1. CS 403: Programming Languages
Lecture 8
Fall 2003
Department of Computer Science
University of Alabama
Joel Jones

2. Announcement Graduate School Preview Day Thursday, September 25
Starts 3PM Ferguson Theater
Breakout sessions 4:15–5
Engineering 307 Ferguson
Reception 5–5:30 in North Ferguson Lobby
More information on flyer
Lecture 8
©2003 Joel Jones

3. Outline Questions Answers to last time’s exercises
Induction and Recursion
Reading for next time
Lecture 8
©2003 Joel Jones

4. Questions
Lecture 8
©2003 Joel Jones

5. Problems (sumsquares n) (list1 x) (list2 x y) (list3 x y z) (atom? x)
(equal l1 l2) – We’ll see this later
Lecture 8
©2003 Joel Jones

6. Why recursion and induction?
Another tool for thinking about problems
Easier to reason about recursive programs (!)
Certain data structures are naturally recursive
Pair Up:
• What data structures are naturally recursive?
Lecture 8
©2003 Joel Jones

7. Induction and Recursion
“When programming recursively, think inductively”
Terms/Concepts
Precondtion or assumption
Postcondition or guarantee
Base case
Recursive or inductive case
Lecture 8
©2003 Joel Jones

8. Exponentiation Version 1
If n ≥ 0, then exp n evaluates to 2n
Pair Up:
• What is the precondition?
• What is the postcondition?
; p. 227 (217)
(define exp (lambda (n)
(if (= n 0) 1
(* 2 (exp (- n 1))))))
Lecture 8
©2003 Joel Jones

9. Exponentiation Version 1 (cont.)
If n ≥ 0, then exp n evaluates to 2n
; p. 227 (217)
(define exp (lambda (n)
(if (= n 0) 1
(* 2 (exp (- n 1))))))
Does this code satisfy the specification?
Pair Up:
• Give an inductive proof. (Reminder of inductive proofs: prove the base case, prove the inductive case)
Lecture 8
©2003 Joel Jones

10. Exponentiation Version 1 (cont.)
Running time as a function of n
T(0) = O(1)
T(n + 1) = O(1) + T(n) So
T(n) = O(n)
Lecture 8
©2003 Joel Jones

11. Exponentiation Version 2
Can we do better? Yes!
; p. 229 (219)
(define square (lambda (n) (* n n)))
(define double (lambda (n) (+ n n)))
(define fast_exp (lambda (n)
(if (= n 0) 1
(if (= 0 (mod n 2))
(square (fast_exp (div n 2)))
(double (fast_exp (- n 1)))))))
Does this code satisfy the specification. Yes, and we can prove it
using complete induction, a form of mathematical induction in
Which we may prove that n > 0 has a desired property by assuming
not only that the predecessor has it, but that all preceeding
numbers have it, and arguing that therefore n must have it.
Here’s how it is done. For n = 0, the argument is exactly as
before. Suppose, then, that n > 0. If n is even, the value of exp n
Is the result of squaring the value of exp (n / 2). Inductively this
Value is 2^(n/2), so squaring it yields 2^(n/2) * 2^(n/2) = 2^(n/2)
= 2^n, as required. If, on the hand, n is odd, the value is the result
of doubling exp (n - 1). Inductively the latter value is 2^(n - 1), so
Doubling it yields 2^n
Lecture 8
©2003 Joel Jones

12. Exponentiation Version 2 (cont.)
Given
T(0) = O(1)
T(2n) = O(1) + T(n)
T(2n + 1) = O(1) + T(2n)
= O(1) + T(n) So
T(n) = O(lg n)
Lecture 8
©2003 Joel Jones

13. But who cares?
The general pattern of base case and recursive case carries through to many scheme functions
And to handling recursive data structures in other languages
(define f (lambda (n)
(if (basecase? n)
(dobase n)
(combine (current n) (f (reduce n)))))
Lecture 8
©2003 Joel Jones

14. Example of induction on lists
(equal l1 l2)
Pair Up:
• Use the pattern above (or a more complicated version) to write equal on two lists)
Lecture 8
©2003 Joel Jones

15. Reading & Questions for Next Class
Online version of the book “Introduction to ML Programming” chapters 25 and 26, at:
I have placed corresponding Scheme code on web site
Lecture 8
©2003 Joel Jones