Emergent sums and differences in graphic lambda calculus

See the page Graphic lambda calculus for background.

Here I want to discuss the treatment of one identity concerning approximate sums and differences in emergent algebras. The identity is the following:

\Delta^{x}_{\varepsilon}(u, \Sigma^{x}_{\varepsilon}(u,v)) = v

The approximate sum (maybe emergent sum would be a better name) \Sigma^{x}_{\varepsilon}(u,w) has the following associated graph in $GRAPH$:

emer_sum_1

The letters in red “x, u, w, \Sigma” are there only for the convenience of the reader.

Likewise, the graph in GRAPH which corresponds to the approximate difference (or emergent difference) \Delta^{x}_{\varepsilon}(u,w) is the following:

emer_diff_1

The graph which corresponds to \Delta^{x}_{\varepsilon}(u, \Sigma^{x}_{\varepsilon}(u,v)) is this one:

emer_dif_sum_1

By a succession of CO-ASSOC moves we arrive to this graph:

emer_dif_sum_2

We are ready to apply an R2 move to get:

emer_dif_sum_3

We use now an ext2 move at the node marked by “1”

emer_dif_sum_4

followed by local pruning

emer_dif_sum_5

Here comes the funny part! We cannot continue unless we work with a graph where at the edges marked by the red letters “x, u” we put two disjoint (not connected by edges) graphs in GRAPH, say X, U:

emer_dif_sum_6

Let us suppose that from the beginning we had X, U connected at the edges marked by the red letters x, u, and proceed further. My claim is that by three  GLOBAL FAN-OUT  moves we can make the following move

emer_dif_sum_7

We use this move and we obtain:

emer_dif_sum_8

As previously, we use an R2 move and another ext2 move to finally obtain this:

emer_dif_sum_9

which is the answer we were looking for. We could use GLOBAL PRUNING to get rid of the part of the graph which ends by a termination gate.

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