Ouroboros predecessor (II): start of the healing process

Let’s continue from the place we stopped in  Ouroboros predecessor, another chemlambda reduction .  After the first reduction stage which involves 6 rewrites, we see that the signal is given for the start of the healing process.

 

pred_round_2

The signal for the healing is given by the beta reduction

L[59,59,23]  A[23,27,14] –beta–>

Arrow[59,14] Arrow[27,59]

The COMB moves are not figured. But they go like this in this case:

Arrow [59,14] Arrow[27,59] –COMB–>

Arrow[27,14]

and then

A[52,54,27] Arrow[27,14] –COMB–>

A[52,54,27]

In the third graph we see the element:

A[19,54,27]

which comes from yet another COMB move

A[52,54,27] Arrow[19,52] –COMB–>

A[19,54,27]

where the Arrow[19,52] comes from the FAN-IN move

FI[6,19,2] FO[2,52,53] –FAN-IN–>

Arrow[6,53] Arrow[19,52]

There are 8 rewrites per reduction step, starting from the 2nd figure. The repeating patterns are:

  • a train of 4 DIST patterns which travel along the external (in our arbitrary embedding in the plane)  ring of FO elements.
  • a “healing” pair which appears first time in the 3rd figure as A[30,50,31] L[31,30, 19]. This healing pair is consumed (by a beta move) each time, but is recreated. In the 4th figure we see it again as A[47,69,46] L[46,47,29], in the 5th figure appears a A[67,89,65] L[65,67,42]. It has the role to switch in the right position the ports of an A element, to form pairs A and FO which heal back the graph.
  • FI nodes in such a pattern so that at any step of the reduction only two FAN-IN moves are possible, one coming from a newly created DIST pattern, the other coming from an older FI node.

The number of nodes, from the 2nd to the 5th figure is the same.

What will happen next?

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