Compilation and Partial Evaluation

Instead of interpreting an expression, which is performing a full evaluation, we can think about compiling it: translating it to a different language which we can later run more easily, more efficiently, on more platforms, etc. Another feature that is usually associated with compilation is that a lot more work was done at the compilation stage, making the actual running of the code faster.

For example, translating an AST into one that has de-Bruijn indexes instead of identifier names is a form of compilation — not only is it translating one language into another, it does the work involved in name lookup before the program starts running.

This is something that we can experiment with now. An easy way to achieve this is to start with our evaluation function:

and change it so it compiles a given expression to a Racket function. (This is, of course, just to demonstrate a conceptual point, it is only the tip of what compilers actually do…) This means that we need to turn it into a function that receives a TOY expression and compiles it. In other words, eval no longer consumes and environment argument which makes sense because the environment is a place to hold run-time values, so it is a data structure that is not part of the compiler (it is usually represented as the call stack).

So we split the two arguments into a compile-time and run-time, which can be done by simply currying the eval function — here this is done, and all calls to eval are also curried:

We also need to change the eval call in the main run function:

Not much has changed so far.

Note that in the general case of a compiler we need to run a program several times, so we’d want to avoid parsing it over and over again. We can do that by keeping a single parsed AST of the input. Now we went one step further by making it possible to do more work ahead and keep the result of the first “stage” of eval around (except that “more work” is really not saying much at the moment):

At this point, even though our “compiler” is not much more than a slightly different representation of the same functionality, we rename eval to compile which is a more appropriate description of what we intend it to do (so we change the purpose statement too):

Not much changed, still. We curried the eval function and renamed it to compile. But when we actually call compile almost nothing happens — all it does is create a Racket closure which will do the rest of the work. (And this closure closes over the given expression.)

Running this “compiled” code is going to be very much like the previous usage of eval, except a little slower, because now every recursive call involves calling compile to generate a closure, which is then immediately used — so we just added some allocations at the recursive call points! (Actually, the extra cost is minimal because the Racket compiler will optimize away such immediate closure applications.)

Another way to see how this is not really a compiler yet is to consider when compile gets called. A proper compiler is something that does all of its work before running the code, which means that once it spits out the compiled code it shouldn’t be used again (except for compiling other code, of course). Our current code is not really a compiler since it breaks this feature. (For example, if GCC behaved this way, then it would “compile” files by producing code that invokes GCC to compile the next step, which, when run, invokes GCC again, etc.)

However, the conceptual change is substantial — we now have a function that does its work in two stages — the first part gets an expression and can do some compile-time work, and the second part does the run-time work, and includes anything inside the (lambda (env) …). The thing is that so far, the code does nothing at the compilation stage (remember: only creates a closure). But because we have two stages, we can now shift work from the second stage (the run-time) to the first (the compile-time).

For example, consider the following simple example:

We can do the same thing here — separate foo it into two stages using currying, and modify bar appropriately:

Now instead of a simple multiplication, lets expand it a little, for example, do a case split on common cases where x is 0, 1, or 2:

This is not much faster, since Racket already optimizes multiplication in a similar way.

Now comes the real magic: deciding what branch of the cond to take depends only on x, so we can push the lambda inside:

We just made an improvement — the comparisons for the common cases are now done as soon as (foo x) is called, they’re not delayed to when the resulting function is used. Now go back to the way this is used in bar and make it call foo once for the given c:

Now foo-c is generated once, and if c happens to be one of the three common cases (as in the last expression), we can avoid doing any multiplication. (And if we hit the default case, then we’re doing the same thing we did before.)

[However, the result runs at roughly the same speed! This heavily depends on what kind of optimizations the compiler can do, in this case, optimizing multiplications (which are essentially a single machine-code instruction) vs optimizing multiple-stage function calls.]

Here is another useful example that demonstrates this:

(Question: when can you do that?)

This is not unique to Racket, it can happen in any language. Racket (or any language with first class function values) only makes it easy to create a local function that is specialized for the flag.

Getting our thing closer to a compiler is done in a similar way — we push the (lambda (env) ...) inside the various cases. (Note that compile* depends on the env argument, so it also needs to move inside — this is done for all cases that use it, and will eventually go away.) We actually need to use (lambda ([env : ENV]) ...) though, to avoid upsetting the type checker:

and with this we shifted a bit of actual work to compile time — the code that checks what structure we have, and extracts its different slots. But this is still not good enough — it’s only the first top-level cases that is moved to compile-time — recursive calls to compile are still there in the resulting closures. This can be seen by the fact that we have those calls to compile in the Racket closures that are the results of our compiler, which, as discussed above, mean that it’s not an actual compiler yet.

For example, consider the Bind case:

At compile-time we identify and deconstruct the Bind structure, then create a the runtime closure that will access these parts when the code runs. But this closure will itself call compile on bound-body and each of the expressions. Both of these calls can be done at compile time, since they only need the expressions — they don’t depend on the environment. Note that compile* turns to run here, since all it does is run a compiled expression on the current environment.

We can move it back up, out of the resulting functions, by making it a function that consumes an environment and returns a “caller” function:

Once this is done, we have a bunch of work that can happen at compile time: we pre-scan the main “bind spine” of the code.

We can deal in a similar way with other occurrences of compile calls in compiled code. The two branches that need to be fixed are:

1. In the If branch, there is not much to do. After we make it pre-compile the cond-expr, we also need to make it pre-compile both the then-expr and the else-expr. This might seem like doing more work (since before changing it only one would get compiled), but since this is compile-time work, then it’s not as important. Also, if expressions are evaluated many times (being part of a loop, for example), so overall we still win.

2. The Call branch is a little trickier: the problem here is that the expressions that are compiled are coming from the closure that is being applied. The solution for this is obvious: we need to change the closure type so that it closes over compiled expressions instead of over plain ones. This makes sense because closures are run-time values, so they need to close over the compiled expressions since this is what we use as “code” at run-time.

Again, the goal is to have no compile calls that happen at runtime: they should all happen before that. This would allow, for example, to obliterate the compiler once it has done its work, similar to how you don’t need GCC to run a C application. Yet another way to look at this is that we shouldn’t look at the AST at runtime — again, the analogy to GCC is that the AST is a data structure that the compiler uses, and it does not exist at runtime. Any runtime reference to the TOY AST is, therefore, as bad as any runtime reference to compile.

When we’re done with this process we’ll have something that is an actual compiler: translating TOY programs into Racket closures. To see how this is an actual compiler consider the fact that Racket uses a JIT to translate bytecode into machine code when it’s running functions. This means that the compiled version of our TOY programs are, in fact, translated all the way down to machine code.

Yet another way to see this is to change the compiler code so instead of producing a Racket closure it spits out the Racket code that makes up these closures when evaluated. For example, change

into

so we get a string that is a Racket program. But since we’re using a Lisp dialect, it’s generally better to use S-expressions instead:

(Later in the course we’ll talk about these “`“s and “,“s. For now, it’s enough to know that “`” is kind of like a quote, and “,” is an unquote.)

Lazy Evaluation: Using a Lazy Racket

PLAI §7 (done with Haskell)

For this part, we will use a new language, Lazy Racket.

As the name suggests, this is a version of the normal (untyped) Racket language that is lazy.

First of all, let’s verify that this is indeed a lazy language:

That went without a problem — the argument expression was indeed not evaluated. In this language, you can treat all expressions as future promises to evaluate. There are certain points where such promises are actually forced, all of these stem from some need to print a resulting value, in our case, it’s the REPL that prints such values:

The expression by itself only generates a promise, but when we want to print it, this promise is forced to evaluate — this forces the addition, which forces its arguments (plain values rather than computation promises), and at this stage we get an error. (If we never want to see any results, then the language will never do anything at all.) So a promise is forced either when a value printout is needed, or if it is needed to recursively compute a value to print:

Note that the error was raised by the internal expression: the outer expression uses *, and + requires actual values not promises.

Another example, which is now obvious, is that we can now define an if function:

Actually, in this language if, and, and or are all function values instead of special forms:

(By now, you should know that these have no value in Racket — using them like this in plain will lead to syntax errors.) There are some primitives that do not force their arguments. Constructors fall in this category, for example cons and list:

Nothing — the definition simply worked, but that’s expected, since nothing is printed. If we try to inspect this value, we can get some of its parts, provided we do not force the bogus one:

The same holds for cons:

Now if this is the case, then how about this:

Everything is fine, as expected — but what is the value of ones now? Clearly, it is a list that has 1 as its first element:

But what do we have in the tail of this list? We have ones which we already know is a list that has 1 in its first place — so following Racket’s usual rules, it means that the second element of ones is, again, 1. If we continue this, we can see that ones is, in fact, an infinite list of 1s:

In this sense, the way define behaves is that it defines a true equation: if ones is defined as (cons 1 ones), then the real value does satisfy

which means that the value is the fixpoint of the defined expression.

We can use append in a similar way:

This looks like it has some common theme with the discussion of implementing recursive environments — it actually demonstrates that in this language, letrec can be used for simple values too. First of all, a side note — here an expression that indicated a bug in our substituting evaluator:

When our evaluator returned 1 for this, we noticed that this was a bug: it does not obey the lexical scoping rules. As seen above, Lazy Racket is correctly using lexical scope. Now we can go back to the use of letrec — what do we get by this definition:

we get an error about xs being undefined.

xs is unbound because of the usual scope that let uses. How can we make this work? — We simply use letrec:

As expected, if we try to print an infinite list will cause an infinite loop, which DrRacket catches and prints in that weird way:

How would we inspect an infinite list? We write a function that returns part of it:

Dealing with infinite lists can lead to lots of interesting things, for example:

To see how it works, see what you know about fibs[n] which will be our notation for the nth element of fibs (starting from 1):

and for all n>2:

so it follows the exact definition of Fibonacci numbers.

Note that the list examples demonstrate that laziness applies to nested values (actually, nested computations) too: a value that is not needed is not computed, even if it is contained in a value that is needed. For example, in:

the if needs to know only whether its first argument (note: it is an argument, since this if is a function) is #f or not. Once it is determined that it is a pair (a cons cell), there is no need to actually look at the values inside the pair, and therefore (+ 1 x) (and more specifically, x) is never evaluated and we see no error.