Danila Fedorin
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---
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title: Compiling a Functional Language Using C++, Part 5 - Execution
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date: 2019-08-06T14:26:38-07:00
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tags: ["C and C++", "Functional Languages", "Compilers"]
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---
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{{< gmachine_css >}}
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We now have trees representing valid programs in our language,
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and it's time to think about how to compile them into machine code,
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to be executed on hardware. But __how should we execute programs__?
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The programs we define are actually lists of definitions. But
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you can't evaluate definitions - they just tell you, well,
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how things are defined. Expressions, on the other hand,
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can be simplified. So, let's start by evaluating
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the body of the function called `main`, similarly
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to how C/C++ programs start.
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Alright, we've made it past that hurdle. Next,
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to figure out how to evaluate expressions. It's easy
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enough with binary operators: `3+2*6` becomes `3+12`,
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and `3+12` becomes `15`. Functions are when things
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get interesting. Consider:
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```
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double (160+3)
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```
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There's many perfectly valid ways to evaluate the program.
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When we get to a function application, we can first evaluate
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the arguments, and then expand the function definition:
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```
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double (160+3)
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double 163
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163+163
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326
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```
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Let's come up with a more interesting program to illustrate
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execution. How about:
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```
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data Pair = { P Int Int }
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defn fst p = {
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case p of {
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P x y -> { x }
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}
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}
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defn snd p = {
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case p of {
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P x y -> { y }
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}
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}
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defn slow x = { returns x after waiting for 1 second }
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defn main = { fst (P (slow 320) (slow 6)) }
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```
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If we follow our rules for evaluating functions,
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the execution will follow the following steps:
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```
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fst (P (slow 320) (slow 6))
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fst (P 320 (slow 6)) <- after 1 second
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fst (P 320 6) <- after 1 second
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320
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```
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We waited for two seconds, even though we really only
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needed to wait one. To avoid this, we could instead
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define our function application to substitute in
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the parameters of a function before evaluating them:
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```
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fst (P (slow 320) (slow 6))
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(slow 320)
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320 <- after 1 second
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```
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This seems good, until we try doubling an expression again:
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```
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double (slow 163)
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(slow 163) + (slow 163)
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163 + (slow 163) <- after 1 second
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163 + 163 <- after 1 second
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326
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```
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With ony one argument, we've actually spent two seconds on the
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evaluation! If we instead tried to triple using addition,
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we'd spend three seconds.
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Observe that with these new rules (called "call by name" in programming language theory),
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we only waste time because we evaluate an expression that was passed in more than 1 time.
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What if we didn't have to do that? Since we have a functional language, there's no way
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that two expressions that are the same evaluate to a different value. Thus,
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once we know the result of an expression, we can replace all occurences of that expression
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with the result:
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```
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double (slow 163)
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(slow 163) + (slow 163)
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163 + 163 <- after 1 second
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326
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```
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We're back down to one second, and since we're still substituting parameters
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before we evaluate them, we still only take one second.
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Alright, this all sounds good. How do we go about implementing this?
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Since we're substituting variables for whole expressions, we can't
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just use values. Instead, because expressions are represented with trees,
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we might as well consider operating on trees. When we evaluate a tree,
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we can substitute it in-place with what it evaluates to. We'll do this
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depth-first, replacing the children of a node with their reduced trees,
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and then moving on to the parent.
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There's only one problem with this: if we substitute a variable that occurs many times
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with the same expression tree, we no longer have a tree! Trees, by definition,
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have only one path from the root to any other node. Since we now have
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many ways to reach that expression we substituted, we instead have a __graph__.
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Indeed, the way we will be executing our functional code is called __graph reduction__.
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### Building Graphs
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Naively, we might consider creating a tree for each function at the beginning of our
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program, and then, when that function is called, substituting the variables
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in it with the parameters of the application. But that approach quickly goes out
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the window when we realize that we could be applying a function
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multiple times - in fact, an arbitrary number of times. This means we can't
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have a single tree, and we must build a new tree every time we call a function.
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The question, then, is: how do we construct a new graph? We could
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reach into Plato's [Theory of Forms](https://en.wikipedia.org/wiki/Theory_of_forms) and
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have a "reference" tree which we then copy every time we apply the function.
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But how do you copy a tree? Copying a tree is usually a recursive function,
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and __every__ time that we copy a tree, we'll have to look at each node
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and decide whether or not to visit its children (or if it has any at all).
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If we copy a tree 100 times, we will have to look at each "reference"
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node 100 times. Since the reference tree doesn't change, __we'd
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be following the exact same sequence of decisions 100 times__. That's
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no good!
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An alternative approach, one that we'll use from now on, is to instead
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convert each function's expression tree into a sequence of instructions
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that you can follow to build an identical tree. Every time we have
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to apply a function, we'll follow the corresponding recipe for
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that function, and end up with a new tree that we continue evaluating.
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### G-machine
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"Instructions" is a very generic term. Specifically, we will be creating instructions
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for a [G-machine](https://link.springer.com/chapter/10.1007/3-540-15975-4_50),
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an abstract architecture which we will use to reduce our graphs. The G-machine
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is stack-based - all operations push and pop items from a stack. The machine
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will also have a "dump", which is a stack of stacks; this will help with
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separating evaluation of various graphs.
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We will follow the same notation as Simon Peyton Jones in
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[his book](https://www.microsoft.com/en-us/research/wp-content/uploads/1992/01/student.pdf)
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, which was my source of truth when implementing my compiler. The machine
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will be executing instructions that we give it, and as such, it must have
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an instruction queue, which we will reference as \\(i\\). We will write
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\\(x:i\\) to mean "an instruction queue that starts with
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an instruction x and ends with instructions \\(i\\)". A stack machine
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obviously needs to have a stack - we will call it \\(s\\), and will
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adopt a similar notation to the instruction queue: \\(a\_1, a\_2, a\_3 : s\\)
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will mean "a stack with the top values \\(a\_1\\), \\(a\_2\\), and \\(a\_3\\),
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and remaining instructions \\(s\\)". Finally, as we said, our stack
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machine has a dump, which we will write as \\(d\\). On this dump,
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we will push not only the current stack, but also the current
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instructions that we are executing, so we may resume execution
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later. We will write \\(\\langle i, s \\rangle : d\\) to mean
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"a dump with instructions \\(i\\) and stack \\(s\\) on top,
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followed by instructions and stacks in \\(d\\)".
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There's one more thing the G-machine will have that we've not yet discussed at all,
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and it's needed because of the following quip earlier in the post:
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> When we evaluate a tree, we can substitute it in-place with what it evaluates to.
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How can we substitute a value in place? Surely we won't iterate over the entire
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tree and look for an occurence of the tree we evaluted. Rather, wouldn't it be
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nice if we could update all references to a tree to be something else? Indeed,
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we can achieve this effect by using __pointers__. I don't mean specifically
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C/C++ pointers - I mean the more general concept of "an address in memory".
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The G-machine has a __heap__, much like the heap of a C/C++ process. We
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can create a tree node on the heap, and then get an __address__ of the node.
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We then have trees use these addresses to link their child nodes.
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If we want to replace a tree node with its reduced form, we keep
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its address the same, but change the value on the heap.
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This way, all trees that reference the node we change become updated,
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without us having to change them - their child address remains the same,
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but the child has now been updated. We represent the heap
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using \\(h\\). We write \\(h[a : v]\\) to say "the address \\(a\\) points
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to value \\(v\\) in the heap \\(h\\)". Now you also know why we used
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the letter \\(a\\) when describing values on the stack - the stack contains
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addresses of (or pointers to) tree nodes.
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_Compiling Functional Languages: a tutorial_ also keeps another component
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of the G-machine, the __global map__, which maps function names to addresses of nodes
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that represent them. We'll stick with this, and call this global map \\(m\\).
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Finally, let's talk about what kind of nodes our trees will be made of.
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We don't have to include every node that we've defined as a subclass of
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`ast` - some nodes we can compile to instructions, without having to build
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them. We will also include nodes that we didn't need for to represent expressions.
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Here's the list of nodes types we'll have:
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* `NInt` - represents an integer.
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* `NApp` - represents an application (has two children).
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* `NGlobal` - represents a global function (like the `f` in `f x`).
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* `NInd` - an "indrection" node that points to another node. This will help with "replacing" a node.
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* `NData` - a "packed" node that will represent a constructor with all the arguments.
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With these nodes in mind, let's try defining some instructions for the G-machine.
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We start with instructions we'll use to assemble new version of function body trees as we discussed above.
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First up is __PushInt__:
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{{< gmachine "PushInt" >}}
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{{< gmachine_inner "Before">}}
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\( \text{PushInt} \; n : i \quad s \quad d \quad h \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "After" >}}
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\( i \quad a : s \quad d \quad h[a : \text{NInt} \; n] \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "Description" >}}
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Push an integer \(n\) onto the stack.
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{{< /gmachine_inner >}}
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{{< /gmachine >}}
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Let's go through this. We start with an instruction queue
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with `PushInt n` on top. We allocate a new `NInt` with the
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number `n` on the heap at address \\(a\\). We then push
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the address of the `NInt` node on top of the stack. Next,
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__PushGlobal__:
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{{< gmachine "PushGlobal" >}}
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{{< gmachine_inner "Before">}}
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\( \text{PushGlobal} \; f : i \quad s \quad d \quad h \quad m[f : a] \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "After" >}}
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\( i \quad a : s \quad d \quad h \quad m[f : a] \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "Description" >}}
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Push a global function \(f\) onto the stack.
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{{< /gmachine_inner >}}
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{{< /gmachine >}}
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We don't allocate anything new on the heap for this one -
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we already have a node for the global function. Next up,
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__Push__:
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{{< gmachine "Push" >}}
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{{< gmachine_inner "Before">}}
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\( \text{Push} \; n : i \quad a_0, a_1, ..., a_n : s \quad d \quad h \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "After" >}}
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\( i \quad a_n, a_0, a_1, ..., a_n : s \quad d \quad h \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "Description" >}}
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Push a value at offset \(n\) from the top of the stack onto the stack.
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{{< /gmachine_inner >}}
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{{< /gmachine >}}
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We define this instruction to work if and only if there exists an address
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on the stack at offset \\(n\\). We take the value at that offset, and
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push it onto the stack again. This can be helpful for something like
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`f x x`, where we use the same tree twice. Speaking of that - let's
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define an instruction to combine two nodes into an application:
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{{< gmachine "MkApp" >}}
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{{< gmachine_inner "Before">}}
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\( \text{MkApp} : i \quad a_0, a_1 : s \quad d \quad h \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "After" >}}
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\( i \quad a : s \quad d \quad h[ a : \text{NApp} \; a_0 \; a_1] \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "Description" >}}
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Apply a function at the top of the stack to a value after it.
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{{< /gmachine_inner >}}
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{{< /gmachine >}}
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We pop two things off the stack: first, the thing we're applying, then
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the thing we apply it to. We then create a new node on the heap
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that is an `NApp` node, with its two children being the nodes we popped off.
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Finally, we push it onto the stack.
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Let's try use these instructions to get a feel for it. In
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order to conserve space, let's use \\(\\text{G}\\) for PushGlobal,
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\\(\\text{I}\\) for PushInt, and \\(\\text{A}\\) for PushApp.
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Let's say we want to construct a graph for `double 326`. We'll
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use the instructions \\(\\text{I} \; 326\\), \\(\\text{G} \; \\text{double}\\),
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and \\(\\text{A}\\). Let's watch these instructions play out:
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{{< latex >}}
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\begin{aligned}
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[\text{I} \; 326, \text{G} \; \text{double}, \text{A}] & \quad s \quad & d \quad & h \quad & m[\text{double} : a_d] \\
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[\text{G} \; \text{double}, \text{A}] & \quad a_1 : s \quad & d \quad & h[a_1 : \text{NInt} \; 326] \quad & m[\text{double} : a_d] \\
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[\text{A}] & \quad a_d, a_1 : s \quad & d \quad & h[a_1 : \text{NInt} \; 326] \quad & m[\text{double} : a_d] \\
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[] & \quad a_2 : s \quad & d \quad & h[\substack{\begin{aligned}a_1 & : \text{NInt} \; 326 \\ a_2 & : \text{NApp} \; a_d \; a_1 \end{aligned}}] \quad & m[\text{double} : a_d] \\
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\end{aligned}
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{{< /latex >}}
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How did we come up with these instructions? We'll answer this question with
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more generality later, but let's take a look at this particular expression
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right now. We know it's an application, so we'll be using MkApp eventually.
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We also know that MkApp expects two values on top of the stack from
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which to make an application. The node on top has to be the function, and the next
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node is the value to be passed into that function. Since a stack is first-in-last-out,
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for the function (`double`, in our case) to be on top, we need
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to push it last. Thus, we push `double` first, then 326. Finally,
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we call MkApp now that the stack is in the right state.
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Having defined instructions to __build__ graphs, it's now time
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to move on to instructions to __reduce__ graphs - after all,
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we're performing graph reduction. A crucial instruction for the
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G-machine is __Unwind__. What Unwind does depends on what
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nodes are on the stack. Its name comes from how it behaves
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when the top of the stack is an `NApp` node that is at
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the top of a potentially long chain of applications: given
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an application node, it pushes its left hand side onto the stack.
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It then __continues to run Unwind__. This is effectively a while loop:
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applications nodes continue to be expanded this way until the left
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hand side of an application is finally something
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that __isn't__ an application. Let's write this rule as follows:
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{{< gmachine "Unwind-App" >}}
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{{< gmachine_inner "Before">}}
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\( \text{Unwind} : i \quad a : s \quad d \quad h[a : \text{NApp} \; a_0 \; a_1] \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "After" >}}
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\( \text{Unwind} : i \quad a_0, a : s \quad d \quad h[ a : \text{NApp} \; a_0 \; a_1] \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "Description" >}}
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Unwind an application by pushing its left node.
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{{< /gmachine_inner >}}
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{{< /gmachine >}}
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Let's talk about what happens when Unwind hits a node that isn't an application. Of all nodes
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we have described, `NGlobal` seems to be the most likely to be on top of the stack after
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an application chain has finished unwinding. In this case we want to run the instructions
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for building the referenced global function. Naturally, these instructions
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may reference the arguments of the application. We can find the first argument
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by looking at offset 1 on the stack, which will be an `NApp` node, and then going
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to its right child. The same can be done for the second and third arguments, if
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they exist. But this doesn't feel right - we don't want to constantly be looking
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at the right child of a node on the stack. Instead, we replace each application
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node on the stack with its right child. Once that's done, we run the actual
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code for the global function:
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{{< gmachine "Unwind-Global" >}}
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{{< gmachine_inner "Before">}}
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\( \text{Unwind} : i \quad a, a_0, a_1, ..., a_{n-1} : s \quad d \quad h[\substack{a : \text{NGlobal} \; n \; c \\ a_k : \text{NApp} \; a_{k-1} \; a_k'}] \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "After" >}}
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\( c \quad a_0', a_1', ..., a_{n-1}', a_{n-1} : s \quad d \quad h[\substack{a : \text{NGlobal} \; n \; c \\ a_k : \text{NApp} \; a_{k-1} \; a_k'}] \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "Description" >}}
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Call a global function.
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{{< /gmachine_inner >}}
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{{< /gmachine >}}
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In this rule, we used a general rule for \\(a\_k\\), in which \\(k\\) is any number
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between 1 and \\(n-1\\). We also expect the `NGlobal` node to contain two parameters,
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\\(n\\) and \\(c\\). \\(n\\) is the arity of the function (the number of arguments
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it expects), and \\(c\\) are the instructions to construct the function's tree.
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The attentive reader will have noticed a catch: we kept \\(a\_{n-1}\\) on the stack!
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This once again goes back to replacing a node in-place. \\(a\_{n-1}\\) is the address of the "root" of the
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whole expression we're simplifying. Thus, to replace the value at this address, we need to keep
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the address until we have something to replace it with.
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There's one more thing that can be found at the leftmost end of a tree of applications: `NInd`.
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We simply replace `NInd` with the node it points to, and resume Unwind:
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{{< gmachine "Unwind-Ind" >}}
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{{< gmachine_inner "Before">}}
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\( \text{Unwind} : i \quad a : s \quad d \quad h[a : \text{NInd} \; a' ] \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "After" >}}
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\( \text{Unwind} : i \quad a' : s \quad d \quad h[a : \text{NInd} \; a' ] \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "Description" >}}
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Replace indirection node with its target.
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{{< /gmachine_inner >}}
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{{< /gmachine >}}
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We've talked about replacing a node, and we've talked about indirection, but we
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haven't yet an instruction to perform these actions. Let's do so now:
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{{< gmachine "Update" >}}
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{{< gmachine_inner "Before">}}
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\( \text{Update} \; n : i \quad a,a_0,a_1,...a_n : s \quad d \quad h \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "After" >}}
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\( i \quad a_0,a_1,...,a_n : s \quad d \quad h[a_n : \text{NInd} \; a ] \quad m \)
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{{< /gmachine_inner >}}
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{{< gmachine_inner "Description" >}}
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Transform node at offset into an indirection.
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{{< /gmachine_inner >}}
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{{< /gmachine >}}
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This instruction pops an address from the top of the stack, and replaces
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a node at the given offset with an indirection to the popped node. After
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we evaluate a function call, we will use `update` to make sure it's
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not evaluated again.
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Now, let's talk about data structures. We have mentioned an `NData` node,
|
|
but we've given no explanation of how it will work. Obviously, we need
|
|
to distinguish values of a type created by different constructors:
|
|
If we have a value of type `List`, it could have been created either
|
|
using `Nil` or `Cons`. Depending on which constructor was used to
|
|
create a value of a type, we might treat it differently. Furthermore,
|
|
it's not always possible to know what constructor was used to
|
|
create what value at compile time. So, we need a way to know,
|
|
at runtime, how the value was constructed. We do this using
|
|
a __tag__. A tag is an integer value that will be contained in
|
|
the `NData` node. We assign a tag number to each constructor,
|
|
and when we create a node with that constructor, we set
|
|
the node's tag accordingly. This way, we can easily
|
|
tell if a `List` value is a `Nil` or a `Cons`, or
|
|
if a `Tree` value is a `Node` or a `Leaf`.
|
|
|
|
To operate on `NData` nodes, we will need two primitive operations: __Pack__ and __Split__.
|
|
Pack will create an `NData` node with a tag from some number of nodes
|
|
on the stack. These nodes will be placed into a dynamically
|
|
allocated array:
|
|
|
|
{{< gmachine "Pack" >}}
|
|
{{< gmachine_inner "Before">}}
|
|
\( \text{Pack} \; t \; n : i \quad a_1,a_2,...a_n : s \quad d \quad h \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "After" >}}
|
|
\( i \quad a : s \quad d \quad h[a : \text{NData} \; t \; [a_1, a_2, ..., a_n] ] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "Description" >}}
|
|
Pack \(n\) nodes from the stack into a node with tag \(t\).
|
|
{{< /gmachine_inner >}}
|
|
{{< /gmachine >}}
|
|
|
|
Split will do the opposite, by popping
|
|
of an `NData` node and moving the contents of its
|
|
array onto the stack:
|
|
|
|
{{< gmachine "Split" >}}
|
|
{{< gmachine_inner "Before">}}
|
|
\( \text{Split} : i \quad a : s \quad d \quad h[a : \text{NData} \; t \; [a_1, a_2, ..., a_n] ] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "After" >}}
|
|
\( i \quad a_1, a_2, ...,a_n : s \quad d \quad h[a : \text{NData} \; t \; [a_1, a_2, ..., a_n] ] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "Description" >}}
|
|
Unpack a data node on top of the stack.
|
|
{{< /gmachine_inner >}}
|
|
{{< /gmachine >}}
|
|
|
|
These two instructions are a good start, but we're missing something
|
|
fairly big: case analysis. After we've constructed a data type,
|
|
to perform operations on it, we want to figure out what
|
|
constructor and values which were used to create it. In order
|
|
to implement patterns and case expressions, we'll need another
|
|
instruction that's capable of making a decision based on
|
|
the tag of an `NData` node. We'll call this instruction __Jump__,
|
|
and define it to contain a mapping from tags to instructions
|
|
to be executed for a value of that tag. For instance,
|
|
if the constructor `Nil` has tag 0, and `Cons` has tag 1,
|
|
the mapping for the case expression of a length function
|
|
could be written as \\([0 \\rightarrow [\\text{PushInt} \; 0], 1 \\rightarrow [\\text{PushGlobal} \; \\text{length}, ...] ]\\).
|
|
Let's define the rule for it:
|
|
|
|
{{< gmachine "Jump" >}}
|
|
{{< gmachine_inner "Before">}}
|
|
\( \text{Jump} [..., t \rightarrow i_t, ...] : i \quad a : s \quad d \quad h[a : \text{NData} \; t \; as ] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "After" >}}
|
|
\( i_t, i \quad a : s \quad d \quad h[a : \text{NData} \; t \; as ] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "Description" >}}
|
|
Execute instructions corresponding to a tag.
|
|
{{< /gmachine_inner >}}
|
|
{{< /gmachine >}}
|
|
|
|
Alright, we've made it through the interesting instructions,
|
|
but there's still a few that are needed, but less shiny and cool.
|
|
For instance: imagine we've made a function call. As per the
|
|
rules for Unwind, we've placed the right hand sides of all applications
|
|
on the stack, and ran the instructions provided by the function,
|
|
creating a final graph. We then continue to reduce this final
|
|
graph. But we've left the function parameters on the stack!
|
|
This is untidy. We define a __Slide__ instruction,
|
|
which keeps the address at the top of the stack, but gets
|
|
rid of the next \\(n\\) addresses:
|
|
|
|
{{< gmachine "Slide" >}}
|
|
{{< gmachine_inner "Before">}}
|
|
\( \text{Slide} \; n : i \quad a_0, a_1, ..., a_n : s \quad d \quad h \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "After" >}}
|
|
\( i \quad a_0 : s \quad d \quad h \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "Description" >}}
|
|
Remove \(n\) addresses after the top from the stack.
|
|
{{< /gmachine_inner >}}
|
|
{{< /gmachine >}}
|
|
|
|
Just a few more. Next up, we observe that we have not
|
|
defined any way for our G-machine to perform arithmetic,
|
|
or indeed, any primitive operations. Since we've
|
|
not defined any built-in type for booleans,
|
|
let's avoid talking about operations like `<`, `==`,
|
|
and so on (in fact, we've omitted them from the grammar so far).
|
|
So instead, let's talk about the [closed](https://en.wikipedia.org/wiki/Closure_(mathematics)) operations,
|
|
namely `+`, `-`, `*`, and `/`. We'll define a special instruction for
|
|
them, called __BinOp__:
|
|
|
|
{{< gmachine "BinOp" >}}
|
|
{{< gmachine_inner "Before">}}
|
|
\( \text{BinOp} \; \text{op} : i \quad a_0, a_1 : s \quad d \quad h[\substack{a_0 : \text{NInt} \; n \\ a_1 : \text{NInt} \; m}] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "After" >}}
|
|
\( i \quad a : s \quad d \quad h[\substack{a_0 : \text{NInt} \; n \\ a_1 : \text{NInt} \; m \\ a : \text{NInt} \; (\text{op} \; n \; m)}] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "Description" >}}
|
|
Apply a binary operator on integers.
|
|
{{< /gmachine_inner >}}
|
|
{{< /gmachine >}}
|
|
|
|
Nothing should be particularly surprising here:
|
|
the instruction pops two integers off the stack, applies the given
|
|
binary operation to them, and places the result on the stack.
|
|
|
|
We're not yet done with primitive operations, though.
|
|
We have a lazy graph reduction machine, which means
|
|
something like the expression `3*(2+6)` might not
|
|
be a binary operator applied to two `NInt` nodes.
|
|
We keep around graphs until they __really__ need to
|
|
be reduced. So now we need an instruction to trigger
|
|
reducing a graph, to say, "we need this value now".
|
|
We call this instruction __Eval__. This is where
|
|
the dump finally comes in!
|
|
|
|
When we execute Eval, another graph becomes our "focus", and we switch
|
|
to a new stack. We obviously want to return from this once we've finished
|
|
evaluating what we "focused" on, so we must store the program state somewhere -
|
|
on the dump. Here's the rule:
|
|
|
|
{{< gmachine "Eval" >}}
|
|
{{< gmachine_inner "Before">}}
|
|
\( \text{Eval} : i \quad a : s \quad d \quad h \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "After" >}}
|
|
\( [\text{Unwind}] \quad [a] \quad \langle i, s\rangle : d \quad h \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "Description" >}}
|
|
Evaluate graph to its normal form.
|
|
{{< /gmachine_inner >}}
|
|
{{< /gmachine >}}
|
|
|
|
We store the current set of instructions and the current stack on the dump,
|
|
and start with only Unwind and the value we want to evaluate.
|
|
That does the job, but we're missing one thing - a way to return to
|
|
the state we placed onto the dump. To do this, we add __another__
|
|
rule to Unwind:
|
|
|
|
{{< gmachine "Unwind-Return" >}}
|
|
{{< gmachine_inner "Before">}}
|
|
\( \text{Unwind} : i \quad a : s \quad \langle i', s'\rangle : d \quad h[a : \text{NInt} \; n] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "After" >}}
|
|
\( i' \quad a : s' \quad d \quad h[a : \text{NInt} \; n] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "Description" >}}
|
|
Return from Eval instruction.
|
|
{{< /gmachine_inner >}}
|
|
{{< /gmachine >}}
|
|
|
|
Just a couple more special-purpose instructions, and we're done!
|
|
|
|
Sometimes, it's possible for a tree node to reference itself.
|
|
For instance, Haskell defines the
|
|
[fixpoint combinator](https://en.wikipedia.org/wiki/Fixed-point_combinator)
|
|
as follows:
|
|
```Haskell
|
|
fix f = let x = f x in x
|
|
```
|
|
|
|
In order to do this, an address that references a node must be present
|
|
while the node is being constructed. We define an instruction,
|
|
__Alloc__, which helps with that:
|
|
|
|
{{< gmachine "Alloc" >}}
|
|
{{< gmachine_inner "Before">}}
|
|
\( \text{Alloc} \; n : i \quad s \quad d \quad h \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "After" >}}
|
|
\( i \quad s \quad d \quad h[a_k : \text{NInd} \; \text{null}] \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "Description" >}}
|
|
Allocate indirection nodes.
|
|
{{< /gmachine_inner >}}
|
|
{{< /gmachine >}}
|
|
|
|
We can allocate an indirection on the stack, and call Update on it when
|
|
we've constructed a node. While we're constructing the tree, we can
|
|
refer to the indirection when a self-reference is required.
|
|
|
|
Lastly, we also define a Pop instruction, which just removes
|
|
some number of nodes from the stack. We want this because
|
|
calling Update at the end of a function modifies a node further up the stack,
|
|
leaving anything on top of the stack after that node as scratch work. We get
|
|
rid of that scratch work simply by popping it.
|
|
|
|
{{< gmachine "Pop" >}}
|
|
{{< gmachine_inner "Before">}}
|
|
\( \text{Pop} \; n : i \quad a_1, a_2, ..., a_n : s \quad d \quad h \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "After" >}}
|
|
\( i \quad s \quad d \quad h \quad m \)
|
|
{{< /gmachine_inner >}}
|
|
{{< gmachine_inner "Description" >}}
|
|
Pop \(n\) nodes from the stack.
|
|
{{< /gmachine_inner >}}
|
|
{{< /gmachine >}}
|
|
|
|
That's it for the instructions. Knowing them, however, doesn't
|
|
tell us what to do with our `ast` structs. We'll need to define
|
|
rules to translate trees into these instructions, and I've already
|
|
alluded to this when we went over `double 326`.
|
|
However, this has already gotten pretty long,
|
|
so we'll do it in the next post: [Part 6 - Compilation]({{< relref "06_compiler_compilation.md" >}}).
|