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Add G-machine graph creation instructions to Part 5

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Danila Fedorin 2019-09-02 17:51:14 -07:00
parent 4d8d806706
commit a1244f201a
2 changed files with 196 additions and 3 deletions
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45
assets/scss/gmachine.scss Normal file
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@ -0,0 +1,45 @@
$basic-border: 1px solid #bfbfbf;
.gmachine-instruction {
display: flex;
border: $basic-border;
border-radius: 2px;
}
.gmachine-instruction-name {
padding: 10px;
border-right: $basic-border;
flex-grow: 1;
flex-basis: 20%;
text-align: center;
}
.gmachine-instruction-sem {
width: 100%;
flex-grow: 4;
}
.gmachine-inner {
border-bottom: $basic-border;
width: 100%;
&:last-child {
border-bottom: none;
}
}
.gmachine-inner-label {
padding: 10px;
font-weight: bold;
}
.gmachine-inner-text {
padding: 10px;
text-align: right;
flex-grow: 1;
}
.gmachine-instruction-name, .gmachine-inner-label, .gmachine-inner {
display: flex;
align-items: center;
}

154
content/blog/05_compiler_execution.md
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@ -4,6 +4,7 @@ date: 2019-08-06T14:26:38-07:00
draft: true
tags: ["C and C++", "Functional Languages", "Compilers"]
---
{{< gmachine_css >}}
We now have trees representing valid programs in our language,
and it's time to think about how to compile them into machine code,
to be executed on hardware. But __how should we execute programs__?
@ -134,12 +135,159 @@ to apply a function, we'll follow the corresponding recipe for
that function, and end up with a new tree that we continue evaluating.
### G-machine
"Instructions" is a very generic term. We will be creating instructions
"Instructions" is a very generic term. Specifically, we will be creating instructions
for a [G-machine](https://link.springer.com/chapter/10.1007/3-540-15975-4_50),
an abstract architecture which we will use to reduce our graphs. The G-machine
is stack-based - all operations push and pop items from a stack. The machine
will also have a "dump", which is a stack of stacks; this will help with
separating function calls.
Besides constructing graphs, the machine will also have operations that will aid
in evaluating graphs.
We will follow the same notation as Simon Peyton Jones in
[his book](https://www.microsoft.com/en-us/research/wp-content/uploads/1992/01/student.pdf)
, which was my source of truth when implementing my compiler. The machine
will be executing instructions that we give it, and as such, it must have
an instruction queue, which we will reference as \\(i\\). We will write
\\(x:i\\) to mean "an instruction queue that starts with
an instruction x and ends with instructions \\(i\\)". A stack machine
obviously needs to have a stack - we will call it \\(s\\), and will
adopt a similar notation to the instruction queue: \\(a\_1, a\_2, a\_3 : s\\)
will mean "a stack with the top values \\(a\_1\\), \\(a\_2\\), and \\(a\_3\\),
and remaining instructions \\(s\\)".
There's one more thing the G-machine will have that we've not yet discussed at all,
and it's needed because of the following quip earlier in the post:
> When we evaluate a tree, we can substitute it in-place with what it evaluates to.
How can we substitute a value in place? Surely we won't iterate over the entire
tree and look for an occurence of the tree we evaluted. Rather, wouldn't it be
nice if we could update all references to a tree to be something else? Indeed,
we can achieve this effect by using __pointers__. I don't mean specifically
C/C++ pointers - I mean the more general concept of "an address in memory".
The G-machine has a __heap__, much like the heap of a C/C++ process. We
can create a tree node on the heap, and then get an __address__ of the node.
We then have trees use these addresses to link their child nodes.
If we want to replace a tree node with its reduced form, we keep
its address the same, but change the value on the heap.
This way, all trees that reference the node we change become updated,
without us having to change them - their child address remains the same,
but the child has now been updated. We represent the heap
using \\(h\\). We write \\(h[a : v]\\) to say "the address \\(a\\) points
to value \\(v\\) in the heap \\(h\\)". Now you also know why we used
the letter \\(a\\) when describing values on the stack - the stack contains
addresses of (or pointers to) tree nodes.
_Compiling Functional Languages: a tutorial_ also keeps another component
of the G-machine, the __global map__, which maps function names to addresses of nodes
that represent them. We'll stick with this, and call this global map \\(m\\).
Finally, let's talk about what kind of nodes our trees will be made of.
We don't have to include every node that we've defined as a subclass of
`ast` - some nodes we can compile to instructions, without having to build
them. We will also include nodes that we didn't need for to represent expressions.
Here's the list of nodes types we'll have:
* NInt - represents an integer.
* NApp - represents an application (has two children).
* NGlobal - represents a global function (like the `f` in `f x`).
* NInd - an "indrection" node that points to another node. This will help with "replacing" a node.
* NData - a "packed" node that will represent a constructor with all the arguments.
With these nodes in mind, let's try defining some instructions for the G-machine.
We start with instructions we'll use to assemble new version of function body trees as we discussed above.
First up is __PushInt__:
{{< gmachine "PushInt" >}}
{{< gmachine_inner "Before">}}
\( \text{PushInt} \; n : i \quad s \quad h \quad m \)
{{< /gmachine_inner >}}
{{< gmachine_inner "After" >}}
\( i \quad a : s \quad h[a : \text{NInt} \; n] \quad m \)
{{< /gmachine_inner >}}
{{< gmachine_inner "Description" >}}
Push an integer \(n\) onto the stack.
{{< /gmachine_inner >}}
{{< /gmachine >}}
Let's go through this. We start with an instruction queue
with `PushInt n` on top. We allocate a new `NInt` with the
number `n` on the heap at address \\(a\\). We then push
the address of the `NInt` node on top of the stack. Next,
__PushGlobal__:
{{< gmachine "PushGlobal" >}}
{{< gmachine_inner "Before">}}
\( \text{PushGlobal} \; f : i \quad s \quad h \quad m[f : a] \)
{{< /gmachine_inner >}}
{{< gmachine_inner "After" >}}
\( i \quad a : s \quad h \quad m[f : a] \)
{{< /gmachine_inner >}}
{{< gmachine_inner "Description" >}}
Push a global function \(f\) onto the stack.
{{< /gmachine_inner >}}
{{< /gmachine >}}
We don't allocate anything new on the heap for this one -
we already have a node for the global function. Next up,
__Push__:
{{< gmachine "Push" >}}
{{< gmachine_inner "Before">}}
\( \text{Push} \; n : i \quad a_0, a_1, ..., a_n : s \quad h \quad m \)
{{< /gmachine_inner >}}
{{< gmachine_inner "After" >}}
\( i \quad a_n, a_0, a_1, ..., a_n : s \quad h \quad m \)
{{< /gmachine_inner >}}
{{< gmachine_inner "Description" >}}
Push a value at offset \(n\) from the top of the stack onto the stack.
{{< /gmachine_inner >}}
{{< /gmachine >}}
We define this instruction to work if and only if there exists an address
on the stack at offset \\(n\\). We take the value at that offset, and
push it onto the stack again. This can be helpful for something like
`f x x`, where we use the same tree twice. Speaking of that - let's
define an instruction to combine two nodes into an application:
{{< gmachine "MkApp" >}}
{{< gmachine_inner "Before">}}
\( \text{MkApp} : i \quad a_0, a_1 : s \quad h \quad m \)
{{< /gmachine_inner >}}
{{< gmachine_inner "After" >}}
\( i \quad a : s \quad h[ a : \text{NApp} \; a_0 \; a_1] \quad m \)
{{< /gmachine_inner >}}
{{< gmachine_inner "Description" >}}
Apply a function at the top of the stack to a value after it.
{{< /gmachine_inner >}}
{{< /gmachine >}}
We pop two things off the stack: first, the thing we're applying, then
the thing we apply it to. We then create a new node on the heap
that is an `NApp` node, with its two children being the nodes we popped off.
Finally, we push it onto the stack.
Let's try use these instructions to get a feel for it. To save some space,
let's assume that \\(m\\) contains \\(\\text{double} : a\_{\\text{double}}\\) and \\(\\text{halve} : a\_{\\text{halve}} \\).
For the same reason, let's also use
* \\(\\text{G}\\) for \\(\\text{PushGlobal}\\)
* \\(\\text{I}\\) for \\(\\text{PushInt}\\)
* \\(\\text{P}\\) for \\(\\text{Push}\\)
* \\(\\text{A}\\) for \\(\\text{MakeApp}\\)
Let's say we want to construct a graph for the expression `double 326`.
The sequence of instructions \\(\\text{I} \; 326, \\text{G} \; \\text{double},
\\text{A}\\) will do the trick. Let's
step through them:
$$
\\begin{align}
[\\text{I} \; 326, \\text{G} \; \\text{double}, \\text{A}] & \\quad s \\quad & h \\quad & m \\\\\\
[\\text{G} \; \\text{double},\\text{A} ] & \\quad a\_0 : s \\quad & h[a\_0 : \\text{NInt} \; 326] \\quad & m \\\\\\
[\\text{A}] & \\quad a\_{\\text{double}}, a\_0 : s \\quad & h[a\_0 : \\text{NInt} \; 326] \\quad & m \\\\\\
[] & \\quad a\_1: s \\quad & h[\; \\begin{aligned} a\_0 & : \\text{NInt} \; 326 \\\ a\_1 & : \\text{NApp} \; a\_{\\text{double}} \; a\_0 \\end{aligned} ] \\quad & m \\\\\\
\\end{align}
$$
We end up with a node, \\(a\_1\\), on top of the stack, which represents the application of `double` to `326`. You can see
how the notation gets unwieldy very quickly, so I'll try to steer clear of more examples like this.