While privileging polynomial time descriptions might help, it isn’t clear how one should distinguish between two Turing machines, one which runs in very short time (say degree 2) and another that is long (say degree 20) but the degree 2 has a much smaller number of states. Which one of those is simpler?
I have given one answer in the post, selected somewhat arbitrarily (but with the desirable property of working more or less correctly for physical theories we have seen so far). I think basing things on circuits rather than TM’s is clearly a better start.
For one thing, I don’t know what “degree 2” means for you. Is that in terms of the size of the universe? The # of the current observation? Both of these have significant problems (the first fails if the universe gets padded with junk, the second fails if your observations don’t begin at the beginning of time).
For a second thing, Turing machines are a horrible way to measure the degree of polynomial run-times. There is basically no reason to believe that you get out a reasonable thing, unlike with circuits. Of course, using circuits introduces other issues, which require new solutions. I suggest one.
Of course, the post just contains my best try. I would be happy to see a better go at it, but I would be very surprised if it was based on any inherently uniform model of computation which is known today.
I have given one answer in the post, selected somewhat arbitrarily (but with the desirable property of working more or less correctly for physical theories we have seen so far). I think basing things on circuits rather than TM’s is clearly a better start.
For one thing, I don’t know what “degree 2” means for you. Is that in terms of the size of the universe? The # of the current observation? Both of these have significant problems (the first fails if the universe gets padded with junk, the second fails if your observations don’t begin at the beginning of time).
For a second thing, Turing machines are a horrible way to measure the degree of polynomial run-times. There is basically no reason to believe that you get out a reasonable thing, unlike with circuits. Of course, using circuits introduces other issues, which require new solutions. I suggest one.
Of course, the post just contains my best try. I would be happy to see a better go at it, but I would be very surprised if it was based on any inherently uniform model of computation which is known today.
I was thinking in terms of number of observations. I agree that this has problems.