An "Algebra" of Birds and Nests

An "Algebra" of Birds and Nests

Here is an attempt to describe the operations upon birds and nests. In the following diagrams the rule is that if X is given to a bird then copies of X appear on every nest reachable from that bird by following the directed arcs.

When you take out a nest with an egg and drop it you produce a bird/nest pair that can be visualized as

Figure 1 - A single bird and her nest (the I pattern)

If the nest N₁is copied we obtain

Figure 2 - A single bird and two nests (the V pattern)

If instead, B₁ is copied we obtain

Figure 3 - Two birds and their nest (the inverted V pattern)

If both B₁ and N₁ are copied we obtain

Figure 4 - Two birds and two nests (the X pattern)

If take two instances of Figure 2 and combine one nest of each (N₂ and N₃) by dropping one on the other you get

andbecomes

Figure 5 - Two birds and their two nests after one nest of each are joined (V+V becomes the W pattern)

If a bird's nest is copied and then combined (N₁ and N₂) you get

becomes

Figure 6- A bird and her two nests after the nests are joined (the V pattern becomes the I pattern)

In a post to the ToonTalk discussion list Jack Waugh suggested that the V pattern become a diamond pattern rather than an I pattern as in Figure 6. His idea is that since then there would be two paths to N_1/2 from B₁that two copies of everything B₁ receives should arrive on N_1/2. One would then have to introduce a new kind of entity to these diagrams -- invisible forwarders of messages. N₁ and N₂ would become forwarders to N_1/2 rather than replaced by N_1/2 as in these diagrams.

Furthermore in the same message, Jack suggests that ToonTalk be extended so if a bird is dropped on a bird they are joined. Then if birds from an inverted V pattern (Figure 3) are joined the sub-graph would either become an I pattern (consistent with Figure 6) or a diamond pattern (consistent with Jack's alternative for Figure 6). Again a diamond pattern requires new entities (bird forwarders) and results in duplicate messages arriving at the apex of the diamond.

Another suggestion Jack makes is that it should be possible to combine a bird and nest thereby eliminating both. This would be a way of connecting the birds of the annihilated nest to reach the nests of the annihilated bird. If the nest was a nest of the bird then it would simply be a way of transforming an I pattern to the null graph.