Then, we could exploit it to compress the description of the full-fidelity/lowest-level history.
I don't think this works if the lowest level laws of physics are very very simple. The laws of physics at the lowest level + initial conditions are sufficient to roll out the whole history, so (in K-complexity) there's no benefit to adding descriptions of the higher levels.
Maybe if lots of noise is constantly being injected into the universe, this would change things. Because then the noise counts as part of the initial conditions. So the K-complexity of the universe-history is large, but high-level structure is common anyway because it's more robust to that noise?
The laws of physics at the lowest level + initial conditions are sufficient to roll out the whole history, so (in K-complexity) there's no benefit to adding descriptions of the higher levels.
Unless high-level structure lets you compress the initial conditions themselves, no?
Self-arguing on the topic
Counter-argument: The initial state had no structure we could exploit for compression, pure chaos.
Counter²-argument: Any given history that ended up well-abstracting corresponds to a specific inhomogeneous distribution of mass in the early universe, which defined the way the galaxies are spread across it. At least that seems to be the step that could already be compressed. If there were a step upstream of it where the state really didn't have any structure, that unstructured state could be generated by describing the post-structure-formation state, describing the "timestamp" of the post-structure-formation state, then running physics in reverse to generate the unstructured state. So unless the later structured state fails to be lower-description-length than the earlier unstructured state, structure/abstractibility should still allow you to compress the initial state's description, even if the structure only appears later.
Counter³-argumnent: The real "initial state" is the initial state of the quantum multiverse from which all possible Everett branches (and so all possible inhomogeneous distributions of mass, etc.) are generated. Its description length could be incredibly low, such as a uniform point singularity with no expensive-to-describe inhomogeneities whatsoever. The bits you later have to spend to describe the state of your universe are effectively spent on pinpointing the specific Everett branch you're in, but the actual algorithm generating the whole Tegmark III multiverse did not have to do that. It just described the simple state from which all possible branches descend.
Counter⁴-argument: My understanding is that under QM/QFT, the universe doesn't start from a singularity; it's a general-relativity thing. QM/QFT require an initial inhomogeneous universal wavefunction to start working.
Counter⁵-argument: Perhaps the real Theory of Everything unifying QFT and GR would have an initial homogeneous singularity from which all possible Everett branches are generated, and this end result seems plausible enough that we may as well assume it right now.
I don't know enough fundamental physics to make a confident call here. Though...
Counter⁶-argument: There seems to be some process which reallocates realityfluid within Tegmark III as well, between Everett branches. I think this is a hint that the "Tegmark III entire is a single program for the purposes of anthropics/from Tegmark IV's point of view" idea is somehow wrong.
Wait, none of that actually helps; you're right. If we can specify the full state of the universe/multiverse at any one moment, the rest of its history can be generated from that moment. To do so most efficiently, we should pick the simplest-to-describe state, and there we would benefit from having some structure. But as long as we have one simple-to-describe state, we can have all the other states be arbitrarily unstructured, with no loss of simplicity. So what we should expect is a history with at least one moment of structure (e. g., the initial conditions) that can then immediately dissolve into chaos.
To impose structure on the entire history, we do have to introduce some source of randomness that interferes in the state-transition process, making it impossible to deterministically compute later states from early ones. I. e., the laws of physics themselves have to be "incomplete"/stochastic, such that they can't be used as the decompression algorithm. I do have some thoughts on why that may (effectively) be the case, but they're on a line of reasoning I don't really trust.
... Alternatively, what if the most compact description of the lowest-level state at any given moment routes through describing the entire multi-level history? I. e., what if even abstractions that exist in the distant future shed some light at the present lowest-level state, and they do so in a way that's cheaper than specifying the lowest-level state manually?
Suppose the state is parametrized by real numbers. As it evolves, ever-more-distant decimal digits become relevant. This means that, if you want to simulate this universe on a non-analog computer (i. e., a computer that doesn't use unlimited-precision reals) from to starting from some initial state , with the simulation error never exceeding some value, the precision with which you have to specify scales with . Indeed, as goes to infinity, so does the needed precision (i. e., the description length).
Given all that, is it plausible that far-future abstractions summarize redundant information stored in the current state? Such that specifying the lowest-level state up to the needed precision is cheaper by describing the future history, rather than by manually specifying the position of every particle (or, rather, the finer details of the universal wavefunction).
... Yes, I think? Like, consider the state , with some high-level system existing in it. Suppose we want to infer from . How much information does tell us about ? Intuitively, quite a lot: for to end up arising, many fine details in the distant past had to line up just right. Thus, knowing about likely gives us more bits about the exact low-level past state than the description length of itself.
Ever-further-in-the-future high-level abstractions essentially serve as compressed information about sets of ever-more-distant decimal-expansion digits of past lowest-level states. As long as an abstraction takes fewer bits to specify than the bits it communicates about the initial conditions, its presence decreases that initial state's description length.
This is basically just the scaled-up version of counter²-argument from the collapsible. If an unstructured state deterministically evolves into a structured state, those future structures are implicit in its at-a-glance-unstructured form. Thus, the more simple-to-describe high-level structures a state produces across its history, the simpler it itself is to describe. So if we want to run a universe from to with a bounded simulation error, the simplest initial conditions would impose the well-abstractibility property on the whole 0-to-n interval. That recovers the property I want.
Main diff with your initial argument: the idea that the description length of the lowest-level state at any given moment effectively scales with the length of history you want to model, rather than being constant-and-finite. This makes it a question of whether any given additional period of future history is cheaper to specify by directly describing the desired future multi-level abstract state, or by packing that information into the initial conditions; and the former seems cheaper.
All that reasoning is pretty raw, obviously. Any obvious errors there?
Also, this is pretty useful. For bounty purposes, I'm currently feeling $20 on this one; feel free to send your preferred payment method via PMs.
The idea that there's a simple state in the future, that still pins down the entire past, seems possible but weird. Most of the time when events evolve into a simple state, it's because information is destroyed. This isn't really a counter-argument, it's just trying to put into words what feels odd.
One thing that's confusing to me: Why K-complexity of the low-level history? Why not, for example, Algorithmic Minimal Sufficient Statistic, which doesn't count the uniform noise? Or memory-bounded K-complexity, which might also favour multi-level descriptions.
I think I prefer frequentist justifications for complexity priors, because they explain why it works even on small parts of the universe.
But this explanation doesn't sit well with me, because under this kind of prior, the fundamental laws of physics being so simple is really surprising.
"If your life choices led you to a place where you had to figure out anthropics before you could decide what to do next, are you really living your life correctly?"
To revisit our premises: Why should we think the end result is achievable at all? Why should it be possible to usefully represent the universe as an easily interpretable symbolic structure?
First, I very much agree with the sentiment quoted above, so we aren't quite doing that here. Most of the actual reason is just: it sure looks like that's the case, empirically. As I'd argued before, human world-models seem autosymbolic, and the entirety of our (quite successful) scientific edifice relies on something-like-this being true. I think the basic case is convincing enough not to require much further justification.
But can we come up with theoretical justifications regarding why the universe might have ended up with this structure?
Here's one one approach: it's because there's an anthropics-based selection pressure on universes to be structured this way. Three lines of reasoning:
I go full anthropics-weirdness in this section, taking the premises of Tegmark IV as granted. I primarily view that type of reasoning as a useful tool for organizing one's thinking and informing one's intuitions in internally consistent ways. The conclusions don't rely on anthropics' metaphysical premises being literally true. (Indeed, I expect this can all be translated into more mundane reasoning about Solomonoff-induction approximation / generalized Occam's razor: reasoning about what theories about the universe's structure we should consider more likely a priori. But the language would grow cumbersome.)
Consider a universe whose initial conditions and physics unroll into a history that has certain robustly reoccurring high-level regularities/emergent dynamics. In other words, it has "abstraction levels", which follow different high-level "laws" and across which "high-level histories" play out. The tower of abstractions can have many levels, of course (fundamental particles to molecules to cells to organisms to populations).
Suppose that the high-level history is generated by simple laws as well. (Chemistry, evolution, microeconomics.) Then, we could exploit it to compress the description of the full-fidelity/lowest-level history. Instead of having to use an encoding that specifies the full state at any given moment, you could point to the high-level history, then "clarify" the low-level states using "correction terms". This would work inasmuch as the high-level history distills the information that is repeated in several low-level variables.
(If we have two random variables , with nonzero mutual information , we have , because sum of individual descriptions double-counts the shared information. Describing the low-level state directly is equivalent to using the sum of individual descriptions.)
Alternate reasoning: Any description of a low-level history necessarily describes the high-level history, so a longer-description high-level history necessarily means a longer-description low-level history. Now consider a history without robust abstraction levels. This is isomorphic to it having a high-level history which takes a random walk through some state-space, which blows up its description length.
(Note that high-level histories don't have to be absolutely robust: abstractions can be leaky or even sometimes destroyed. Governments fall apart, the orbital dynamics of star systems change. In this frame, it only means you have to spend bits on "manually encoding" discontinuous jumps through the high-level state-space ("abstraction leaks", high-level state momentarily becoming dependent on low-level details), or on switching the function generating the high-level history ("abstraction break", the high-level emergent dynamics fundamentally changing). As long as each high-level dynamic stays around for a nontrivial interval and largely explains (an aspect of) the low-level dynamics within this interval, it still serves to compress the low-level history.)
All this naturally works better if we have many layers of abstraction: intermediate levels compress lower levels and are in turn compressed by higher levels.
So: if we assume some process in Tegmark 4 that picks which histories to implement, instead of implementing all histories blindly and uniformly, the histories with "tall" abstraction towers would be picked more often / have more realityfluid / are anthropically selected-for. (In our case, this process is maybe the interference patterns between Everett branches.)
I. e.: this is the same logic I initially used to argue that aiming for a low-description-length representation of our universe would recover a well-structured representation. If reality "prefers" universes that are simple, it prefers universes that can be compressed well, which means it prefers well-abstracting universes.
Sidenote: This potentially sheds light on the induction problem. If large intervals of history have to approximately agree with the output of a simple program, then looking at the past data and inferring its generators would yield you some generators which are still "active" in your time. Hence, compressing the past lets you predict the future.
If you sampled an agent from the anthropic prior, what sort of agent would you expect to see?
Consider "coarse" agents: agents that are implemented on some high-level abstract variables, "live within" high-level histories, and frequently discard low-level details of their observations.
The observation-history of a coarse agent would correspond to a set of lower-level observation-histories; to a set of lower-level agents which are approximately similar to each other. Mechanistically, the observation-streams would "diverge" the moment two agents perceive some detail that differs between their universes, but would then "merge again" once they forget this detail. Coarse agents would then necessarily hoard more realityfluid than any "precise" agent. (See a more gradual introduction of the ideas here. Also, note that this doesn't give you any weird anthropic superpowers. Like, this idea feels weird, but as far as I can, it still adds up to normality.)
Note, however, that such agents could only live in well-abstracting universes. After all, as a premise, we're assuming that they're able to discard detailed information about their observations and still navigate their universe well. This naturally implies that there are some high-level laws under which the agents live, instead of being exposed to the raw low-level dynamics.
This also means we can expand the arguments above. We can count all approximately similar universal histories as "the same" history, for the purposes of adding up the realityfluid of the embedded agents. That biases the anthropic prior towards well-abstracting histories even further: they're not only simpler by themselves, but they also embed a simpler type of agent. (The relevant notion of "approximately similar" here is provided by the particulars of the agent's vantage point/at what abstraction level it's embedded.)
Consider:
Therefore, if a universe's dynamics aren't (approximately) compressible, embedded agents cannot (even approximately) model them, cannot navigate that universe, and cannot survive/thrive in it.
I'm not entirely confident in the specifics of that theoretical reasoning; my anthropics-related arguments may contain errors. Still, I think the overall intuitive story is convincing: histories with higher levels of organization are meaningfully simpler than histories without them, and inasmuch as we put any stock in arguments for simplicity, we should expect to find ourselves in a universe with a well-abstracting history.