is a decision to take the issue persists, it's only if you always go left (or always go right) that the issue doesn't exist.
> As long as there is a decision to take the issue persists, it's only if you always go left (or always go right) that the issue doesn't exist.
That's not true. There are two problems.
1. The math is set up to rule out a lot of obvious solutions. Write out A_t(x) for the decision rule "always walk right at a constant rate." For any rate, A_t(x) becomes a spike at 1 for t > some threshold that depends on the rate.
2. The math rules out randomness. A_t(x) can't be defined for the decision rule, "flip a coin and walk right half the time, left the other half," because you wind up at 0 with probability 1/2 and 1 with probability 1/2 for t > some threshold and the problem is defined so A_t(x) must resolve to a single point.
Once you do that, the only solutions left are kind of mind fucks. But you can't draw any general conclusions from it.
