First, thank you for writing this. I would ask that you continue to think & refine and share back what you discover, prove, or disprove.
To me, it seems quite likely that B will have a lot of regularity to it. It will not be good code from the human perspective, but there will be a lot of structure I think, simply because that structure is in T and the environment.
I’m interested to see if we can (i) do more than claim this is likely and (ii) unpack reasons that might require that it be the case.
One argument for (ii) would go like this. Assume the generating process for A has a preference for shorter length programs. So we can think of a A as a tending to find shorter description lengths that match task T.
Claim: shorter (and correct) descriptions reflect some combination of environmental structure and compression.
by ‘environmental structure’ I mean the laws underlying the task.
by ‘compression’ I mean using information theory embodied in algorithms to make the program smaller
I think this claim is true, but let’s not answer that too quickly. I’d like to probe this question more deeply.
Are there more than two factors (environmental structure & compression)?
Is it possible that the description gets the structure wrong but makes up for it with great compression? I think so. One can imagine a clever trick by which a small program expands itself into something like a big ball of mud that solves the task well.
Any expansion process takes time and space. This makes me wonder if we should care not only about description length but also run time and space. If we pay attention to both, it might be possible to penalize programs that expand into a big ball of mud.
However, penalizing run time and space might be unwise, depending on what we care about. One could imagine a program that start with first principles and derives higher-level approximations that are good enough to model the domain. It might be worth paying the cost of setting up the approximations because they are used frequently. (In other words, the amortized cost of the expansion is low.)
Broadly, what mathematical tools can we use on this problem?
First, thank you for writing this. I would ask that you continue to think & refine and share back what you discover, prove, or disprove.
I’m interested to see if we can (i) do more than claim this is likely and (ii) unpack reasons that might require that it be the case.
One argument for (ii) would go like this. Assume the generating process for A has a preference for shorter length programs. So we can think of a A as a tending to find shorter description lengths that match task T.
Claim: shorter (and correct) descriptions reflect some combination of environmental structure and compression.
by ‘environmental structure’ I mean the laws underlying the task.
by ‘compression’ I mean using information theory embodied in algorithms to make the program smaller
I think this claim is true, but let’s not answer that too quickly. I’d like to probe this question more deeply.
Are there more than two factors (environmental structure & compression)?
Is it possible that the description gets the structure wrong but makes up for it with great compression? I think so. One can imagine a clever trick by which a small program expands itself into something like a big ball of mud that solves the task well.
Any expansion process takes time and space. This makes me wonder if we should care not only about description length but also run time and space. If we pay attention to both, it might be possible to penalize programs that expand into a big ball of mud.
However, penalizing run time and space might be unwise, depending on what we care about. One could imagine a program that start with first principles and derives higher-level approximations that are good enough to model the domain. It might be worth paying the cost of setting up the approximations because they are used frequently. (In other words, the amortized cost of the expansion is low.)
Broadly, what mathematical tools can we use on this problem?