PL: Lecture #19  Tuesday, March 29th
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Side Note: Similarity to Generators and Channels

Using infinite lists is similar to using channels — a tool for synchronizing threads and (see a Rob Pike’s talk), and generators (as they exist in Python). Here are examples of both, note how similar they both are, and how similar they are to the above definition of primes. (But note that there is an important difference, can you see it? It has to be with whether a stream is reusable or not.)

First, the threads & channels version:

#lang racket

(define-syntax-rule (bg expr ...) (thread (lambda () expr ...)))

(define nats
  (let ([out (make-channel)])
    (define (loop i) (channel-put out i) (loop (add1 i)))
    (bg (loop 1))
    out))

(define (divides? n m)
  (zero? (modulo m n)))

(define (filter pred c)
  (define out (make-channel))
  (define (loop)
    (let ([x (channel-get c)])
      (when (pred x) (channel-put out x))
      (loop)))
  (bg (loop))
  out)

(define (sift n c)
  (filter (lambda (x) (not (divides? n x))) c))

(define (sieve c)
  (define out (make-channel))
  (define (loop c)
    (define first (channel-get c))
    (channel-put out first)
    (loop (sift first c)))
  (bg (loop c))
  out)

(define primes
  (begin (channel-get nats) (sieve nats)))

(define (take n c)
  (if (zero? n) null (cons (channel-get c) (take (sub1 n) c))))

(take 10 primes)

And here is the generator version:

#lang racket

(require racket/generator)

(define nats
  (generator ()
    (define (loop i)
      (yield i)
      (loop (add1 i)))
    (loop 1)))

(define (divides? n m)
  (zero? (modulo m n)))

(define (filter pred g)
  (generator ()
    (define (loop)
      (let ([x (g)])
        (when (pred x) (yield x))
        (loop)))
    (loop)))

(define (sift n g)
  (filter (lambda (x) (not (divides? n x))) g))

(define (sieve g)
  (define (loop g)
    (define first (g))
    (yield first)
    (loop (sift first g)))
  (generator () (loop g)))

(define primes
  (begin (nats) (sieve nats)))

(define (take n g)
  (if (zero? n) null (cons (g) (take (sub1 n) g))))

(take 10 primes)

Call by Need vs Call by Name

Finally, note that on requiring different parts of the primes, the same calls are not repeated. This indicates that our language implements “call by need” rather than “call by name”: once an expression is forced, its value is remembered, so subsequent usages of this value do not require further computations.

Using “call by name” means that we actually use expressions which can lead to confusing code. An old programming language that used this is Algol. A confusing example that demonstrates this evaluation strategy is:

#lang algol60
begin
  integer procedure SIGMA(x, i, n);
    value n;
    integer x, i, n;
  begin
    integer sum;
    sum := 0;
    for i := 1 step 1 until n do
      sum := sum + x;
    SIGMA := sum;
  end;
  integer q;
  printnln(SIGMA(q*2-1, q, 7));
end

x and i are arguments that are passed by name, which means that they can use the same memory location. This is called aliasing, a problem that happens when pointers are involved (eg, pointers in C and reference arguments in C++). The code, BTW, is called “Jensen’s device”.

Example of Feature Embedding

Another interesting behavior that we can now observe, is that the TOY evaluation rule for with:

eval({with {x E1} E2}) = eval(E2[eval(E1)/x])

is specifying an eager evaluator only if the language that this rule is written in is itself eager. Indeed, if we run the TOY interpreter in Lazy Racket (or other interpreters we have implemented), we can verify that running:

(run "{bind {{x {/ 1 0}}} 1}")

is perfectly fine — the call to Racket’s division is done in the evaluation of the TOY division expression, but since Lazy Racket is lazy, then if this value is never used then we never get to do this division! On the other hand, if we evaluate

(run "{bind {{x {/ 1 0}}} {+ x 1}}")

we do get an error when DrRacket tries to display the result, which forces strictness. Note how the arrows in DrRacket that show where the computation is are quite confusing: the computation seem to go directly to the point of the arithmetic operations (arith-op) since the rest of the evaluation that the evaluator performed was already done, and succeeded. The actual failure happens when we try to force the resulting promise which contains only the strict points in our code.

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:

{bind {{x y}}
  {bind {{y 2}}
    {+ x y}}}

then the result should be an error because we cannot substitute the y expression in 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:

{bind {{x {+ 4 5}}}
  {bind {{y {+ x x}}}
    {bind {{z y}}
      {bind {{x 4}}
        z}}}}

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:

{bind {{x {+ 4 5}}}
  {bind {{y {+ x x}}}
    {bind {{z y}}
      {bind {{x 4}}
        z}}}}
-->
{bind {{y {+ {+ 4 5} {+ 4 5}}}}
  {bind {{z y}}
    {bind {{x 4}}
      z}}}
-->
{bind {{z {+ {+ 4 5} {+ 4 5}}}}
  {bind {{x 4}}
    z}}
-->
{bind {{x 4}}
  {+ {+ 4 5} {+ 4 5}}}
-->
{+ {+ 4 5} {+ 4 5}}
-->
{+ 9 9}
-->
18

And what about lazy evaluation using environments:

{bind {{x {+ 4 5}}}
  {bind {{y {+ x x}}}
    {bind {{z y}}
      {bind {{x 4}}
        z}}}}        []
-->
{bind {{y {+ x x}}}
  {bind {{z y}}
    {bind {{x 4}}
      z}}}            [x:={+ 4 5}]
-->
{bind {{z y}}
  {bind {{x 4}}
    z}}              [x:={+ 4 5}, y:={+ x x}]
-->
{bind {{x 4}}
  z}                  [x:={+ 4 5}, y:={+ x x}, z:=y]
-->
z                    [x:=4, y:={+ x x}, z:=y]
-->
y                    [x:=4, y:={+ x x}, z:=y]
-->
{+ x x}              [x:=4, y:={+ x x}, z:=y]
-->
{+ 4 4}              [x:=4, y:={+ x x}, z:=y]
-->
8                    [x:=4, y:={+ x x}, z:=y]

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:

(define-type VAL
  [RktV  Any]
  [FunV  (Listof Symbol) SLOTH ENV]
  [ExprV                SLOTH ENV] ;*** new: expression and scope
  [PrimV ((Listof VAL) -> VAL)])

(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:

[(Bind names exprs bound-body)
(eval bound-body (extend names (map eval* exprs) env))]

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

(: eval* : SLOTH -> VAL)
(define (eval* expr) (eval expr env))

to:

(: eval* : SLOTH -> VAL)
(define (eval* expr) (ExprV expr env))

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.)


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)

int foo(int x, str y) {
  ...do some work...
}

with

// rename `foo':
int real_foo(int x, str y) {
  ...same work...
}

// `foo' is a delayed constructor, instead of a plain function
struct delayed_foo {
  int x;
  str y;
}
delayed_foo foo(int x, str y) {
  return new delayed_foo(x, y);
}

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:

int force_foo(delayed_foo promise) {
  return real_foo(promise.x, promise.y);
}

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:

int real_foo(int x, str y) {
  ...same work...
}

struct delayed_foo {
  int  x;
  str  y;
  bool is_computed;
  int  result;
}
delayed_foo foo(int x, str y) {
  return new delayed_foo(x, y, false, 0);
}

int force_foo(delayed_foo promise) {
  if (!promise.is_computed) {
    promise.result = real_foo(promise.x, promise.y);
    promise.is_computed = true;
  }
  return promise.result;
}

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


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

{bind {{x 1}} x}

will return:

(ExprV (Num 1) ...the-global-environment...)

and the same goes for eval-uating

{{fun {x} x} 1}

Similarly, evaluating

{bind {{x {+ 1 2}}} x}

should return

(ExprV (Call (Id +) (Num 1) (Num 2)) ...the-global-environment...)

But what about evaluating an expression like this one:

{bind {{x 2}}
  {+ x x}}

?

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?

(: eval* : SLOTH -> VAL)
(define (eval* expr) (ExprV expr env))
(: real-eval* : SLOTH -> VAL)
(define (real-eval* expr) (eval expr env))
(cases expr
  ...
  [(Call fun-expr arg-exprs)
  (let ([fval (eval fun-expr env)]
        ;; move: [arg-vals (map eval* arg-exprs)]
        )
    (cases fval
      [(PrimV proc) (proc (map real-eval* arg-exprs))] ; change
      [(FunV names body fun-env)
        (eval body (extend names (map eval* arg-exprs) fun-env))]
      ...))]
  ...)

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:

[(If cond-expr then-expr else-expr)
(eval* (if (cases (real-eval* cond-expr)
              [(RktV v) v] ; Racket value => use as boolean
              [else #t])  ; other values are always true
          then-expr
          else-expr))]

…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:

[(If cond-expr then-expr else-expr)
(eval* (if (cases (real-eval* cond-expr)
              [(RktV v) v] ; Racket value => use as boolean
              [(ExprV expr env) (eval expr env)] ; force a promise
              [else #t])  ; other values are always true
          then-expr
          else-expr))]

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

(define (real-eval* expr) (eval expr env))

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:

(define (real-eval* expr)
  (let ([val (eval expr env)])
    (cases val
      [(ExprV expr env) (eval expr env)]
      [else val])))

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

{bind {{x true}}
  {bind {{y x}}
    {bind {{z y}}
      {if z
        {foo}
        {bar}}}}}

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

(: strict : VAL -> VAL)
;; forces a (possibly nested) ExprV promise,
;; returns a VAL that is not an ExprV
(define (strict val)
  (cases val
    [(ExprV expr env) (strict (eval expr env))] ; loop back
    [else val]))

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:

[(Call fun-expr arg-exprs)
(let ([fval (strict (eval* fun-expr))]          ;*** strict!
      [arg-vals (map eval* arg-exprs)])
  (cases fval
    [(PrimV proc) (proc (map strict arg-vals))] ;*** strict!
    [(FunV names body fun-env)
      (eval body (extend names arg-vals fun-env))]
    [else (error 'eval "function call with a non-function: ~s"
                  fval)]))]

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)):

[(If cond-expr then-expr else-expr)
(eval* (if (cases (strict (eval* cond-expr))
              [(RktV v) v] ; Racket value => use as boolean
              [else #t])  ; other values are always true
          then-expr
          else-expr))]

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.

(: run : String -> Any)
;; evaluate a SLOTH program contained in a string
(define (run str)
  (let ([result (strict (eval (parse str) global-environment))])
    (cases result
      [(RktV v) v]
      [else (error 'run "evaluation returned a bad value: ~s"
                  result)])))

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:

;; Test laziness
(test (run "{{fun {x} 1} {/ 9 0}}") => 1)
(test (run "{{fun {x} 1} {{fun {x} {x x}} {fun {x} {x x}}}}") => 1)
(test (run "{bind {{x {{fun {x} {x x}} {fun {x} {x x}}}}} 1}") => 1)

[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.]