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Summary: We consider how and why to write procedures
that may take different numbers of parameters.
Contents:
A procedure's arity is the number of parameters it takes.
For instance, the arity of the cons procedure is two and the
arity of the predicate char-uppercase? is one. You'll probably
have noticed that while some of Scheme's built-in procedures always take
the same number of parameters others have variable arity -- that
is, they can take any number of parameters. All one can say about the
arity of some procedures -- such as list, +,
or string-append -- is that it is some non-negative integer.
Still other Scheme procedures, such as + and
display, require at least a certain number of parameters, but
will accept one or more additional parameters if they are provided. For
example, the arity of + is 1 or more
, and the arity of
display is 1 or 2
. These procedures, too, are said to
have variable arity, because their arity varies from one call to another.
All of the procedures you've written so far have had a fixed arity. Is
it possible to define your own new new variable-arity procedures in Scheme?
Yes! You can create variable-arity procedures by using alternate forms of
the lambda-expression.
In all of the programmer-defined procedures that we have seen so far,
the keyword lambda has been followed by a list of
parameters:
names for the values that will be supplied by the caller when the
procedure is invoked. If, instead, what follows lambda
is a single identifier (not a list, but a simple identifier that
is not enclosed in parentheses) then the procedure denoted by the
lambda-expression will accept any number of parameters,
and the identifier following lambda will name a list of
all the parameters supplied in the call.
This variant takes the following form.
Here's one of the simplest variable-arity procedures, one that simply
reports on how many parameters it is called with:
(define num-parameters
(lambda params
(length params)))
As the following examples show, we can call this procedure with
different numbers of parameters and still get a result.
> (num-parameters)
0
> > (num-parameters 1 2 3 4 5)
5
> (num-parameters (list 1 2 3))
1
Here's a slightly more complicated example: We'll define a display-line
procedure that takes any number of parameters and prints out each one (by
applying the display procedure to it), then terminates the
output line (by invoking newline). Note that in the
lambda-expression, the identifier parameters
denotes a list of all the items to be printed:
;;; Procedure:
;;; display-line
;;; Parameters:
;;; val1 ... valn, 0 or more values.
;;; Purpose:
;;; Displays the strings terminated by a carriage return.
;;; Produces:
;;; [Nothing]
;;; Preconditions:
;;; 0 or more values given as parameters.
;;; Postconditions:
;;; All of the values have been displayed.
;;; The output is now at the beginning of a new line.
(define display-line
(lambda parameters
(let kernel ((rest parameters))
(if (null? rest)
(newline)
(begin
(display (car rest))
(kernel (cdr rest)))))))
When display-line is invoked, however, the caller does not
assemble the items to be printed into a list, but just supplies them as
parameters:
> (display-line "+--" "Here is a string!" "--+")
+--Here is a string!--+
> (display-line "ratio = " 35/15)
ratio = 7/3
If you wish to require some fixed minimum number of parameters
while permitting (but not requiring) additional ones, you can use yet
another form of the lambda-expression, in which a dot is
placed between the last two identifiers in the parameter list. This form
looks like the following
(lambda (ARG1 ARG2 ... ARGN . REMAINING-ARGS)
BODY)
All the identifiers to the left of this dot correspond to required
parameters. The identifier to the right of the dot designates the list
of all of the remaining parameters, the ones that are optional.
For instance, we can define a procedure called
display-separated-line that always takes at least one
argument, separator, but may take any number of additional
parameters. Display-separated-line will print out each of the
additional parameters (by invoking display) and terminate the
line, just as display-line does, but with the difference that
a copy of separator will be displayed between any two of the
remaining values. Here is some sample output:
> (display-separated-line "..." "going" "going" "gone")
going...going...gone
> (display-separated-line ":-" 5 4 3 2 1 'done)
5:-4:-3:-2:-1:-done
> (display-separated-line #\space "+--" "Here is a string!" "--+")
+-- Here is a string! --+
> (display-separated-line (integer->char 9) 1997 'foo 'wombat 'quux)
1997 foo wombat quux
; (integer->char 9) is the tab character.
And here is the definition of the procedure:
;;; Procedure:
;;; display-separated-line
;;; Parameters:
;;; separator, a string
;;; val1 ... valn, 0 or more additional values.
;;; Purpose:
;;; Displays the values separated by the separator and followed
;;; by a carriage return.
;;; Preconditions:
;;; The separator is a string.
;;; Postconditions:
;;; All the values have been displayed.
;;; The output is now at the beginning of a new line.
(define display-separated-line
(lambda (separator . parameters)
(if (null? parameters)
(newline)
(let kernel ((rest parameters))
(display (car rest))
(if (null? (cdr rest))
(newline)
(begin
(display separator)
(kernel (cdr rest))))))))
We often use variable-arity procedures to provide more general versions
of fixed-arity procedures. For example, you may recall writing a
procedure, (intersect lst1 lst2) that computes the intersection
of two lists. Let's look at a variant of that procedure,
(set-difference initial others), that, given two lists, removes all
elements others from initial. We document it first.
;;; Procedure:
;;; set-difference
;;; Parameters:
;;; initial, a list
;;; others, a list
;;; Purpose:
;;; Removes all elements of others from initial.
;;; Produces:
;;; newset, a set
;;; Preconditions:
;;; (none)
;;; Postconditions:
;;; All values in newset appear in initial.
;;; No values in newset appear in others.
;;; All values in initial appear in either newset or others.
How do we implement intersect?
Hmmm ... it sounds like we're going to be removing values from a list.
Since we've written procedures that remove elements from a list in the
past, we should start by writing a more general, higher-order,
remove procedure that removes all values for which a
predicate holds. (Remember, when you find yourself writing the same
code again and again, you should write the higher-order equivalent.)
;;; Procedure:
;;; remove
;;; Parameters:
;;; pred?, a unary predicate
;;; lst, a list
;;; Purpose:
;;; Remove all values from lst for which predicate holds.
;;; Produces:
;;; newlst, a list.
;;; Preconditions:
;;; pred? can be applied to every element of lst.
;;; Postconditions:
;;; (pred? (list-ref newlst i)) is #f for all reasonable i.
;;; Every element of newlst appears in lst.
;;; For all values, v, in lst for which pred? is #f, v appears
;;; in newlst.
(define remove
(lambda (pred? lst)
(cond
((null? lst) null)
((pred? (car lst)) (remove pred? (cdr lst)))
(else (cons (car lst) (remove pred? (cdr lst)))))))
The simple version of set-difference is then
fairly straightforward: We want to remove every element of
initial that is a member of others. We've
defined the remove part above. Next, is a member
of others
can be written (right-second member
others). Putting it together, we get
(define set-difference
(lambda (initial others)
(remove (right-section member others) initial)))
Now, this version of set-difference accepts only two lists.
What if we want a
set-difference
procedure that takes one or more lists l1,
l2, ..., ln as parameters and
returns a list containing all of the elements of l1 that
are not also elements of any of l2, ...,
ln. For example,
> (set-difference (list 'a 'b 'c 'd 'e 'f 'g)
(list 'a 'e) (list 'b) (list 'a 'f 'h))
(c d g)
We get the result because the elements 'c, 'd,
and 'g of the first argument are not elements of any of the
subsequent lists. This improved set-difference procedure
allows the caller to supply any number of lists of values to be pruned
out
of the initial list.
Here's the definition:
;;; Procedure:
;;; set-difference
;;; Parameters:
;;; initial, a set
;;; other-1 ... other-n, zero or more additional sets
;;; Purpose:
;;; Removes elements of the additional sets from the original set.
;;; Produces:
;;; newset, a set
;;; Preconditions:
;;; All the sets are represented as lists.
;;; Postconditions:
;;; All values in newset appear in initial.
;;; No values in newset appear in other-1 ... other-n.
;;; All values in initial appear in either newset or other-1...other-n.
;;; Does not affect any of the parameters. (Yes, this is a standard
;;; postcondition, but I considered it important to restate since this
;;; procedure is described as "removing" elements from its parameters;
;;; it doesn't.)
(define set-difference
(lambda (initial . others)
(let kernel ((set initial)
(remaining others))
(if (null? remaining) set
(kernel (remove (right-section member? (car remaining)) set)
(cdr remaining))))))
In English: Call the initial list initial and collect all of
the other parameters into a list called others. Using list
recursion, process the values in others: In the base case, where
remaining is null, just return set. In any
other case, separate remaining into its car and its cdr,
remove all elements in the car from the set (just as we did in the
original set-difference), and then repeat with
the cdr.
The dot notation can be used to specify any number of initial values.
Thus, a parameter list of the form
(first-value second-value . remaining-values)
indicates that the first two parameters are required, while additional
parameters will be collected into a list named
remaining-values.
You will have the opportunity to explore such procedures in the
laboratory.
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