Extras

Concurrency


           A[x] -> B.y
           P[z] -> Q.w
                ✅


              A[x] -> B.y
              P[z] -> Q.w
              C[m] -> D.k


              A[x] -> B.y
                   ✅
              C[m] -> D.k


                ✅
           P[z] -> Q.w
           C[m] -> D.k

Asynchrony

Asyncrony


              A[x] -> B.y
              P[z] -> A.w
              A[m] -> D.k

Asyncrony


              A[x] -> B.y
              P[z] -> A.w
              A[m] -> D.k


                ✅
           P[z] -> A.w
           A[m] -> D.k

Asyncrony


              A[x] -> B.y
              P[z] -> A.w
              A[m] -> D.k


             ✅ -> B.y
           P[z] -> A.w
           A[m] -> D.k


             ✅ -> ✅
           P[z] -> A.w
           A[m] -> D.k

Asyncrony


              A[x] -> B.y
              P[z] -> A.w
              A[m] -> D.k


                ✅ -> B.y
                   ✅
              A[m] -> D.k


                ✅
           P[z] -> A.w
           A[m] -> D.k

Asyncrony


             A[x] -> B.y
             P[z] -> A.w
                  ❌


              A[x] -> B.y
              P[z] -> A.w
              A[m] -> D.k


              A[x] -> B.y
                   ✅
              A[m] -> D.k


                ✅
           P[z] -> A.w
           A[m] -> D.k

Projection


        let x@A = crunch()
        let y@B = crunch()
        A[x] -> B.x
        B[y] -> A.y


        let x@A = crunch()
        let y@B = crunch()
        A[x] -> B.x
        B[y] -> A.y


        x = crunch()


        let x@A = crunch()
        let y@B = crunch()
        A[x] -> B.x
        B[y] -> A.y


        x = crunch()


        y = crunch()


        let x@A = crunch()
        let y@B = crunch()
        A[x] -> B.x
        B[y] -> A.y


        x = crunch()
        send("A",x)


        y = crunch()
        x = receive("A")


        let x@a = crunch()
        let y@b = crunch()
        A[x] -> B.x
        B[y] -> A.y


        x = crunch()
        send("B",x)
        y = receive("B")


        y = crunch()
        x = receive("A")
        send("A",y)

Confluence

project(p,c) = (succ,EP)

Where:

(p) is a process name
(c) is a choreography
(EP) is a single endpoint
(succ) is a boolean signaling success


    Definition project_def:
    ...
    ∧ project proc (Com p1 v1 p2 v2 c) =
        if proc = p1 ∧ proc = p2
        then (F,Nil)
        else if proc = p1
             then Send p2 v1 <Γ> project proc c
             else if proc = p2
                  then Receive p1 v2 <Γ> project proc c
                  else project proc c
    ...