One of the major themes of this series of

this series of lectures has been going on

the data side from one level of

description to another.

From data to data prime with some kind of

coarse graining prescription and then

asking the question, okay, if this was

your model for the data at this scale what

is the corresponding model prime for the

data at this scale here and this

relationship was the one that we

understood as the renormalization

relationship. And that goes all the way

from how a Markov chain coarse grains and

flows indeed to the south manifold of the

higher dimensional space that the Markov

chains originally lived in. It applies

just as well to how electrodynamics

changes as you go from the finer

grain scale, where

let's say you can observe electrons on,

let's say a scale of 1 millimeter, up to

a scale of, let's say, a meter, and that

renormalization there, as I indicated

could be understood as changing, not the

laws of electrodynamics, but just one of

its parameters, the electron charge as you

moved to different distances. This

operation here we've left somewhat

ambiguous. In each of the talks I told you

a coarse graining operation that we were

going to use. And we did the Markov chains

and said "okay, you have some finite time

resolution" when we came to study the

icing model I said "okay, look here's how

we're going to decimate, we have our grid

and what we're going to do is we're going

to take every other particle as you go

along the grid in these directions we're

going to average over every other particle

like that, or rather, trace over every

particle like that." The one time where we

really started to ask which coarse

graining do we want to use was when we

came to do the CAs, we looked at Israeli

and Goldenfeld's work, where we found is

that they were simultaneously solving for

the model, the g, that came from the f,

but also, solving for the projection

function that took the supercells and

mapped them into groups, into single cell

examples. And so I'll draw an example here

of how Goldenfeld and Israeli's projection

might work in some case, in fact it takes

blank spaces to a blank cell, but if

there's one filled-in cell, it always

takes it to a filled-in cell at the coarse

grain level description. What Israeli and

Goldenfeld were doing were simultaneously

solving for these two objects. And when

they did that, one of the things that we

talked quite a bit about was that they

found that in fact, Rule 110 could indeed

be efficiently coarse-grained. And that's

kind of remarkable, right? It's sort of

like saying, "you know, like, yeah, you

know your clock speed is 5 GHz and you

have you know, memory of, you know, 16 GB

but actually I can do what you think you

want to do, I could do it in half the

memory and half the time." Now, when we

actually came to look at what the coarse

graining was doing for Rule 110, we were

much less impressed. And for example, one

of the kinds of coarse grainings that

Israeli and Goldenfeld discovered was the

Garden of Eden coarse graining, which

turned out to be incredibly trivial. What

it did was it took a certain subset of

supercells that could never be produced by

Rule 110, not in fact blocks of two, they

had to go to a longer set of blocks to

find them. But they found these Garden of

Eden supercells and then projected the

whole world into Garden of Eden versus not

Garden of Eden, you know post-fall, right?

And then by projecting them into those two

spaces, then they could actually map

Rule 110 onto Rule 0. And yet they

satisfied what they wanted, and which

seems like a natural thing to satisfy,

which is the commutation of the diagram.

Right, the diagram they wanted to commute

was if you evolve on the fine grain scale

and project. If you use the f operation

twice and then project, it's the same as

using the projection and then the g

operation once. So these commuted and yet,

the answer was somewhat unsatisfying.

In the case of the icing model, we had

this goal, our goal was secretly to figure

out what was going on with phase

transitions in the sort-of two-dimensional

grid where all of a sudden at some

critical point you found that the whole

system coordinated. And so in the end they

said "you know, look, this was not the

world's greatest coarse graining, because

you couldn't quite get a solution, but it

was good enough." Always what's happening

in each of these stories, the Markov

chain, the cellular automata, the icing

model, the Krohn-Rhodes Theorem is that

secretly we have some idea of what we want

the data to do for us, and therefore, we

have some idea of what we want this

projection operator to be. And in a subset

of the cases, we also had an idea about g,

so if you think about the icing model case

we really didn't like that term that was

the quartet, sigma 1, sigma 2, sigma 3 ...

We actually just neglected it. And we

didn't like it because it made

calculations hard. So secretly, we also

have a little bit of a constraint on g,

but in general, what we were doing was

picking a p that we hoped did what we want

And that goes all the way back to the

Alice in Wonderland story that we began

with. Here's an image, here's the coarse

graining, do you like it?