Summary: Algorithm designers regularly find it useful to name the values their algorithms process. We consider why and how to name new values within an algorithm.
So far we've seen three ways in which a value can be associated with a name in Scheme:
quotient, are predefined. When DrScheme starts up, these names are already bound to the procedures they denote.
There are often times when it seems that you repeat work that should only have to be done once. For example, consider the problem of squaring the average of a list of numbers. We can write
(* (/ (sum lst) (length lst)) (/ (sum lst) (length lst)))
But that's inefficent because we repeat the work of summing the list and computing the length of the list. How can we name the common computation so that we do it only once? If we rely only on the Scheme we know so far, we can write a helper function that takes the average as a parameter.
;;; Procedure: ;;; square ;;; Parameters: ;;; val, a number ;;; Purpose: ;;; Compute val*val ;;; Produces: ;;; result, a number ;;; Preconditions: ;;; val must be a number ;;; Postconditions: ;;; result is the same "type" of number as val (e.g., if ;;; val is an integer, so is the result; if val is exact, ;;; so is the result). The one exception is that the ;;; square of a complex number may be real. (Of course, ;;; every real is complex.) ;;; result = val*val ;;; Citations: ;;; Based on code created by John David Stone dated March 17, 2000 ;;; and contained in the Web page ;;; http://www.math.grin.edu/~stone/courses/scheme/spring-2000/procedure-definitions.xhtml ;;; Changes to ;;; Parameter names ;;; Formatting ;;; Comments (define square (lambda (val) (* value val)))
Now, we can simply write
(square (/ (sum lst) (length lst)))
In that case,
value names the average of the sum of lst
and the length of lst.
But that's a lot of extra work. It's inconvenient to have to write (and document!) a procedure that we're just going to use once.
let expressions as an alternative way to create
local bindings. A
let-expression contains a binding
list and a body. The body can be any expression, or sequence of
expressions, to be evaluated with the help of the local name bindings. The
binding list is a pair of structural parentheses enclosing zero or more
binding specifications; a binding specification, in turn, is a pair
of structural parentheses enclosing a name and an expression.
Here's the general form of a
(let ((name1 exp1) (name2 exp2) ... (namen expn)) body1 body2 ... bodym)
When Scheme encounters a
let-expression, it begins by
evaluating all of the expressions inside its binding specifications. Then
the names in the binding specifications are bound to those values. Next,
the expressions making up the body of the
evaluated, in order. The value of the last expression in the body becomes
the value of the entire
let-expression. Finally, the local
bindings of the names are cancelled. (Names that were unbound before the
let-expression become unbound again; names that had different
bindings before the
let-expression resume those earlier
Here's how we'd solve the earlier problem with
(let ((average (/ (sum lst) (length lst)))) (* average average))
Here's another example of a binding list, taken from a
let-expression in a real Scheme program:
(let ((next (car source)) (stuff null)) ...)
This binding list contains two binding specifications -- one in which the
value of the expression
(car source) is bound to the name
next, and the other in which the empty list is bound to the
stuff. Notice that binding lists and binding
specifications are not procedure calls; their role in a
let-expression simply to give names to certain values while
the body of the expression is being evaluated. The outer parentheses in a
binding list are
structural, like the outer parentheses in a
cond-clause -- they are there to group the pieces of the
binding list together.
let-expression often simplifies an expression that
contains two or more occurrences of the same subexpression. The programmer
can compute the value of the subexpression just once, bind a name to it,
and then use that name whenever the value is needed again. Sometimes this
speeds things up by avoiding such redundancies as the recomputation of
In other cases, there is little difference in speed, but the code may
be a little clearer. For instance, consider the
procedure that removes all copies of a value from a list. In the past,
we might have written that procedure as follows.
;;; Procedure: ;;; remove-all ;;; Parameters: ;;; item, a value ;;; ls, a list of values ;;; Purpose: ;;; Removes all copies of item from ls and its sublists. ;;; Produces: ;;; newls, a list ;;; Preconditions: ;;; ls is a list. It may be empty. ;;; Postconditions: ;;; No values equal to item appear in newls. ;;; Every value not equal to item that appeared in ls also ;;; appears in newls. ;;; Every value that appears in newls also appears in ls. ;;; If a preceded b in ls and neither a nor b equals item, ;;; then a precedes b in newls. (define remove-all (lambda (item ls) (cond ; If the list is empty, removing the element still gives ; us the empty list ((null? ls) null) ; If the first element of the list matches, skip over it. ((equal? item (car ls)) (remove-all item (cdr ls))) ; Otherwise, preseve the first element and remove item ; from the remainder of ls (else (cons (car ls) (remove-all item (cdr ls)))))))
Here is an alternative definition of the
procedure which some people find clearer.
(define remove-all (lambda (item ls) ; If the list is empty, removing the element still gives ; us the empty list (if (null? ls) null (let ( ; Name the car of the list first-element. (first-element (car ls)) ; Recurse on the rest of the list and name it ; rest-of result. (rest-of-result (remove-all item (cdr ls)))) ; If the first element of the list matches, skip over it. (if (equal? first-element item) rest-of-result ; Otherwise, preserve the first element and attach ; it to the rest. (cons first-element rest-of-result))))))
Sometimes we may want to name a number of interrelated things.
For example, suppose we wanted to square the average of a list
of numbers (well, it's something that people do sometimes). Since
computing the average involves summing values, we may want to name two
different things: the total and the average (mean). We can nest one
let-expression inside another to name both things.
(let ((total (+ 8 3 4 2 7))) (let ((mean (/ total 5))) (* mean mean)))
One might be tempted to try to combine the binding lists for the nested
;; Combining the binding lists doesn't work! (let ((total (+ 8 3 4 2 7)) (mean (/ total 5))) (* mean mean))
This wouldn't work (try it and see!), and it's important to understand why
not. The problem is that, within one binding list, all of the
expressions are evaluated before any of the names are bound.
Specifically, Scheme will try to evaluate both
(+ 8 3 4 2 7)
(/ total 5) before binding either of the names
(/ total 5) can't
be computed until
total has a value, an error occurs. You have
to think of the local bindings coming into existence simultaneously rather
than one at a time.
Because one often needs sequential rather than simultaneous binding, Scheme
provides a variant of the
let-expression that rearranges the
order of events: If one writes
let* rather than
let, each binding specification in the binding list is
completely processed before the next one is taken up:
;; Using let* instead of let works! (let* ((total (+ 8 3 4 2 7)) (mean (/ total 5))) (* mean mean))
The star in the keyword
let* has nothing to do with
multiplication. Just think of it as an oddly shaped letter. It
means "do things in sequence, rather than all at once". I have
no idea why they've chosen to do that.
One can use a
let*-expression to create a
local name for a procedure:
(define hypotenuse-of-right-triangle (let ((square (lambda (n) (* n n)))) (lambda (first-leg second-leg) (sqrt (+ (square first-leg) (square second-leg))))))
Regardless of whether
square is defined outside this
definition, the local binding gives it the appropriate meaning within the
lambda-expression that describes what
Note that there are two places one might define
locally. We can define it before the lambda (as above) or below
the lambda (as below). In the first case, the definition is done
only once. In the second case, it is done every time the
procedure is executed. As you practice more with
you'll find times that each choice is appropriate.
(define hypotenuse-of-right-triangle (lambda (first-leg second-leg) (let ((square (lambda (n) (* n n)))) (sqrt (+ (square first-leg) (square second-leg))))))
I usually create these pages
on the fly, which means that I rarely
proofread them and they may contain bad grammar and incorrect details.
It also means that I tend to update them regularly (see the history for
more details). Feel free to contact me with any suggestions for changes.
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