me: “…ideas about predictive models needing to contain as many data elements as the thing they’re predicting in order to be accurate…”
Mr. Cawley: “…Sorry, an unsound idea, or one misapplied. Predictive models do not need to contain as many data elements as the thing they are predicting in order to be accurate. There are all kinds of simplifying substitutions and shortcuts in formal and real behaviors. Even for every single detail. I have a really accurate description of the future value of every single cell of rule 0 after the initial condition for every initial condition regardless of size or number of steps, using just one element. When the behavior is simple it can be fully predicted without ‘one to one and onto’ modeling…”
I’m going to respond to this, four years later, because I’m poking around this forum again and, frankly, I get a little annoyed when members of the inner NKS clique take a superior tone with me (JC says I “miss basic points at the outset of the whole subject”…he’s “plowing the sea” by participating in this thread with me…he’ll “give it a try…and see if any of it sticks”…sorry you seem to have such a low opinion of me, JC; I admire your philosophical and analytical perspectives on this site and others). PJL took a similar tone with me in person at the D.C. NKS conference and Cawley does it with me in this thread. You all should be aware, as people who clearly want to promote an NKS slant in the world, that when you approach outsiders like me with that kind of tone, it’s a turn-off to your whole group. That said, I am clearly very interested in thinking about these ideas and participating with you within the context of this forum, so I’ll move on to the content of my rebuttal to part of what JC writes above:
I may not have been as clear in my reference, in 2006, as I should have been, to the idea I’m talking about it, which I heard via — I don’t know — some popularized Hawking book. The idea is that to predict an irreducible system (of the type most oft discussed in this domain) that, being there’s no shortcut-style, reductive description of the system (unlike there usually is in math and physics—math and physics *are*, essentially, reductive descriptions), that as you build a simulation of this complex system, you end up needing to make your system more and more complex (using more and more “elements”—physical elements, conceptual elements)…and that there’s a dynamic that starts to illustrate itself, wherein if you’re creating a simulation of what’s going to happen next in a complex universe, the more and more accurately you want to do that—in cases where there is no reductive description of the history or unfolded dynamics of the world—you approach a situation wherein it’s less and less like a simulation that you can run beforehand, and more and more like an exact copy of the thing you’re trying to simulate in the first place…which, when time is part of the universe…means that, less and less, you get the benefit of being able to predict events with your simulation…since the simulation takes as long to run as the universe itself.
In the part of your response that I quoted, you’re talking about simple systems, clearly, systems that can be reductively described. In my proposition about classifying one’s own complexity, or classifying a system that you cannot predict, clearly I am not talking about that kind of system.
I wasn’t as rigorous as I should have been in my original post, perhaps. What I was trying to get at, was that—I’ll make a weaker and more articulated assertion here—when one wants to figure out exactly how complex an observed system is, there are limits inherent in that: if you “cannot predict” the system such that you have no exact reductive description of its unfolded dynamics, then there are elements in the unfolded history that, since you can’t predict them, you don’t understand well enough to eliminate the possibility that they contain complex elements. If you can’t predict a system completely, if you can’t reduce it completely, then setting an upper bound for its complexity seems to me to be at best a dicey matter! (a functionally-capping upper bound…an upper bound that is lower than the highest upper bound in your classification scheme, Class IV in the case of NKS)
If I’m a teacher and I give you a test, and I have a model that allows me to always guess right before you take the test, about what you will answer on the test, then I can claim to classify your test-taking behavior in a wholly-more-secure way than if I can’t predict what you will answer on the test…because in the former case, since your behavior doesn’t deviate from my reductive description, it would be significantly harder to say that there’s anything in your behavior that’s eluding me than if your behavior deviates from my best reductive description (prediction). If you’re doing something I don’t understand, something I can’t predict, then you may very well be doing something that is highly complex, sensible, meaningful, etc., that, if I understood it, or could recognize it, or describe it, might affect my classification of your complexity (upward). I might be filling in ovals on a multiple-choice test to spell out “this class is boring” in a compressed binary format, completely ignoring the questions being asked of me. That’s an example of a system whose output (my answers on the test) looks Class III to you, but is really Class IV. So while I obviously recognize that there is a taxonomic difference between Class III and Class IV systems, the example I just gave should be sufficient reason to doubt that, in general, behavior that looks random, cannot contain complex, intelligent, or universal behavior.
Distinct from that question, in my mind, is the question: if I know the rules of the system and its initial state and I see every part of the output of the system from step 0, can an intelligent, non-random system produce behavior that looks random (Class III) from the very beginning. My example of the student differs from this in that I wasn’t observing the student from step 0, didn’t see its initial state, etc. In that example, perhaps obviously, only part of the output of the system is random. Is there a Class III-looking CA, or some other simple system, that looks random from step 0, but that actually contains nonrandom, meaningful, behavior? I certainly don’t know, or else I would post the damn thing here. It is hard for me to imagine something like an ECA that could do this…organize itself through time, having instantly assumed a random-looking output. It seems to me that there might usually be some initialization period during which the thing had to decide to, for example, write compressed, binary-encoded messages in multiple-choice answers on a test. (To be more demanding of the test example, it would have to move in the direction of there being the lookup-table part of a compressed message encoded somewhere in my test…the decision to be cryptic would have to be somewhere, right?, in the rule or in the system output(?)…and then, would it be possible for that decision or nature to itself be so cryptic that it looked random to me…(?)…that, frankly, is hard for me to imagine.) But, myself, I do not see reason enough to cast out the possibility that this could happen, that this kind of system could exist.
For one, and this is quite general, but I think relevant here: the way we’re viewing CA output is part of why it seems to have form, or to be random, to us. Even the 2d grid, widely-regarded as simple, probably one of the least-presumptive output visualization mechanisms our species can think of, contains assumptions and mappings that inform our ability to see the behavior of the system. It may be that different visualization or perception mechanisms for CAs (and other systems, obviously), when used, would force, say, the 256 ECAs into different Class I-Class IV categories. Maybe rule 110, when viewed through my network-unrolling methodology, looks like a different class than it does in the 2d-grid perception mechanism.
For another, I happen to have seen, and have posted here, years ago, systems very much like ECAs except with a denser connectivity, if you will—the “water” systems, which are like ECAs except with two rows of memory, while not fulfilling the requirements I’ve given above of a system that appears random from step 0 while actually containing highly-complex, nonrandom order, look a whole lot more like TV snow, on the whole, than any of the ECAs, while clearly not being purely random in their behavior. That doesn’t, of course, mean that there are systems with no detectable initialization period that look completely random and yet contain decidedly nonrandom and meaningful behavior, but to me it’s one reason to wonder if perhaps there might be such systems.
I suppose, in a way, that some classic PRNGs are non-CA examples of systems whose output, from step 0, even with visibility into the system rule, does not demonstrate a visible initialization period in which the system is organizing itself into a state where it can slip secret messages past me in the mail, and yet, those systems demonstrate decidedly nonrandom (cyclic) behavior, even while most people’s way of perceiving the system makes the system look completely random, through and through. It’s not intelligent behavior, as far as I know, so I don’t find that example very satisfying, myself.
Is there a system where I can know the rule, see initial state and output from step 0, that looks random from step 0 (when using the 2d grid visualization, by which we’d say it’s Class III), yet is meaningfully nonrandom when viewed in a different way? I don’t know. I’ve looked through quite a lot of CA-like systems, programmatically searching for such an example, without finding one.
You’re right, Mr. Cawley, you can classify anything you please. =) (I hope you keep doing so.) And I like the way you all classify things, Class I-IV and such. There’s still a nagging question, though, in my brain, about whether I can be sure that every Class III system is, in fact, not possibly a universal system that is just hard for me to see. Short a satisfying example, however, I certainly defer to you that what looks unintelligently random, is exactly that.