While your initial exercises in Scheme have been numeric, Scheme is not limited to numerical computation, but can also operate on pure symbols.
Scheme's ancestor, Lisp, was originally developed to aid in experiments in artificial intelligence. At the time, a leading theory suggested that intelligence emphasizes symbolic manipulation. Hence, it is sensible that Lisp and Scheme include symbols as a basic type.
When we want to refer to something as a value involved in a computation, rather than as the name of some other value, we put an apostrophe (usually pronounced ``quote'') in front of it. In effect, by quoting the symbol, we're telling Scheme to take it literally and without further interpretation or evaluation:
> 'sample sample
Note that you can quote many different things. You can even quote Scheme expressions.
> '(+ 2 3) (+ 2 3)
In addition to ``unstructured'' data types such as symbols and numbers, Scheme supports lists, which are structures that contain other values as elements. There is one list -- the empty list -- that contains no elements at all. Any other list is constructed by attaching some value, called the car of the list, to a previously constructed list, which is called the cdr of the new list.
Scheme's name for the empty list is a pair of parentheses with nothing
(). When we refer to the empty list in a Scheme
program, we have to put an apostrophe before the left parenthesis, so that
Scheme won't mistake the parentheses for a procedure call:
> '() ()
Since this conventional name for the empty list is not very readable, our
implementation of Scheme also provides a built-in name,
for the empty list. I follow this usage and recommend it.
> null ()
The ``constructor'' procedure for non-empty lists is called
cons. It takes two arguments and returns a list that is just
like the second argument, except that the first argument has been added at
the beginning, as a new first element. By repeated applications of
cons, we can build up a list of any size:
> (define singleton (cons 'sample null)) > singleton (sample) > (define doubleton (cons 'another-element singleton)) > doubleton (another-element sample) > (define tripleton (cons 'yet-another-element doubleton)) > tripleton (yet-another-element another-element sample) > (cons 'senior (cons 'junior (cons 'sophomore (cons 'freshling null)))) (senior junior sophomore freshling)
cons procedure never returns an empty list, since it
always adds an element at the beginning of another list.
As you may have noted from the discussion of atoms, there is another way to create lists. You can
For example, the value of the expression
'(38 72 apple -1/3 sample)
is a five-element list consisting of two numbers, a symbol, another number, and finally another symbol. Note that the apostrophe blocks the evaluation of the whole list, so that it is not necessary to quote separately the symbols that occur as elements of the list.
In a list literal like this one, the apostrophe must be present so
that Scheme does not misinterpret the left parenthesis as the beginning of
a procedure call. Sometimes that apostrophe is all that distinguishes two
different, correctly formed expressions. For instance,
(+ 5 3) is a procedure call that has the value 8,
'(+ 5 3) is a list literal denoting a list
of three elements -- the symbol
+ and the numbers 5 and 3.
> > (+ 5 3) 8 > '(+ 5 3) (+ 5 3)
Yet another way to create a list is to invoke a procedure named
list. This procedure takes all of its arguments, however many
of them there may be, and packs them into a list. (Behind the scenes,
cons once for each element of the
completed list, to hook that element onto the previously created cdr.)
Just as the addition procedure
+ sums its arguments and
returns the result, so the
list procedure collects its
arguments and returns the resulting list:
> (list 38 72 'apple -1/3 'sample) (38 72 apple -1/3 sample) > (define a 2) > (define b 3) > (list a b) (3 10)
It is possible, and indeed common, for a list to be an element of another list. For instance, the expression
(list 'alpha 'beta (list 'gamma-1 'gamma-2) 'delta)
creates a four-element list: Its first element is the symbol
alpha, its second is the symbol
beta, its third
is a two-element list comprising the symbols
gamma-2, and its fourth is the symbol
It is possible for all of the elements of a list to be lists. It is possible for a list that is an element of another list to have lists as its elements, and so on -- lists can be embedded within lists to any desired level of nesting. This idea is subtler and more powerful than it may initially seem to be.
To recover elements from a list, one commonly uses the built-in Scheme
car, which takes one argument (a non-empty list)
and returns its first element, and
cdr, which takes one
argument (a non-empty list), and returns a list just like the one it was
given, except that the first element has been removed. In a sense,
cdr are the inverses of
cons; if you think of a non-empty list as having been
assembled by a call to the
gives you back the first argument to
gives you back the second one.
> (car (cons 'apple (cons 'orange null))) apple > (cdr (cons 'apple (cons 'orange null))) (orange)
If you want the second rather than the first element of a list, you can
cdr to extract it:
> (define sample (cons 'apple (cons 'orange null))) > (car (cdr sample)) orange
The idea is that the procedure call
(cdr sample) computes a
list just like
sample except that the symbol
apple is gone, and then
car gives you the first
element of that computed list. Similarly,
(car (cdr (cdr
longer-list))) is the third element of
Just as Scheme provides many built-in procedures that perform simple operations on numbers, there are several built-in procedures that operate on lists. Here are four that are very frequently used:
length procedure takes one argument, which must be a
list, and computes the number of elements in the list. (An element that
happens to be itself a list nevertheless contributes 1 to the total that
length computes, regardless of how many elements it happens
reverse procedure takes a list and returns a new list
containing the same elements, but in the opposite order.
> (reverse '(a b c)) (c b a)
append procedure takes any number of arguments, each of
which is a list, and returns a new list formed by stringing together all of
the elements of the argument lists, in order, to form one long list.
list-ref procedure takes two arguments, the first of which
is a list and the second a non-negative integer less than the length of the
list. It recovers an element from the list by skipping over the number of
initial elements specified by the second argument (applying
that many times) and extracting the next element (by invoking
(list-ref sample 0) is the same as
(list-ref sample 1) is the same as
(car (cdr sample)), and so on.
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/lists.html
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