# How to Simulate Nested Parallelism on a Sequential Machine

So I have sets of functions $A$, $B$, $C$, $D$, $E$, and $F$. I want to run them in a nested way. I also want to run some of them in parallel, and some of them in sequence. Here is how that might look:

In Parallel A {
In Sequence B {
In Parallel C {
In Sequence D {
In Parallel E {
In Sequence F
}
}
}
}
}


Now I start getting lost on how this would look. Here is my attempt at explaining it...

### Part I

This part I can sort of understand.

Say we have $n$ $A$ processes running in parallel with $\land$.

$$A = a_1 \land a_2 \land \dots \land a_n$$

Within $a_1$, say we have 5 steps running in sequence with $\to$, and $a_2$ is 10 steps, etc. We have:

\begin{align} \land\ a_1 &= b_1 \to b_2 \to \dots \to b_5\\ \land\ a_2 &= b_1 \to b_2 \to \dots \to b_{10}\\ \land\ a_3 &= b_1 \to b_2\\ \dots\\ \land\ a_n &= b_1 \to b_2 \to \dots \to b_n \end{align}

Now, I would like for "all of $A$ to run at the same time". This means that each $a \in A$ is a whole process. That is, if $a_1$ takes 5 steps and $a_2$ takes 10 steps, the whole process won't start over until 10 steps later.

$$A_{1,s=10} \to A_{2,s=10} \to \dotsc \to A_{n,s=10}$$

That means it will sort of look like this:

\begin{align} A_1 &: s\ s\ s\ s\ s\ s\ s\ s\ s\ s\\ A_2 &: s\ s\ s\ s\ s\ w\ w\ w\ w\ w\\ A_3 &: s\ s\ w\ w\ w\ w\ w\ w\ w\ w\\ \dots\\ A_n &: s_n \dots w_n \end{align}

Where $s$ is a step and $w$ is waiting.

### Part II

This part is where I get lost.

Now I want to be able to just apply this reasoning to the nested parallel and sequential processes. Informally, so $A$ has started, this means $a_1$ has started, which means $b_1$ has started. Now $b_1$ wont complete until all of the $C$ nested within it have completed, but they are running in parallel.

$$b_1 = c_1 \land c_2 \land \dots \land c_n$$

So it's like:

| a1. . . . .           | a1. . . . .
| b1. . b2. .           | b1. . b2. .
| c .   c . .           | c .   c . .
| c . . c . .           | c . . c . .
| c .   c . .           | c .   c . .
| c .   c . .           | c .   c . .
| c .   c .             | c .   c .

| a2. . . . . . . . . . | a2. . . . . . . . . .
| b1. . . . b2. . . . . | b1. . . . b2. . . . .
| c . . .   c . . . . . | c . . .   c . . . . .
| c . .     c .         | c . .     c .
| c . . . . c . . .     | c . . . . c . . .
| c . .     c . . .     | c . .     c . . .
| c .       c . . . .   | c .       c . . . .


Then it goes into further nesting all the way to $F$, which would be hard to draw and is really hard to think about.

### Part III

This is where I get quite lost.

Now to model them sequentially.

I have just been wanting to do this:

for a in A:
for b in B:
for c in C:
run c...


But that isn't right. So then I've tried:

for a in A:
start a
for b in B:
start b
for c in C:
start c
for aa in start a
... hmm


Now that's not going to work.

So then it's like, if the functions were all flattened somehow, then we could just iterate through them. But I am already really confused by this point.

### Question

My question is, how I simulate this on a sequential machine.

Any help would be greatly appreciated.