Naming Values with Local Bindings

When writing programs and algorithms, it is useful to name values we compute along the way. For example, in an algorithm that, given a color, computes a grey color with the same brightness as the original color, it is useful to name that brightness. 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 names are bound to values in Scheme.

As you develop more and longer procedures, you will find that there are many times you want to create local names for values that are not parameters. We will consider such names in this reading.

You may have already noted that there are times when it seems that you repeat work that should only have to be done once. For example, consider the problem of computing a grey that has the same brightness as a given color. We can certainly write the following:

;;; Procedure:
;;;   rgb-greyscale
;;; Parameters:
;;;   color, an RGB color
;;; Purpose:
;;;   Compute a greyscale version of color.
;;; Produces:
;;;   grey, an RGB color
;;; Postconditions:
;;;   grey is likely to be interpreted as the same brightness as color.
(define rgb-greyscale 
  (lambda (color)
    (rgb-new (+ (* 0.30 (rgb-red color))
                (* 0.59 (rgb-green color))
                (* 0.11 (rgb-blue color)))
             (+ (* 0.30 (rgb-red color))
                (* 0.59 (rgb-green color))
                (* 0.11 (rgb-blue color)))
             (+ (* 0.30 (rgb-red color))
                (* 0.59 (rgb-green color))
                (* 0.11 (rgb-blue color))))))

However, that code has some difficulties. First, it's a bit hard to read. Why are we using those weird formulae? Are the three formulae the same? If not, why not? Second, it's hard to correct the code. If we change our formula, we'll need to change it in three places, since the formula is repeated three times. Finally, this repetition of code will lead to inefficiencies. Why should we compute the same product and the same sum three times? If we rely only on the Scheme we know so far, we can solve the first two problems by writing a separate procedure to compute the grey component.

;;; Procedure:
;;;   rgb-brightness255
;;; Parameters:
;;;   color, an RGB color
;;; Purpose:
;;;   Computes the brightness of color on a 0 (dark) to 255 (bright) scale.
;;; Produces:
;;;    brightness, an integer
;;; Preconditions:
;;;   color is a valid RGB color.  That is, each component is between
;;;     0 and 255, inclusive.
;;; Postconditions:
;;;   If color1 is likely to be perceived as brighter than color2,
;;;     then (rgb-brightness255 color1) > (rgb-brightness255 color2).
(define rgb-brightness255
  (lambda (color)
    (+ (* .30 (rgb-red color))
       (* .59 (rgb-green color))
       (* .11 (rgb-blue color)))))

Once we've defined rgb-brightness255, all we need to do is call it three times to build a single shade of grey.

(define rgb-greyscale
  (lambda (color)
    (rgb-new (rgb-brightness255 color)
             (rgb-brightness255 color)
             (rgb-brightness255 color))))

This change certainly makes it easier to read rgb-greyscale. In addition, the procedure easier to update. If we decide to compute brightness differently, such as by averaging the three components, we only need to change one piece of code.

(define rgb-brightness255
  (lambda (color)
    (/ (+ (rgb-red color) (rgb-green color) (rgb-blue color)) 3)))

Similarly, if we want to ensure that the value produced by rgb-brightness255 is an exact integer (the values we expect for the components of RGB colors), we need only make one change, rather than three.

(define rgb-brightness255
  (lambda (color)
    (inexact->exact
     (round
      (+ (* .30 (rgb-red color))
         (* .59 (rgb-green color))
         (* .11 (rgb-blue color)))))))

Although we have solved the first two deficiencies of the original code, we are still repeating work. Can we avoid the repetition of work? Certainly, we can write a procedure that makes a shade of grey from a single brightness value.

;;; Procedure:
;;;   rgb-grey
;;; Parameters:
;;;   level, an integer
;;; Purpose:
;;;   Produces a shade of grey.
;;; Produces:
;;;   grey, a color.
;;; Preconditions:
;;;   level is an integer between 0 and 255, inclusive.
;;; Postconditions:
;;;   Each component of grey is level .
(define rgb-grey
  (lambda (level)
    (rgb-new level level level)))

Now, we can simply write

(define rgb-greyscale
  (lambda (color)
    (rgb-grey (rgb-brightness255 color))))

But getting to this version took a lot of extra work. We had to write (and document!) two procedures that we may never use again.

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 any sequence of expressions, to be evaluated with the help of the local name bindings. The binding list takes the form of a parentheses enclosing zero or more binding expressions of the form (name value).

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 and without helpers.

(define rgb-greyscale
  (lambda (color)
    (let ((component (+ (* 0.30 (rgb-red color)) 
                        (* 0.59 (rgb-green color))
                        (* 0.11 (rgb-blue color)))))
      (rgb-new component component component))))

Again, if we want to make the change to the computation of the brightness component (e.g., by rounding and making it exact), we need change our code only once.

(define rgb-greyscale
  (lambda (color)
    (let ((component (inexact->exact (round (+ (* 0.30 (rgb-red color)) 
                                               (* 0.59 (rgb-green color))
                                               (* 0.11 (rgb-blue color)))))))
      (rgb-new component component component))))

Here's another example of a binding list, taken from a let-expression in a real Scheme program:

(let ((black (rgb-new 0 0 0))
      (complement (rgb-complement rgb)))
  ...)

This binding list contains two binding specifications -- one in which the value of the expression (rgb-new 0 0 0) is bound to the name black, and the other in which the complement of rgb (presumably, an RGB color) is bound to the name complement.

Note that even though binding lists and binding specifications start with parentheses, the 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” -- 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 recomputation of values. In other cases, there is little difference in speed, but the code may be a little clearer.

You may have missed it, but there are a few subtle and important issues with the the use of let rather than define to name values and procedures. One difference has to do with the availability (or “scope”) of the name. Values named by define are available essentially everywhere in your program. In contrast, values named by let are available only within the let expression. (In case you were wondering, the term “scope” has nothing to do with the mouthwash.)

In addition, local variables (given by let) and global variables (given by define) affect previous uses of the name differently (or at least appear to). When we do a new top-level define, we permanently replace the old value associated with the name. That value is no longer accessible. In contrast, when we use let to override the value associated with a name, as soon as the let binding is finished, the previous association is restored.

Finally, there's a benefit to using let instead of define according to the principle of “information hiding”. Evidence suggests that programs work better if the various pieces do not access values relevant primarily to the internal workings of other pieces. If you use define for your names, they are accessible (and therefore changable) everywhere. Hence, you enforce this separation by policy or obscurity. In contrast, if you use let to define your local names, these names are completely inaccessible to other pieces of code. We return to this issue in our discussion of the ordering of let and lambda below.

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 (+ (rgb-red color) (rgb-green color) (rgb-blue color))))
    (let ((mean (/ total 3)))
      (* 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 (+ (rgb-red color) (rgb-green color) (rgb-blue color))
        (mean (/ total 3)))
    (* 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 (+ (rgb-red color) (rgb-green color) (rgb-blue color)))
         (mean (/ total 3)))
    (* 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”. While someone probably knows the reason to use * for that meaning, the authors of this text do not.

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. (SamR says, “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 produce does correctly compute the desired result. However, it is a bit hard to read. For clarity, you might want to name some of the values.

(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 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 hours-per-day 24)
(define minutes-per-hour 60)
(define seconds-per-minute 60)
(define seconds-per-year 
  (* days-per-year hours-per-day minutes-per-hour 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 will 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. For the first example, in which we bound the name black, one might write

   (let ((black (rgb-new 0 0 0)))
     (lambda (rgb ...)
       ...

However, it is not always possible to move the bindings outside of the lambda. In particular, if your let-bindings use parameters, then you need to keep them within the body of the lambda. Consider the other half of the initial example. In that case, we were computing the complement of rgb, which we assumed was a parameter. In that case, we cannot write

   (let ((black (rgb-new 0 0 0))
         (complement (rgb-complement rgb)))
     (lambda (rgb ...)
       ...

but must instead write

   (let ((black (rgb-new 0 0 0)))
     (lambda (rgb ...)
       (let ((complement (rgb-complement rgb)))
         ...

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.

Scheme also lets you create local bindings by nesting define statements within each other. We recommend that you limit your use of define to top-level definitions, using just let and let* for internal definitions. While you can use one define within another, the semantics are a bit complicated, and such use can lead to unexpected results and therefore confusion. (We also find internal define statements inelegant.)