A New Kind of Science: The NKS Forum > Pure NKS > Errors in rule equivalences table?
Author
Philippe Capdepuy
University of Hertfordshire
UK

Registered: Jul 2006
Posts: 2

Errors in rule equivalences table?

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.

Last edited by Philippe Capdepuy on 07-27-2006 at 11:51 AM

Report this post to a moderator | IP: Logged

07-27-2006 10:34 AM
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

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...

Report this post to a moderator | IP: Logged

07-27-2006 06:25 PM
Sean Lynch
Rowan University
New Jersey

Registered: Jul 2006
Posts: 13

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.

Report this post to a moderator | IP: Logged

07-27-2006 06:45 PM
Sean Lynch
Rowan University
New Jersey

Registered: Jul 2006
Posts: 13

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.

Report this post to a moderator | IP: Logged

07-27-2006 07:01 PM
Todd Rowland
Wolfram Research
Maryland

Registered: Oct 2003
Posts: 103

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

Report this post to a moderator | IP: Logged

07-27-2006 07:07 PM
Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

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

Report this post to a moderator | IP: Logged

07-27-2006 07:28 PM
Todd Rowland
Wolfram Research
Maryland

Registered: Oct 2003
Posts: 103

I got rule 105 and rule 150 confused.

Report this post to a moderator | IP: Logged

07-27-2006 08:25 PM
Philippe Capdepuy
University of Hertfordshire
UK

Registered: Jul 2006
Posts: 2

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.

Report this post to a moderator | IP: Logged

07-27-2006 11:17 PM
Emmanuel Garces

Registered: Jul 2005
Posts: 7

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.

Attachment: rule_150_into_rule_105.nb