No, the domain of U need not be finite. (1) The paper explicitly says that U need not be computable. (2) The model of computation implicit in the paper permits computable functions to have infinite domain (it’s basically Turing machines). (3) The way the paper proposes that a computable U should be implemented involves only looking at a finite amount of the information in the argument of U. (Kinda.)
But yes: the paper’s model does permit a non-computable decision process. (Neither does it require that U be uncomputable.)
There is a countable infinite of computable functions. (Because there is a countable infinity of programs, because a program is a finite string or something of the kind.) An environment is defined to be a function from sequences of actions to sequences of perceptions, and there are uncountably many (more precisely: continuum-many) of those. So the situation envisaged by the paper is as follows: our agent assigns a nonzero probability to each computable environment; if the sum of those probabilities is less than 1, it also assigns a nonzero probability to some sets (though not necessarily any singleton sets) of other possible environments.
Suppose you get an expression for the expected utility in some situation, and it takes the form of an infinite series: 1 − 1 + 1 − 1 + 1 − 1 + 1 - … (etc.). Then you might want to say “well, there’s no actual expected utility, but we can say that the expected utility is some kind of fuzzy thing taking values between 0 and 1”—but unless there’s some reason why the order of summation you used there is the only possible one, you could rearrange the terms a little (swap terms 2 and 3, terms 6 and 7, etc.) and get 1 + 1 − 1 − 1 + 1 + 1 − 1 − 1 + … whose partial sums vary between 0 and 2 instead. Or you could rearrange the terms more drastically and get partial sums that become arbitrarily large-and-positive and arbitrarily large-and-negative. Or you could just swap the first two terms, getting a series whose values vary between −1 and +1 instead of between 0 and +1.
Perhaps it’s possible to construct some theory of “fuzzy convergence” for series like this when there’s a single right order to add up the terms in. But it’s far from obvious that you could base any usable sort of decision theory on utilities defined as the sums of such series. In any case, in this instance there’s one term for each possible computable environment, and it seems very obvious that there isn’t a canonical way to order those, and that different orderings will give substantially different sequences of partial sums.
I was unhappy with having an uncomputable decision process, but I realize now that isn’t important—we can approximate it, but only if it exists. The paper is about whether it exists atall.
No, the domain of U need not be finite. (1) The paper explicitly says that U need not be computable. (2) The model of computation implicit in the paper permits computable functions to have infinite domain (it’s basically Turing machines). (3) The way the paper proposes that a computable U should be implemented involves only looking at a finite amount of the information in the argument of U. (Kinda.)
But yes: the paper’s model does permit a non-computable decision process. (Neither does it require that U be uncomputable.)
There is a countable infinite of computable functions. (Because there is a countable infinity of programs, because a program is a finite string or something of the kind.) An environment is defined to be a function from sequences of actions to sequences of perceptions, and there are uncountably many (more precisely: continuum-many) of those. So the situation envisaged by the paper is as follows: our agent assigns a nonzero probability to each computable environment; if the sum of those probabilities is less than 1, it also assigns a nonzero probability to some sets (though not necessarily any singleton sets) of other possible environments.
Suppose you get an expression for the expected utility in some situation, and it takes the form of an infinite series: 1 − 1 + 1 − 1 + 1 − 1 + 1 - … (etc.). Then you might want to say “well, there’s no actual expected utility, but we can say that the expected utility is some kind of fuzzy thing taking values between 0 and 1”—but unless there’s some reason why the order of summation you used there is the only possible one, you could rearrange the terms a little (swap terms 2 and 3, terms 6 and 7, etc.) and get 1 + 1 − 1 − 1 + 1 + 1 − 1 − 1 + … whose partial sums vary between 0 and 2 instead. Or you could rearrange the terms more drastically and get partial sums that become arbitrarily large-and-positive and arbitrarily large-and-negative. Or you could just swap the first two terms, getting a series whose values vary between −1 and +1 instead of between 0 and +1.
Perhaps it’s possible to construct some theory of “fuzzy convergence” for series like this when there’s a single right order to add up the terms in. But it’s far from obvious that you could base any usable sort of decision theory on utilities defined as the sums of such series. In any case, in this instance there’s one term for each possible computable environment, and it seems very obvious that there isn’t a canonical way to order those, and that different orderings will give substantially different sequences of partial sums.
I was unhappy with having an uncomputable decision process, but I realize now that isn’t important—we can approximate it, but only if it exists. The paper is about whether it exists atall.