The function U in PdB’s paper doesn’t take integers as arguments, it takes infinite sequences of “perceptions”. Provided there are at least two possible perceptions at each time step, there are uncountably many such sequences. How are you proposing to change the model to make the domain of U countable, while still making it be a model of agents acting in time?
The domain of U must not only be countable; it must be finite.
Remember, an agent needs to compute this sum once for every possible action!
Both the number of possible worlds, and the number of possible actions, need to be finite; or the utility function needs to be solvable analytically (unlikely); or else the agent will take an infinite amount of time to make a decision.
So, the model of an agent acting in time, where we compute utility at each timestep in the future, does not seem to produce a computable decision process. Is that right?
I think you are mistaken about what those requirements are. What he needs is a probability measure on the space of environments; he doesn’t insist that each individual possible environment have nonzero probability.
You’re right. Let’s try to figure out how many possible environments there are.
“Let p:H->R be our probability distribution on the hypothesis set. We require that there exist some computable function p-bar:N->Q such that for all psi_n in S_I, 0 < p-bar(n) ⇐ p(psi_n)”.
p(psi_n) must be non-zero for all psi_n in S_I. S is the set of mu-recursive functions on a single argument. A mu-recursive function is one that can be computed by Turing machines. S_I is the set of computable environments f such that for every input that has been tested, f(i) = h(i), where h is the true environment.
How large is S_I? It could be up to the size of S. Is the number of computable functions less than R (2^N)?
In any case, if you try to sum an infinite series with infinitely many terms of absolute value > 1, then the range of values in your partial sums depends on the order of summation. In particular, even if the U(k) were bounded, that wouldn’t enable you to say things like “the expected utility is a divergent series but its value oscillates only between −13 and +56” unless there were One True Order of summation for the terms of the series. Which I can’t see that there is or could be.
If I construct a countable series so as to have a 1-1 pairing between terms with value 1 and terms with value −1, does that do no good? e.g. U(n) = 1 if n is even, −1 if n is odd.
Is there a way to enter subscripts and superscripts in this thing?
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.
The domain of U must not only be countable; it must be finite. Remember, an agent needs to compute this sum once for every possible action! Both the number of possible worlds, and the number of possible actions, need to be finite; or the utility function needs to be solvable analytically (unlikely); or else the agent will take an infinite amount of time to make a decision.
So, the model of an agent acting in time, where we compute utility at each timestep in the future, does not seem to produce a computable decision process. Is that right?
You’re right. Let’s try to figure out how many possible environments there are.
“Let p:H->R be our probability distribution on the hypothesis set. We require that there exist some computable function p-bar:N->Q such that for all psi_n in S_I, 0 < p-bar(n) ⇐ p(psi_n)”.
p(psi_n) must be non-zero for all psi_n in S_I. S is the set of mu-recursive functions on a single argument. A mu-recursive function is one that can be computed by Turing machines. S_I is the set of computable environments f such that for every input that has been tested, f(i) = h(i), where h is the true environment.
How large is S_I? It could be up to the size of S. Is the number of computable functions less than R (2^N)?
If I construct a countable series so as to have a 1-1 pairing between terms with value 1 and terms with value −1, does that do no good? e.g. U(n) = 1 if n is even, −1 if n is odd.
Is there a way to enter subscripts and superscripts in this thing?
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.
There are numerical methods for estimating integrals, you know.