Fundamentals of Computer Science I (CS151.02 2007S)

Lists in Scheme

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Summary: We consider lists, the most important compound data type in Lisp.


Introducing Lists

One of Scheme's great strengths is that it includes a powerful data structure, the list, as a built-in data type. In contrast to Scheme's unstructured data types, such as symbols and numbers, lists are structures that contain other values as elements. A list is an ordered collection of values. In Scheme, lists can be heterogeneous, in that they may contain different kinds of values.

As we will see throughout the semester, lists provide an elegant and easy-to-use mechanism for organizing collections of information.

In order to work with lists, you need to know how to create lists, update lists (or at least create new lists based on other lists), and extract values from lists.

Creating Lists with list

The easiest 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. 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)
(2 3)
> (list 'a a 'b b)
(a 2 b 3)

Constructing Lists with cons

Although list is a convenient way to build lists, the list operation is, itself, composed of other, more primitive, operations. Behind the scenes, list invokes cons once for each element of the completed list, to hook that element onto the previously created list.

We have to start somewhere. As you might guess, the simplest list is the empty list, that contains no elements at all. Any other list is constructed by attaching some value, called the car of the new 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 between them: (). 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, null, for the empty list. We follow this usage and recommend it.

> null

If, for some reason, the implementation of Scheme that you're using does not include null, you can always define it yourself.

(define 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
> (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 'third-year (cons 'second-year (cons 'freshling null))))
(senior third-year second-year freshling)

The cons procedure never returns an empty list, since it always adds an element at the beginning of another list.

Warning: Cons-ing to Non-Lists

Although we have defined cons in such a way that it expects a list as its second parameter, it is possible to use a value other than a list as the second parameter. When you do so, you build something that is a bit like a list, but also a bit different. Most Scheme environments indicate the difference by putting a period before the final value.

> (cons 'a (cons 'b null))
(a b)
> (cons 'a (cons 'b 'null))
(a b . null)

If you are trying to create a list and you see the period, you have probably done something wrong, so you'll need to think about your algorithm.

Constructing List Literals

Warning! This section discusses a technique that we do not recommend using. We include it because it reveals a lot about the workings of Scheme.

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, whereas '(+ 5 3) is a list literal denoting a list of three elements, the symbol + and the numbers 5 and 3.

> (+ 5 3)
> '(+ 5 3)
(+ 5 3)

While list literals seem like a convenient way to create lists, experience shows that they can also lead to problems. We recommend that you generally avoid using list literals.

Nested Lists

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-1 and gamma-2, and its fourth is the symbol delta.

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.

Taking Lists Apart

To recover elements from a list, one commonly uses the built-in Scheme procedures 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, car and cdr are the inverses of cons; if you think of a non-empty list as having been assembled by a call to the cons procedure, car gives you back the first argument to cons and cdr gives you back the second one.

> (car (cons 'apple (cons 'orange null)))
> (cdr (cons 'apple (cons 'orange null)))

If you want the second rather than the first element of a list, you can combine car and cdr to extract it:

> (define sample (cons 'apple (cons 'orange null)))
> (car (cdr sample))

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 longer-list, and so on.

Many implementations of Scheme provide these variants, using names like cadr (for the car of the cdr) and cddr for the (cdr of the cdir).

Common List Procedures

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:


The 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 to contain.)


The 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)


The 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.


The 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 cdr that many times) and extracting the next element (by invoking car). So (list-ref sample 0) is the same as (car sample), (list-ref sample 1) is the same as (car (cdr sample)), and so on.




Disclaimer: 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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Samuel A. Rebelsky,

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