Lexical Scope using Racket Closures

PLAI §11 (without the last part about recursion)

An alternative representation for an environment.

We’ve already seen how first-class functions can be used to implement “objects” that contain some information. We can use the same idea to represent an environment. The basic intuition is — an environment is a mapping (a function) between an identifier and some value. For example, we can represent the environment that maps 'a to 1 and 'b to 2 (using just numbers for simplicity) using this function:

An empty mapping that is implemented in this way has the same type:

We can use this idea to implement our environments: we only need to define three things — EmptyEnv, Extend, and lookup. If we manage to keep the contract to these functions intact, we will be able to simply plug it into the same evaluator code with no other changes. It will also be more convenient to define ENV as the appropriate function type for use in the VAL type definition instead of using the actual type:

Now we get to EmptyEnv — this is expected to be a function that expects no arguments and creates an empty environment, one that behaves like the empty-mapping function defined above. We could define it like this (changing the empty-mapping type to return a VAL):

but we can skip the need for an extra definition and simply return an empty mapping function:

(The un-Rackety name is to avoid replacing previous code that used the EmptyEnv name for the constructor that was created by the type definition.)

The next thing we tackle is lookup. The previous definition that was used is:

How should it be modified now? Easy — an environment is a mapping: a Racket function that will do the searching job itself. We don’t need to modify the contract since we’re still using ENV, except a different implementation for it. The new definition is:

Note that lookup does almost nothing — it simply delegates the real work to the env argument. This is a good hint for the error message that empty mappings should throw —

Finally, Extend — this was previously created by the variant case of the ENV type definition:

keeping the same type that is implied by this variant means that the new Extend should look like this:

The question is — how do we extend a given environment? Well, first, we know that the result should be mapping — a symbol -> VAL function that expects an identifier to look for:

Next, we know that in the generated mapping, if we look for id then the result should be val:

If the name that we’re looking for is not the same as id, then we need to search through the previous environment:

But we know what lookup does — it simply delegates back to the mapping function (which is our rest argument), so we can take a direct route instead:

To see how all this works, try out extending an empty environment a few times and examine the result. For example, the environment that we began with:

behaves in the same way (if the type of values is numbers) as

The new code is now the same, except for the environment code:

More Closures (on both levels)

Racket closures (= functions) can be used in other places too, and as we have seen, they can do more than encapsulate various values — they can also hold the behavior that is expected of these values.

To demonstrate this we will deal with closures in our language. We currently use a variant that holds the three pieces of relevant information:

We can replace this by a functional object, which will hold the three values. First, change the VAL type to hold functions for FunV values:

And note that the function should somehow encapsulate the same information that was there previously, the question is how this information is going to be done, and this will determine the actual type. This information plays a role in two places in our evaluator — generating a closure in the Fun case, and using it in the Call case:

we can simply fold the marked functionality bit of Call into a Racket function that will be stored in a FunV object — this piece of functionality takes an argument value, extends the closure’s environment with its value and the function’s name, and continues to evaluate the function body. Folding all of this into a function gives us:

where the values of bound-body, bound-id, and val are known at the time that the FunV is constructed. Doing this gives us the following code for the two cases:

And now the type of the function is clear:

And again, the rest of the code is unmodified:

Types of Evaluators

What we did just now is implement lexical environments and closures in the language we implement using lexical environments and closures in our own language (Racket)!

This is another example of embedding a feature of the host language in the implemented language, an issue that we have already discussed.

There are many examples of this, even when the two languages involved are different. For example, if we have this bit in the C implementation of Racket:

then the special semantics of evaluating a Racket and form is being inherited from C’s special treatment of &&. You can see this by the fact that if there is a bug in the C compiler, then it will propagate to the resulting Racket implementation too. A different solution is to not use && at all:

and we can say that this is even better since it evaluates the second expression in tail position. But in this case we don’t really get that benefit, since C itself is not doing tail-call optimization as a standard feature (though some compilers do so under some circumstances).

We have seen a few different implementations of evaluators that are quite different in flavor. They suggest the following taxonomy.

• A syntactic evaluator is one that uses its own language to represent expressions and semantic runtime values of the evaluated language, implementing all the corresponding behavior explicitly.

• A meta evaluator is an evaluator that uses language features of its own language to directly implement behavior of the evaluated language.

While our substitution-based FLANG evaluator was close to being a syntactic evaluator, we haven’t written any purely syntactic evaluators so far: we still relied on things like Racket arithmetics etc. The most recent evaluator that we have studied, is even more of a meta evaluator than the preceding ones: it doesn’t even implement closures and lexical scope, and instead, it uses the fact that Racket itself has them.

With a good match between the evaluated language and the implementation language, writing a meta evaluator can be very easy. With a bad match, though, it can be very hard. With a syntactic evaluator, implementing each semantic feature will be somewhat hard, but in return you don’t have to worry as much about how well the implementation and the evaluated languages match up. In particular, if there is a particularly strong mismatch between the implementation and the evaluated language, it may take less effort to write a syntactic evaluator than a meta evaluator. As an exercise, we can build upon our latest evaluator to remove the encapsulation of the evaluator’s response in the VAL type. The resulting evaluator is shown below. This is a true meta evaluator: it uses Racket closures to implement FLANG closures, Racket function application for FLANG function application, Racket numbers for FLANG numbers, and Racket arithmetic for FLANG arithmetic. In fact, ignoring some small syntactic differences between Racket and FLANG, this latest evaluator can be classified as something more specific than a meta evaluator:

• A meta-circular evaluator is a meta evaluator in which the implementation and the evaluated languages are the same.

(Put differently, the trivial nature of the evaluator clues us in to the deep connection between the two languages, whatever their syntactic differences may be.)

Feature Embedding

We saw that the difference between lazy evaluation and eager evaluation is in the evaluation rules for with forms, function applications, etc:

is eager, and

is lazy. But is the first rule really eager? The fact is that the only thing that makes it eager is the fact that our understanding of the mathematical notation is eager — if we were to take math as lazy, then the description of the rule becomes a description of lazy evaluation.

Another way to look at this is — take the piece of code that implements this evaluation:

and the same question applies: is this really implementing eager evaluation? We know that this is indeed eager — we can simply try it and check that it is, but it is only eager because we are using an eager language for the implementation! If our own language was lazy, then the evaluator’s implementation would run lazily, which means that the above applications of the eval and the subst functions would also be lazy, making our evaluator lazy as well.

This is a general phenomena where some of the semantic features of the language we use (math in the formal description, Racket in our code) gets embedded into the language we implement.

Here’s another example — consider the code that implements arithmetics:

what if it was written like this:

Would it still implement unlimited integers and exact fractions? That depends on the language that was used to implement it: the above syntax suggests C, C++, Java, or some other relative, which usually come with limited integers and no exact fractions. But this really depends on the language — even our own code has unlimited integers and exact rationals only because Racket has them. If we were using a language that didn’t have such features (there are such Scheme implementations), then our implemented language would absorb these (lack of) features too, and its own numbers would be limited in just the same way. (And this includes the syntax for numbers, which we embedded intentionally, like the syntax for identifiers).

The bottom line is that we should be aware of such issues, and be very careful when we talk about semantics. Even the language that we use to communicate (semi-formal logic) can mean different things.

Aside: read “Reflections on Trusting Trust” by Ken Thompson (You can skip to the “Stage II” part to get to the interesting stuff.)

(And when you’re done, look for “XcodeGhost” to see a relevant example, and don’t miss the leaked document on the wikipedia page…)

Here is yet another variation of our evaluator that is even closer to a meta-circular evaluator. It uses Racket values directly to implement values, so arithmetic operations become straightforward. Note especially how the case for function application is similar to arithmetics: a FLANG function application translates to a Racket function application. In both cases (applications and arithmetics) we don’t even check the objects since they are simple Racket objects — if our language happens to have some meaning for arithmetics with functions, or for applying numbers, then we will inherit the same semantics in our language. This means that we now specify less behavior and fall back more often on what Racket does.

We use Racket values with this type definition:

And the evaluation function can now be:

Note how the arithmetics implementation is simple — it’s a direct translation of the FLANG syntax to Racket operations, and since we don’t check the inputs to the Racket operations, we let Racket throw type errors for us. Note also how function application is just like the arithmetic operations: a FLANG application is directly translated to a Racket application.

However, this does not work quite as simply in Typed Racket. The whole point of typechecking is that we never run into type errors — so we cannot throw back on Racket errors since code that might produce them is forbidden! A way around this is to perform explicit checks that guarantee that Racket cannot run into type errors. We do this with the following two helpers that are defined inside eval:

Note that Typed Racket is “smart enough” to figure out that in evalF the result of the recursive evaluation has to be either Number or (VAL -> VAL); and since the if throws out on numbers, we’re left with (VAL -> VAL) functions, not just any function.

Recursion, Recursion, Recursion

PLAI §9

There is one major feature that is still missing from our language: we have no way to perform recursion (therefore no kind of loops). So far, we could only use recursion when we had names. In FLANG, the only way we can have names is through with which not good enough for recursion.

To discuss the issue of recursion, we switch to a “broken” version of (untyped) Racket — one where a define has a different scoping rules: the scope of the defined name does not cover the defined expression. Specifically, in this language, this doesn’t work:

In our language, this translation would also not work (assuming we have if etc):

And similarly, in plain Racket this won’t work if let is the only tool you use to create bindings:

In the broken-scope language, the define form is more similar to a mathematical definition. For example, when we write:

it is actually shorthand for

we can then replace defined names with their definitions:

and this can go on, until we get to the actual code that we wrote:

This means that the above fact definition is similar to writing:

which is not a well-formed definition — it is meaningless (this is a formal use of the word “meaningless”). What we’d really want, is to take the equation (using = instead of :=)

and find a solution which will be a value for fact that makes this true.

If you look at the Racket evaluation rules handout on the web page, you will see that this problem is related to the way that we introduced the Racket define: there is a hand-wavy explanation that talks about knowing things.

The big question is: can we define recursive functions without Racket’s magical define form?

Note: This question is a little different than the question of implementing recursion in our language — in the Racket case we have no control over the implementation of the language. As it will eventually turn out, implementing recursion in our own language will be quite easy when we use mutation in a specific way. So the question that we’re now facing can be phrased as either “can we get recursion in Racket without Racket’s magical definition forms?” or “can we get recursion in our interpreter without mutation?”.

Recursion without the Magic

PLAI §22.4 (we go much deeper)

Note: This explanation is similar to the one you can find in “The Why of Y”, by Richard Gabriel.

To implement recursion without the define magic, we first make an observation: this problem does not come up in a dynamically-scoped language. Consider the let-version of the problem:

This works fine — because by the time we get to evaluate the body of the function, fact is already bound to itself in the current dynamic scope. (This is another reason why dynamic scope is perceived as a convenient approach in new languages.)

Regardless, the problem that we have with lexical scope is still there, but the way things work in a dynamic scope suggest a solution that we can use now. Just like in the dynamic scope case, when fact is called, it does have a value — the only problem is that this value is inaccessible in the lexical scope of its body.

Instead of trying to get the value in via lexical scope, we can imitate what happens in the dynamically scoped language by passing the fact value to itself so it can call itself (going back to the original code in the broken-scope language):

except that now the recursive call should still send itself along:

The problem is that this required rewriting calls to fact — both outside and recursive calls inside. To make this an acceptable solution, calls from both places should not change. Eventually, we should be able to get a working fact definition that uses just

The first step in resolving this problem is to curry the fact definition.

Now fact is no longer our factorial function — it’s a function that constructs it. So call it make-fact, and bind fact to the actual factorial function.

We can try to do the same thing in the body of the factorial function: instead of calling (self self), just bind fact to it:

This works fine, but if we consider our original goal, we need to get that local fact binding outside of the (lambda (n) ...) — so we’re left with a definition that uses the factorial expression as is. So, swap the two lines:

But the problem is that this gets us into an infinite loop because we’re trying to evaluate (self self) too ea(ge)rly. In fact, if we ignore the body of the let and other details, we basically do this:

And this expression has an interesting property: it reduces to itself, so evaluating it gets stuck in an infinite loop.

So how do we solve this? Well, we know that (self self) should be the same value that is the factorial function itself — so it must be a one-argument function. If it’s such a function, we can use a value that is equivalent, except that it will not get evaluated until it is needed, when the function is called. The trick here is the observation that (lambda (n) (add1 n)) is really the same as add1 (provided that add1 is a one-argument function), except that the add1 part doesn’t get evaluated until the function is called. Applying this trick to our code produces a version that does not get stuck in the same infinite loop:

Continuing from here — we know that

(remember how we derived fun from a with), so we can turn that let into the equivalent function application form:

And note now that the (lambda (fact) …) expression is everything that we need for a recursive definition of fact — it has the proper factorial body with a plain recursive call. It’s almost like the usual value that we’d want to define fact as, except that we still have to abstract on the recursive value itself. So lets move this code into a separate definition for fact-step:

We can now proceed by moving the (make-fact make-fact) self application into its own function which is what creates the real factorial:

Rewrite the make-fact definition using an explicit lambda:

and fold the functionality of make-fact and make-real-fact into a single make-fact function by just using the value of make-fact explicitly instead of through a definition:

We can now observe that make-real-fact has nothing that is specific to factorial — we can make it take a “core function” as an argument:

and call it make-recursive:

We’re almost done now — there’s no real need for a separate fact-step definition, just use the value for the definition of fact:

turn the let into a function form:

do some renamings to make things simpler — make and self turn to x, and core to f:

or we can manually expand that first (lambda (x) (x x)) application to make the symmetry more obvious (not really surprising because it started with a let whose purpose was to do a self-application):

And we finally got what we were looking for: a general way to define any recursive function without any magical define tricks. This also work for other recursive functions:

A convenient tool that people often use on paper is to perform a kind of a syntactic abstraction: “assume that whenever I write (twice foo) I really meant to write (foo foo)”. This can often be done as plain abstractions (that is, using functions), but in some cases — for example, if we want to abstract over definitions — we just want such a rewrite rule. (More on this towards the end of the course.) The broken-scope language does provide such a tool — rewrite extends the language with a rewrite rule. Using this, and our make-recursive, we can make up a recursive definition form:

In other words, we’ve created our own “magical definition” form. The above code can now be written in almost the same way it is written in plain Racket:

Finally, note that make-recursive is limited to 1-argument functions only because of the protection from eager evaluation. In any case, it can be used in any way you want, for example,

is a function that returns itself rather than calling itself. Using the rewrite rule, this would be:

which is the same as:

in plain Racket.