Cellular Automata Lecture

CINVESTAV

Professor: Harold V. McIntosh.

**Enbedding Rule 110 in a five-cell neighborhood**

One of the items in the discussion on Saturday afternoon was wherher new
things
were to be learned about
Rule 110, and one of them was to create some mutants
by embedding it is a five-cell neighborhood. However since we haven't really
discussed
cellular automata and weren't planning to emphasize them, it might
be a good idea to quickly run through their history.

Automata as such even existed in antiquity; Hero's steam engine, which may
actually have been used to lift fuel in the lighthouse of Alexandria. The
idea
is not just to have machines, but machines capable of performing a variety of
actions and even to decide on their sequence. At one time some ingenious
mechanisms were constructed; one might think of Swiss music boxes, and even
Babbage's mechanical calculator. His Analytical Engine, which was to be
programmable, was never constructed, and the whole series of projects is as
an interesting lesson in the management of abbitious projects as were any of
its technical accomplishments.

With electrical circuitry, in the form of relays and later vacuum tubes, it
finally became possible to create large scale calculators and think of
computers. The difference is that calculators perform individual operations
whereas computers can be programmed and decide their own sequences.
John von Neumann
got interested in the extent to which computers could really act
like living organisms, which can
reproduce
, particularly like humans who
could
make calculations and think about them. He finally settled on cellular
automata, a purely mathematical construct, after deciding that physical
models would be too cumbersome. It helps to recall that this was in an era
in which people knew about genes and chromosomes, and even DNA, but
Watson
and Crick were still ten or fifteen years from deciphering the molecular
structure of DNA.

So von Neumann worked on
self reproducing cellular automata, giving lectures
in various places and on various occasions, which set off a wave of interest
in the topic and produced, among other things, Moore's Garden of Eden
theorem.

With a lapse of about fifteen years,
John Conway
was reviewing the field and
decided to see what could be done with a binary automaton. Von Neumann's was
two dimensional with a five cell neighborhood and 29 states (4 x 7 + 1 for
what were really seven operations like reading, writing, and moving, but in
four directions. Plus a quiescent state.) Conway found an interesting rule
in a nine-cell neighborhood which was publicized in
Scientific American and
set off a new burst of activity which ended up with a self-reproducing
configuration and numerous interesting details. For reasons having probably
to do with the spirit of the times (little green trisexual microbes) his
invention was called
"The Game of Life," or simply "Life."

Of course, neither
von Neumann's nor Conway's automata were ever actually
constructed, being extraordinarily large and cumbersome, but the details were
sufficiently clear and convincing that their operation is generally accepted.
Their studies both depended on selecting one particular rule set for the
automaton and following out the consequences in minute detail. The rule was
chosen among various posibilities to do the job (von Neumann) or for its
intrinsic elegance (Conway), and thereafter no attempt was made to change
the rule. Little, anyway; there were some
``Alien LIfe'' 's.
.

In contrast, another fifteen years later,
Stephen Wolfram had greater access
to a computer than had previously been the custom, and set out to try all the
possibilities, albeit with smaller neighborhoods and in one dimension. One
result of this activity was to decide that there were two kinds of rules ---
those for which nothing happened, and those for which everything happened.
But he may have been influenced by some current fashions in differential
equation and dynamical systems theory, so he postulated
four classes of
automata. Class I: fixed points; Class II, limit cycles; Class III, chaos;
and Class IV, ``Islands of chaos in a sea of tranquility.'' Nowadays we
combine Classes I and II, and consider Class IV as a boundary or transition
between this class and Class III.

Just to be precocious, he also decided that the automata of Class IV would be
``capable of Universal Computation.'' And then he went on to include ``Birds
singing in the trees, the wind sighing in the boughs, brooks burbling on
their way to the sea, ...., all these are performing universal
computations.''

But for all the hoopla, this Rule 110 deserves to be taken seriously, and is
the subject of a story of its own. It should be mentioned that the
nomenclature
{Rule 110} is another invention of Wolfram's, to make a list of the images of
all the neighborhoods in order, then treat the list as a number and convert
it to decimal. It works for binary automata with three neighbors and a few
others, but eventually any description of a rule becomes hard, nay,
impossible,
to implement with short words or expressions.

Cellular automata are subject to a phenomon called shift-periodicity, which
makes them a special object of study. Automata in general are objects with
states, interconnected in ways which could be called neighborhoods. But there
need be no uniformity, either in the number of states nor in the connections.
Such symmetry, especially when spread out over a crystallographic lattice,
when present splits the behavioral studies into local parts, which can then
be copied globally. Shift periodicity results when the configuration of the
whole automaton repeats after a certain time, although it may reappear at
another place in the lattice, which is the shift. If a small part of the
configuration is observed to move that way, it is called a glider. The name
even applies to moving pieces where the rest of the configuration changes
less noticably, but their detection is usually possible in the simpler
context.

In fact, it seems that Conway coined the term {glider} from crystallographic
glide planes in his excitement over observing them for the first time in the
rule he was formulating.

Gliders are the most visible aspect of Class IV automata, and of course
gliders can collide with one another, to either disappear, cross over each
other, or initiate some other consequences. When collisions are clean enough
it is possible to think about computing, as for example in the
two-dimensional
automaton called WireWorld. But having gliders is a long way from concluding
that computing is going on.

The reason that the particular rule, Rule 110, has aroused such interest, is
that it is about the simplest automataton meeting the requirements for
computation if, in fact, it does. First, it is a binary rule, and second, it
has two neighbors. A one-neighbor automaton is trivial, whereas two neighbors
take their rules from the Boolean functions of two variables, whose behavior
is thoroughly familiar. Third, among the three neighbor automata, it is one
which has a large number of recognizable gliders.

Actually it has gliders of two kinds. The general gliders do not seem to be
especially noteworthy, but one set moves relative to a background of small
triangles in a way which is complicated yet holding hope of understanding it.
There are between eight and twelve gliders, some of which exist in alternate
forms. They have periods in the tens, twenties, and thirties, which means
that
it takes a reasonable time for them to go through their cycles. They move at
different velocities, but well under light velocity, which means that their
shifts are around half as far as their periods, but still an appreciable
distance. One family is static with a period seven, which coincides with the
period of the background. The background has a 14 x 7 unit cell, which is
noticably large, although the background triangle fits into a 5 x 5 square.

The triangles are, in fact, a defining feature of Rule 110. Altering the rule
in the sense of Wuensche's mutations always produces a defect in the
triangles, which are right isosceles triangles with the right angle at top
left. If it were not for a supplementary condition of never aligning two top
margins and allowing them to touch, Rule 110 would be an exercise in tiling
the plane (or at least a half-plane) with integer triangles on a square grid.

Another characteristic of Rule 110 is that it has a high membrane and
macrocell
index, meaning that all but one of the rules respect the spine of the
triangle,
meaning its left edge. The exception, or course, is 111 -> 0, needed in
defining the top edge. So the only way to interrupt a vertical column of 1's
is to place 1's on both sides of it. Were it not for this exception, a
vertical line would be a membrane, and all evolution would be confined to
its interior. So it could be said that we have a semipermeable membrane, and
the macrocells never get high enough to be well defined.

Of course, one might suspect that further tampering might degrade the tiling,
but it is easy enough to do, and is one of the first new suggestions that we
have seen in quite a while. And there is one interesting artifact - we can
have something akin to 30 degree triangles. Or more strictly, they are still
45 degree triangles, but with the hypotenuse on top rather than on the right.

When **Rule 110** is simply embedded into a (2,2) automaton, by ignoring the
outer
cells to the left and to the right, the result is Rule **3CFC3CFC** (expressed in
hexadecimal which is shorter and more convenient than decimal). There are 32
rules at Hamming Distance 1 from this base rule, whose basins are one of the
things which Wuensche's program computes; but it also shows sample evolutions
for them and other things as well.

No matter whether these mutants are useful rules, examining them can increase
one's understanding of Rule 110 itself. And who knows? Maybe there is
something interesting there.

The mutations procede by selecting a neighborhood and flipping the action of
Rule 110, but according to the margins of the original neighborhood within
the
extended neighborhood. This is just one of many ways of introducing a flip.

**I. Neighborhood 000 -> 0, which is responsible for conserving the interior of
triangles. Flipping it essentially destroys large triangles. **

Well, it took the major part of a day or two just to run out samples of all this. NXLCAU22 is notably slower than NXLCAU21, especially in generating screens of evolution. So trying to get a lot of cases on one page will require running things overnight, or on a machine nobody is using or whatever.

Also, the analyses given to Rule 110 in terms of the de Bruijn diagram and the subset diagram are so slow that only a couple of generations are possible. But we need 10, at least. Of course there is no reason to bother if the rule doesn't look promising in the first place. If one does look promising, it would be interesting to know if the prohibition on large T's holds, whether it has the same glider structure, and so on. .

The cycle, or basis, diagrams run faster, more or less as for the three neighbor automata, but they haven't been especially useful in analyzing Rule 110, and probably won't be for the mutants either.

- hvm

**Solutions of simple wave equations**

The simplest wave equations are partial differential equations, equating
derivatives of one function of two variables to another. One way to work
with partial differential equations is to try separation of variables, in
which f(x,y) = X(x)Y(y). In German, this is called an Ansatz, and with a
little algebra and luck, you get a function of one variable alone equal to
a function of the other variable alone. That is only possible if both are
constant, whose common variable is called the variable of separation. In
the case of Wave equations, it is a frequency, but for Dirac or Schroedinger
euations it is called an energy, due to its meaning in the applications.

When distance and time are the variables and there is no explicit dependence
on the time, some time derivative of the time function is proportional to
itself, so the solution is a complex exponential. In the simplest equations
the same thing happens for the space variable, which leaves a product of two
exponentials, one in time with an energy exponential, the other in space with
a wave number exponential. The two names mean the same thing, but different
words are used to distinguish between them when it comes to solving the rest
of the equation, or more complicated equations.

If the equation is linear, any linear combination of solutions is a solution,
so it is worth looking for a basis and thinking in terms of linear algebra
with eigenvalues and eigenvectore. Due to the simplicity of the time part,
this usually means finding a basis for the space part, multiplying by the
time exponential, and summing (or integrating, as the case may be).

In the meantime, it is worth noting that f(x,y) = phi(x-y) will solve the
equation where two derivatives are equal, and phi(alpha x - beta t) will do
the job when there are some constant coefficients. When they are not constant,
pretending that they are is one way to get an approximate solution. The point
of this observation is that solutions can be constant along lines of a fixed
slope, which leads to the idea of things moving at constant velocity, even
though the thing that actually moves (the envelope of the solution) can be
fairly complicated.

When coefficients are not constant, motion occurs at different velocities for
different parts of the solution, and that is what is called dispersion; the
connection between frequence and wave number is called a dispersion relation
(or, for light, an index of refraction; think of prisms where different colors
move at different velocities and rays get bent by the prism).

With electrical, acoustical, or optic waves, the mathematics has all become
relatively familiar, so that there are not only methods of solution, but ways
of talking about them.

For the Schroedinger equation, the only real difference arises in working
with probababilities, and even then the problem is not so much with using
them as with understanding why, but that lies with the philosophical
foundations, not with applications. Even then, it would not be so bad if
solutions were normalizable. For bound states they are, and that can be used
as a sort of boundary condition. However, there is a whole theory of linear
ordinary differential equations which lets you treat everything form the same
systematic theory. Trouble is, neither quantum mechanics nor differential
equations are often taught that way.
Amongst other things complicating the applications is the fact that higher
order differential equations can be reduced to lower order differential
equations by introducing new variables and working with systems of equations.
The easiest substitution is to make the higher derivatives into new variables
and end up with their defining equations and the original equation forming
the system, in which only first derivatives appear. The process can also be
reversed; by calculating lots of derivatives, everything can sometimes be
crammed into one single equation of high order.

What happens is that the linear algebra approach gives a basis for solutions
via the Sturm-Liouville theory, but the basis is for the

Because of the way Dirac derived his equation, the starting point is a pair of
first order equations rather than a single second order equation, and the two
components are not derivatives of one another. You need the square root of the
sum of their squares, not one of them separately, to fit the initial equation
when solving the space-time equation by separation of variables. There are
quite a few ways to pick components, all giving the same length vector, amd
that is what is behind the paper of Wigner and Newton, as well as the paper of
Foldy and Wouthuysen. However, they don't quite see it that way, and they also
work with 3 + 1 dimensions, rather than 1 + 1, so spin gets involved as well.

To read these papers, it would be best to begin with simple wave packets,
studying the formation of a packet from the interference of just a few waves,
and getting a clear idea of the difference between phase velocity (the f(x -
t), above) and group velocity (which is how the combination behaves). The
page of Malincrodt has nice examples in Java, and they were on demonstration
at the meeting on Saturday. Because of the importance of Wuensche's visit, it
may be that few people saw them, but the should be on exhibition around the
CIEA; in principle everyone could download them for themselves. It would be
worth gerring the source code, if possible, because we will want to work with
ever more complicated combinations as time goes on.

We will see more about free particle wave packets when we begin to look at
the Heisenberg uncertainty relation and at coherent states, but for the moment
experimentation with the very simplest is in order.
Joining two or three sectors in each of which the potential is constant
gives a very popular form of demonstration, mainly because of the interesting
effects which arise when a wave packet makes the transition from one region
to another. Diffraction and refliction take place, with a lot of squiggles
visible in the wave packets. The article of Goldberg, Schey and Schwartz,
``Computer-Generated Motion Pictures of One-Dimensional Quantum Mechanical
Trasnsmission and Reflection Problems,'' is a classic, having been written
at a time when computer graphics were still very expensive and required
substantial work to create. Even the numerical analysis was a feat in itself;
not so much for originality as for carrying it out adequately in a practical
setting. We used to have a copy of the film, back at the ESFM, but eventually
it got loaned out without being returned so it is hard to know where it is now,
if it even still exists.

Such pictures formed an important part of Saxon's book on introductory
quantum mechanics. By now it is out of print, although I think we still have
our copy. I'm just not sure who's using it right now. But anyway, it is now
very easy to create one's own computer graphics, as well to find (hundreds) of
web sites. It is one thing to look at the pretty pictures; another to be sure
that you understand the velocities and dispersions, the meaning of the
ripples (including their wave length) where the collisions occur, how much
of the packet is transmitted, trapped, or transmitted, and so on.

The recent article by Nitta, Kudo and Minowa, ``Motion of a wave packet in the
Klein paradox,'' representa a similar computation, but in the context of the
Dirac Equation. In the article they remark on the long time lapsed since
Goldberg, Schey an Schwartz without anyone having made a similar study for
the Dirac equation.

Since I have been combing the literature for items to present in the course, I
can say that they seem to be right. To begin with, hardly anyone seems to be
willing to consider the Dirac equation in one dimension, where there is no
spin. But that is actually an advantage, because {spin} and {negative energy
states} are actually two quite different phenomona, which just happen to
coexist in the three dimensional Dirac equation.

However, Bernd Thaller, author of the book ``Visual Quantum Mechanics,'' has
some one-dimensional Dirac items on his website, promised for volume II of his
book. I hope that he has better luck meeting deadlines than Wolfram in his
``New Science,'' but nevertheless he seems to be two years overdue.

Nitta, Kudo and Minowa show three different ways to try to form an initial
wave packet, to give an idea that it is not such a well understood enterprise.
The article of Foldy and Wouthuysen goes to the heart of the matter, although
they too choose to work in three dimensions rather than one, thereby making
the analysis more complicated, and is probably the reason they didn't go
ahead and show some examples.

Part of what they do depends on the fact that you can not only form linear
combinations of the separated wave functions, but you can form the linear
combination with a matrix. They choose the option of diagonalizing the
operator which picks out states belonging to negative eigenvalues, which
procedes without further ado for plane waves. But it is still not all that
clear just what they have done, and we need to try out some variations on
the theme. Here also you get this Zitterbewegung, without the necessity of
having a collision with a potential step, which lack is also is a feature in
Thaller's demonstrations.

The reason for the Newton - Wigner paper is that they get similar results
form rather general relativistic arguments --- one idea is to try to find all
the Lorentz-invariant differential equations and see if they have some use or
other. Foldy-Wouthuysen worked out details; it all depends on a paper of Pryce,
and it may go back to the thirties. It is a simple fact that understanding the
Dirac equation has been a long time in coming, and we may still not know it so
very well.

The remaining paper in the collection, Chisholm's ``Generalizing the Heisenberg
Uncertainty Relation,'' is a recent version of attempts to understand the
spreading of wave packets in a more general context than an exercise in the
general properties of the Fourier Transform, although perhaps it offers new
insight into that as well. As time goes on, we will want to look at several
more articles of this general nature.

---

I have continued examining all the Hamming-distance-1 variants on Rule 110 as
a (2,2) rule, but I am not sure that there is an improved Rule 110 there. It
is worth recalling that Rule 110 itself is a mutant of the two-neighbor XOR
subject to turning into an OR when the left cell is 1, which is what creates
the semipermeable membrane. But as the earlier analysis showe, Rule 110 is
ideal for tiling the plane with isosceles integer right reiangles, and variants
don't seem to do any better. In fact, in my survey of Rule 110, these variants
were already considered and discarded.

I gusee that Wuensche's programs are pretty good for isolating Class IV Rules,
but Rule 110 seems to lie in a class by itself. In any event, NXLCAU22 won't
go very far in making the analysis via de Bruijn diagrams, subset diagram, etc
which NXLCAU made for Rule 110. So those who haven't seen Rule 110, or
automata in general, are welcome to play around with the program. Those who
have already worked with it exhaustively may prefer to concentrate on the
details of the gliders rather than provinf that there are gliders. But for
my part, I still haven't thought much about how to turn the superstable -
superneutral criterion into an equivalent when there is an ether background.
Which is clearly essential for Rule 110.

In that respect, several of the mutants seem to prefer T1 ethers, and have
gliders therein, but it is not clear whether there is a sufficient variety.

- hvm

In the end, it turns out that the Zitterbewegung is a sort of Gibbs phenomonon, and this can be deduced from Sakurai's book on Advanced Quantum Mechanics. What a simple solution to something which has been bothering people for 75 years or so, as is evident from the articles of Newton-Wigner and Foldy-Wouthuysen which everyone has to study. It is not that I think they (the two articles) are particularly understandable, especially if one does not have a background in quantum mechanics. Nevertheless they can be read from the point of view of how the authors seem to understand their subject, and how well they have succeeded in presenting it. The algebra is very nice, but I never thought that it was particularly clear what it all meant.

It will be interesting to see what Bernd Thaller does with the topic, in that second edition that we are waiting for.

It will be interesting to see what Bernd Thaller does with the topic, in that second edition that we are waiting for.

The Wolfram counter has incremented by two more months in at least one on-line bookstore.

I got a bounce on the last message from someone who had exceeded the Yahoo! quota. That may be from trying to download things found on the internet. Would it be better for everyone to try and get a CIEA address?

- hvm