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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.
Contents:
When writing programs and algorithms, it is useful to name
values we compute along the way. For example, in an algorithm that
computes the roots of a quadratic polynomial, we often name the
coefficient of the quadratic term, a, the coefficient of the
linear term, b, and the constant term, c. In addition,
since there is a common term (the square root of b squared
minus four a c
), we might want to name that term,
too. When we associate a name with a value, we say that we
bind that name to the value.
So far we've seen three ways in which Scheme permits the algorithm
writer to bind a name to a value:
- The names of built-in procedures, such as
cons and
quotient, are predefined. When DrScheme starts up,
these names are already bound to the procedures they denote.
- The programmer can introduce a new binding by means of a definition.
A definition may introduce a new equivalent for an old name, or it may give
a name to a newly constructed value.
- When a programmer-defined procedure is called, the parameters of
the procedure are bound to the values of the corresponding arguments in the
procedure call. Unlike the other two kinds of bindings, parameter bindings
are local -- they apply only within the body of the procedure.
Scheme discards these bindings when it leaves the procedure and returns to
the point at which the procedure was called.
There are often times when it seems that you repeat work that should
only have to be done once. For example, consider again the problem
of computing the roots of a quadratic (assuming, of course, that
it has two roots). We can write the following:
(define roots
(lambda (a b c)
(list (/ (+ (- b) (sqrt (- (* b b) (* 4 a c)))) (* 4 a c))
(/ (- (- b) (sqrt (- (* b b) (* 4 a c)))) (* 4 a c)))))
But that's inefficient because we repeat lot of work.
(It's also a bit dangerous, since we might have gotten it wrong
in one place or the other, and we have to look closely to tell.)
How can we name the common computations so that we do them only once?
If we rely only
on the Scheme we know so far, we can write a helper function that
takes the three common parts as a parameter
;;; Procedure:
;;; roots-helper
;;; Parameters:
;;; negative-b, a number
;;; square-root-of, a number
;;; two-a, a number
;;; Purpose:
;;; Compute the roots of a quadratic.
;;; Produces:
;;; (root1 root2), a list of two numbers
;;; Preconditions:
;;; negative-b must be (- b), where b is the coefficient of the
;;; linear term
;;; square-root-of must be (sqrt (- (* b b) (* 4 a c)), where
;;; a is the coefficient of the quadratic term, b is the
;;; coefficient of the linear term, and c is the constant term.
;;; two-a must be (* 2 a), where a is as above.
;;; Postconditions:
;;; root1 and root2 are the roots of a quadratic. That is,
;;; (+ (* a root1 root1) (* b root1) c) = 0
;;; (+ (* a root2 root2) (* b root2) c) = 0
(define roots-helper
(lambda (negative-b square-root-of two-a)
(list (/ (+ negative-b square-root-of) two-a)
(/ (- negative-b square-root-of) two-a))))
Now, we can simply write
(define roots
(lambda (a b c)
(roots-helper (- b) (sqrt (- (* b b) (* 4 a c))) (* 2 a))))
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. It's
also not particularly clear in the call.
Scheme provides
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.
That precise definition may have been a bit confusing, so
here's the general form of a let expression
(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 let-expression are
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
bindings.)
Here's how we'd solve the earlier problem with let.
(define roots
(lambda (a b c)
(let ((negative-b (- b))
(square-root-of (sqrt (- (* b b) (* 4 a c))))
(two-a (* 2 a)))
(list (/ (+ negative-b square-root-of) two-a)
(/ (- negative-b square-root-of) two-a)))))
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
name 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.
Using a 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 re0computation of
values.
In other cases, there is little difference in speed, but the
code may be a little clearer. For instance, consider the following
remove-all 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, preserve 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 remove-all
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
let-expressions, thus:
;; 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)
and (/ total 5) before binding either of the names
total and mean; since (/ 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.
In the examples above, we've tended to do the naming within the body
of the procedure. That is, we write
(define proc
(lambda (params)
(let (...)
exp)))
However, Scheme also lets us choose an alternate ordering. We can
instead put the let before (outside of) the
lambda.
(define proc
(let (...)
(lambda (params)
exp)))
Why would we ever choose to do so? Let us consider an example. Suppose
that we regularly need to convert years to seconds. (When you have
sons between the ages of 5 and 12, you'll understand.) You might begin
with
(define years-to-seconds
(lambda (years)
(return (* years 365.24 24 60 60))))
This is, of course, correct. However, it is a bit hard to read. You
might want to name some of the values for clarity.
(define years-to-seconds
(lambda (years)
(let* ((days-per-year 365.24)
(hours-per-day 24)
(minutes-per-hour 60)
(seconds-per-minute 60)
(seconds-per-year (* days-per-year hours-per-day
minutes-per-hour seconds-per-minute)))
(* years seconds-per-year))))
> (years-to-seconds 10)
315567360.0
We have clearly clarified the code, although we have also lengthened
it a bit. However, as we noted before, a second goal of naming is to
avoid recomputation of values. Unfortunately, even though the number
of seconds per year never changes, we compute it every time that
someone calls years-to-seconds. How can we avoid this
recomputation? One strategy is to move the bindings to define
statements.
(define days-per-year 365.24)
...
(define seconds-per-year (* days-per-year ... seconds-per-minute))
(define years-to-seconds
(lambda (years)
(* years seconds-per-year)))
However, such a strategy is a bit dangerous. After all, there is nothing
to prevent someone else from changing the values here.
(define days-per-year 360) ; Some strange calendar, perhaps in Indiana
...
> (years-to-seconds 10)
311040000
What we'd like to do is to declare the values once, but keep them
local to years-to-seconds. The strategy is to move the
let outside the lambda.
(define years-to-seconds
(let* ((days-per-year 365.24)
(hours-per-day 24)
(minutes-per-hour 60)
(seconds-per-minute 60)
(seconds-per-year (* days-per-year hours-per-day
minutes-per-hour seconds-per-minute)))
(lambda (years)
(* years seconds-per-year))))
> (years-to-seconds 10)
315567360.0
As you'll see in the lab, it is possible to empirically verify that
the bindings occur only once in this case, and each time the procedure
is called in the prior case.
So, one moral of this story is whenever possible, move your
bindings outside the lambda. However, it is not
always possible to do so. For example, if your let-bindings use
parameters (as in the earlier quadratic example), then you need to
keep them within the body of the lambda.
As you may have noted, let behaves somewhat like
define in that programmers can use it to name values.
But we've used define to name more than values; we've
also used it to name procedures. Can we also use let
for procedures?
Yes, one can use a let- or let*-expression to
create a local name for a procedure. And we name procedures locally
for the same reason that we name values, because it speeds and clarifies
the code.
(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 also defined outside
this definition, the local binding gives it the appropriate
meaning within the lambda-expression that describes what
hypotenuse-of-right-triangle does.
Note, once again, that there are two places one might define
square locally. We can define it before the lambda (as
above) or after 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.
(define hypotenuse-of-right-triangle
(lambda (first-leg second-leg)
(let ((square (lambda (n) (* n n))))
(sqrt (+ (square first-leg) (square second-leg))))))
So, which we should you do it? If the helper procedure you're defining
uses any of the parameters of the main procedure, it needs to come after
the lambda. Otherwise, it is generally a better idea to
do it before the lambda. As you practice more with let,
you'll find times that each choice is appropriate.
Unfortunately, you cannot use let to define recursive
procedures. In a subsequent reading, you'll learn another variant
of let that supports recursive procedures. Once you've
learned that technique your helpers can all be local.
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