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:

The approximate sum (maybe emergent sum would be a better name) has the following associated graph in $GRAPH$:

The letters in red “” are there only for the convenience of the reader.

Likewise, the graph in which corresponds to the approximate difference (or emergent difference) is the following:

The graph which corresponds to is this one:

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

We are ready to apply an R2 move to get:

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

followed by local pruning

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

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

We use this move and we obtain:

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

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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