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

[Ben Franksen]

Am 01.07.20 um 18:45 schrieb James Cook:
>> Just to check I understood the idea (up to this point): roughly
>> speaking, you represent "unrepresentable" states by formally commuting
>> patches.
>>
>> That is, if we regard the primitive states as vertices and primitive
>> (named) patches as edges in a graph, then you extend the graph with new
>> nodes and edges.
> 
> That's exactly the intuition.
> 
> If you had n patches that all commuted, this graph would be a
> hypercube, i.e. the nodes are the 2^n subsets of patches and each edge
> adds one patch.
> 
> I think my theory always extends this graph exactly to that hypercube,
> filling in exactly those nodes and edges that are missing. I haven't
> thought carefully about that point of view so maybe that's not true.
> 
>> The construction works by inductively taking any pair of incommutable
>> patches (in sequence, i.e. with a common middle state) and then
>> "formally" commuting it. The new node is represented as that pair of
>> patches. Is that, essentially, the idea?
> 
> Yes, that's exactly what happens when there are only two patches
> involved and they don't commute.

Okay, good. Now, you go on to define

> Definition: Given any primitive patch theory, the /extended patch
> universe/ is the set of all patch addresses. A primitive patch P going
> from context a to b is embedded as (a, b, [P], name of P, {}, {}).

Here you cleverly rename your concept. Before this point you talk about
patch /sequence/ addresses, now it's only patch addresses. Perhaps just
an ommision, but I suspect a bit of wishful thinking here.

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 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?

Cheers
Ben