These notes cover the following topics:
Simple examples of recursion:
(define (factorial n)
(if (= 0 n)
1
(* n (factorial (- n 1)))))
;; (double-each '(1 3 4)) => (2 6 8)
(define (double-each s)
(if (null? s)
()
(cons (* 2 (car s))
(double-each (cdr s)))))
For numbers, the base case is usually a small integer constant, and a smaller version of the problem is something like n-1.
For lists, the base case is usually the empty list, and a smaller version of the
problem is usually the rest or "cdr" of the list. Here is a template for most
recursive functions you'll write:
(define (fn args)
(if base-case
base-value
(compute-result (fn (smaller args)))))
Many recursive functions build up their answer bit by bit. We'll call these augmenting recursive functions.
;; general form
(define (func x)
(if end-test
end-value
(augmenting-function augmenting-value (func reduced-x))))
Factorial is the classic example:
(define (factorial n)
(if (= n 0)
1
(* n (factorial (- n 1)))))
Many Scheme functions operate on lists, so we often see an important special case of augmenting recursion where our augmenting function is cons and our base case is the empty list. These are sometimes called mapping functions, because they map a function onto every element of a list. Notice that because we are using cons, we are actually constructing a brand new list and returning it. In other words, we are not changing the original list.
(define (double-each s)
(if (null? s)
()
(cons (* 2 (car s)) (double-each (cdr s)))))
With cons augmenting recursion, our base value (the value returned when we
reach the base case) isn't always ():
;; still cons as augmenting-function, but end-value is not ()
(define (my-append x y)
(if (null? x) y
(cons (car x) (my-append (cdr x) y))))
A final important kind of augmenting recursion over lists is the class of reducing functions, because they reduce a list of elements to a single element.
;; sum the elements of a list
(define (sumlist x)
(if (null? x) 0
(+ (car x) (sumlist (cdr x)))))
In tail recursion, we don't build up a solution, but rather, just return a recursive call on a smaller version of the problem. Double-test tail recursion is the most common form:
;; general form:
(define (func x)
(cond (end-test-1 end-value-1)
(end-test-2 end-value-2)
(else (func reduced-x))))
;; example: are all elements of a list positive?
(define (all-positive x)
(cond ((null? x) #t)
((<= (car x) 0) #f)
(else (all-positive (cdr x)))))
;; (all-positive '(3 5 6)) => #t ;; (all-positive '(3 5 -6)) => #f
Some tail recursive functions will have more than one argument. All but one of the arguments are usually passed along unaltered.
;; example: member:
(define (my-member e x)
(cond ((null? x) #f)
((equal? e (car x)) #t)
(else (my-member e (cdr x)))))
A less commonly seen form is single-test tail recursion.
Scheme compilers handle tail recursion very efficiently, as efficiently as a program that just uses loops instead of recursion. (In particular, tail recursive functions don't use stack space for every recursive call.)
;; test whether two lists have the same length
(define (same-length x y)
(cond ((and (null? x) (null? y)) #t)
((null? x) #f)
((null? y) #f)
(else (same-length (cdr x) (cdr y)))))
;; do these two objects have the same shape?
(define (same-shape x y)
(cond ((and (pair? x) (pair? y))
(and (same-shape (car x) (car y))
(same-shape (cdr x) (cdr y))))
((or (pair? x) (pair? y)) #f)
(else #t)))
A final important template is conditional augmentation, where we don't necessarily augment on every step. These functions are sometimes called filtering functions, because they remove elements from a list that don't pass some test.
define (func x)
(cond (end-test end-value)
(aug-test (augmenting-function augmenting-value (func reduced-x)))
(else (func reduced-x))))
;; example: remove all non-positive numbers from a list
(define (positive-numbers x)
(cond ((null? x) ())
((> (car x) 0) (cons (car x) (positive-numbers (cdr x))))
(else (positive-numbers (cdr x)))))
;; variation on list-consing recursion
(define (insert x s)
(cond ((null? s) (list x))
((< x (car s)) (cons x s))
(else (cons (car s) (insert x (cdr s))))))
;; augmenting recursion
(define (isort s)
(if (null? s)
()
(insert (car s) (isort (cdr s)))))
Applicative programming uses higher order functions -- functions that take other functions as arguments, or that return functions as the result. Scheme gets much of its expressiveness and capacity for abstraction by supporting an applicative programming style. Scheme supports functions as first class citizens.
;; here is a general function that adds N to every element of a list (define (addN-to-list alist n) (cond ((null? alist) ()) (else (cons (+ n (car alist)) (addN-to-list (cdr alist) n))))) ;; here is a function that takes the cdr of every element of a list ;; (the argument must be a list of lists) (define (getcdrs alist) (cond ((null? alist) ()) (else (cons (cdr (car alist)) (getcdrs (cdr alist))))))
If we think about what these functions are doing at an abstract level, we see that they are applying some function to every element of a list, and returning the list of resulting values. Scheme provides us with a built in function to support this idiom.
map is a built in Scheme function that takes a function and a list as an argument, and
returns the list that results by applying that function to every element of the list.
(map function list) ;; general form (map null? '(3 () () 5)) => (#f #t #t #f) (map round '(3 3.3 4.6 5)) => (3 3 5 5) (map cdr '((1 2) (3 4) (5 6))) => ((2) (4) (6))
We often find ourselves mapping little functions over lists that would be annoying to write a separate definition for. For instance, if we wanted to map the double function, we'd have to do the following:
(define (double x) (* 2 x)) (map double '(3 4 5)) => (6 8 10)
Scheme provides us with the special form lambda to define an
anonymous function. lambda really is special, because it actually
creates a lexical closure, which is a bundling of code and a lexical
environment. We'll see why this is important later.
(map (lambda (x) (* 2 x)) '(3 4 5)) ;; the right way to define addN-to-list (define (addN-to-list alist n) (map (lambda (x) (+ n x)) alist))
Please note that in the addN-to-list function, the body of the
lambda function can refer to the variable n, which is in the
lexical scope of the lambda expression.
map can also be used with functions that take more than one argument. Examples:
(map + '(1 2 3) '(10 11 12)) => (11 13 15) (map (lambda (x y) (list x y)) '(a b c) '(j k l)) => ((a j) (b k) (c l))
Suppose we wanted to define the 1-argument version of map ourselves.
We can do it like this:
(define (new-map func alist)
(if (null? alist)
()
(cons (func (car alist)) (new-map func (cdr alist)))))
Now that we have the code for new-map, we can talk about what makes LAMBDA special. Remember that a lambda expresssion evaluates to a lexical closure, which is a coupling of code and a lexical environment (a scope, essentially). The lexical environment is necessary because the code needs a place to look up the definitions of symbols it references. For example, look again at addN-to-list below.
(define (addN-to-list alist n) (new-map (lambda (x) (+ n x)) alist))
When the lambda expression is used in new-map, it needs to know
where to look up the variable name n. It can get the right value for
n,
because it retains its lexical environment.
You can have arbitrarily nested scopes (scopes within scopes within scopes ...). Further, since function names are bound like any other variable, function names also obey the scope rules.
(define x 100)
(define y 200)
(define (strange y) (+ y 300))
(define (weird x)
(let ((strange (lambda (y) (+ y 10))))
(strange x)))
(strange 42) => ???
(weird 42) => ???