Anomalous Presumptions

Austin Henderson in his comment on Dancing toward the singularity starts by remarking on an issue that often troubles people when dealing with reflexive (or reflective) systems:

On UI, when the machine starts modeling us then we have to incorporate that modeling into our usage of it. Which leads to “I think that you think that ….”. Which is broken by popping reflective and talking about the talk. Maybe concurrently with continuing to work. In fact that may the usual case: reflecting *while* you are working.

We need the UIs to support his complexity. You talk about the ability to “support rapid evolution of conventions between the machine and the human,”. …. As for the “largely without conscious human choice” caveat, I think that addresses the other way out of the thinking about thinking infinite regress: practice, practice, practice.

I think our systems need to be reflexive. Certainly our social systems need to be reflective. But then what about the infinite regress that concerns Austin?

There are many specific tricks, but really they all boil down to the same trick: take the fixed point. Fixed points make recursive formal systems, such as lambda calculus, work. They let us find stable structure in dynamic systems. They are great.

Fixed points are easy to describe, but sometimes hard to understand. The basic idea is that you know a system is at a fixed point when you apply a transformation f to the system, and nothing happens. If the state of the system is x, then at the fixed point, f(x) = x — nothing changes. If the system isn’t at a fixed point, then f(x) = x’ — when you apply f to x, you “move” the system to x’.

A given system may have a unique fixed point — for example, well behaved expressions in the lambda calculus have a unique least fixed point. Or a system may have many fixed points, in which case it will get stuck at the one it gets to first. Or it may have no fixed points, in which case it just keeps changing each time you apply f.

Now suppose we have a reflective system. Let’s say we’re modeling a computer system we’re using (as we must to understand it). Let’s also say that at the same time, the system is modeling us, with the goal of (for example) showing us what we want to see at each point. We’d like our behavior and the system’s behavior to converge to a fixed point, where our models don’t change any more — which is to say, we understand each other. If we never reached a fixed point, we’d find it very inconvenient — the system’s behavior would keep changing, and we’d have to keep “chasing” it. This sort of inconvenience does arise, for example, in lists that try to keep your recent choices near the top.

Actually, of course, we probably won’t reach a truly fixed point, just a “quiescent” point that changes much more slowly than it did in the initial learning phase. As we learn new aspects of the system, as our needs change, and perhaps even as the system accumulates a lot more information about us, our respective models will adjust relatively slowly. I don’t know if there is a correct formal name for this sort of slowly changing point.

People model each other in interactions, and we can see people finding fixed points of comfortable interaction, that drift and occasionally change suddenly when they discover some commonality or difference. People can also get locked into very unpleasant fixed points with each other. This might be a good way to think about the sort of pathologies that Ronald Laing called “knots”.

Fixed points are needed within modeling systems, as well as between them. The statistical modeling folks have recently (say the last ten years) found that many models containing loops, which they previous thought were intractable, are perfectly well behaved with the right analysis — they provably converge to the (right) fixed points. This sort of reliably convergent feedback is essential in lots of reasoning paradigms, including the set of compression / decompression algorithms that come closest to the the Shannon bounds on channel capacity.

Unfortunately we typically aren’t taught to analyze systems in terms of this sort of dynamics, and we don’t have good techniques for designing reflexive systems — for example, UIs that model the user and converge on stable, but not excessively stable fixed points. If I’m right that we’re entering an era where our systems will model everything they interact with, including us, we’d better get used to reflexive systems and start working on those design ideas.

Filed by Jed at 12:24 am under Language, System design
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