1. David Evans http://www.cs.virginia.edu/~evans CS200: Computer Science University of Virginia Computer Science Lecture 5: Recursion Beware the Lizards!

2. 25 January 2002CS 200 Spring 20022 Menu PS1 Comments FIBO Example

3. 25 January 2002CS 200 Spring 20023 Problem Set 1 Without Evaluation Rules, Question 2 was “guesswork” Now you know the Evaluation Rules, Question 2 should be obvious

4. 25 January 2002CS 200 Spring 20024 2d (100 + 100) Evaluation Rule 3. Application. a. Evaluate all the subexpressions 100 100 b. Apply the value of the first subexpression to the values of all the other subexpressions Error: 100 is not a procedure, we only have apply rules for procedures!

5. 25 January 2002CS 200 Spring 20025 2h (if (not "cookies") "eat" "starve") Evaluation Rule 4-if. Evaluate Expression 0. If it evaluates to #f, the value of the if expression is the value of Expression 1. Otherwise, the value of the if expression is the value of Expression 2. Evaluate (not "cookies")

6. 25 January 2002CS 200 Spring 20026 Evaluate (not "cookies") Evaluation Rule 3. Application. a. Evaluate all the subexpressions “cookies” The quotes really matter here! Without them what would cookies evaluate to? b. Apply the value of the first subexpression to the values of all the other subexpressions (not v) evaluates to #t if v is #f, otherwise it evaluates to #f.(SICP, p. 19) So, (not “cookies”) evaluates to #f

7. 25 January 2002CS 200 Spring 20027 2h (if (not "cookies") "eat" "starve") Evaluation Rule 4-if. Evaluate Expression 0. If it evaluates to #f, the value of the if expression is the value of Expression 1. Otherwise, the value of the if expression is the value of Expression 2. Evaluate (not "cookies") => #f So, value of if is value of Expression 2 => “starve”

8. 25 January 2002CS 200 Spring 20028 (+ (abs (- (get-red color1) (get-red sample))) (abs (- (get-blue color1) (get-blue sample))) (abs (- (get-green color1) (get-green sample)))) (+ (abs (- (get-red color2) (get-red sample))) (abs (- (get-blue color2) (get-blue sample))) (abs (- (get-green color2) (get-green sample)))) (define (closer-color? sample color1 color2) (< )) closer-color?

9. 25 January 2002CS 200 Spring 20029 (+ (abs (- (get-red color2) (get-red sample))) (abs (- (get-blue color2) (get-blue sample))) (abs (- (get-green color2) (get-green sample)))) (define (closer-color? sample color1 color2) (< )) (+ (abs (- (get-red color1) (get-red sample))) (abs (- (get-blue color1) (get-blue sample))) (abs (- (get-green color1) (get-green sample))))

10. 25 January 2002CS 200 Spring 200210 (+ (abs (- (get-red ) (get-red ))) (abs (- (get-blue ) (get-blue ))) (abs (- (get-green ) (get-green )))) (+ (abs (- (get-red color2) (get-red sample))) (abs (- (get-blue color2) (get-blue sample))) (abs (- (get-green color2) (get-green sample)))) (define (closer-color? sample color1 color2) (< )) color1 sample color1 sample (lambda ( )

11. 25 January 2002CS 200 Spring 200211 (+ (abs (- (get-red ) (get-red ))) (abs (- (get-blue ) (get-blue ))) (abs (- (get-green ) (get-green )))) (+ (abs (- (get-red color2) (get-red sample))) (abs (- (get-blue color2) (get-blue sample))) (abs (- (get-green color2) (get-green sample)))) (define (closer-color? sample color1 color2) (< )) (color-difference color1 sample) colora colorb colora colorb (lambda (colora colorb) (define color-difference )) (color-difference color2 sample)

12. 25 January 2002CS 200 Spring 200212 (define color-difference (lambda (colora colorb) (+ (abs (- (get-red colora) (get-red colorb))) (abs (- (get-green colora) (get-green colorb))) (abs (- (get-blue colora) (get-blue colorb)))))) (define (closer-color? sample color1 color2) (< (color-difference color1 sample) (color-difference color2 sample))) What is you want to use square instead of abs?

13. 25 January 2002CS 200 Spring 200213 (define color-difference (lambda (colora colorb) (+ (abs (- (get-red colora) (get-red colorb))) (abs (- (get-green colora) (get-green colorb))) (abs (- (get-blue colora) (get-blue colorb)))))) (define (closer-color? sample color1 color2) (< (color-difference color1 sample) (color-difference color2 sample)))

14. 25 January 2002CS 200 Spring 200214 (define color-difference (lambda (cf) (lambda (colora colorb) (+ (cf (- (get-red colora) (get-red colorb))) (cf (- (get-green colora) (get-green colorb))) (cf (- (get-blue colora) (get-blue colorb))))))) (define (closer-color? sample color1 color2) (< (color-difference color1 sample) (color-difference color2 sample)))

15. 25 January 2002CS 200 Spring 200215 (define color-difference (lambda (cf) (lambda (colora colorb) (+ (cf (- (get-red colora) (get-red colorb)) (cf (- (get-green colora) (get-green colorb)) (cf (- (get-blue colora) (get-blue colorb)))))))) (define (closer-color? sample color1 color2) (< ((color-difference square) color1 sample) ((color-difference square) color2 sample)))

16. 25 January 2002CS 200 Spring 200216 Next: Fibonacci Questions on PS1 or PS2?

17. 25 January 2002CS 200 Spring 200217 Fibonacci’s Problem Filius Bonacci, 1202 in Pisa: Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?

18. 25 January 2002CS 200 Spring 200218 Rabbits From http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html

19. 25 January 2002CS 200 Spring 200219 Fibonacci Numbers GEB p. 136: These numbers are best defined recursively by the pair of formulas FIBO (n) = FIBO (n – 1) + FIBO (n – 2) for n > 2 FIBO (1) = FIBO (2) = 1 Can we turn this into a Scheme function? Note: SICP defines Fib with Fib(0)= 0 and Fib(1) = 1 for base case. Same function except for Fib(0) is undefined in GEB version.

20. 25 January 2002CS 200 Spring 200220 Defining Recursive Procedures 1.Be optimistic. –Assume you can solve it. –If you could, how would you solve a bigger problem. 2.Think of the simplest version of the problem, something you can already solve. (This is the base case.) 3.Combine them to solve the problem. From last lecture:

21. 25 January 2002CS 200 Spring 200221 Defining FIBO 1.Be optimistic - assume you can solve it, if you could, how would you solve a bigger problem. 2.Think of the simplest version of the problem, something you can already solve. 3.Combine them to solve the problem. These numbers are best defined recursively by the pair of formulas FIBO (n) = FIBO (n – 1) + FIBO (n – 2) for n > 2 FIBO (1) = FIBO (2) = 1

22. 25 January 2002CS 200 Spring 200222 Defining fibo ;;; (fibo n) evaluates to the nth Fibonacci ;;; number (define (fibo n) (if (or (= n 1) (= n 2)) 1 ;;; base case (+ (fibo (- n 1)) (fibo (- n 2))))) FIBO (1) = FIBO (2) = 1 FIBO (n) = FIBO (n – 1) + FIBO (n – 2) for n > 2

23. 25 January 2002CS 200 Spring 200223 Fibo Results > (fibo 2) 1 > (fibo 3) 2 > (fibo 4) 3 > (fibo 10) 55 > (fibo 100) Still working after 4 hours… Why can’t our 100,000x Apollo Guidance Computer calculate (fibo 100) ?

24. 25 January 2002CS 200 Spring 200224 Tracing Fibo > (require-library "trace.ss") > (trace fibo) (fibo) > (fibo 3) |(fibo 3) | (fibo 2) | 1 | (fibo 1) | 1 |2 2 This turns tracing on

25. 25 January 2002CS 200 Spring 200225 > (fibo 5) |(fibo 5) | (fibo 4) | |(fibo 3) | | (fibo 2) | | 1 | | (fibo 1) | | 1 | |2 | |(fibo 2) | |1 | 3 | (fibo 3) | |(fibo 2) | |1 | |(fibo 1) | |1 | 2 |5 5 (fibo 5) = (fibo 4) + (fibo 3) (fibo 3) + (fibo 2) + (fibo 2) + (fibo 1) (fibo 2) + (fibo 1) + 1 + 1 + 1 1 + 1 2 + 1 + 2 3 + 2 = 5 To calculate (fibo 5) we caluculated: (fibo 4)1 time (fibo 3)2 times (fibo 2)3 times (fibo 1)2 times = 8 calls to fibo = (fibo 6) How many calls to calculate (fibo 100) ?

26. 25 January 2002CS 200 Spring 200226 fast-fibo (define (fast-fibo n) (define (fibo-worker a b count) (if (= count 1) b (fibo-worker (+ a b) a (- count 1)))) (fibo-worker 1 1 n))

27. 25 January 2002CS 200 Spring 200227 Fast-Fibo Results > (fast-fibo 1) 1 > (fast-fibo 10) 55 > (time (fast-fibo 100)) cpu time: 0 real time: 0 gc time: 0 354224848179261915075

28. 25 January 2002CS 200 Spring 200228 Beware the Bunnies!! ;;; The Earth's mass is 6.0 x 10^24 kg > (define mass-of-earth (* 6 (expt 10 24))) ;;; A typical rabbit's mass is 2.5 kilograms > (define mass-of-rabbit 2.5) > (/ (* mass-of-rabbit (fast-fibo 100)) mass-of-earth) 0.00014759368674135913 > (/ (* mass-of-rabbit (fast-fibo 110)) mass-of-earth) 0.018152823441189517 > (/ (* mass-of-rabbit (fast-fibo 119)) mass-of-earth) 1.379853393132076 > (exact->inexact (/ 119 12)) 9.916666666666666 According to Fibonacci’s model, after less than 10 years, rabbits would out-weigh the Earth!

29. 25 January 2002CS 200 Spring 200229 Charge PS1 Selected Answers on web –Read through the comments on your assignments and check the answers on the web Beware the Bunnies! Next time: –GEB Chapter 5 –More Higher Order Procedures