One somewhat silly reason: for any simple English hypothesis, we can convert it to code by running a simulation of a human and giving them the hypothesis as input, then asking them to predict what will happen next. Therefore the English and code-complexity can differ by at most a constant.
This gives a very loose bound, since it will probably take a lot of bits to specify a human mind. In practice I think the two complexities will usually not differ by too much, because coding languages were designed to be understandable by humans and have syntax similar to human languages. There are some difficult cases like descriptions of visual objects, but even here neural networks should be able to bridge the gap to an extent(in the limit this just becomes the ‘encode a human brain’ strategy)
Regarding ‘levels of abstraction’, I’m not sure if this is a big obstacle, as most programming languages have built-in mechanisms for changing levels of abstraction. e.g. functional programming languages allow you to treat functions as objects.
One somewhat silly reason: for any simple English hypothesis, we can convert it to code by running a simulation of a human and giving them the hypothesis as input, then asking them to predict what will happen next.
Some things are quick for people to do and some things are hard. Some ideas have had multiple people continuously arguing for centuries. I think this either means you can’t apply a simulation of a person like this, or some inputs have unbounded overhead.
because coding languages were designed to be understandable by humans and have syntax similar to human languages.
You should include all levels of abstraction in your reasoning, like raw bytecode. It’s both low level and can be written by humans. It’s not necessarily fun but it’s possible. What about things people design at a transistor level?
e.g. functional programming languages allow you to treat functions as objects.
I use Haskell and have no idea what you’re talking about.
I think abstraction in programming is different to what you mean; e.g. dictionary might a simple data structure to use, but a list of tuples is harder to use even though you can implement a dictionary using a list of tuples. the abstraction layer (the implementation) is what makes complex operations on list of tuples into simple operations on dictionaries.
Some things are quick for people to do and some things are hard. Some ideas have had multiple people continuously arguing for centuries. I think this either means you can’t apply a simulation of a person like this, or some inputs have unbounded overhead.
Solomonoff induction is fine with inputs taking unboundedly long to run. There might be cases where the human doesn’t converge to a stable answer even after an indefinite amount of time. But if a “simple” hypothesis can have people debating indefinitely about what it actually predicts, I’m okay with saying that it’s not actually simple(or that it’s too vague to count as a hypothesis), so it’s okay if SI doesn’t return an answer in those cases.
You should include all levels of abstraction in your reasoning, like raw bytecode. It’s both low level and can be written by humans. It’s not necessarily fun but it’s possible. What about things people design at a transistor level?
Why do you need to include those things? Solomonoff induction can use any Turing-complete programming language for its definition of simplicity, there’s nothing special about low-level languages.
I use Haskell and have no idea what you’re talking about.
I mean you can pass functions as arguments to other functions and perform operations on them.
Regarding dictionary/list-of-tuples, the point is that you only have to write the abstraction layer *once*. So if you had one programming language with dictionaries built-in and other without, the one with dictionaries gets at most a constant advantage in code-length. In general two different universal programming languages will have at most a constant difference, as johnswentworth mentioned. This means that SI is relatively insensitive to the choice of programming language: as you see more data, the predictions of 2 versions of Solomonoff induction with different programming languages will converge.
for any simple English hypothesis, we can convert it to code by running a simulation of a human and giving them the hypothesis as input, then asking them to predict what will happen next. Therefore the English and code-complexity can differ by at most a constant.
Some things are quick for people to do and some things are hard. Some ideas have had multiple people continuously arguing for centuries. I think this either means you can’t apply a simulation of a person like this, or some inputs have unbounded overhead.
Solomonoff induction is fine with inputs taking unboundedly long to run. There might be cases where the human doesn’t converge to a stable answer even after an indefinite amount of time. But if a “simple” hypothesis can have people debating indefinitely about what it actually predicts, I’m okay with saying that it’s not actually simple(or that it’s too vague to count as a hypothesis), so it’s okay if SI doesn’t return an answer in those cases.
Yeah okay, I think that’s fair.
My issue generally (which is in my reply to johnswentworth) is the overhead is non-negligible if you’re going to invoke a human. In that case we can’t conclude that simplicity would carry over from the english representation to the code representation. So this argument doesn’t answer the question.
You do say it’s a loose bound, but I don’t think it’s useful. One big reason is that the overhead would dwarf any program we’d ever run, and pretty much every program would look identical b/c of the overhead. For simplicity to carry over we need relatively small overhead (even like the entire python runtime is only like 20mb extra via py2exe, much smaller than a mind and definitely not simple).
Maybe it’s worth mentioning the question in OP: I read it as: “why would stuff the simplicity an idea had in one form (code) necessarily correspond to simplicity when it is in another form (english)? or more generally: why would the complexity of an idea stay roughly the same when the idea is expressed through different abstraction layers?” After that there’s implications for Occam’s Razor. Particularly it’s relevant b/c occam’s razor would give different answers when comparing ideas at different levels of abstraction, and if that’s the case we can’t be sure that ideas which are simple in english will be simple in code and we don’t have a reason for Occam’s Razor applying to SI.
Does that line up with what you think OP is about? If not we might be talking cross-purposes.
I mean you can pass functions as arguments to other functions and perform operations on them.
Ahh okay; first class functions.
Re “perform operations on [functions]”: you can make new functions and partially or fully apply functions, but that’s about it. (that does mean you can partially apply functions and pass them on, though, which is super useful)
So if you had one programming language with dictionaries built-in and other without, the one with dictionaries gets at most a constant advantage in code-length.
I agree with you that the theoretical upper bound on the minimum overhead is the size of a compiler/interpreter.
I think we might disagree on this, though: the compiler/interpreter includes data such as initial conditions (e.g. binary extensions, dynamic libraries, etc). I think this is an issue b/c there’s no upper bound on that. If you invoke a whole person then it’s an issue b/c for that person to solve more and more complex problems (or a wider and wider array) those initial conditions are going to grow correspondingly. Our estimates for the data requirements to store a mind are like 1020 bits. I’d expect the minimum required data to drop as problems got “simpler”, but my intuition is that pattern is not the same pattern as what Occam’s Razor gives us (e.g. minds taking less data can still think about what Thor would do).
I read it as: “why would stuff the simplicity an idea had in one form (code) necessarily correspond to simplicity when it is in another form (english)? or more generally: why would the complexity of an idea stay roughly the same when the idea is expressed through different abstraction layers?”
I think that the argument about emulating one Turing machine with another is the best you’re going to get in full generality. You’re right that we have no guarantee that the explanation that looks simplest to a human will also look the simplest to a newly-initialized SI, because the ‘constant factor’ needed to specify that human could be very large.
I do think it’s meaningful that there is at most a constant difference between different versions of Solomonoff induction(including “human-SI”). This is because of what happens as the two versions update on incoming data: they will necessarily converge in their predictions, differing at most on a constant number of predictions.
So while SI and humans might have very different notions of simplicity at first, they will eventually come to have the same notion, after they see enough data from the world. If an emulation of a human takes X bits to specify, it means a human can beat SI at binary predictions at most X times(roughly) on a given task before SI wises up. For domains with lots of data, such as sensory prediction, this means you should expect SI to converge to giving answers as good as humans relatively quickly, even if the overhead is quite large*.
Our estimates for the data requirements to store a mind are like 10^20 bits
The quantity that matters is how many bits it takes to specify the mind, not store it(storage is free for SI just like computation time). For the human brain this shouldn’t be too much more than the length of the human genome, about 3.3 GB. Of course, getting your human brain to understand English and have common sense could take a lot more than that.
*Although, those relatively few times when the predictions differ could cause problems. This is an ongoing area of research.
I think that the argument about emulating one Turing machine with another is the best you’re going to get in full generality.
In that case I especially don’t think that argument answers the question in OP.
I’ve left some details in another reply about why I think the constant overhead argument is flawed.
So while SI and humans might have very different notions of simplicity at first, they will eventually come to have the same notion, after they see enough data from the world.
I don’t think this is true. I do agree some conclusions would be converged on by both systems (SI and humans), but I don’t think simplicity needs to be one of them.
If an emulation of a human takes X bits to specify, it means a human can beat SI at binary predictions at most X times(roughly) on a given task before SI wises up.
Uhh, I don’t follow this. Could you explain or link to an explanation please?
The quantity that matters is how many bits it takes to specify the mind, not store it(storage is free for SI just like computation time).
I don’t think that applies here. I think that data is part of the program.
For the human brain this shouldn’t be too much more than the length of the human genome, about 3.3 GB.
You would have to raise the program like a human child in that case^1. Can you really make the case you’re predicting something or creating new knowledge via SI if you have to spend (the equiv. of) 20 human years to get it to a useful state?
How would you ask multiple questions? Practically, you’d save the state and load that state in a new SI machine (or whatever). This means the data is part of the program.
Moreover, if you did have to raise the program like any other newborn, you have to use some non-SI process to create all the knowledge in that system (because people don’t use SI, or if they do use SI, they have other system(s) too).
1: at least in terms of knowledge; though if you used the complete human genome arguably you’d need to simulate a mother and other ppl too, but they have to be good simulations after the first few years, which is a regressive problem. So it’s probably easier to instantiate it in a body and raise it like a person b/c human people are already suitable. You also need to worry about it becoming mistaken (intuitively one disagrees with most people on most things we’d use an SI program for).
Uhh, I don’t follow this. Could you explain or link to an explanation please?
Intuitive explanation: Say it takes X bits to specify a human, and that the human knows how to correctly predict whatever sequence we’re applying SI to. SI has to find the human among the other 2^X programs of length X. Say SI is trying to predict the next bit. There will be some fraction of those 2^X programs that predict it will be 0, and some fraction predicting 1. There fractions define SI’s probabilities for what the next bit will be. Imagine the next bit will be 0. Then SI is predicting badly if greater than half of those programs predict a 1. But then, all those programs will be eliminated in the update phase. Clearly, this can happen at most X times before most of the weight of SI is on the human hypothesis(or a hypothesis that’s just as good at predicting the sequence in question)
The above is a sketch, not quite how SI really works. Rigorous bounds can be found here, in particular the bottom of page 979(“we observe that Theorem 2 implies the number of errors of the universal predictor is finite if the number of errors of the informed prior is finite...”). In the case where the number of errors is not finite, the universal and informed prior still have the same asymptotic rate of growth of error (error of universal prior is in big-O class of error of informed prior)
I don’t think this is true. I do agree some conclusions would be converged on by both systems (SI and humans), but I don’t think simplicity needs to be one of them.
When I say the ‘sense of simplicity of SI’, I use ‘simple program’ to mean the programs that SI gives the highest weight to in its predictions(these will by definition be the shortest programs that haven’t been ruled out by data). The above results imply that, if humans use their own sense of simplicity to predict things, and their predictions do well at a given task, SI will be able to learn their sense of simplicity after a bounded number of errors.
How would you ask multiple questions? Practically, you’d save the state and load that state in a new SI machine (or whatever). This means the data is part of the program.
I think you can input multiple questions by just feeding a sequence of question/answer pairs. Actually getting SI to act like a question-answering oracle is going to involve various implementation details. The above arguments are just meant to establish that SI won’t do much worse than humans at sequence prediction(of any type) -- so, to the extent that we use simplicity to attempt to predict things, SI will “learn” that sense after at most a finite number of mistakes(in particular, it won’t do any *worse* than ‘human-SI’, hypotheses ranked by the shortness of their English description, then fed to a human predictor)
One somewhat silly reason: for any simple English hypothesis, we can convert it to code by running a simulation of a human and giving them the hypothesis as input, then asking them to predict what will happen next. Therefore the English and code-complexity can differ by at most a constant.
This gives a very loose bound, since it will probably take a lot of bits to specify a human mind. In practice I think the two complexities will usually not differ by too much, because coding languages were designed to be understandable by humans and have syntax similar to human languages. There are some difficult cases like descriptions of visual objects, but even here neural networks should be able to bridge the gap to an extent(in the limit this just becomes the ‘encode a human brain’ strategy)
Regarding ‘levels of abstraction’, I’m not sure if this is a big obstacle, as most programming languages have built-in mechanisms for changing levels of abstraction. e.g. functional programming languages allow you to treat functions as objects.
Some things are quick for people to do and some things are hard. Some ideas have had multiple people continuously arguing for centuries. I think this either means you can’t apply a simulation of a person like this, or some inputs have unbounded overhead.
You should include all levels of abstraction in your reasoning, like raw bytecode. It’s both low level and can be written by humans. It’s not necessarily fun but it’s possible. What about things people design at a transistor level?
I use Haskell and have no idea what you’re talking about.
I think abstraction in programming is different to what you mean; e.g. dictionary might a simple data structure to use, but a list of tuples is harder to use even though you can implement a dictionary using a list of tuples. the abstraction layer (the implementation) is what makes complex operations on list of tuples into simple operations on dictionaries.
Solomonoff induction is fine with inputs taking unboundedly long to run. There might be cases where the human doesn’t converge to a stable answer even after an indefinite amount of time. But if a “simple” hypothesis can have people debating indefinitely about what it actually predicts, I’m okay with saying that it’s not actually simple(or that it’s too vague to count as a hypothesis), so it’s okay if SI doesn’t return an answer in those cases.
Why do you need to include those things? Solomonoff induction can use any Turing-complete programming language for its definition of simplicity, there’s nothing special about low-level languages.
I mean you can pass functions as arguments to other functions and perform operations on them.
Regarding dictionary/list-of-tuples, the point is that you only have to write the abstraction layer *once*. So if you had one programming language with dictionaries built-in and other without, the one with dictionaries gets at most a constant advantage in code-length. In general two different universal programming languages will have at most a constant difference, as johnswentworth mentioned. This means that SI is relatively insensitive to the choice of programming language: as you see more data, the predictions of 2 versions of Solomonoff induction with different programming languages will converge.
Yeah okay, I think that’s fair.
My issue generally (which is in my reply to johnswentworth) is the overhead is non-negligible if you’re going to invoke a human. In that case we can’t conclude that simplicity would carry over from the english representation to the code representation. So this argument doesn’t answer the question.
You do say it’s a loose bound, but I don’t think it’s useful. One big reason is that the overhead would dwarf any program we’d ever run, and pretty much every program would look identical b/c of the overhead. For simplicity to carry over we need relatively small overhead (even like the entire python runtime is only like 20mb extra via py2exe, much smaller than a mind and definitely not simple).
Maybe it’s worth mentioning the question in OP: I read it as: “why would stuff the simplicity an idea had in one form (code) necessarily correspond to simplicity when it is in another form (english)? or more generally: why would the complexity of an idea stay roughly the same when the idea is expressed through different abstraction layers?” After that there’s implications for Occam’s Razor. Particularly it’s relevant b/c occam’s razor would give different answers when comparing ideas at different levels of abstraction, and if that’s the case we can’t be sure that ideas which are simple in english will be simple in code and we don’t have a reason for Occam’s Razor applying to SI.
Does that line up with what you think OP is about? If not we might be talking cross-purposes.
Ahh okay; first class functions.
Re “perform operations on [functions]”: you can make new functions and partially or fully apply functions, but that’s about it. (that does mean you can partially apply functions and pass them on, though, which is super useful)
I agree with you that the theoretical upper bound on the minimum overhead is the size of a compiler/interpreter.
I think we might disagree on this, though: the compiler/interpreter includes data such as initial conditions (e.g. binary extensions, dynamic libraries, etc). I think this is an issue b/c there’s no upper bound on that. If you invoke a whole person then it’s an issue b/c for that person to solve more and more complex problems (or a wider and wider array) those initial conditions are going to grow correspondingly. Our estimates for the data requirements to store a mind are like 1020 bits. I’d expect the minimum required data to drop as problems got “simpler”, but my intuition is that pattern is not the same pattern as what Occam’s Razor gives us (e.g. minds taking less data can still think about what Thor would do).
I think that the argument about emulating one Turing machine with another is the best you’re going to get in full generality. You’re right that we have no guarantee that the explanation that looks simplest to a human will also look the simplest to a newly-initialized SI, because the ‘constant factor’ needed to specify that human could be very large.
I do think it’s meaningful that there is at most a constant difference between different versions of Solomonoff induction(including “human-SI”). This is because of what happens as the two versions update on incoming data: they will necessarily converge in their predictions, differing at most on a constant number of predictions.
So while SI and humans might have very different notions of simplicity at first, they will eventually come to have the same notion, after they see enough data from the world. If an emulation of a human takes X bits to specify, it means a human can beat SI at binary predictions at most X times(roughly) on a given task before SI wises up. For domains with lots of data, such as sensory prediction, this means you should expect SI to converge to giving answers as good as humans relatively quickly, even if the overhead is quite large*.
The quantity that matters is how many bits it takes to specify the mind, not store it(storage is free for SI just like computation time). For the human brain this shouldn’t be too much more than the length of the human genome, about 3.3 GB. Of course, getting your human brain to understand English and have common sense could take a lot more than that.
*Although, those relatively few times when the predictions differ could cause problems. This is an ongoing area of research.
In that case I especially don’t think that argument answers the question in OP.
I’ve left some details in another reply about why I think the constant overhead argument is flawed.
I don’t think this is true. I do agree some conclusions would be converged on by both systems (SI and humans), but I don’t think simplicity needs to be one of them.
Uhh, I don’t follow this. Could you explain or link to an explanation please?
I don’t think that applies here. I think that data is part of the program.
You would have to raise the program like a human child in that case^1. Can you really make the case you’re predicting something or creating new knowledge via SI if you have to spend (the equiv. of) 20 human years to get it to a useful state?
How would you ask multiple questions? Practically, you’d save the state and load that state in a new SI machine (or whatever). This means the data is part of the program.
Moreover, if you did have to raise the program like any other newborn, you have to use some non-SI process to create all the knowledge in that system (because people don’t use SI, or if they do use SI, they have other system(s) too).
1: at least in terms of knowledge; though if you used the complete human genome arguably you’d need to simulate a mother and other ppl too, but they have to be good simulations after the first few years, which is a regressive problem. So it’s probably easier to instantiate it in a body and raise it like a person b/c human people are already suitable. You also need to worry about it becoming mistaken (intuitively one disagrees with most people on most things we’d use an SI program for).
Intuitive explanation: Say it takes X bits to specify a human, and that the human knows how to correctly predict whatever sequence we’re applying SI to. SI has to find the human among the other 2^X programs of length X. Say SI is trying to predict the next bit. There will be some fraction of those 2^X programs that predict it will be 0, and some fraction predicting 1. There fractions define SI’s probabilities for what the next bit will be. Imagine the next bit will be 0. Then SI is predicting badly if greater than half of those programs predict a 1. But then, all those programs will be eliminated in the update phase. Clearly, this can happen at most X times before most of the weight of SI is on the human hypothesis(or a hypothesis that’s just as good at predicting the sequence in question)
The above is a sketch, not quite how SI really works. Rigorous bounds can be found here, in particular the bottom of page 979(“we observe that Theorem 2 implies the number of errors of the universal predictor is finite if the number of errors of the informed prior is finite...”). In the case where the number of errors is not finite, the universal and informed prior still have the same asymptotic rate of growth of error (error of universal prior is in big-O class of error of informed prior)
When I say the ‘sense of simplicity of SI’, I use ‘simple program’ to mean the programs that SI gives the highest weight to in its predictions(these will by definition be the shortest programs that haven’t been ruled out by data). The above results imply that, if humans use their own sense of simplicity to predict things, and their predictions do well at a given task, SI will be able to learn their sense of simplicity after a bounded number of errors.
I think you can input multiple questions by just feeding a sequence of question/answer pairs. Actually getting SI to act like a question-answering oracle is going to involve various implementation details. The above arguments are just meant to establish that SI won’t do much worse than humans at sequence prediction(of any type) -- so, to the extent that we use simplicity to attempt to predict things, SI will “learn” that sense after at most a finite number of mistakes(in particular, it won’t do any *worse* than ‘human-SI’, hypotheses ranked by the shortness of their English description, then fed to a human predictor)