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Is anyone using Mathematica to work on t'Hooft's new book on CA?

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Is anyone using Mathematica to work on t'Hooft's new book on CA? "The Cellular Automaton Interpretation of Quantum Mechanics":

t'Hooft's new book on CA

POSTED BY: Lee Bloomquist
Answer
8 months ago

I haven't seen this book, but at the 2015 Wolfram Summer School, Sibesh Kar was working on trying to understand those ideas from the point of view of simple computer experiments like those found in Stephen Wolfram's A New Kind of Science.

There were many directions this project could have gone in because of the wide variety of things t'Hooft was writing about.

It seemed like the type of cellular automaton rule with separate updates on the even cells and the odd cells was a reasonable intersection. t'Hooft has some papers showing how those updates can be analyzed like they were quantum mechanical operators, and at the same time it is possible to find simple rules which are candidates for universal behavior according to Wolfram's Principle of Computational Equivalence.

Here is an example of something Sibesh found, which also happens to be a reversible rule {57,99},

Even Odd update using ECAs 57 and 99

with this update definition

EvenOddUpdate[even_, odd_][init_] := 
 Riffle[CellularAutomaton[odd,  Riffle[init[[1 ;; ;; 2]], 
    #]][[1 ;; ;; 2]], #] &[ CellularAutomaton[even, init][[2 ;; ;; 2]]]

and running this evolution

ArrayPlot[NestList[EvenOddUpdate[57, 99], RandomInteger[1, 400], 400]]
POSTED BY: Todd Rowland
Answer
8 months ago

Thank you Todd, in this Fqxi essay submission there are "streams" and "Chu spaces." how would you suggest implementing these? LB

POSTED BY: Lee Bloomquist
Answer
8 months ago

Lee,

I'll take a look but first meet me halfway and tell me a little about these things and how you think an implementation might work.

Since this isn't a physics forum it is safe to assume that nobody is an expert, and you are free to keep things at an elementary level.

POSTED BY: Todd Rowland
Answer
8 months ago

The starting point is the "non-well founded" set.

https://plato.stanford.edu/entries/nonwellfounded-set-theory/

Jon Barwise and Lawrence Moss have written a book, "Vicious Circles,"

https://web.stanford.edu/group/cslipublications/cslipublications/site/1575860082.shtml

from which I got this idea of time:

properTime = (clockTime, properTime)

Next I modeled clockTime using nonstandard analysis:

clockTime = (nonstandardFuture, standardPresent, nonstandardPast)

Here clockTime is a "monad" as that term is used in nonstandard analysis. Around the standard part of the monad, a real number, there are nonstandard real numbers infinitely close.

H. Jerome Keisler has written a book on nonstandard analysis which you can download for free.

https://www.math.wisc.edu/~keisler/

Now I must confess that I don't yet have a license to Mathematica. So I'm probably putting the cart before the horse to bother you like this. I apologize.

LB

POSTED BY: Lee Bloomquist
Answer
8 months ago

Ok, there is a lot to say here, but I should first of all say that I am not really an expert. I think that there has been some implementation of nonstandard analysis in the Wolfram language, although I cannot remember where.

About using Wolfram language, you can get started without a license at the Wolfram Open Cloud.

About Chu spaces, there is a concrete definition given on page 4 of the Samson Abramsky paper that your FQXI paper references. There are two sets and an evaluation function from their Tuples ( a third set is implicit as the range of that function).

What I recommend is to start with the simplest case. Maybe have X, A, and K be two element sets and then find all e that satisfy the axioms. There are many ways to do functions in Wolfram language, but you'd want to pick a way that makes it easy to enumerate, e.g. using an array of values. You will want a way to visualize it. Having a multiplication table (like in your essay) is a fine thing. This could all be done using functions like Table for enumeration, and Select for picking out cases with the right properties, and ArrayPlot.

This follows a general scheme inspired by Wolfram's book. Enumerate the simplest cases and then see what they do.

In terms of the interpretation, it does not matter that it is too simple for what you want. The point is to be objective and you have to understand the simplest case first. Going to three element sets is not unreasonable.

In the further case that you want X to be a space and A to be a collection of sets in that space (maybe closed under intersection) that is also something you can enumerate. Then you can make more elaborate visualizations if you have one for X.

POSTED BY: Todd Rowland
Answer
8 months ago

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