In the frame of chemlambda and g-patterns, here is how not to do the beta move. We pass from chemlambda to a slightly enlarged version, see the graphical formalism of projective conical spaces, which would correspond to an only local moves version of the whole GLC, with the emergent algebra nodes and moves.

Then we do emergent algebra moves instead.

Look, instead of the beta move (see here all moves with g-patterns)

L[a,d,k] A[k,b,c]

<–BETA–>

Arrow[a,c] Arrow[b,d]

lets do for an epsilon arbitrary the epsilon beta move

Remark that I don’t do the beta move, really. In g-patterns the epsilon beta move does not replace the LEFT pattern by another, only it **ADDS TO IT**.

L[a,d,k] A[k,b,c]

— epsilon BETA –>

FO[a,e,f] FO[b,g,h]

L[f,i,k] A[k,h,j]

epsilon[g,i,d] epsilon[e,j,c]

Here, of course, epsilon[g,i,d] is the new graphical element corresponding to a dilation node of coefficient epsilon.

Now, when epsilon=1 then we may apply only ext2 move and LOC pruning (i.e. emergent algebra moves)

and we get back the original g-pattern.

But if **epsilon goes to 0** then, only by emergent algebra moves:

that’s it the BETA MOVE is performed!

What is the status of the first reduction from the figure? Hm, in the figure appears a node which has a “0” as decoration. I should have written instead a limit when epsilon goes to 0… For the meaning of the node with epsilon=0 see the post Towards qubits: graphic lambda calculus over conical groups and the barycentric move. However, I don’t take the barycentric move BAR, here, as being among the allowed moves. Also, I wrote “epsilon goes to 0”, not “epsilon=0”.

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epsilon can be a complex number…

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**Questions/exercises: **

- the beta move pattern is still present after the epsilon BETA move, what happens if we continue with another, say a mu BETA move, for a mu arbitrary?
- what happens if we do a reverse regular BETA move after a epsilon beta move?
- why consider “epsilon goes to 0” instead of “epsilon = 0”?
- can we do the same for other moves, like DIST, for example?

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