Hm, but then in your example you only have one bit of action. The model is more like “you have a certain amount of doors (the bits in Y) and as many switches (the corresponding bits in A) and you can set them to open or closed as you please as long as you can guess the password”. There are as many possible future states as there are combinations of the switches you can set, essentially 2H(A) (without including the chaotic effects of unknown laws and/or parts of the world).
Of course there’s a bit of a sleight of hand in how we’re just allowing your “free will” to set from the outside the bits in A, at zero cost and in a completely acausal manner. A proper world would have to also contain your own mind and memory stored in a part of the tape, with the evolution laws accordingly making it function. At which point the entire thing would look entirely deterministic, your actions/optimizations just a natural outcome of the dynamics of the world, and you could indeed argue that there is no optimization, just one state evolving the only way it can. And you’d be right, but insofar as we know, that’s true of the real world as well (quantum shenanigans aside; those make things potentially more complicated if you believe that the wavefunction collapse is a real and not just apparent phenomenon).
After reading your post in more depth, I can confidently say I’m confused. I don’t get at which step you get more bits out of optimization than you get in. Yet I think I understand this passage:
Y; then there’s no limit to how many bits you can copy, and the only information you need to leverage is the knowledge of the secret “key”. We know this works because this is exactly how locks work in real life too. If I possess someone’s bank account password, or the combination to a safe, or the US nuclear codes, then I’m able to produce disproportionate effects to the tiny size of that knowledge, just because I can also rely on the state of the world and its laws being set up in such a way that I can use that knowledge as a pivot to trigger much bigger effects.
To give another example of what you’re saying: if you know someone who’s the top player in an MMO with huge amounts of resources in the game, equivalent to e.g. 1kb of optimization power, then knowing their 40 bit acount details gets you 1kb of optimization power. This is analogous to your bank account password setup.
I disagree though that you get 1kb of optimization from only 40 bits of observation. Remove every bit of knowledge you have about the game and what the account’s is doing and so on, then consider what you could do with 1 bit more info: not much. Certainly not affecting the world w/ 960 bits of optimization. So your existing knowledge about the game is what let’s you use the account’s resources to optimize the world. This is key: if you start out with O bits of observation, and seeing some extra bits o let’s you apply A bits of optimization power, then you have O+o bits of observation are worth A bits of optimization.
Maybe you adressed this point in the rest of the post, but if so, I didn’t understand it.
So your existing knowledge about the game is what let’s you use the account’s resources to optimize the world.
This I say explicitly in the post. See conclusions:
besides the observation region O, we also need to know a certain amount of bits detailing the rules of the world. In its uncompressed form, this knowledge is always greater than the amount of optimization (for example, with CCSWAP gates, we would need four addresses for each optimized bit);
The bit about locks and lock-like processes comes from the original question I linked at the top; you should go check that (and the accepted answer) for full context. Essentially I set out in this post to actually disprove that answer, because I didn’t find it satisfying; I ended up finding that the answer itself is not per se wrong, but it applies with constraints to a reversible system. If you consider uncompressed knowledge about the rules of the world (in your example, the source code of the game), put together with the password, you always have more bits of knowledge than of optimization. However, since compression is a thing (for example, you don’t actually know the source code when stealing an account), I doubt that’s a hard theorem. All I can say on that is “there are situations in which n bits of knowledge can be turned into N>n bits of optimization, but only as long as the world is already set up to allow you to do so”. It would be interesting to have a reasonable definition of what a “natural” set of laws would be like and whether lock-like maps could ever spontaneously occur within it.
Hm, but then in your example you only have one bit of action. The model is more like “you have a certain amount of doors (the bits in Y) and as many switches (the corresponding bits in A) and you can set them to open or closed as you please as long as you can guess the password”. There are as many possible future states as there are combinations of the switches you can set, essentially 2H(A) (without including the chaotic effects of unknown laws and/or parts of the world).
Of course there’s a bit of a sleight of hand in how we’re just allowing your “free will” to set from the outside the bits in A, at zero cost and in a completely acausal manner. A proper world would have to also contain your own mind and memory stored in a part of the tape, with the evolution laws accordingly making it function. At which point the entire thing would look entirely deterministic, your actions/optimizations just a natural outcome of the dynamics of the world, and you could indeed argue that there is no optimization, just one state evolving the only way it can. And you’d be right, but insofar as we know, that’s true of the real world as well (quantum shenanigans aside; those make things potentially more complicated if you believe that the wavefunction collapse is a real and not just apparent phenomenon).
After reading your post in more depth, I can confidently say I’m confused. I don’t get at which step you get more bits out of optimization than you get in. Yet I think I understand this passage:
To give another example of what you’re saying: if you know someone who’s the top player in an MMO with huge amounts of resources in the game, equivalent to e.g. 1kb of optimization power, then knowing their 40 bit acount details gets you 1kb of optimization power. This is analogous to your bank account password setup.
I disagree though that you get 1kb of optimization from only 40 bits of observation. Remove every bit of knowledge you have about the game and what the account’s is doing and so on, then consider what you could do with 1 bit more info: not much. Certainly not affecting the world w/ 960 bits of optimization. So your existing knowledge about the game is what let’s you use the account’s resources to optimize the world. This is key: if you start out with O bits of observation, and seeing some extra bits o let’s you apply A bits of optimization power, then you have O+o bits of observation are worth A bits of optimization.
Maybe you adressed this point in the rest of the post, but if so, I didn’t understand it.
This I say explicitly in the post. See conclusions:
The bit about locks and lock-like processes comes from the original question I linked at the top; you should go check that (and the accepted answer) for full context. Essentially I set out in this post to actually disprove that answer, because I didn’t find it satisfying; I ended up finding that the answer itself is not per se wrong, but it applies with constraints to a reversible system. If you consider uncompressed knowledge about the rules of the world (in your example, the source code of the game), put together with the password, you always have more bits of knowledge than of optimization. However, since compression is a thing (for example, you don’t actually know the source code when stealing an account), I doubt that’s a hard theorem. All I can say on that is “there are situations in which n bits of knowledge can be turned into N>n bits of optimization, but only as long as the world is already set up to allow you to do so”. It would be interesting to have a reasonable definition of what a “natural” set of laws would be like and whether lock-like maps could ever spontaneously occur within it.