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

Posted 8 years ago

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
6 Replies
POSTED BY: Todd Rowland

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

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

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

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
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