I didn’t miss your point; instead, I’ve tried to explain your point to you, and perhaps didn’t do a really good job. You’re confusing two different things which are both spelled out in the paper you referenced on page 1120, but in your comment you linked me to you’ve only recounted the first one, perhaps thinking that the second one is the same thing, which it is not.
The first thing is setting up probabilities over programs of a given fixed length L, so that padding helps us take into account also shorter programs. In the paper, this is the paragraph starting “This sum is still only over programs of length ℓ...”. This is what you talked about in your other comment, and this is what I talked about in my first paragraph of the reply above.
The second thing is a way of setting up probabilities over programs of any length without bound, using a UTM-with-random-noise model as justification for the formula. This is an inherently different way of assigning probabilities, where in particular we do not use padding, but on the contrary only take into account minimal programs. This is what the paper talks about in the next two paragraphs, the second of which, starting “This is a highly technical explanation...” explains the UTM-with-random-noise. I talk about this second thing in the third paragraph of my reply above.
Hopefully I made it clearer. If you still think I missed your point, let’s leave it at that.
I didn’t miss your point; instead, I’ve tried to explain your point to you, and perhaps didn’t do a really good job. You’re confusing two different things which are both spelled out in the paper you referenced on page 1120, but in your comment you linked me to you’ve only recounted the first one, perhaps thinking that the second one is the same thing, which it is not.
The first thing is setting up probabilities over programs of a given fixed length L, so that padding helps us take into account also shorter programs. In the paper, this is the paragraph starting “This sum is still only over programs of length ℓ...”. This is what you talked about in your other comment, and this is what I talked about in my first paragraph of the reply above.
The second thing is a way of setting up probabilities over programs of any length without bound, using a UTM-with-random-noise model as justification for the formula. This is an inherently different way of assigning probabilities, where in particular we do not use padding, but on the contrary only take into account minimal programs. This is what the paper talks about in the next two paragraphs, the second of which, starting “This is a highly technical explanation...” explains the UTM-with-random-noise. I talk about this second thing in the third paragraph of my reply above.
Hopefully I made it clearer. If you still think I missed your point, let’s leave it at that.