2010-02-26
- Recursive Environments
- Recursion: Scheme's `letrec'
- Implementing Recursion using `letrec'
- Implementing `rec' Using a Cyclic Structure
- Boxes and Mutation

========================================================================
>>> Recursive Environments

What we really need for recursion, is a special kind of an environment,
one that can refer to itself.  So instead of doing (note: `call's
removed for readability):

  {with {fact {fun {n}
                {if {zero? n} 1 {* n {fact {- n 1}}}}}}
    {fact 5}}

which does not work for the usual reasons, we want to use some

  {rec {fact {fun {n}
               {if {zero? n} 1 {* n {fact {- n 1}}}}}}
    {fact 5}}

that will do the necessary magic.

One way to achieve this is using the Y combinator as we have seen -- a
kind of a "constructor" for recursive functions.  We can do that in a
similar way to the `rewrite' rule that we have seen in Schlac --
translate the above expression to:

  {with {fact {make-rec
                {fun {fact}
                  {fun {n}
                    {if {zero? n} 1 {* n {fact {- n 1}}}}}}}}
    {fact 5}}

or even:

  {with {fact {{fun {f} {{fun {x} {f {x x}}} {fun {x} {f {x x}}}}}
               {fun {fact}
                 {fun {n}
                   {if {zero? n} 1 {* n {fact {- n 1}}}}}}}}
    {fact 5}}

Now, we will see how it can be used in *our* code to implement a
recursive environment.

If we look at what `with' does in

  {with {fact {fun {n}
                {if {zero? n} 1 {* n {fact {- n 1}}}}}}
    {fact 5}}

then we can say that to evaluate this expression, we evaluate the body
expression in an extended environment that contains `fact', even if a
bogus one that is good for 0 only -- the new environment is created with
something like this:

  extend("fact", make-fact-closure(), env)

so we can take this whole thing as an operation over `env'

  add-fact(env) := extend("fact", make-fact-closure(), env)

This gives us the first-level fact.  But `fact' itself is still
undefined in `env', so it cannot call itself.  We can try this:

  add-fact(add-fact(env))

but that still doesn't work, and it will never work no matter how far we
go:

  add-fact(add-fact(add-fact(add-fact(add-fact(...env...)))))

What we really want is infinity: a place where add-fact works and the
result is the same as what we've started with -- we want to create a
"magical" environment that makes this possible:

  let magic-env = ???
  such that:
     add-fact(magic-env) = magic-env

which basically gives us the illusion of being at the infinity point.
This magic-env thing is exactly the *fixed-point* of the `add-fact'
operation.  We can use:

  magic-env = rec(add-fact)

and following the main property of the Y combinator, we know that:

  magic-env = rec(add-fact)           ; def. of magic-env
            = add-fact(rec(add-fact)) ; Y(f) = f(Y(f))
            = add-fact(magic-env)     ; def. of magic-env

What does all this mean?  It means that if we have a fixed-point
operator at the level of the implementation of our environments, then we
can use it to implement a recursive binder.  In our case, this means
that a fixed-point in Scheme can be used to implement a recursive
language.  But we have that -- Scheme does have recursive functions, so
we should be able to use that to implement our recursive binder.

There are two ways that make it possible to write recursive functions in
Scheme.  One is to define a function, and use its name to do a recursive
call -- using the Scheme formal rules, we can see that we said that we
mark that we now *know* that a variable is bound to a value.  This is
essentially a side-effect -- we modify what we know, which corresponds
to modifying the global environment.  The second way is a new form:
`letrec'.  This form is similar to `let', except that the scope that is
established includes the named expressions -- it is exactly what we want
`bindrec' to do.  A third way is using recursive local definitions, but
that is equivalent to using `letrec', more on this soon.

========================================================================
>>> Recursion: Scheme's `letrec'

So we want to add recursion to our language, practically.  We already
know that Scheme makes it possible to write recursive functions, which
is possible because of the way it implements its "global environment":
our evaluator can only *extend* an environment, while Scheme *modifies*
its global environment.  This means that whenever a function is defined
in the global environment, the resulting closure will have it as its
environment "pointer", but the global environment was not extended -- it
stays the same, and was just modified with one additional binding.

But Scheme has another, a bit more organized way of using recursion:
there is a special local-binding construct that is similar to `let', but
allows a function to refer to itself.  It is called `letrec':

  (letrec ([fact (lambda (n)
                   (if (zero? n)
                     1
                     (* n (fact (- n 1)))))])
    (fact 5))

Some people may remember that there was a third way for creating
recursive functions: using local definition in function bodies.  For
example, we have seen things like:

  (define (length list)
    (define (helper list len)
      (if (null? list)
        len
        (helper (cdr list) (+ len 1))))
    (helper list 0))

This looks like the same kind of environment magic that happens with a
global `define' -- but actually, Scheme defines the meaning of internal
definitions using `letrec' -- so the above code is exactly the same as:

  (define (length list)
    (letrec ([helper (lambda (list len)
                       (if (null? list)
                         len
                         (helper (cdr list) (+ len 1))))])
      (helper list 0)))

The scoping rules for a `letrec' is that the scope of the bound name
covers both the body *and* the named expression.  Furthermore, multiple
names can be bound to multiple expressions, and the scope of each name
covers all named expression as well as the body.  This makes it easy to
define mutually recursive functions, such as:

  (letrec ([even? (lambda (n) (if (zero? n) #t (odd?  (- n 1))))]
           [odd?  (lambda (n) (if (zero? n) #f (even? (- n 1))))])
    (even? 99))

But it is not a required functionality -- it could be done with a single
recursive binding that contains several procedures:

  (letrec ([even+odd
            (list (lambda (n)
                    (if (zero? n) #t ((second even+odd) (- n 1))))
                  (lambda (n)
                    (if (zero? n) #f ((first  even+odd) (- n 1)))))])
    ((first even+odd) 99))

This is basically the same problem we face if we want to use the Y
combinator for mutually recursive bindings.  The above solution is
inconvenient, but it can be improved using more `let's to have easier
name access.  For example:

  (letrec ([even+odd
            (list (lambda (n)
                    (let ([even? (first  even+odd)]
                          [odd?  (second even+odd)])
                      (if (zero? n) #t (odd? (- n 1)))))
                  (lambda (n)
                    (let ([even? (first  even+odd)]
                          [odd?  (second even+odd)])
                      (if (zero? n) #f (even? (- n 1))))))])
    (let ([even? (first  even+odd)]
          [odd?  (second even+odd)])
      (even? 99)))

========================================================================
>>> Implementing Recursion using `letrec'

We will see how to add a similar construct to our language -- for
simplicity, we will add a `rec' form that handles a single binding:

  {rec {fact {fun {n}
               {if {= 0 n}
                 1
                 {* n {fact {- n 1}}}}}}
    {fact 5}}

Using this, things can get a little tricky.  What should we get if we
do:

  {rec {x x} x}

?  Currently, it seems like there is no point in using any expression
except for a *function* expression in a `rec' expression, so we will
handle only these cases.

(BTW, under what circumstances would non-function values be useful in a
letrec?)

========================================================================

One way to achieve this is to use the same trick that we have recently
seen: instead of re-implementing Scheme features, we can use existing
features and hope that if Scheme has the right functionality, then it
can be re-used to in our evaluator.

Previously, we have seen a way to implement environments using Scheme
closures:

  ;; Define a type for functional environments
  (define-type ENV = (Symbol -> VAL))

  (: EmptyEnv : (-> ENV))
  (define (EmptyEnv)
    (lambda (id) (error 'lookup "no binding for ~s" id)))

  (: lookup : (Symbol ENV -> VAL))
  (define (lookup name env)
    (env name))

  (: Extend : (Symbol VAL ENV -> ENV))
  (define (Extend id v rest-env)
    (lambda (name)
      (if (eq? name id)
        v
        (rest-env name))))

We can use this implementation, and create circular environments using
Scheme's `letrec'.  The code for handling a `with' expressions is:

      [(With bound-id named-expr bound-body)
       (eval bound-body
             (Extend bound-id (eval named-expr env) env))]

It looks like we should be able to handle `Rec' in a similar way:

      [(Rec bound-id named-expr bound-body)
       (eval bound-body
             (Extend bound-id (eval named-expr env) env))]

but this won't work because the named expression is evaluated
prematurely, in the previous environment.  Instead, we will move
everything that needs to be done, including evaluation, to a separate
`extend-rec' function:

      [(Rec bound-id named-expr bound-body)
       (eval bound-body
             (extend-rec bound-id named-expr env))]

Now, the `extend-rec' function needs to provide the new, "magically
circular" environment.  Following what we know about the arguments to
`extend-rec', and the fact that it returns a new environment (= a lookup
procedure), we can sketch a rough definition:

  (: extend-rec : (Symbol FLANG ENV -> ENV)) ; FLANG, not VAL!
  (define (extend-rec id expr rest-env)
    (lambda (name)
      (if (eq? name id)
        ... something that uses expr to get a value ...
        (rest-env name))))

What should the missing expression be?  It can simply evaluate the
object given itself:

  (: extend-rec : (Symbol FLANG ENV -> ENV))
  (define (extend-rec id expr rest-env)
    (lambda (name)
      (if (eq? name id)
        (eval expr ...this environment...)
        (rest-env name))))

But how do we get "this environment", before it is defined?  Well, the
environment is itself a Scheme *function*, so we can use Scheme's
`letrec' to make the function refer to itself recursively:

  (: extend-rec : (Symbol FLANG ENV -> ENV))
  (define (extend-rec id expr rest-env)
    (letrec ([rec-env (lambda (name)
                        (if (eq? name id)
                          (eval expr rec-env)
                          (rest-env name)))])
      rec-env))

But we need to add types to this.  One way to do so is to use `letrec:'
which is a version of `letrec' that has type annotations, but it's a
little more convenient to just use internal definitions:

  (: extend-rec : (Symbol FLANG ENV -> ENV))
  (define (extend-rec id expr rest-env)
    (: rec-env : (Symbol -> VAL))
    (define (rec-env name)
      (if (eq? name id)
        (eval expr rec-env)
        (rest-env name)))
    rec-env)

This works, but there are several problems:

1. First, we no longer do a simple lookup in the new environment,
   instead, we evaluate the expression on every such lookup.  This seems
   like a technical point, because we do not have side-effects in our
   language (also because we said that we want to handle only function
   expressions).  Still, it wastes space since each evaluation will
   allocate a new closure.

2. Second, a related problem -- what happens if we try to run this:

     {rec {x x} x}

   ?  Well, we do that stuff to extend the current environment, then
   evaluate the body in the new environment, this body is a single
   variable reference:
     (eval (Id 'x) the-new-env)
   so we look up the value:
     (lookup 'x the-new-env)
   which is:
     (the-new-env 'x)
   which goes into the function which implements this environment, there
   we see that `name' is the same as `name1', so we return:
     (eval expr rec-env)
   but the `expr' here is the original named-expression which is itself
   (Id 'x), and we're in an infinite loop.

We can try to get over these problems using another binding.  Scheme
allows several bindings in a single `letrec' expression, so we change
`rec-extend' to use the newly-created environment:

  (: extend-rec : (Symbol FLANG ENV -> ENV))
  (define (extend-rec id expr rest-env)
    (letrec ([rec-env (lambda (name)
                        (if (eq? name id)
                          val
                          (rest-env name)))]
             [val (eval expr rec-env)])
      rec-env))

and switching to typed internal definitions:

  (: extend-rec : (Symbol FLANG ENV -> ENV))
  (define (extend-rec id expr rest-env)
    (: rec-env : (Symbol -> VAL))
    (define (rec-env name)
      (if (eq? name id)
        val
        (rest-env name)))
    (: val : VAL)
    (define val (eval expr rec-env))
    rec-env)

This seems to work fine with `fact', but it raises more questions,
beginning with: what is the meaning of:

  (letrec ([x ...]
           [y ...x...])
    ...)

or equivalently, an internal block of

  (define x ...)
  (define y ...x...)

?  Well, DrScheme seems to do the "right thing" in this case, but what
about:

  (letrec ([y ...x...]
           [x ...])
    ...)

?  As a hint, see what happens when we now try to evaluate the
problematic

  {rec {x x} x}

expression (and compare that with the result that you'd get from
Scheme).

It should be clear now why we want to restrict usage to just binding
recursive functions.  There are no problems with such definitions
because when we evaluate a `fun' expression, there is no evaluation of
the body, which is the only place where there are potential references
to the same procedure that is defined -- a function's body is *delayed*,
and executed only when the function is applied later.

But the biggest question that is still open: we just implemented a
circular environment using Scheme's own circular environment
implementation, and that does not explain how they are actually
implemented.  The cycle of pointers that we've implemented depends on
the cycle of pointers that Scheme uses, and that is a black box we want
to open up.

The complete code, for reference:

  ----------------------------------------------------------------------
  #lang pl

  #|
  The grammar:
    <FLANG> ::= <num>
              | { + <FLANG> <FLANG> }
              | { - <FLANG> <FLANG> }
              | { * <FLANG> <FLANG> }
              | { / <FLANG> <FLANG> }
              | { with { <id> <FLANG> } <FLANG> }
              | { rec { <id> <FLANG> } <FLANG> }
              | <id>
              | { fun { <id> } <FLANG> }
              | { call <FLANG> <FLANG> }

  Evaluation rules:
    eval(N,env)                = N
    eval({+ E1 E2},env)        = eval(E1,env) + eval(E2,env)
    eval({- E1 E2},env)        = eval(E1,env) - eval(E2,env)
    eval({* E1 E2},env)        = eval(E1,env) * eval(E2,env)
    eval({/ E1 E2},env)        = eval(E1,env) / eval(E2,env)
    eval(x,env)                = lookup(x,env)
    eval({with {x E1} E2},env) = eval(E2,extend(x,eval(E1,env),env))
    eval({rec {x E1} E2},env)  = ???
    eval({fun {x} E},env)      = <{fun {x} E},env>
    eval({call E1 E2},env1)
             = eval(Ef,extend(x,eval(E2,env1),env2))
                               if eval(E1,env1)=<{fun {x} Ef},env2>
             = error!          otherwise
  |#

  (define-type FLANG
    [Num  (n Number)]
    [Add  (lhs FLANG) (rhs FLANG)]
    [Sub  (lhs FLANG) (rhs FLANG)]
    [Mul  (lhs FLANG) (rhs FLANG)]
    [Div  (lhs FLANG) (rhs FLANG)]
    [Id   (name Symbol)]
    [With (name Symbol) (named FLANG) (body FLANG)]
    [Rec  (name Symbol) (named FLANG) (body FLANG)]
    [Fun  (name Symbol) (body FLANG)]
    [Call (fun-expr FLANG) (arg-expr FLANG)])

  (: parse-sexpr : (Sexpr -> FLANG))
  ;; to convert s-expressions into FLANGs
  (define (parse-sexpr sexpr)
    (match sexpr
      [(number: n)    (Num n)]
      [(symbol: name) (Id name)]
      [(cons 'with more)
       (match sexpr
         [(list 'with (list (symbol: name) named) body)
          (With name (parse-sexpr named) (parse-sexpr body))]
         [else (error 'parse-sexpr "bad `with' syntax in ~s" sexpr)])]
      [(cons 'rec more)
       (match sexpr
         [(list 'rec (list (symbol: name) named) body)
          (Rec name (parse-sexpr named) (parse-sexpr body))]
         [else (error 'parse-sexpr "bad `rec' syntax in ~s" sexpr)])]
      [(cons 'fun more)
       (match sexpr
         [(list 'fun (list (symbol: name)) body)
          (Fun name (parse-sexpr body))]
         [else (error 'parse-sexpr "bad `fun' syntax in ~s" sexpr)])]
      [(list '+ lhs rhs) (Add (parse-sexpr lhs) (parse-sexpr rhs))]
      [(list '- lhs rhs) (Sub (parse-sexpr lhs) (parse-sexpr rhs))]
      [(list '* lhs rhs) (Mul (parse-sexpr lhs) (parse-sexpr rhs))]
      [(list '/ lhs rhs) (Div (parse-sexpr lhs) (parse-sexpr rhs))]
      [(list 'call fun arg) (Call (parse-sexpr fun) (parse-sexpr arg))]
      [else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))

  (: parse : (String -> FLANG))
  ;; parses a string containing a FLANG expression to a FLANG AST
  (define (parse str)
    (parse-sexpr (string->sexpr str)))

  ;; Types for environments, values, and a lookup function

  (define-type VAL
    [NumV (n Number)]
    [FunV (name Symbol) (body FLANG) (env ENV)])

  ;; Define a type for functional environments
  (define-type ENV = (Symbol -> VAL))

  (: EmptyEnv : (-> ENV))
  (define (EmptyEnv)
    (lambda (id) (error 'lookup "no binding for ~s" id)))

  (: lookup : (Symbol ENV -> VAL))
  (define (lookup name env)
    (env name))

  (: Extend : (Symbol VAL ENV -> ENV))
  (define (Extend id v rest-env)
    (lambda (name)
      (if (eq? name id)
        v
        (rest-env name))))

  (: extend-rec : (Symbol FLANG ENV -> ENV))
  (define (extend-rec id expr rest-env)
    (: rec-env : (Symbol -> VAL))
    (define (rec-env name)
      (if (eq? name id)
        val
        (rest-env name)))
    (: val : VAL)
    (define val (eval expr rec-env))
    rec-env)

  (: arith-op : ((Number Number -> Number) VAL VAL -> VAL))
  ;; gets a Scheme numeric binary operator, and uses it within a NumV
  ;; wrapper
  (define (arith-op op val1 val2)
    (: NumV->number : (VAL -> Number))
    (define (NumV->number v)
      (cases v
        [(NumV n) n]
        [else (error 'arith-op "expects a number, got: ~s" v)]))
    (NumV (op (NumV->number val1) (NumV->number val2))))

  (: eval : (FLANG ENV -> VAL))
  ;; evaluates FLANG expressions by reducing them to values
  (define (eval expr env)
    (cases expr
      [(Num n) (NumV n)]
      [(Add l r) (arith-op + (eval l env) (eval r env))]
      [(Sub l r) (arith-op - (eval l env) (eval r env))]
      [(Mul l r) (arith-op * (eval l env) (eval r env))]
      [(Div l r) (arith-op / (eval l env) (eval r env))]
      [(With bound-id named-expr bound-body)
       (eval bound-body
             (Extend bound-id (eval named-expr env) env))]
      [(Rec bound-id named-expr bound-body)
       (eval bound-body
             (extend-rec bound-id named-expr env))]
      [(Id name) (lookup name env)]
      [(Fun bound-id bound-body)
       (FunV bound-id bound-body env)]
      [(Call fun-expr arg-expr)
       (let ([fval (eval fun-expr env)])
         (cases fval
           [(FunV bound-id bound-body f-env)
            (eval bound-body
                  (Extend bound-id (eval arg-expr env) f-env))]
           [else (error 'eval "`call' expects a function, got: ~s"
                              fval)]))]))

  (: run : (String -> Number))
  ;; evaluate a FLANG program contained in a string
  (define (run str)
    (let ([result (eval (parse str) (EmptyEnv))])
      (cases result
        [(NumV n) n]
        [else (error 'run
                     "evaluation returned a non-number: ~s" result)])))

  ;; tests
  (test (run "{call {fun {x} {+ x 1}} 4}")
        => 5)
  (test (run "{with {add3 {fun {x} {+ x 3}}}
                {call add3 1}}")
        => 4)
  (test (run "{with {add3 {fun {x} {+ x 3}}}
                {with {add1 {fun {x} {+ x 1}}}
                  {with {x 3}
                    {call add1 {call add3 x}}}}}")
        => 7)
  (test (run "{with {identity {fun {x} x}}
                {with {foo {fun {x} {+ x 1}}}
                  {call {call identity foo} 123}}}")
        => 124)
  (test (run "{with {x 3}
                {with {f {fun {y} {+ x y}}}
                  {with {x 5}
                    {call f 4}}}}")
        => 7)
  (test (run "{call {with {x 3}
                      {fun {y} {+ x y}}}
                    4}")
        => 7)
  (test (run "{call {call {fun {x} {call x 1}}
                          {fun {x} {fun {y} {+ x y}}}}
                    123}")
        => 124)
  ----------------------------------------------------------------------

========================================================================
>>> Implementing `rec' Using a Cyclic Structure

Looking at the arrows in the environment diagrams, what we're really
looking for is a closure that has an environment pointer which is the
same environment in which it was defined.  This will make it possible
for `fact' to be bound to a closure that can refer to *itself* since its
environment is the same one in which it is defined.  However, so far we
have no tools that makes it possible to do this.

What we need is to create a "cycle of pointers", and so far we do not
have a way of achieving that: when we create a closure, we begin with an
environment which is saved in the slot's environment slot, but we want
that closure to be the value of a binding in that same environment.

========================================================================
>>> Boxes and Mutation

To actually implement a circular structure, we will now use
*side-effects*, using a new kind of Scheme value which supports
mutation: a box.  A box value is built with the `box' constructor:

  (define my-thing (box 7))

the value is retrieved with the `unbox' function,

  (* 6 (unbox my-thing))

and finally, the value can be changed with the `set-box!' function.

  (set-box! my-thing 17)
  (* 6 (unbox my-thing))

An important thing to note is that `set-box!' is much like `display'
etc, it returns a value that is not printed in the Scheme REPL, because
there is no point in using the result of a `set-box!', it is called for
the side-effect it generates.  (Languages like C blur this distinction
between returning a value and a side-effect with its assignment
statement.)

As a side note, we now have side effects of two kinds: mutation of
state, and I/O (at least the O part).  (Actually, there is also infinite
looping that can be viewed as another form of a side effect.)  This
means that we're now in a completely different world, and lots of new
things can make sense now.  A few things that you should know about:

* We never used more than one expression in a function body because
  there was no point in it, but now there is.  To evaluate a sequence of
  Scheme expressions, you wrap them in a `begin' expression.

* In most places you don't actually need to use `begin' -- these are
  places that are said to have an "implicit `begin'": the body of a
  function (or any lambda expression), the body of a `let' (and
  `let'-relatives), the consequence positions in `cond', `match', and
  `cases' clauses and more.  One of the common places where a `begin' is
  used is in an `if' expression (and some people prefer using `cond'
  instead when there is more than a single expression).

* `cond' without an `else' in the end can make sense, if all you're
  using it it for is side-effects.

* `if' could get a single expression which is executed when the
  condition is true (and an unspecified value is used otherwise), but
  our language (as well as the default PLT language) always forbids this
  -- there are convenient special forms for a one-sided `if's: `when' &
  `unless', and they can have any number of expressions (they have an
  implicit `begin').  They have an advantage of saying "this code does
  some side-effects here" more explicit.

* There is a function called `for-each' which is just like `map', except
  that it doesn't collect the list of results, it is used only for
  performing side effects.

When any one of these things is used (in Scheme or other languages), you
can tell that side-effects are involved, because there is no point in
any of them otherwise.  In addition, any name that ends with a `!'
("bang") is used to mark a function that changes state (usually a
function that only changes state).

So how do we create a cycle?  Simple, boxes can have any value, and they
can be put in other values like lists, so we can do this:

  #lang pl untyped
  (define foo (list 1 (box 3)))
  (set-box! (second foo) foo)

and we get a circular value.  (Note how it is printed.)  And with types:

  #lang pl
  (: foo : (List Number (Boxof Any)))
  (define foo (list 1 (box 3)))
  (set-box! (second foo) foo)

Obviously, `Any' is not too great -- it is the most generic type, so it
provides the least information.  For example, notice that

  (unbox (second foo))

returns the right list, which is equal to `foo' itself -- but if we try
to do the same once again:

  (unbox (second (unbox (second foo))))

we get a type error, because the result of the first `unbox' is `Any',
so typed scheme knows nothing about it.  It is not too surprising that a
type constructor that can help in this case is `Rec' which we have
already seen -- it allows a type that can refer to itself:

  #lang pl
  (: foo : (Rec this (List Number (Boxof (U #f this)))))
  (define foo (list 1 (box #f)))
  (set-box! (second foo) foo)

This diesn't work for technical reasons: typed scheme needs to infer the
right type for `#f' -- an usually it does a good job, but with boxes
things are tricker: if T1 is a subtype of T2, then (Boxof T1) is a
*supertype* of (Boxof T2).  To see why this happens this, consider this:

  (: b : (Boxof Integer))
  (define b (box 2))
  (: foo : (List Number (Boxof Number)))
  (define foo (list 1 b))

We get a (surprising) type error, because otherwise we could do this:

  (set-box! (second foo) 3.7)
  (unbox b)

which would violate the previous type declaration for `b'.  We can
easily fix this specific error by replacing `(Boxof Integer)' with
`(Boxof Number)' or the other way, but the real question is how to deal
with a circular structure that we want to get with boxes.  The same
solution can work in our case, if we use a name for the type:

  #lang pl
  (define-type FOO = (Rec this (List Number (Boxof (U #f this)))))
  (: foo : FOO)
  (define foo (list 1 (box (ann #f : (U #f FOO)))))
  (set-box! (second foo) foo)

where `ann' is used to force typed scheme to use the declared type for
`#f' rather than try to guess it.  However, we will never encounter this
problem in practice.

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