These points are well taken. I agree re: log-score. We were trying to compare to the most straightforward/naive reward maximization setup. Like in a game where you get +1 point for correct answers and −1 point for incorrect. But I take your point that other scores lead to different (in this case better) results.
Re: cheating. Yes! That is correct. It was too much to explain in this post, but ultimately we would like to argue that we can augment Turing Machines with a heat source (ie a source of entropy) such that it can generate random bits. Under that type of setup the “random number generator” becomes much simpler/easier than having to use a deterministic algorithm. In addition, the argument will go, this augmented Turing machine is in better correspondence with natural systems that we want to understand as computing.
Which leads to your last point, which I think is very fundamental. I disagree a little bit, in a specific sense. While it is true that “randomness” comes from a specific type of “laziness,” I think it’s equally true that this laziness actually confers computational powers of a certain sort. For now, I’ll just say that this has to do with abstraction and uncertainty, and leave the explanation of that for another post.
These points are well taken. I agree re: log-score. We were trying to compare to the most straightforward/naive reward maximization setup. Like in a game where you get +1 point for correct answers and −1 point for incorrect. But I take your point that other scores lead to different (in this case better) results.
Re: cheating. Yes! That is correct. It was too much to explain in this post, but ultimately we would like to argue that we can augment Turing Machines with a heat source (ie a source of entropy) such that it can generate random bits. Under that type of setup the “random number generator” becomes much simpler/easier than having to use a deterministic algorithm. In addition, the argument will go, this augmented Turing machine is in better correspondence with natural systems that we want to understand as computing.
Which leads to your last point, which I think is very fundamental. I disagree a little bit, in a specific sense. While it is true that “randomness” comes from a specific type of “laziness,” I think it’s equally true that this laziness actually confers computational powers of a certain sort. For now, I’ll just say that this has to do with abstraction and uncertainty, and leave the explanation of that for another post.