Does this theory handle Drescher’s example of raising my hand because I want the universe a billion years ago to be such that I would raise my hand a billion years hence?
ETA: To be exact, you have a fixed state a billion years ago, a computation which runs on that state to determine “Will you raise your hand a billion years hence?”, and you can know the initial state without knowing the output of the function, but then determine that the function outputs “Yes” iff your decision diagonal outputs “Raise hand”, so if your values U maximize at “Yes” of this function on that data, then you can (will) exert logical control over the value of this fixed mathematical function in which a copy of you is embedded.
That’s what life is all about, actually. You could just regard the universe as a big mathematical function containing a copy of you, over which you’re exerting logical control.
ETA2: You’d have to ask Gary Drescher whether he knows of anyone else who’s reductionist enough to realize that you can control the output of a fixed deterministic mathematical function if that function happens to be one in which you are embedded. As far as I know, it’s just Gary Drescher.
ETA3: “Logical control” and “Thou art math” is essentially the same idea as timeless control and thou art physics, it’s just even more fun.
Nice. A while ago I also noticed that you can control any mathematical structure if it knows about you and you know about it (i.e. there is logical dependence), which generalizes the notion of trade with other possible worlds, control of the past, etc. If that other mathematical structure is interpreted as an agent, it can be made to behave as you prefer, if in return you behave is it prefers. Thus, it’s possible for us to have and realize preferences over mathematical structures, in particular by trading with them in this manner.
At the same time there are all sorts of weird limitations of what’s possible to affect this way, for example you can control something faster than light (logical control), but only with info that is already in the logical dependence, which excludes the info that only one side has. For example, if you send away a perfect simulation of your mind on a spaceship, you can “control” what happens of the spaceship if neither of you receives observations from outside, as both computations will be identical. If some info from a year ago is sent to the spaceship, and both you and the simulation observe it (simultaneously), you remain synchronized, but now you learned something new. This way, streams of observations can be sent in both directions, continuously updating both copies. These observations, being identical, are added to logical dependence between you and the simulation, and so can be used in logical control. Thus, the whole state of knowledge in shared, and the conclusions of the whole algorithm of mind can be used for control.
On the other hand, if you know something above and beyond this shared knowledge (like recent observations), you can’t use this knowledge or any conclusions reached from this knowledge in logical control. You can’t update on non-shared knowledge and retain ability to handle logical dependence. This seems related to non-updating in counterfactual mugging: you need to exercise control over the other possible world, and so you can’t update on the observation that is particular to your possible world and use the whole algorithm that includes this update to control the other world. You can “update” if you can factor your state of knowledge into what’s dependent to what and what can be used for control of what though.
Eliezer, does the formalism on Pearl’s graphs allow to capture this idea? So far, I’m not sure how much insight can be gained from studying it (and your TDT), so I leave it to after I finish learning basics of logic.
I think you could use a non-updated Pearl graph for your updateless decision theory, but the part where you (instead of updating) decide which computational processes are similar or dissimilar to you, would be a logical problem, I think, not the domain of causal graphs.
Not-updating is the same kind of simplified denotational behemoth as a GLUT. Much of the usefulness of probabilistic graphical models comes from the fact that they compress the probability distribution into smaller representations and allow manipulation and specification of these distributions in terms of the compact representations. If I just start copying a lot of the graphical models, it won’t capture the structure of the problem, so instead of being updateless, the decision theory must update what it can, or represent a lot of partially dependent states of knowledge in a single structure, allowing to extract decisions unaffected by the knowledge that doesn’t belong to them.
I suspect that expectation maximization/probability won’t play an important role in this structure, as the structure of graphical models seems to capture the same objects as logical dependence must (where do you get the causal graphs from?), and so a structure that can work with logical (in)dependence may already contain the structure captured by probabilistic graphical models, subsuming the latter.
Just as a matter of terminology, I prefer to say that we can choose (or that we have a choice about) the output, rather than that we control it. To me, control has too strong a connotation of cause.
It’s tricky, of course, because the concepts of choice-about and causal-influence-over are so thoroughly conflated that most people will use the same word to refer to both without distinction. So my terminology suggestion is kind of like most materialsts’ choice to relinquish the word soul to refer to something extraphysical, retaining consciousness to refer to the actual physical/computational process. (Causes, unlike souls, are real, but still distinct from what they’re often conflated with.)
Again, this is just terminology, nothing substantive.
EDIT: In the (usual) special case where a means-end link is causal, I agree with you that we control something that’s ultimately mathematical, even in my proposed sense of the term.
Hm. To me, “choose” sounds like invoking the idea of multiple possibilities, while “control” sounds more determinism-compatible. Of course that is a mere matter of terminology.
Though I’m not sure what you mean by “in the special case where a means-end link is causal”—my thesis was that if you are uncertain about the output of your decision computation, and you factor the universe the Pearlian way, then your logical decision will end up being, in the graph, the logical cause of box B containing a million dollars. You mean the special case where a means-end link is physical? But what is physics except math? Or are we assuming that the local causal relations in physics are more privileged as ontologically basic causes, whereas “logical causality” is just a convenient way of factoring uncertainty and a winning way to construe counterfactuals? (That last one may have some justice to it.)
I agree that “choose” connotes multiple alternatives, but they’re counterfactual antecedents, and when construed as such, are not inconsistent with determinism.
I don’t know about being ontologically basic, but (what I think of as) physical/causal laws have the important property that they compactly specify the entirety of space-time (together with a specification of the initial conditions).
Does this theory handle Drescher’s example of raising my hand because I want the universe a billion years ago to be such that I would raise my hand a billion years hence?
Yes. That’s a logical dependence.
ETA: To be exact, you have a fixed state a billion years ago, a computation which runs on that state to determine “Will you raise your hand a billion years hence?”, and you can know the initial state without knowing the output of the function, but then determine that the function outputs “Yes” iff your decision diagonal outputs “Raise hand”, so if your values U maximize at “Yes” of this function on that data, then you can (will) exert logical control over the value of this fixed mathematical function in which a copy of you is embedded.
That’s what life is all about, actually. You could just regard the universe as a big mathematical function containing a copy of you, over which you’re exerting logical control.
ETA2: You’d have to ask Gary Drescher whether he knows of anyone else who’s reductionist enough to realize that you can control the output of a fixed deterministic mathematical function if that function happens to be one in which you are embedded. As far as I know, it’s just Gary Drescher.
ETA3: “Logical control” and “Thou art math” is essentially the same idea as timeless control and thou art physics, it’s just even more fun.
Nice. A while ago I also noticed that you can control any mathematical structure if it knows about you and you know about it (i.e. there is logical dependence), which generalizes the notion of trade with other possible worlds, control of the past, etc. If that other mathematical structure is interpreted as an agent, it can be made to behave as you prefer, if in return you behave is it prefers. Thus, it’s possible for us to have and realize preferences over mathematical structures, in particular by trading with them in this manner.
At the same time there are all sorts of weird limitations of what’s possible to affect this way, for example you can control something faster than light (logical control), but only with info that is already in the logical dependence, which excludes the info that only one side has. For example, if you send away a perfect simulation of your mind on a spaceship, you can “control” what happens of the spaceship if neither of you receives observations from outside, as both computations will be identical. If some info from a year ago is sent to the spaceship, and both you and the simulation observe it (simultaneously), you remain synchronized, but now you learned something new. This way, streams of observations can be sent in both directions, continuously updating both copies. These observations, being identical, are added to logical dependence between you and the simulation, and so can be used in logical control. Thus, the whole state of knowledge in shared, and the conclusions of the whole algorithm of mind can be used for control.
On the other hand, if you know something above and beyond this shared knowledge (like recent observations), you can’t use this knowledge or any conclusions reached from this knowledge in logical control. You can’t update on non-shared knowledge and retain ability to handle logical dependence. This seems related to non-updating in counterfactual mugging: you need to exercise control over the other possible world, and so you can’t update on the observation that is particular to your possible world and use the whole algorithm that includes this update to control the other world. You can “update” if you can factor your state of knowledge into what’s dependent to what and what can be used for control of what though.
Eliezer, does the formalism on Pearl’s graphs allow to capture this idea? So far, I’m not sure how much insight can be gained from studying it (and your TDT), so I leave it to after I finish learning basics of logic.
I think you could use a non-updated Pearl graph for your updateless decision theory, but the part where you (instead of updating) decide which computational processes are similar or dissimilar to you, would be a logical problem, I think, not the domain of causal graphs.
Not-updating is the same kind of simplified denotational behemoth as a GLUT. Much of the usefulness of probabilistic graphical models comes from the fact that they compress the probability distribution into smaller representations and allow manipulation and specification of these distributions in terms of the compact representations. If I just start copying a lot of the graphical models, it won’t capture the structure of the problem, so instead of being updateless, the decision theory must update what it can, or represent a lot of partially dependent states of knowledge in a single structure, allowing to extract decisions unaffected by the knowledge that doesn’t belong to them.
I suspect that expectation maximization/probability won’t play an important role in this structure, as the structure of graphical models seems to capture the same objects as logical dependence must (where do you get the causal graphs from?), and so a structure that can work with logical (in)dependence may already contain the structure captured by probabilistic graphical models, subsuming the latter.
Just as a matter of terminology, I prefer to say that we can choose (or that we have a choice about) the output, rather than that we control it. To me, control has too strong a connotation of cause.
It’s tricky, of course, because the concepts of choice-about and causal-influence-over are so thoroughly conflated that most people will use the same word to refer to both without distinction. So my terminology suggestion is kind of like most materialsts’ choice to relinquish the word soul to refer to something extraphysical, retaining consciousness to refer to the actual physical/computational process. (Causes, unlike souls, are real, but still distinct from what they’re often conflated with.)
Again, this is just terminology, nothing substantive.
EDIT: In the (usual) special case where a means-end link is causal, I agree with you that we control something that’s ultimately mathematical, even in my proposed sense of the term.
Hm. To me, “choose” sounds like invoking the idea of multiple possibilities, while “control” sounds more determinism-compatible. Of course that is a mere matter of terminology.
Though I’m not sure what you mean by “in the special case where a means-end link is causal”—my thesis was that if you are uncertain about the output of your decision computation, and you factor the universe the Pearlian way, then your logical decision will end up being, in the graph, the logical cause of box B containing a million dollars. You mean the special case where a means-end link is physical? But what is physics except math? Or are we assuming that the local causal relations in physics are more privileged as ontologically basic causes, whereas “logical causality” is just a convenient way of factoring uncertainty and a winning way to construe counterfactuals? (That last one may have some justice to it.)
I agree that “choose” connotes multiple alternatives, but they’re counterfactual antecedents, and when construed as such, are not inconsistent with determinism.
I don’t know about being ontologically basic, but (what I think of as) physical/causal laws have the important property that they compactly specify the entirety of space-time (together with a specification of the initial conditions).
Is there a formulation of this example that isn’t purely metaphysical, i.e. where you could actually detect the difference?