## Blue cell

It’s not immediately obvious how the blue calculus lets you do object references, so here’s a worked example.

I’m going to define a process, called Reference, that takes a new variable and turns it into an object reference. Because it’s easier to type, I’m going to use \ for lambda and v for nu, and when I apply a term to something other than a variable, you’re supposed to do the embedding from lambda calculus into blue calculus in your head. I’m leaving it out because the embedding behaves the same as lambda calculus and things are confusing enough as it is.

Reference = \ref.vc.( c | c = \x.(ref <= Pair c (x | cx)) )

Notice the Pair combinator (Pair = \xyz.zxy); it’s being applied to two processes that function like write and read, respectively, and is waiting for a third; if the third input is K, we get Kxy = x, and if it’s KI, we get KIxy = Iy = y.

You create an object by feeding it a variable for the reference and an initial state:

Reference ref u = vc.(cu | c = \x.(ref <= Pair c (x | cx)))

Now if you have a reference with some state in parallel with a writer process, it updates the state:

(Reference ref u | ref K w)

= ( vc.(cu | c = \x.(ref <= Pair c (x | cx))) | ref K w )

= vc.(cu | c = \x.(ref <= Pair c (x | cx)) | ref K w ) [by scope migration]

= vc.(\x.(ref <= Pair c (x | cx)) u | c = \x.(ref <= Pair c (x | cx)) | ref K w ) [by resource fetching on c]

= vc.(ref <= Pair c (u | cu) | c = \x.(ref <= Pair c (x | cx)) | ref K w ) [by small beta reduction on u]

= vc.(c = \x.(ref <= Pair c (x | cx)) | Pair c (u | cu) K w ) [by resource fetching on ref]

= vc.(c = \x.(ref <= Pair c (x | cx)) | K c (u | cu) w ) [by definition of Pair = \xyz.zxy]

= vc.(c = \x.(ref <= Pair c (x | cx)) | cw ) [by definition of K = \xy.x]

= vc.(cw | c = \x.(ref <= Pair c (x | cx))) [by symmetry of |]

= Reference ref w

Similarly, if you put it in parallel with a reader process,

(Reference ref u | ref KI w x y … z )

= vc.(c = \x.(ref <= Pair c (x | cx)) | K I c (u | cu) w x y … z) [by the same as above]

= vc.(c = \x.(ref <= Pair c (x | cx)) | I (u | cu) w x y … z) [by the definition of K]

= vc.(c = \x.(ref <= Pair c (x | cx)) | (u | cu) w x y … z) [by the definition of I]

= vc.(c = \x.(ref <= Pair c (x | cx)) | uwxy…z | cuwxy…z) [by distributivity]

= vc.(c = \x.(ref <= Pair c (x | cx)) | uwxy…z | \x.(ref <= Pair c (x | cx)) uwxy…z) [by resource fetching on c]

= vc.(c = \x.(ref <= Pair c (x | cx)) | uwxy…z | (ref <= Pair c (u | cu)) wxy…z) [by small beta reduction on u]

= vc.(c = \x.(ref <= Pair c (x | cx)) | uwxy…z | (ref <= Pair c (u | cu))) [by small beta reduction on w, x, y, …, z]

= vc.(c = \x.(ref <= Pair c (x | cx)) | uwxy…z | \x.(ref <= Pair c (x | cx)) u) [by extensionality]

= vc.(c = \x.(ref <= Pair c (x | cx)) | uwxy…z | cu) [by extensionality]

= vc.(cu | c = \x.(ref <= Pair c (x | cx)) | uwxy…z) [by extensionality]

= (Reference ref u | uwxy…z)

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