Implementing Laziness (in plain Racket)

PLAI §8

Generally, we know how lazy evaluation works when we use the substitution model. We even know that if we have:

then the result should be an error because we cannot substitute the binding of x into the body expression because it will capture the y — changing the binding structure. As an indication, the original expression contains a free reference to y, which is exactly why we shouldn’t substitute it. But what about:

Evaluating this eagerly returns 18, we therefore expect any other evaluation (eager or lazy, using substitutions or environments) to return 18 too, because any of these options should not change the meaning of numbers, of addition, or of the scoping rules. (And we know that no matter what evaluation strategy we choose, if we get to a value (no infinite loop or exception) then it’ll always be the same value.) For example, try using lazy evaluation with substitutions:

And what about lazy evaluation using environments:

We have a problem! This problem should be familiar now, it is very similar to the problem that led us down the mistaken path of dynamic scoping when we tried to have first-class functions. In both cases, substitution always worked, and it looks like in both cases the problem is that we don’t remember the environment of an expression: in the case of functions, it is the environment at the time of creating the closure that we want to capture and use when we go back later to evaluate the body of the function. Here we have a similar situation, except that we don’t need a function to defer computation: most expressions get evaluated at some time in the future, so every time we defer such a computation we need to remember the lexical environment of the expression.

This is the major point that will make things work again: every expression creates something like a closure — an object that closes over an expression and an environment at the (lexical) place where that expression was used, and when we actually want to evaluate it later, we need to do it in the right lexical context. So it is like a closure except it doesn’t need to be applied, and there are no arguments. In fact it is also a form of a closure — instead of closing over a function body and an environment, it closes over any expression and an environment. (As we shall see, lazy evaluation is tightly related to using nullary functions: thunks.)

Sloth: A Lazy Evaluator

So we implement this by creating such closure values for all expressions that are not evaluated right now. We begin with the Toy language, and rename it to “Sloth”. We then add one more case to the data type of values which implements the new kind of expression closures, which contains the expression and its environment:

(Intuition#1: ExprV is a delayed evaluation and therefore it has the two values that are ultimately passed to eval. Intuition#2: laziness can be implemented with thunks, so we hold the same information as a FunV does, only there’s no need for the argument names.)

Where should we use the new ExprV? — At any place where we want to be lazy and defer evaluating an expression for later. The two places in the interpreter where we want to delay evaluation are the named expressions in a bind form and the argument expressions in a function application. Both of these cases use the helper eval* function to do their evaluations, for example:

To delay these evaluations, we need to change eval* so it returns an expression closure instead of actually doing the evaluation — change:

to:

Note how simple this change is — instead of an eval function call, we create a value that contains the parts that would have been used in the eval function call. This value serves as a promise to do this evaluation (the eval call) later, if needed. (This is exactly why a Lazy Racket would make this a lazy evaluator: in it, all function calls are promises.)

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Side note: this can be used in any case when you’re using an eager language, and you want to delay some function call — all you need to do is replace (using a C-ish syntax)

with

now all calls to foo return a delayed_foo instance instead of an integer. Whenever we want to force the delayed promise, we can use this function:

You might even want to make sure that each such promise is evaluated exactly once — this is simple to achieve by adding a cache field to the struct:

As we will see shortly, this corresponds to switching from a call-by-name lazy language to a call-by-need one.

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Back to our Sloth interpreter — given the eval* change, we expect that eval-uating:

will return:

and the same goes for eval-uating

Similarly, evaluating

should return

But what about evaluating an expression like this one:

?

Using what we have so far, we will get to evaluate the body, which is a (Call …) expression, but when we evaluate the arguments for this function call, we will get ExprV values — so we will not be able to perform the addition. Instead, we will get an error from the function that racket-func->prim-val creates, due to the value being an ExprV instead of a RktV.

What we really want is to actually add two values, not promises. So maybe distinguish the two applications — treat PrimV differently from FunV closures?

This still doesn’t work — the problem is that the function now gets a bunch of values, where some of these can still be ExprVs because the evaluation itself can return such values… Another way to see this problem is to consider the code for evaluating an If conditional expression:

…we need to take care of a possible ExprV here. What should we do? The obvious solution is to use eval if we get an ExprV value:

Note how this translates back the data structure that represents a delayed eval promise back into a real eval call…

Going back to our code for Call, there is a problem with it — the

will indeed evaluate the expression instead of lazily deferring this to the future, but this evaluation might itself return such lazy values. So we need to inspect the resulting value again, forcing the promise if needed:

But we still have a problem — programs can get an arbitrarily long nested chains of ExprVs that get forced to other ExprVs.

What we really need is to write a loop that keeps forcing promises over and over until it gets a proper non-ExprV value.

Note that it’s close to real-eval*, but there’s no need to mix it with eval. The recursive call is important: we can never be sure that eval didn’t return an ExprV promise, so we have to keep looping until we get a “real” value.

Now we can change the evaluation of function calls to something more manageable:

The code is fairly similar to what we had previously — the only difference is that we wrap a strict call where a proper value is needed — the function value itself, and arguments to primitive functions.

The If case is similar (note that it doesn’t matter if strict is used with the result of eval or eval* (which returns an ExprV)):

Note that, like before, we always return #t for non-RktV values — this is because we know that the value there is never an ExprV. All we need now to get a working evaluator, is one more strictness point: the outermost point that starts our evaluation — run — needs to use strict to get a proper result value.

With this, all of the tests that we took from the Toy evaluator run successfully. To make sure that the interpreter is lazy, we can add a test that will fail if the language is strict:

[In fact, we can continue and replace all eval calls with ExprV, leaving only the one call in strict. This doesn’t make any difference, because the resulting promises will eventually be forced by strict anyway.]

Getting more from Sloth

As we’ve seen, using strict in places where we need an actual value rather than a delayed promise is enough to get a working lazy evaluator. Our current implementation assumes that all primitive functions need strict values, therefore the argument values are all passed through the strict function — but this is not always the case. Specifically, if we have constructor functions, then we don’t need (and usually don’t want) to force the promises. This is basically what allows us to use infinite lists in Lazy Racket: the fact that list and cons do not require forcing their arguments.

To allow some primitive functions to consume strict values and some to leave them as is, we’re going to change racket-func->prim-val and add a flag that indicates whether the primitive function is strict or not. Obviously, we also need to move the strict call around arguments to a primitive function application into the racket-func->prim-val generated function — which simplifies the Call case in eval (we go from (proc (map strict arg-vals)) back to (proc arg-vals)). The new code for racket-func->prim-val and its helper is:

We now need to annotate the primitives in the global environment, as well as add a few constructors:

Note that this last change raises a subtle type issue: we’re actually abusing the Racket list and cons constructors to hold Sloth values. One way in which this becomes a problem is the current assumption that a primitive function always returns a Racket value (it is always wrapped in a RktV) — but this is no longer the case for first and rest: when we use

in Sloth, the resulting value will be

This leads to two problems: first, if we use Racket’s first and rest, they will complain (throw a runtime error) since the input value is not a proper list (it’s a pair that has a non-list value in its tail). To resolve that, we use the more primitive car and cdr functions to implement Sloth’s first and rest.

The second problem happens when we try and grab the first value of this

we will eventually get back the ExprV and wrap it in a RktV:

and finally run will strip off the RktV and return the ExprV. A solution to this is to make our first and rest functions return a value without wrapping it in a RktV — we can identify this situation by the fact that the returned value is already a VAL instead of some other Racket value. We can identify such values with the VAL? predicate that gets defined by our define-type, implemented by a new wrap-in-val helper:

Note that we don’t need to worry about the result being an ExprV — that will eventually be taken care of by strict.

The Sloth Implementation

The complete Sloth code follows. It can be used to do the same fun things we did with Lazy Racket.

Shouldn’t there be more ExprV promises?

You might notice that there are some apparently missing promises. For example, consider our evaluation of Bind forms:

The named expressions are turned into expression promises via eval*, but shouldn’t we change the first eval (the one that evaluates the body) into a promise too? This is a confusing point, and the bottom line is that there is no need to create a promise there. The main idea is that the eval function is actually called from contexts that actually need to be evaluated. One example is when we force a promise via strict, and another one is when run calls eval. Note that in both of these cases, we actuallly need a (forced) value, so creating a promise in there doesn’t make any difference.

To see this differently, consider how bind might be used within the language. The first case is when bind is the topmost expression, or part of a bind “spine”:

In these cases we evaluate the bind expression when we need to return a result for the whole run, so adding an ExprV is not going to make a difference. The second case is when bind is used in an expression line a function argument:

Here there is also no point in adding an ExprV to the Bind case, since the evaluation of the whole argument (the Bind value) will be wrapped in an ExprV, so it is already delayed. (And when it get forced, we will need to do the bind evaluation anyway, so again, it adds no value.)

A generalization of this is that when we actually call eval (either directly or via strict), there is never any point in making the result that it returns a promise.

(And if you’ll follow this carefully and look at all of the eval calls, you will see that this means that neither of the eval*s in the If case are needed!)