Your points about the difficulty of getting uniform results in this framework are interesting. My inclination is to regard this as a failure of the framework. The LI paper introduced the idea of “e.c. traders,” and the goal of not being exploitable (in some sense) by such traders; these weren’t well-established notions which the paper simply proven some new theorems about. So they are up for critique as much as anything else in the paper (indeed, they are the only things up for critique, since I’m not disputing that the theorems themselves follow from the premises). And if our chosen framework only lets us prove something that is too weak, while leaving the most obvious strengthening clearly out of reach, that suggests we are not looking at the problem (the philosophical problem, about how to think about logical induction) at the right level of “resolution.”
As I said to Vadim earlier, I am not necessarily pessimistic about the performance of some (faster?) version of LIA with a “good” ordering for T^k. But if such a thing were to work, it would be for reasons above and beyond satisfying the LI criterion, and I wouldn’t expect the LI criterion to do much work in illuminating its success. (It might serve as a sanity check—too weak, but its negation would be bad—but it might not end up being the kind of sanity check we want, i.e. the failures it does not permit might be just those required for good and/or fast finite-time performance.
I don’t necessarily think this is likely, but I won’t know if it’s true or not until the hypothetical work on LIA-like algorithms is done.)
Your points about the difficulty of getting uniform results in this framework are interesting. My inclination is to regard this as a failure of the framework. The LI paper introduced the idea of “e.c. traders,” and the goal of not being exploitable (in some sense) by such traders; these weren’t well-established notions which the paper simply proven some new theorems about. So they are up for critique as much as anything else in the paper (indeed, they are the only things up for critique, since I’m not disputing that the theorems themselves follow from the premises). And if our chosen framework only lets us prove something that is too weak, while leaving the most obvious strengthening clearly out of reach, that suggests we are not looking at the problem (the philosophical problem, about how to think about logical induction) at the right level of “resolution.”
As I said to Vadim earlier, I am not necessarily pessimistic about the performance of some (faster?) version of LIA with a “good” ordering for T^k. But if such a thing were to work, it would be for reasons above and beyond satisfying the LI criterion, and I wouldn’t expect the LI criterion to do much work in illuminating its success. (It might serve as a sanity check—too weak, but its negation would be bad—but it might not end up being the kind of sanity check we want, i.e. the failures it does not permit might be just those required for good and/or fast finite-time performance. I don’t necessarily think this is likely, but I won’t know if it’s true or not until the hypothetical work on LIA-like algorithms is done.)