So you mention that one advantage of this method is that it can reason about things faster—because it uses less computational power.
One of his “desiderata”, his principles of construction, is that the robot gives equal plausibility assignments to logically equivalent statements, which gets us into all this trouble when we try to build a robot we can ask about theorems. But I’m keeping this desideratum; I’m building fully Jaynesian robots.
Unless I’m misunderstanding something, I don’t think you’ll be able to keep this with limited computational resources. All (ETA: syntactically) true statements will be assigned the same probability, which makes this kind of probability useless. And by hiding logical information, as you have, you are already breaking this.
I think a better way to think about this is as an update on new computations. (For simplicity, assume that N>2.) You start with the prior that P(Prime(N)|N=n) = 1/log(n). Then you compute (i.e. prove) from the definition of “prime” that all primes are odd, and express this as P(Odd(N)|Prime(P), N=n)=1. Similarly, compute that P(Odd(N)|N=n)=1/2. Then you can compute the Bayesian update. Any new logical information has to come from a computation of some kind.
So if you are given a large number, say 442537, you have a prior probability of it being prime at about 18%. You then compute that it is odd, and then update on this information, giving you a new probability of about 35%, etc…
Does this apply to Boolean logic to, in addition to deductions using the axioms of the system?
The archetype I’ve got in my head is Jaynes’s derivation of P(A or B) = P(A) + P(B) - P(A and B).
The first step is that P(A or B) = P(~(~A and ~B)). So if we’re updating on computations rather than facts, what we really have is only P(A or B|C) = P(A|C) + P(B|C) - P(A and B|C), where C is the computation that (A or B) is the same as ~(~A and ~B)).
Does that make sense, or is that a different kind of thing?
Unless I’m misunderstanding something, I don’t think you’ll be able to keep this with limited computational resources. All true statements will be assigned the same probability, which makes this kind of probability useless. And by hiding logical information, as you have, you are already breaking this.
No. It sounds like you imagine that the robot knows the axioms of number theory? It doesn’t.
The idea is that you’ve got some system you’re interested in, but the robot’s knowledge underdetermines that system. From the info you’ve fed it, it can’t prove all the true things about that system. So, there are things true about the system that it doesn’t assign probability of 1 to. One way of thinking about it is that the robot doesn’t know precisely what system you want it to think about. I mean, I’ve left out all facts about ordering for example, how’s the robot to know we’re even talking about numbers?
EDIT: however I’m realizing now that it still has to know boolean logic which means it assigns probabilities of 0 or 1 to the answers to 3-SAT problems, which are NP-complete. So, yeah, it’s still got useless 0⁄1 probabilities that you can’t calculate in reasonable time.
re: updating on computations, I have to give that more thought. It’s easy to respond to clarify my thoughts, and I didn’t want to keep you waiting unnecessarily, but it’ll take me more time to adapt them.
So you mention that one advantage of this method is that it can reason about things faster—because it uses less computational power.
Unless I’m misunderstanding something, I don’t think you’ll be able to keep this with limited computational resources. All (ETA: syntactically) true statements will be assigned the same probability, which makes this kind of probability useless. And by hiding logical information, as you have, you are already breaking this.
I think a better way to think about this is as an update on new computations. (For simplicity, assume that N>2.) You start with the prior that P(Prime(N)|N=n) = 1/log(n). Then you compute (i.e. prove) from the definition of “prime” that all primes are odd, and express this as P(Odd(N)|Prime(P), N=n)=1. Similarly, compute that P(Odd(N)|N=n)=1/2. Then you can compute the Bayesian update. Any new logical information has to come from a computation of some kind.
So if you are given a large number, say 442537, you have a prior probability of it being prime at about 18%. You then compute that it is odd, and then update on this information, giving you a new probability of about 35%, etc…
Does this seem to clarify the concept?
Yeah, this makes sense.
Does this apply to Boolean logic to, in addition to deductions using the axioms of the system?
The archetype I’ve got in my head is Jaynes’s derivation of P(A or B) = P(A) + P(B) - P(A and B).
The first step is that P(A or B) = P(~(~A and ~B)). So if we’re updating on computations rather than facts, what we really have is only P(A or B|C) = P(A|C) + P(B|C) - P(A and B|C), where C is the computation that (A or B) is the same as ~(~A and ~B)).
Does that make sense, or is that a different kind of thing?
No. It sounds like you imagine that the robot knows the axioms of number theory? It doesn’t.
The idea is that you’ve got some system you’re interested in, but the robot’s knowledge underdetermines that system. From the info you’ve fed it, it can’t prove all the true things about that system. So, there are things true about the system that it doesn’t assign probability of 1 to. One way of thinking about it is that the robot doesn’t know precisely what system you want it to think about. I mean, I’ve left out all facts about ordering for example, how’s the robot to know we’re even talking about numbers?
EDIT: however I’m realizing now that it still has to know boolean logic which means it assigns probabilities of 0 or 1 to the answers to 3-SAT problems, which are NP-complete. So, yeah, it’s still got useless 0⁄1 probabilities that you can’t calculate in reasonable time.
re: updating on computations, I have to give that more thought. It’s easy to respond to clarify my thoughts, and I didn’t want to keep you waiting unnecessarily, but it’ll take me more time to adapt them.