If your argument is, “if it is possible for humans to produce some (verbal or mechanical) output, then it is possible for a program/machine to produce that output”, then, that’s true I suppose?
I don’t see why you specified “finite depth boolean circuit”.
While it does seem like the number of states for a given region of space is bounded, I’m not sure how relevant this is. Not all possible functions from states to {0,1} (or to some larger discrete set) are implementable as some possible state, for cardinality reasons.
I guess maybe that’s why you mentioned the thing along the lines of “assume that some amount of wiggle room that is tolerated” ?
One thing you say is that the set of superintelligences is a subset of the set of finite-depth boolean circuits. Later, you say that a lookup table is implementable as a finite-depth boolean circuit, and say that some such lookup table is the aligned superintelligence. But, just because it can be expressed as a finite-depth boolean circuit, it does not follow that it is in the set of possible superintelligences. How are you concluding that such a lookup table constitutes a superintelligence? It seems
Now, I don’t think that “aligned superintelligence” is logically impossible, or anything like that, and so I expect that there mathematically-exists a possible aligned-superintelligence (if it isn’t logically impossible, then by model existence theorem, there exists a model in which one exists… I guess that doesn’t establish that we live in such a model, but whatever).
But I don’t find this argument a compelling proof(-sketch).
Not all possible functions from states to {0,1} (or to some larger discrete set) are implementable as some possible state, for cardinality reasons
All cardinalities here are finite. The set of generically realizable states is a finite set because they each have a finite and bounded information content description (a list of instructions to realize that state, which is not greater in bits than the number of neurons in all the human brains on Earth).
Yes, I knew the cardinalities in question were finite. The point applies regardless though. For any set X, there is no injection from 2^X to X. In the finite case, this is 2^n > n for all natural numbers n.
If there are N possible states, then the number of functions from possible states to {0,1} is 2^N , which is more than N, so there is some function from the set of possible states to {0,1} which is not implemented by any state.
Not if the point of the argument is to establish that a superintelligence is compatible with achieving the best possible outcome.
Here is a parody of the issue, which is somewhat unfair and leaves out almost all of your argument, but which I hope makes clear the issue I have in mind:
“Proof that a superintelligence can lead to the best possible outcome: Suppose by some method we achieved the best possible outcome. Then, there’s no properties we would want a superintelligence to have beyond that, so let’s call however we achieved the best possible outcome, ‘a superintelligence’. Then, it is possible to have a superintelligence produce the best possible outcome, QED.”
In order for an argument to be compelling for the conclusion “It is possible for a superintelligence to lead to good outcomes.” you need to use a meaning of “a superintelligence” in the argument, such that the statement “It is possible for a superintelligence to lead to good outcomes”, when interpreted with that meaning of “a superintelligence”, produces the meaning you want that sentence to have? If I argue “it is possible for a superintelligence, by which I mean computer with a clock speed faster than N, to lead to good outcomes”, then, even if I convincingly argue that a computer with a clock speed faster than N can lead to good outcomes, that shouldn’t convince people that it is possible for a superintelligence, in the sense that they have in mind (presumably not defined as “a computer with a clock speed faster than N”), is compatible with good outcomes.
Now, in your argument you say that a superintelligence would presumably be some computational process. True enough! If you then showed that some predicate is true of every computational process, you would then be justified in concluding that that predicate is (presumably) true of every possible superintelligence. But instead, you seem to have argued that a predicate is true of some computational process, and then concluded that it is therefore true of some possible superintelligence. This does not follow.
The problem with this is that people use the word “superintelligence” without a precise definition. Clearly they mean some computational process. But nobody who uses the term colloquially defines it.
So, I will make the assertion that if a computational process achieves the best possible outcome for you, it is a superintelligence. I don’t think anyone would disagree with that.
If you do, please state what other properties you think a “superintelligence” must have other than being a computational process achieves the best possible outcome.
If your argument is, “if it is possible for humans to produce some (verbal or mechanical) output, then it is possible for a program/machine to produce that output”, then, that’s true I suppose?
I don’t see why you specified “finite depth boolean circuit”.
While it does seem like the number of states for a given region of space is bounded, I’m not sure how relevant this is. Not all possible functions from states to {0,1} (or to some larger discrete set) are implementable as some possible state, for cardinality reasons.
I guess maybe that’s why you mentioned the thing along the lines of “assume that some amount of wiggle room that is tolerated” ?
One thing you say is that the set of superintelligences is a subset of the set of finite-depth boolean circuits. Later, you say that a lookup table is implementable as a finite-depth boolean circuit, and say that some such lookup table is the aligned superintelligence. But, just because it can be expressed as a finite-depth boolean circuit, it does not follow that it is in the set of possible superintelligences. How are you concluding that such a lookup table constitutes a superintelligence? It seems
Now, I don’t think that “aligned superintelligence” is logically impossible, or anything like that, and so I expect that there mathematically-exists a possible aligned-superintelligence (if it isn’t logically impossible, then by model existence theorem, there exists a model in which one exists… I guess that doesn’t establish that we live in such a model, but whatever).
But I don’t find this argument a compelling proof(-sketch).
Until I wrote this proof, it was a live possibility that aligned superintelligence is in fact logically impossible.
All cardinalities here are finite. The set of generically realizable states is a finite set because they each have a finite and bounded information content description (a list of instructions to realize that state, which is not greater in bits than the number of neurons in all the human brains on Earth).
Yes, I knew the cardinalities in question were finite. The point applies regardless though. For any set X, there is no injection from 2^X to X. In the finite case, this is 2^n > n for all natural numbers n.
If there are N possible states, then the number of functions from possible states to {0,1} is 2^N , which is more than N, so there is some function from the set of possible states to {0,1} which is not implemented by any state.
I never said it had to be implemented by a state. That is not the claim: the claim is merely that such a function exists.
Isn’t it enough that it achieves the best possible outcome? What other criteria do you want a “superintelligence” to have?
Not if the point of the argument is to establish that a superintelligence is compatible with achieving the best possible outcome.
Here is a parody of the issue, which is somewhat unfair and leaves out almost all of your argument, but which I hope makes clear the issue I have in mind:
“Proof that a superintelligence can lead to the best possible outcome: Suppose by some method we achieved the best possible outcome. Then, there’s no properties we would want a superintelligence to have beyond that, so let’s call however we achieved the best possible outcome, ‘a superintelligence’. Then, it is possible to have a superintelligence produce the best possible outcome, QED.”
In order for an argument to be compelling for the conclusion “It is possible for a superintelligence to lead to good outcomes.” you need to use a meaning of “a superintelligence” in the argument, such that the statement “It is possible for a superintelligence to lead to good outcomes”, when interpreted with that meaning of “a superintelligence”, produces the meaning you want that sentence to have? If I argue “it is possible for a superintelligence, by which I mean computer with a clock speed faster than N, to lead to good outcomes”, then, even if I convincingly argue that a computer with a clock speed faster than N can lead to good outcomes, that shouldn’t convince people that it is possible for a superintelligence, in the sense that they have in mind (presumably not defined as “a computer with a clock speed faster than N”), is compatible with good outcomes.
Now, in your argument you say that a superintelligence would presumably be some computational process. True enough! If you then showed that some predicate is true of every computational process, you would then be justified in concluding that that predicate is (presumably) true of every possible superintelligence. But instead, you seem to have argued that a predicate is true of some computational process, and then concluded that it is therefore true of some possible superintelligence. This does not follow.
The problem with this is that people use the word “superintelligence” without a precise definition. Clearly they mean some computational process. But nobody who uses the term colloquially defines it.
So, I will make the assertion that if a computational process achieves the best possible outcome for you, it is a superintelligence. I don’t think anyone would disagree with that.
If you do, please state what other properties you think a “superintelligence” must have other than being a computational process achieves the best possible outcome.