On the algebra of processes III

THE ALGEBRA OF SEQUENTIAL PROCESSES AS COSPANS(GRAPH/A): Infinite state case

Let us now assume that the graphs may be infinite. We will however make some finiteness assumptions
which correspond to discrete aspects of the processes.

Consider a cospan of graphs labelled in alphabet A,  X ← G → Y. First we will take the alphabet A to be finite. We could make a less drastic assumption but I want to concentrate attention on X, Y, and G.
A consequence of the fact that A is finite is that G breaks up into subgraphs G_a, one for each element a of A, but all with the same vertex set vert(G).

In considering infinite state processes with discrete control we assume that transitions occur at thresh-hold values. The assumption is that the set of states (or vertices) of G breaks up into a finite disjoint union of sets,
 for example  vert(G)=U₁+U₂+...+U_n.

What happens to a graph G when you break up its vertex set as a disjoint union  U₁+U₂+..+U_n?  For each i,j you get the set of edges from U_i to U_j which we shall call G(i,j). There are domain and codomain functions U_i ←  G(i,j)  →  U_j; that is a span from U_i to U_j. So a graph breaks up into a matrix of spans. This is just the fact that Span(Sets) has direct sums and a graph is an endo-arrow in Span(Sets).

What I have said about the graph G also happens to each graph G_a.

From this matrix of spans we can draw a finite graph with n-vertices  U₁ ,  U₂, ..., U_n, and edges labelled by spans of sets and letters of A. (We omit edges labelled by empty spans.)

I haven't spoken of the sets X and Y. The sets also break up (by extensivity) X=X₁+X₂+...+X_m, Y₁+Y₂+...+Y_k. The functions X → G and Y →  G also break up into functions, and we make one further assumption that all the parts of the functions  X → G and Y →  G are the identity function.

At the end of these assumptions a cospan of labelled graphs between  discrete graphs looks  like  a labelled version of the finite cospans we considered in :Processes II, that is something like:
where the U's are possibly infinite sets, R, S and T are spans of sets, a and b are labels.

What this has to do with simple (Turing equivalent) sequential programming I will describe in the next post.