[Errors in rule equivalences table?] - A New Kind of Science: The NKS Forum

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Errors in rule equivalences table?

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Posted by: Philippe Capdepuy

Hello all,

I was quite surprised while having a look at the rule equivalences table p.883 that some rules have no equivalent, more precisely: 23, 51,77,105,150,178,204,232.

Shouldn't these be:
rule conj. refl. c.r.
23 232 232 23
51 204 204 51
77 178 178 77
105 150 150 105
150 105 105 150
178 77 77 178
204 51 51 204
232 23 23 232

?

Am I mistaken? Is there a reason for this?

I searched on the forum if the issue had already been raised but I couldn't find anything.

Thanks.


Posted by: Jason Cawley

You are correct, it is just incompleteness in the table in the book, presumably from a sloppy code line generating the table.

Here is the right way to generate it in Mathematica -

rules[x_] := 
 With[{digits = IntegerDigits[x, 2, 8]}, {x, 
   FromDigits[digits /. {0 -> 1, 1 -> 0}, 2],
   FromDigits[Reverse[digits], 2],
   FromDigits[Reverse[digits /. {0 -> 1, 1 -> 0}], 2]}]

TableForm[Partition[rules /@ (Range[256] - 1), 8], 
TableDepth -> 2]


Thanks for catching that...


Posted by: Sean Lynch

The rule equivalences are given by

1-Reverse[list], for switch black/white

list[[{1, 5, 3, 7, 2, 6, 4, 8}]], for switch left/right

and both operations for switch black/white and left/right.

The equivalences you give would be due to

Reverse[list], which does not give equivalent rules.

This can be seen by comparing the evolution of the rules you mention. You can see that they don't correspond to switching either black/white, right/left, or both.


Posted by: Sean Lynch

Ok, sorry about that. I was working on my previous post while Jason posted his.

But now I'm confused because the rules you mention don't seem to be equivalent.

For example rule 204 is the identity rule but rule 51 doesn't have the same property.


Posted by: Todd Rowland

Sean is right.

Here are invariant versions of the rules.

23 is the opposite of the majority

51 is the opposite of the center cell

77 is the center cell unless all cells are the same

105 is the color of which there are one or three in the neighborhood

150 is the color of which there are zero or two in the neighborhood

178 is the center cell only when all cells are the same

204 is the identity, i.e. the center cell

232 is the majority rule


Posted by: Jason Cawley

Sorry, I seem to have confused things rather than clarified them.


Posted by: Todd Rowland

I got rule 105 and rule 150 confused.


Posted by: Philippe Capdepuy

Oh ok I'm sorry about this. I thought black/white and left/right were applied to the rules' binary representations, and not their output.
Thanks very much.


Posted by: Emmanuel Garces

Rule 105 and rule 105 don't seem to be equivalent by symmetry or color switching. But It's easy to see that the output of rule 150 is embedded into the output of rule 105.

That implies that rule 105 is able to emulate the behaviour of rule 150 by pushing out the appropriate cells.

It is also observed that rule 150 can be emulated by itself by the same pushing out procedure.

This is interesting, because this proposes another kind of relations between rules.




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