Perceiving a finite-state system, Part 2

I had argued that the perception of a finite-state system is different from the perception of a continuous-state system. I had argued that a perception involves a representation, and I pointed out a difference in what was captured in the two kinds of representation (i.e., the representation of a continuous-state system as a continuous-state system by another continuous-state system, versus, the representation of a finite-state system as a finite-state system by another finite-state system). The difference I pointed out was that slight variations that do not constitute changes of state in the finite-state system but do constitute changes of state in the continuous-state system, are captured in the continuous representation, but not in the finite representation.

But that's not the only difference. Finite-state systems are not merely state spaces but have an underlying continuous-state system exhibiting the tendency to cluster. This means that when a represented state falls far from the center of a cluster, then when it is represented, the representing state typically falls closer to the center of the corresponding cluster. The result of this over several generations of copies (copies and copies of copies and copies of copies of copies) is that nth generation finite representations faithfully capture the state of the original, while nth-generation continuous representation are subject to gradual degradation and drift. So while continuous representation but not finite representation captures the continuous state (at least over the short run, subject to drift if too many generations of copies are made), finite representation captures the finite state better than does continuous representation (over the long run).

But I have compared some key benefits and drawbacks of finite versus continuous representation, without actually saying what the difference is between them. I'll do that now, by applying the criterion that I already elaborated earlier (which is to find a point, find a radius d, and so on). So, we apply the criterion to the represented space and to the representing space, and then we consider the combined space, the union of the represented and representing space, and apply the criterion to that, re-using the points and circles that we used in applying the first two criteria.

For example, suppose we have two continuous-state systems that each yield a finite-state system with two states - this means that the underlying continuous systems exhibit two clusters. So, we can define points n1 and n2 in the first system and N1 and N2 in the second system, and a radius d in the first system and a radius D in the second system, such that 99% of the states of the first system fall inside of circles c1 and c2 (defined as centering on n1 and n2 and having radius d), and 99% of the states of the second system fall inside circles C1 and C2 (defined as centering on N1 and N2 and having radius D), and finally, 99% of the combined states [i.e., (x,X) where x is a state in the first system and X is a state in the second system and X was created by the representing mechanism, e.g., by the scribe, in response to x] fall into c1xC1 U c2xC2 (i.e., x is in c1 and X is in C1, or x is in c2 and X is in C2, 99% of the time).

This doesn't really distinguish a scribe from a Xerox machine, provided that the represented state is clear (i.e. is close to the apex of its cluster). However, a scribe has the ability to clarify as he transcribes, something that a Xerox machine does not. This ability to clarify exhibits an awareness (of sorts) of the finite states that the Xerox machine does not have. A scribe may have an internal sense of how clear or unclear his transcription is. This can be implemented as a continuous measure - maybe an alarm, since the point is to get him to strive for clarity - that goes up the further his output gets from the apex of a cluster.

Perceiving a system as a finite-state system

I described a criterion for the existence of a finite-state system with n states given some continuous-state system, but the criterion was mathematical. What I want to do now is consider how we really identify clusters in a continuous-state system. To recap, I had argued that a real-world finite-state system must always have an underlying continuous-state system, because the laws of the world and the possible states of the world are continuous. But this underlying continuous-state system exhibits clustering of states. Typically, we can easily look at such a system and immediately recognize the finite clusters. The continuum of states of the continuous-state system yields a finite set of states. Previously I defined a mathematical criterion for the existence of a finite-state system, but now I want to consider how that system, and its states, are recognized.

We ourselves are continuous-state systems, our language is a continuous-state system, and so on. This is so even though our language is also a finite-state system. We are material beings. We exist in the universe. Therefore we must be continuous-state systems. There must be a continuum of states into which a human can enter.

One might think that in order even start to think about the finite states of a finite-state system, we must ourselves somehow represent those finite states to ourselves, internally, by means of signs. Possibly words - possibly we think in language. Possibly by other means. One might think that - I think that. And I also think that the representation must be itself performed in a finite-state system. One needs a second finite-state system in order to represent the first finite-state system.

But of course underlying the second finite-state system is a second continuous-state systems. So we have two continuous-state systems: the first one underlies the first, represented finite-state system, and the second underlies the second, representing finite-state system.

What is it about this pair of continuous-state systems that constitutes the representation of one finite-state system by another finite-state system? Why can't we say that the second continuous-state system represents the first continuous-state system?

To answer that question, I think we can't say it, because it isn't happening. We have to ask, what would it be for a continuous-state system to represent another continuous-state system? I.e., to represent it as a continuous-state system. One would need there to be at least some fine-grained representation of the states in the first continuous-state system by corresponding states in the second continuous-state system. But this isn't necessarily happening.

An example of representation of one continuous-state system by another is a Xerox copy (a photocopy). Let's say that a page with a pattern of black and white is a state. There is a continuum of such states, a continuum of such patterns of black and white. The Xerox machine maps that state-space back into itself, because it both reads and produces pages with patterns of black and white on them.

An example of representation of one finite-state system by another is transcription. Someone speaks, and the sound he produces can be seen in two ways: as a particular sound, which he can probably never exactly reproduce, or as a stream of phonemes, which he can easily reproduce, because the variations in the sound that he can't avoid are not enough to prevent him from repeating a stream of phonemes.

A sound is a state of a continuous-state system, and a stream of phonemes (yay long) is a state of a finite-state system. A second person, a scribe, can write down what he hears, producing both a pattern of black and white on a piece of paper (and one distinguishable from anything else he will ever produce), and also a text, which he can easily copy over so there can be two documents with exactly the same text.

I will ignore the difficulties produced by the fact that the language may not be perfectly phonetic. In fact it would probably have been better for me to concentrate only on transcription from one written document to another, rather than on transcription from speech to text. So the difficulty of the incommensurability of phonemes and alphabet can be overcome.

Variations in the speaker's utterance are unlikely to make it into the written document, and meanwhile the scribe is likely to introduce variations in his document that do not trace back to any variations in the utterance. This is in contrast to the faithful reproduction by a Xerox machine of slight variations in writing. Write the same thing a slightly different way, with slightly different letter forms, and the Xerox will make a copy faithfully displaying those differences. But a transcribed text is unlikely to display slight differences in the utterance.

So the particular sound that the person makes is not represented in the text. Only the words that he utters are represented.

Are finite-state systems really there?

OK, so we've taken a continuous-state system, and identified some clusters in it by performing the analysis mentioned (picking points etc.). But what is so special about this particular mathematical analysis we applied? Didn't I just pull that analysis out of my hat? Isn't it true that no matter what I wanted to see, say, the full text of War and Peace, I could have concocted an analysis of the continuous-state system that would yield me what I wanted?

We'd like to say that the clusters are already intrinsically there, or something like that. But what makes the clusters we found more "really there" than the text of War and Peace? What distinguishes the analysis I applied to derive the clusters, from some cockamamie analysis concocted to produce War and Peace?

One answer is that it's really useful to us to pick out these clusters, and quite a bit less useful to us to pick out War and Peace. That is to say, the analysis that I actually applied is generally going to be a lot more useful, not only here but when approaching a wide variety of continuous-state systems, than one concocted to produce War and Peace in just this case, which might not even be generalizable to another continuous-state system. If it is generalizable, then chances are either it will produce War and Peace again, in which case it quickly becomes obvious that the analysis is not about the continuous-state system at all, it teaches nothing about it, but rather uses it as an excuse to emit yet another copy of War and Peace. Chances are either that, or else the analysis will produce some random text that says nothing, like a book picked at random off the shelves of Borges's Library of Babel. The idea here is that the reality of the concept is based on the value of having it.

But in turn, the concept must have its value for some reason. There is probably an explanation of that value in terms of other things.

Emergence, Vagueness

We might say that the finite-state system emerges out of the underlying continuous-state system. So there's a link between this topic and emergence.

We can also notice that a handwritten note can be ambiguous as to what word was written down. It looks kind of like an a, kind of like a u, we might think. There's a continuum of possible marks between a clear a and a clear u. Where is the border between them? What is the least clear a that is still an a? Is there one?

Clarity shading off into unclarity. This is vagueness. So there's a link between this topic and vagueness.

Criteria of existence

We're looking for criteria of existence of finite-state (or discrete-state) systems, couched in terms of the underlying continuous-state systems. We want to answer the question: given that there is suchandsuch continuous-state system, does this instantiate a finite-state system, and if so, then which one?

One criterion of existence is:

1) that there is a finite number n of points in the continuous-state system, and

2) there is some small distance d, much smaller than the distance between these points,

3) such that close to 100% (say, >99%) of the states in a typical run of the continuous-state system fall inside of the union of the n circular regions of radius d around the n points, and finally,

4) n is the smallest number of points for which this is the case.

We can say then that the system displays "clustering" of states around these n points, and that each cluster (around a single point, with radius <= d) corresponds to (at most) one state of the finite-state system.

Notice the following (this is my intuition), that in any particular application of the criterion, where we might come up with some specific n points and some radius d that, in accordance with the criterion, correspond to n clusters:

1) The selection of points is somewhat arbitrary (we could have picked any points sufficiently close to the points actually picked, and still in all likelihood have gotten n clusters out of it).

2) The selection of d is somewhat arbitrary (we could have picked any radius sufficiently close to d and still gotten n clusters).

3) The cutoff of 99% is somewhat arbitrary (we might have said 98% or 99.5% and still come up with the same conclusion, that there are n clusters.

4) The choice of circles (as opposed to triangles etc.) is similarly arbitrary.

5) Finally we will have no problem matching the corresponding clusters that we discover through such second applications to that discovered through the initial application, because the regions around the picked points will (in all likelihood) overlap even though the points don't match.

So while we need to make some arbitrary choices in order to apply the criterion to some real continuous-state system, nevertheless our conclusion is somewhat independent of our choice. We pick those points, but the clusters are not really linked to the points we picked; the points are a ladder that we climb to get to the clusters, but then we can kick the ladder away.

On the other hand, in some instances the choices we make may matter. But that may not be a weakness of the theory; rather, it may reveal an ambiguity which is really there. There are ambiguities in the world, and it is best not to paper them over. How many people are there on the Earth? Well, it depends on what cutoff you pick for "person", since there are person-candidates at all stages of development, from sperm and cell, to grandparent.

Real-world finite-state systems

(I think) there are two ways of looking at any real-world finite-state system.

1) As a finite-state system.

2) As an infinite-state system.

The difference is in what counts as "states". Physical reality admits of a continuum of states (or appears to). Given two points in space, an object can be located halfway between them, a third of the way from one to the other, three-fifths of the way between them, and so on and so forth, ad infinitum. The number of states in this set of states (positions between two points) is infinite. By extension, any physical device, being physical, ought to admit of a continuum of states. How is this possible? One answer is that each finite state (meaning, state in the finite-state system) must include infinitely many slightly different continuous states (meaning, states in the infinite- or continuous-state system).

The relationship between a real-world finite-state system, like a hand calculator for example or an abacus or more importantly language, and its underlying infinite-state system, is a puzzle that has baffled me for years.