As you've noted, we often find it useful to write helper procedures that accompany our main procedures. For example, we might use a helper to recurse on part of a larger structure or to act as the kernel of a husk-and-kernel procedure.
One problem with this technique is that we should restric access to the
helper procedure. In particular, only the procedure that uses the helper
(unless it's a very generic helper) should be able to access the helper.
We know how to restrict access to variables (using
let*). Can we do the same for procedures?
Yes. It is possible for a
let-expression to bind an
identifier to a non-recursive procedure:
> (let ((square (lambda (n) (* n n)))) (square 12)) 144
Like any other binding that is introduced in a
this binding is local. Within the body of the
it supersedes any previous binding of the same identifier, but as soon as the
value of the
let-expression has been computed, the local
However, it is not possible to bind an identifier to a recursively defined procedure in this way:
> (let ((count-down (lambda (n) (if (zero? n) '() (cons (- n 1) (count-down (- n 1))))))) (count-down 10)) reference to undefined identifier: count-down
The difficulty is that when the
evaluated, the identifier
count-down has not yet been bound,
so the value of the
lambda-expression is a procedure that
includes an unbound identifier. Binding this procedure value to the
count-down creates a new environment, but does not
affect the behavior of procedures that were constructed in the old
environment. So, when the body of the
this procedure, we get the unbound-identifier error.
let* wouldn't help in this case,
since even under
be completely evaluated before the binding is established. What we need is
some variant of
let that binds the identifier to some kind of a
place-holder and adds the binding to the environment first, then
computes the value of the
lambda-expression in the new
environment, and then finally substitutes that value for the place-holder.
This will work in Scheme, so long as the procedure is not actually invoked
until we get into the body of the expression. The keyword associated with
this ``recursive binding'' variant of
> (letrec ((count-down (lambda (n) (if (zero? n) '() (cons (- n 1) (count-down (- n 1))))))) (count-down 10)) (9 8 7 6 5 4 3 2 1 0)
letrec-expression constructs all of its place-holder
bindings simultaneously (in effect), then evaluates all of the
lambda-expressions simultaneously, and finally replaces all of
the place-holders simultaneously. This makes it possible to include
binding specifications for mutually recursive procedures (which invoke each
other) in the same binding list:
> (letrec ((up-sum (lambda (ls) (if (null? ls) 0 (+ (down-sum (cdr ls)) (car ls))))) (down-sum (lambda (ls) (if (null? ls) 0 (- (up-sum (cdr ls)) (car ls)))))) (up-sum (list 1 23 6 12 7))) -21 ;; which is 1 - 23 + 6 - 12 + 7.
We can use
letrec expressions to separate
the husk and the kernel of a recursive procedure without having to define
;;; Find the position of a given value in a given list. ;;; Parameters: ;;; sought, a value. ;;; stuff, a list. ;;; Returns: ;;; (1) -1, if sought is not an element of stuff ;;; (2) the position of the first appearance of sought in stuff. That is, ;;; a number, k, such that none of the first k elements of stuff is ;;; is sought but the next one after the first k elements is sought. ;;; Preconditions: ;;; None. ;;; Postconditions: ;;; Affect neither value nor list. (define index (lambda (sought stuff) (index-kernel sought stuff 0))) (define index-kernel (lambda (sought rest num-bypassed) (cond ((null? rest) -1) ((equal? (car rest) sought) num-bypassed) (else (index-kernel sought (cdr rest) (+ num-bypassed 1))))))
This works, but it's more stylish to construct the kernel procedure inside
letrec expression, so that the extra identifier can be bound
to it locally:
(define index (lambda (sought ls) (letrec ((kernel (lambda (rest bypassed) (cond ((null? rest) -1) ((equal? (car rest) sought) bypassed) (else (kernel (cdr rest) (+ bypassed 1))))))) (kernel ls 0))))
Notice, too, that since the recursive kernel procedure is now entirely
inside the body of the
index procedure, it is not necessary to
pass the value of
sought to the kernel as a parameter.
Instead, the kernel can treat
sought as if it were a constant,
since its value doesn't change during any of the recursive calls.
The same approach can be used to perform precondition tests efficiently, by
placing them with the husk in the body of a
and omitting them from the kernel. For instance, here's how to introduce
precondition tests into the
greatest-of-list procedure from
the lab on preconditions
(define greatest-of-list (lambda (ls) (letrec ((all-real? (lambda (ls) (or (null? ls) (and (real? (car ls)) (all-real? (cdr ls)))))) (kernel (lambda (rest) (if (null? (cdr rest)) (car rest) (max (car rest) (kernel (cdr rest))))))) (if (or (not (list? ls)) (null? ls) (not (all-real? ls))) (error "greatest-of-list: The argument must be a non-empty list of reals.") (kernel ls)))))
Embedding the kernel inside the definition of
rather than writing a separate
procedure has another advantage: It is impossible for an incautious user to
kernel procedure directly, bypassing the
precondition tests. The only way to get at the recursive
procedure to which
kernel is bound is to invoke the procedure
within which the binding is established.
We've recycled the name
kernel in this example to drive home
the point that local bindings in separate procedures don't interfere with
one another. Even if both procedures were active at the same time -- if,
for instance, one issued the call
(index (greatest-of-list (list
5 3 7)) (list 18 6 14 7 2)) -- the correct
procedure would be invoked in each case, because the correct local
binding would supersede all others.
Many programmers use
letrec-expressions in writing
most of these husk-and-kernel procedures. When there is only one recursive
procedure to bind, however, a contemporary Scheme programmer might well use
yet another variation of the
let-expression -- the ``named
let has the same syntax as a regular
let-expression, except that there is an identifier between the
let and the binding list. The named
binds this extra identifier to a kernel procedure whose parameters are the
same as the variables in the binding list and whose body is the same as the
body of the
let-expression. Here's the basic form
(let name ((var1 val1) (var2 val2) ... (varn valn)) body)
So, for example, one might write
index procedure as follows:
(define index (lambda (sought ls) (let kernel ((rest ls) (bypassed 0)) (cond ((null? rest) -1) ((equal? (car rest) sought) bypassed) (else (kernel (cdr rest) (+ bypassed 1)))))))
When we enter the named
let, the identifier
is bound to the value of
ls and the identifier
bypassed is bound to 0, just as if we were entering an
let-expression. In addition, however, the identifier
kernel is bound to a procedure that has
bypassed as parameters and the body of the named
let as its body. As we evaluate the
cond-expression, we may encounter a recursive call to the
kernel procedure -- in effect, we re-enter the body of the
rest now re-bound to the former
(cdr rest) and
bypassed to the former
(+ bypassed 1).
As another example, here's a version of
that uses a named
(define sum (lambda (ls) (let kernel ((rest ls) (running-total 0)) (if (null? rest) running-total (kernel (cdr rest) (+ (car rest) running-total))))))
Scheme programmers seem to be mixed in their reaction to the named let. Some find it clear and elegant, others find it murky and too special-purpose. My colleagues like to use it. I'll admit that I don't.
March 2, 1997 (John Stone)
March 17, 2000 (John Stone)
24 October 2000 (Sam Rebelsky)
http://www.cs.grinnell.edu/~stone/courses/scheme/local-binding-and-recursion.xhtmlto this course web.
Disclaimer Often, these pages were created "on the fly" with little, if any, proofreading. Any or all of the information on the pages may be incorrect. Please contact me if you notice errors.
This page may be found at http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2000F/Readings/letrec.html
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