1. CS 403: Programming Languages
Lecture 14
Fall 2003
Department of Computer Science
University of Alabama
Joel Jones

2. Overview Higher order functions for polymorphism Lecture 14
©2003 Joel Jones

3. Higher-order functions for polymorphism
Implementing sets
Constructing mono-morphic sets requires definition of equality—simple if primitive elements
->(define emptyset ‘())
->(define member? (lambda (x s) (exists? ((curry equal? x) s)))
->(define add-element (lambda (x s) (if (member? X s) s (cons x s))))
->(define union (lambda (s1 s2) (foldl add-element s1 s2)))
->(define set-from-list (lambda (l) (foldl add-element ‘() l)))
Problem is equality for complex types
Lecture 14
©2003 Joel Jones

4. HOF for Polymorphism: Parameter per Function
(define member? (lambda (x s eqfun)
(exists? ((curry eqfun) x) s)))
(define add-element (lambda (x s eqfun)
(if (member? x s eqfun) s (cons s x))))
So uses will look like:
(member? X s equal?) or
(add-element x s =alist?)
This part is really difficult for the students. I need to have them read ahead of time and do labs. Look at the Little Schemer to see if they have any helpful exercises.
Lecture 14
©2003 Joel Jones

5. Association Lists (define bind (lambda (x y alist) (if (null? alist)
(list1 (list2 x y))
(if (equal? x (caar alist))
(cons (list2 x y) (cdr alist))
(cons (car alist) (bind x y (cdr alist))))))
(define find (lambda (x alist)
(if (null? alist) ‘()
(car (cdar alist))
(find x (cdr alist))))))
Lecture 14
©2003 Joel Jones

6. HOF for Polymorphism: Part of “object”
->(define mk-set (lambda (eqfun elements)
(cons eqfun elements)))
->(define eqfun-of (lambda (set) (car set)))
->(define elements-of (lambda (set) (cdr set)))
->(define emptyset (lambda (eqfun)
(mk-set eqfun ‘())))
Lecture 14
©2003 Joel Jones

7. HOF for Polymorphism: Part of “object”
Pair Up:
• What was type of the function “exists?” ?
(define exists? (lambda (p? L)…)
(define member? (lambda (x s)
(exists? (curry (eqfun-of s)) x)
(elements-of s))))
(define add-element? (lambda (x s)
(if (member? x s) s
(mk-set (eqfun-of s)
(cons x (elements-of s))))))
Lecture 14
©2003 Joel Jones

8. HOF for Polymorphism: Part of “object”
So uses will look like:
(define alist-empty (emptyset =alist?))
(define s (add-element ‘((U Thant) (I Ching)) alist-empty)
returns
(<procedure> ((U Thang) (I Ching)))
(define s (add-element ‘((Hello Dolly))))
(<procedure> ((Hello Dolly) (U Thang) (I Ching)))
(member? ‘((I Ching))) #t
Advantage is that only constructor (here, emptyset) requires extra parameter
Lecture 14
©2003 Joel Jones

9. Sorting (define insert (lambda (x l) (if (null? l) (list1 x)
(if (< x (car l))
(cons x l)
(cons (car l) (insert x (cdr l)))))))
(define insertion-sort (lambda (l)
(if (null? l)
‘()
(insert (car l) (sort (cdr l))))))
(insertion-sort ‘( ))
-> ( )
Lecture 14
©2003 Joel Jones