Scott Garrabrant

Finite Factored Sets

Sure, if you want to send me an email and propose some times, we could set up a half hour chat. (You could also wait until I post all the math over the next couple weeks.)

Finite Factored Sets

Looks like you copied it wrong. Your B only has one 4.

Finite Factored Sets

I have not thought much about applying to things other than finite sets. (I looked at infinite sets enough to know there is nontrivial work to be done.) I do think it is good that you are thinking about it, but I don't have any promises that it will work out.

What I meant when I think that this can be done in a categorical way is that I think I can define a nice symmetric monodical category of finite factored sets such that things like orthogonality can be given nice categorical definitions. (I see why this was a confusing thing to say.)

Finite Factored Sets

If I understand correctly, that definition is not the same. In particular, it would say that you can get nontrivial factorizations of a 5 element set: {{{0,1},{2,3,4}},{{0,2,4},{1,3}}}.

Finite Factored Sets

When I prove it, I prove and use (a slight notational variation on) these two lemmas.

- If , then for all .
- .

(These are also the two lemmas that I have said elsewhere in the comments look suspiciously like entropy.)

These are not trivial to prove, but they might help.

Finite Factored Sets

I think that you are pointing out that you might get a bunch of false positives in your step 1 after you let a thermodynamical system run for a long time, but they are are only approximate false positives.

Finite Factored Sets

I think my model has macro states. In game of life, if you take the entire grid at time t, that will have full history regardless of t. It is only when you look at the macro states (individual cells) that my time increases with game of life time.

Finite Factored Sets

As for entropy, here is a cute observation (with unclear connection to my framework): whenever you take two independent coin flips (with probabilities not 0,1, or 1/2), their xor will always be high entropy than either of the individual coin flips.

Finite Factored Sets

Wait, I misunderstood, I was just thinking about the game of life combinatorially, and I think you were thinking about temporal inference from statistics. The reversible cellular automaton story is a lot nicer than you'd think.

if you take a general reversible cellular automaton (critters for concreteness), and have a distribution over computations in general position in which initial conditions cells are independent, the cells may not be independent at future time steps.

If all of the initial probabilities are 1/2, you will stay in the uniform distribution, but if the probabilities are in general position, things will change, and time 0 will be special because of the independence between cells.

There will be other events at later times that will be independent, but those later time events will just represent "what was the state at time 0."

For a concrete example consider the reversible cellular automaton that just has 2 cells, X and Y, and each time step it keeps X constant and replaces Y with X xor Y.

Fixed, Thanks.