[darcs-users] How to extend a patch theory to fully commute

Ben Franksen ben.franksen at online.de
Thu Jul 2 07:34:33 UTC 2020


Am 02.07.20 um 00:29 schrieb James Cook:
> Definition: A /patch address/ is a patch sequence address where the
> sequence (nj) has length one.
> 
> Sorry, there are a lot of definitions, but maybe you missed that one?

Oops. Yes, sorry. I was jumping around, trying to get a grip on the idea.

>> Indeed, given a minimal patch sequence address in its general form (a,
>> b, (Qi), (nj), X, Y), is this now a new *singular* patch in the extended
>> theory, regardless of how long the sequence Qi is? I don't see how how
>> this can work. Isn't that universe much larger than what was intended?
> 
> I intended only to include patch addresses, i.e. where (nj) has length
> one. On the other hand, (Qi) is not restricted in length.

Okay, thanks.

>> I think calculating this for a small but non-trivial example would be
>> highly instructive. Say we have primitive patches A;B;C, where neither
>> A;B nor B;C commute. What does the extended universe look like here?
> 
> I think that primitive patch theory has four contexts. Call them a, b,
> c, d: a A b B c C d.
> 
> The new patch theory should have 2^3=8 contexts. Here are the
> remaining four. (I'm using patches and their names interchangeably.)
> 
> (1) After just applying B: (a, c, [A, B], {B}, {A}). (This is the
> simple case of switching two patches.)
> 
> (2) After just applying C: (a, d, [A, B, C], {C}, {A, B}). (We can't
> get C without A and B, so we need everything here.)
> 
> (3) After applying A and C (missing B): (b, d, [B, C], {C}, {B}).
> (This is another example of the simple case of switching two patches.)
> 
> (4) After applying B and C: (a, d, [A, B, C], {B, C}, {A}). (This
> situation is similar to (2), but with everything reversed.)
> 
> The new patch theory should have 12 patches (a cube has 12 edges).
> Three are primitive, so there are 9 left to add.
> 
> (a) B in the sequence B;A;C. (In other words, commute A and B and look
> at B.) This patch's starting context is a, its ending context is (1)
> above, and as a canonical patch address, it is represented as: (a, c,
> [A, B], [B], {}, {A}). The first three elements of the tuple tell us
> this patch lives somewhere on the square of edges between a and c, and
> the last three elements locate the patch within the square: the patch
> is named B; nothing comes before the patch; and A comes after the
> patch.
> 
> (b) C in the sequence C;A;B (or C;B;A) is (a, d, [A, B, C], [C], {}, {A, B}).
> 
> (c) A in the sequence B;A;C is (a, c, [A, B], {B}, {})
> 
> (d) A in the sequence C;A;B is (a, d, [A, B, C], {C}, {B}).
> 
> (e) B in the sequence C;B;A is (a, d, [A, B, C], [B], {C}, {A}).
> 
> (f) C in the sequence A;C;B is (b, d, [B, C], {}, {B}).
> 
> (g) C in the sequence B;C;A is (a, d, [A, B, C], {B}, {A}).
> 
> (h) B in the sequence A,C;B is (b, d, [B, C], [B], {C}, {}).
> 
> (i) A in the sequence B;C;A (or C;B;A) is (a, d, [A, B, C], [A], {B, C}, {}).

Thanks for writing this down. I think you should add this example to
your text as it is a good illustration.

I cannot comment yet on the rest because I still haven't read those
later sections.

Cheers
Ben



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